Dynamic Fiscal Policy Dirk Krueger1 Department of Economics University of Pennsylvania April 2007 *I would like to thank Victor Rios Rull and Jesus Ferandez Villaverde for many helpful discussions. © by Dirk Krueger ii Contents Preface ix I Introduction 1 1 Empirical Facts of Government Economic Activity 5 1.1 Data on Government Activity in the U.S.............. 5 1.2 The Structure of Government Budgets............... 10 1.3 Fiscal Variables and the Business Cycle .............. 13 1.4 Government Deficits and Government Debt............ 17 2 A Two Period Benchmark Model 23 2.1 The Model .............................. 23 2.2 Solution of the Model ........................ 25 2.3 Comparative Statics......................... 30 2.3.1 Income Changes ....................... 30 2.3.2 Interest Rate Changes.................... 31 2.4 Borrowing Constraints........................ 35 3 The Life Cycle Model 39 3.1 Solution of the General Problem.................. 41 3.2 Important Special Cases....................... 43 3.2.1 Equality of ß = ^..................... 43 3.2.2 Two Periods and log-Utility................. 46 3.2.3 The Relation between ß and and Consumption Growth 47 3.3 Empirical Evidence.......................... 49 3.4 Potential Explanations........................ 50 II Positive Theory of Government Activity 55 4 Dynamic Theory of Taxation 57 4.1 The Government Budget Constraint................ 58 4.2 The Timing of Taxes: Ricardian Equivalence ........... 60 4.2.1 Historical Origin....................... 60 iii iv CONTENTS 4.2.2 Derivation of Ricardian Equivalence............ 61 4.2.3 Discussion of the Crucial Assumptions........... 64 4.3 An Excursion into the Fiscal Situation of the US......... 70 4.3.1 Two Measures of the Fiscal Situation ........... 70 4.3.2 Main Assumptions...................... 71 4.3.3 Main Results......................... 72 4.3.4 Interpretation......................... 72 4.4 Consumption, Labor and Capital Income Taxation........ 75 4.4.1 The U.S. Federal Personal Income Tax........... 75 4.4.2 Theoretical Analysis of Consumption Taxes, Labor Income Taxes and Capital Income Taxes........... 87 5 Unfunded Social Security Systems 99 5.1 History of the German Social Security System........... 99 5.2 History of the US Social Security System ............. 101 5.3 The Current US System....................... 102 5.4 Theoretical Analysis......................... 106 5.4.1 Pay-As-You-Go Social Security and Savings Rates .... 106 5.4.2 Welfare Consequences of Social Security.......... 108 5.4.3 The Insurance Aspect of a Social Security System .... 110 6 Social Insurance 113 6.1 International Comparisons of Unemployment Insurance .....113 6.2 Social Insurance: Theory ......................118 6.2.1 A Simple Intertemporal Insurance Model .........118 6.2.2 Solution without Government Policy............118 6.2.3 Public Unemployment Insurance..............122 III Optimal Fiscal Policy 125 7 Optimal Fiscal Policy with Commitment 127 7.1 The Ramsey Problem ........................127 7.2 Main Results in Optimal Taxation.................127 8 The Time Consistency Problem 129 9 Optimal Fiscal Policy without Commitment 131 IV The Political Economics of Fiscal Policy 133 10 Intergenerational Conflict: The Case of Social Security 135 11 Intragenerational Conflict: The Mix of Capital and Labor Income Taxes 137 List of Figures 1.1 Trade Balance as Fraction of GDP, 1950-2005 ........... 9 1.2 Government Spending as Fraction of GDP, 1950-2005 ...... 10 1.3 Unemployment Rate and Government Spending.......... 15 1.4 Unemployment Rate and Tax Receipts............... 16 1.5 Unemployment Rate and Government Deficit........... 17 1.6 US Government Debt ........................ 21 1.7 Government Debt as a Fraction of GDP, 1950-2005 ........ 22 2.1 Optimal Consumption Choice.................... 29 2.2 A Change in Income......................... 32 2.3 An Increase in the Interest Rate .................. 34 2.4 Borrowing Constraints........................ 37 3.1 Life Cycle Profiles, Model...................... 47 3.2 Consumption over the Life Cycle.................. 50 4.1 U.S. Marginal Income Taxes, Individuals Filing Single...... 82 4.2 U.S. Average Tax Rate for Individuals Filing Single ....... 83 5.1 Social Security Replacement Rate and Marginal Benefits .... 106 6.1 The U.S. Unemployment Rate.................... 114 v LIST OF FIGURES List of Tables 1.1 Components of GDP, 2004 ...................... 7 1.2 Federal Government Budget, 2005 ................. 11 1.3 State and Local Budgets, 2002 ................... 12 1.4 Federal Government Deficits as fraction of GDP, 2003 ...... 18 1.5 Government Debt as Fraction of GDP, 2003 ............ 20 2.1 Effects of Interest Rate Changes on Consumption......... 33 4.1 Federal Government Budget, 2005 ................. 58 4.2 Fiscal and Generational Imbalance................. 72 4.3 Fiscal and Generational Imbalance................. 72 4.4 Fiscal and Generational Imbalance................. 73 4.5 Marginal Tax Rates in 2003, Households Filing Single...... 81 4.6 Marginal Tax Rates in 2003, Married Households Filing Jointly . 84 4.7 Labor Supply, Productivity and GDP, 1993-96 .......... 95 4.8 Labor Supply, Productivity and GDP, 1970-74 .......... 95 4.9 Actual and Predicted Labor Supply, 1993-96 ............ 97 4.10 Actual and Predicted Labor Supply, 1970-74 ............ 98 5.1 Social Security Tax Rates...................... 103 6.1 Length of Unemployment Spells................... 115 6.2 Unemployment Rates, OECD.................... 116 6.3 Long-Term Unemployment by Age, OECD............. 116 6.4 Unemployment Benefit Replacement Rates ............ 117 vii LIST OF TABLES Preface In these notes we study fiscal policy in dynamic economic models in which households are rational, forward looking decision units. The government (that is, the federal, state and local governments) affect private decisions of individual households in a number of different ways. Households that work pay income and social security payroll taxes. Income from financial assets is in general subject to taxes as well. Unemployed workers receive temporary transfers from the government in the form of unemployment insurance benefits, and possibly welfare payments thereafter. When retired, most households are entitled to social security benefits and health care assistance in the form of medicare. The presence of all these programs may alter private decisions, thus affect aggregate consumption, saving and thus current and future economic activity. In addition, the government is an important independent player in the macro economy, purchasing a significant fraction of Gross Domestic Product (GDP) on its own, and absorbing a significant fraction of private domestic (and international saving) for the finance of its budget deficit. We attempt to analyze these issues in a unified theoretical framework, at the base of which lies a simple intertemporal decision problem of private households. We then introduce, step by step, fiscal policies like the ones mentioned above to analytically derive the effects of government activity on the private sector. Consequently these notes are organized in the following way. In the first part we first give an overview over the empirical facts concerning government economic activity and then develop the simple intertemporal consumption choice model. In the second part we then analyze the impact on the economy of given fiscal policies, without asking why those policies would or should be enacted. This positive analysis contains the study of the timing and incidence of consumption, labor and capital income taxes, and the study of social security and unemployment insurance. In the third part (yet to be written) we then turn to an investigation on how fiscal policy should be carried out if the government is benevolent and wants to maximize the happiness of its citizens. It turns out to be important for this study that the government can commit to future policies (i.e. is not allowed to change its mind later, after, say, a certain tax reform has been enacted). Since this is a rather strong assumption, we then identify what the government can and should do if it knows that, in the future, it has an incentive to change its policy. ix X PREFACE Finally, in part 4 (again yet to be written) we will discuss how government policies are formed when, instead of being benevolent, the government decides on policies based on political elections or lobbying by pressure groups. This area of research, called political economy, has recently made important advances in explaining why economic policies, such as the generosity of unemployment benefits, differ so vastly between the US and some continental European countries. We will study some of the successful examples in this new field of research. Part I Introduction i 3 In the first part of these notes we want to accomplish two things. First, we want to get a sense on what the government does in modern societies by looking at the data describing government activity. Then we want to construct and analyze the basic intertemporal household decision problem which we will use extensively to study the impact of fiscal policy on private decisions of individual households, and thus the entire macro economy. We start with the simplest version of the model in which households live for only two periods, and then extend it to the standard life cycle model invented by Franco Modigliani, Albert Ando and Richard Brumberg. 4 Chapter 1 Empirical Facts of Government Economic Activity Before proposing theories for the effect and the optimal conduct of fiscal policies it is instructive to study what the government actually does in modern societies. For the most part we will constraint our discussion to the US. 1.1 Data on Government Activity in the U.S. We start our tour of the data by looking at the different components of Gross Domestic Product (GDP) as measured in the National Income and Product Accounts (NIPA). Nominal GDP is computed by summing up the total spending on goods and services by the different sectors of the economy. Formally, let C = Consumption / = (Gross) Investment G = Government Purchases X = Exports M = Imports Y = Nominal GDP Then the well-known spending decomposition of GDP is given by Y = C + I + G+(X-M) Let us turn to a brief description of the components of GDP, acting as a reminder from your intermediate macroeconomics classes. 5 6CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY • Consumption (C) is defined as spending of households on all goods, such as durable goods (cars, TV's, Furniture), nondurable goods (food, clothing, gasoline) and services (massages, financial services, education, health care). The only form of household spending that is not included in consumption is spending on new houses.1 Spending on new houses is included in fixed investment, to which we turn next. • Gross Investment (/) is defined as the sum of all spending of firms on plant, equipment and inventories, and the spending of households on new houses. It is broken down into three categories: residential fixed investment (the spending of households on the construction of new houses), nonresidential fixed investment (the spending of firms on buildings and equipment for business use) and inventory investment (the change in inventories of firms). • Government spending (G) is the sum of federal, state and local government purchases of goods and services. Note that government spending does not equal total government outlays: transfer payments to households (such as welfare, social security or unemployment benefit payments) or interest payments on public debt are part of government outlays, but not included in government spending G. • As an open economy, the US trades goods and services with the rest of the world. Exports (X) are deliveries of US goods and services to the rest of the world, imports (M) are deliveries of goods and services from other countries of the world to the US. The quantity (X — M) is also referred to as net exports or the trade balance. We say that a country (such as the US) has a trade surplus if exports exceed imports, i.e. if X — M > 0. A country has a trade deficit if X — M < 0, which was the case for the US in recent years. In Table I we show the composition of nominal GDP for 2004, broken down to the different spending categories discussed above.2 The numbers are in billion US dollars. We see that government spending amounts to 18.9 percent of total GDP, with roughly two thirds of this coming from purchases of US states and roughly one third stemming from purchases of the federal government. Thus an important point to notice about US government activity is that, due to its federal structure, in this country a large share of government spending is done on the state and local level, rather than the federal level. However, due to recent increases in expenditures for defence and homeland security the share of GDP ^"What about purchases of old houses? Note that no production has occured (since the house was already built before). Hence this transaction does not enter this years' GDP. Of course, when the then new house was first built it entered GDP in the particular year. 2 As with most of the data in this class, the ones underlying the table come from the Economic Report of the President, which is available online at http://www.gpoaccess.gov/eop/index.html. The most recent report is the one from 2006, which contains final NIPA data until 2004; the data for 2005 are still subject to revisions and hence not used in this version of the notes. 1.1. DATA ON GOVERNMENT ACTIVITY IN THE U.S. 7 in billion $ in % of Tot. Nom. GDP Total Nom. GDP 11,734.3 100.0% Consumption 8,214.2 70.0% Durable Goods 987.8 8.4% Nondurable Goods 2,368.3 20.2% Services 4,858.2 41.4% Gross Investment 1,928.1 16.4% Nonresidential 1,198.6 10.2% Residential 673.8 5.7% Changes in Inventory 55.4 0.5% Government Purchases 2,215.9 18.9% Federal Government 827.6 7.1% State and Local Government 1,388.3 11.8% Net Exports -624.0 -5.3% Exports 1,173.8 10.0% Imports 1,797.8 15.3% Table 1.1: Components of GDP, 2004 that goes to federal government spending has increased. Also, it is important to remember that government spending only includes the purchase of goods and services by the government (for national defense or the construction of new roads), but not transfer payments such as unemployment insurance and social security benefits. As such, the fraction of G/Y is a first, but fairly incomplete measure of the "size of government". Table 1.1 also shows other important facts for the US economy which are not directly related to fiscal policy, but will be of some interest in this course. First, about 70% of GDP goes to private consumption expenditures; this share of GDP has been rising substantially in the 1990's and continues to do so. Within consumption we see that the US economy is now to a large extent a service economy, with almost 60% of overall private consumption expenditures (and thus 41% of overall GDP) going to such services as hair-cuts, entertainment services, financial services (banking, tax advise etc.) and so forth. The "traditional" manufacturing sector supplying consumer durable goods such as cars and furniture, now only accounts for about 12% of total consumption expenditures and 8% of total GDP. With respect to investment we note that the bulk of it is investment of firms into machines and factory structures (called nonresidential fixed investment), whereas the construction and purchases of new family homes, called residential fixed investment (for some historical reason this item is not counted in consumer durables consumption), amounts to about 35% of total investment and 5.7% of overall GDP, a number that has risen significantly in recent years, but is expected to decline for 2006 and 2007, due to the recent downturn in the U.S. real estate market. Finally, changes in inventory, have been slightly positive in 2004, but quantitatively small (as is usually the case). SCHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY Finally, the table shows one of the two important deficits the popular economic discussion centers around in recent years. We will talk about the US federal government budget deficit in detail below. The other deficit, the trade deficit (also called net exports or the trade balance), the difference between US exports of goods and services and the value of goods and services the US imports, amounted to about 5.3% of GDP. This means that in 2004 the US population bought $624 billion worth of goods more from abroad than US firms sold to other countries. As a consequence in 2004 on net foreigners acquired (roughly) $624 billion in net assets in the US (buying shares of US firms, government debt, taking over US firms etc.).3 Figure 1.1, which plots the trade balance as a fraction of GDP, shows that the US trade balance was not always negative. In fact, it was mostly positive in the period before the 1980's, before turning sharply negative in the 1990's. Since 1989 the US, traditionally a net lender to the world, has become a net borrower: the net wealth position of the US has become negative in 1989. The US appetite for foreign goods and services also means that, in order to pay for 3In order to make this argument precise we need some more definitions. We already denned what the trade balance is: it is the total value of exports minus the total value of imports of the US with all its trading partners. A closely related concept is the current account balance. The current account balance equals the trade balance plus net unilateral transfers Current Account Balance = Trade Balance + Net Unilateral Transfers Unilateral transfers that the US pays to countries abroad include aid to poor countries, interest payments to foreigners for US government debt, and grants to foreign researchers or institutions. Net unilateral transfers equal transfers of the sort just described received by the US, minus transfers paid out by the US. Usually net unilateral transfers are negative for the US, but small in size (less than 1% of GDP). So for the purpose of this class we can use the trade balance and the current account balance interchangeably. We say that the US has a current account deficit if the current account balance is negative and a current account surplus if the current account balance is positive. The current account balance thus (roughly) keeps track of import and export flows between countries. The capital account balance keeps track of borrowing and lending of the US with abroad. It equals to the change of the net wealth position of the US. The US owes money to foreign countries, in the form of government debt held by foreigners, loans that foreign banks made to US companies and in the form of shares that foreigners hold in US companies. Foreign countries owe money to the US for exactly the same reason The net wealth position of the US is the difference between what the US is owed and what it owes to foreign countries. Thus Capital Account Balance this year = Net wealth position at end of this year — Net wealth postion at end of last year Note that a negative capital account balance means that the net wealth position of the US has decreased: in net terms, wealth has flown out of the US. The reverse is true if the capital account balance is positive: wealth flew into the US. The current account and the capital account balance are intimately related: they are always equal to each other. This is an example of an accounting identity. Current Account Balance this year = Capital Account Balance this year The reason for this is simple: if the US imports more than it exports, it has to borrow from the rest of the world to pay for the imports. But this change in the net asset position is exactly what the capital account balance captures. 1.1. DATA ON GOVERNMENT ACTIVITY IN THE U.S. Trade Balance as Fraction of GDP a u c TO (0 m 73 (0 1950 1960 1970 1980 Year 1990 2000 Figure 1.1: Trade Balance as Fraction of GDP, 1950-2005 these goods, US consumers have to (directly or indirectly through the companies that import the goods) acquire foreign currency for dollars, which puts pressure on the exchange rate between the dollar and foreign currencies. As of late, the dollar has lost significant value against other major currencies, such as the Euro and the Yen. This may have many reasons, but the persistently large trade deficit is surely among them. After this little digression we turn back to the size of government spending activity, as a share of GDP. In figure 1.2 we show how this share has developed over time. We observe a substantial decline in the share of GDP devoted to government spending, both due to sharp declines of this ratio in the late 60's and early seventies, as well as the 1990's. The 1980's, in contrast, saw a mild increase in government spending, as a share of GDP, partly due to increased spending on national defense. 10 CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY Government Spending as Fraction of GDP 251-1-1-1-1-1- 15'-1-1-1-1- 1950 1960 1970 1980 1990 2000 Year Figure 1.2: Government Spending as Fraction of GDP, 1950-2005 1.2 The Structure of Government Budgets We start our discussion with the federal budget. The federal budget surplus is defined as Budget Surplus = Total Federal Tax Receipts —Total Federal Outlays Federal outlays, in turn consist of Total Federal Outlays = Federal Purchases of Goods and Services +Transfers +Interest Payments on Fed. Debt +Other (small) Items 1.2. THE STRUCTURE OF GOVERNMENT BUDGETS 11 2005 Federal Budget (in billion $) Receipts 2153.9 Individual Income Taxes 927.2 Corporate Income Taxes 278.3 Social Insurance Receipts 794.1 Other 154.2 Outlays 2472.2 National Defense 495.3 International Affairs 34.6 Health 250.6 Medicare 298.6 Income Security 345.8 Social Security 523.3 Net Interest 184.0 Other 339.9 Surplus -318.3 Table 1.2: Federal Government Budget, 2005 The entity "government spending" that we considered so far equals to federal, state and local purchases of goods and services, but does not include transfers, such as social security benefits, unemployment insurance and welfare payments. The US federal budget had a deficit every year since 1969 since 1997, then small surpluses between 1998 and 2001, before the increased expenditures for homeland security, the recession and the large Bush tax cuts sent the federal budget into deficit again since 2002. Further substantial deficits are projected for the near future. How can the federal government spend more than it takes in? Simply by borrowing, i.e. issuing government bonds that are bought by private banks and households, both in the US and abroad. The total federal government debt that is outstanding is the accumulation of past budget deficits. The federal debt and the deficit are related by Fed. debt at end of this year = Fed. debt at end of last year +Fed. budget deficit this year Hence when the budget is in deficit, the outstanding federal debt increases, when it is in surplus (as in 1998-2001), the government pays back part of its outstanding debt. Now let us look at the federal government budget for the latest year we have final data for, 2005. See Table 1.2. We see that the bulk of the federal government's receipts comes from income taxes and social security and unemployment contributions paid by private households, and, to a lesser extent from corporate income taxes (taxes on profits of private companies). The role of indirect business taxes (i.e. sales taxes) 12 CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY 2004 State and Local Budgets (in billion $) Total Revenue 1581.7 Personal Taxes 247.2 Taxes on Production and Sales 758.8 Corporate Income Taxes 41.5 Contributions for Soc. Ins. 19.7 Asset Income 77.1 Transfers from Federal Gov. 439.8 Surplus of Gov. Enterprises -2.5 Total Expenditures 1587.5 Govt Spending 1117.7 Social Insurance Benefits 380.5 Interest Payments 88.9 Subsidies 0.5 Surplus -5.9 Table 1.3: State and Local Budgets, 2002 which are included in the "Other" category is relatively minor for the federal budget as most of sales taxes go to the states and cities in which they are levied. On the outlay side the two biggest posts are national defense, which constitutes about two thirds of all federal government purchases (G) and transfer payments, mainly social security benefits (about $822 billion if one includes Medicare) and unemployment (about $346 billion). About 10 — 15% of federal outlays go as transfers to states and cities to help finance projects like highways, bridges and the like. A sizeable fraction (7.4%) of the federal budget is devoted to interest payments on the outstanding federal government debt. The outstanding government debt at the end of 2005 was $7, 932 billion, or about 64% of GDP. In other words, if the federal government could expropriate all production in the US (or equivalently all income of all households) for the whole year of 2005, it would need 64% of this in order to repay all debt at once. The ratio between total government debt (which, roughly, equals federal government debt) and GDP is called the (government) debt-GDP ratio, and is the most commonly reported statistics (apart from the budget deficit as a fraction of GDP) measuring the indebtedness of the federal government. It makes sense to report the debt-GDP ratio instead of the absolute level of the debt because the ratio relates the amount of outstanding debt to the governments' tax base and thus ability to generate revenue, namely GDP. Let's have a brief look at the budget on the state and local level. The latest official final numbers stem from the fiscal year 2004. Table 1.3 summarizes the main facts. The main difference between the federal and state and local governments is the type of revenues and outlays that the different levels of government have, 1.3. FISCAL VARIABLES AND THE BUSINESS CYCLE 13 and the fact that states usually have a balanced budget amendment: they are by law prohibited from running a deficit, and immediate action is required should a deficit arise. 2004 was one of the rare occasions where the aggregated state and local budgets indeed showed a small deficit, partly due to a substantial budget deficit in California. The only state in the US that currently does not have a balanced budget amendment is Vermont. The main observations from the receipts side are that the main source of state and local government revenues stems from indirect sales taxes. A substantial part of revenues on the state and local level comes about from transfers from the federal government; these transfers are intended to help finance large infrastructure projects and expenditures for homeland security on the state level. Income taxes, although not unimportant for state and local governments, do not nearly comprise as an important share of total revenue as it does for the federal government. On the outlay side the single most important category is expenditures for government consumption. On the state and local level a large share of this goes to expenditures for public education, in the form of direct purchases of education material and, more importantly, the pay of public school teachers. All payments to state universities and public subsidies to private schools or universities are also part of these outlays. Also part of this category are expenditures for public infrastructure programs such as roads. An important share of expenditures is used for social insurance, which is comprised mainly of retirement benefits for state employees as well as financial transfers to poor families in the form of welfare and other assistance payments. Finally the state and local governments have to service interest payments on bonds issued to finance certain large projects and they give (small) subsidies to attract businesses to their states or communities. 1.3 Fiscal Variables and the Business Cycle In this section we briefly document to what extent actual fiscal policy is correlated with the business cycle. Since we only look at data, all the statements we can make are about correlations, not about causality.4 4Remember from basic statistics that the correlation coefficient between two time series {xt,yt}J=1 is given by corr(x, y) Cov(x, y) where Cov(x,y) Std(x) Std(y) Std{x) ■ Std(y) 1 f (xt -ä)(yt -y) T \ t=i 14CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY In Figure 1.3 we plot the unemployment rate as prime indicator of business cycle and purchases of the government (federal, state and local) as a fraction of GDP over time. As already discussed above, one feature that appears in the data is that government spending, as a fraction of GDP, has declined over time (see the right scale). One also can detect that in recessions (in times where the unemployment rises, see the left scale) government spending as a fraction of GDP increases. This is consistent with the view that government spending is being used to a certain degree -successfully or not- to smooth out business cycles.5 A similar, even more accentuated picture appears if one plots government transfers (such as unemployment compensation and welfare) against the unemployment rate. The fact that government transfers are countercyclical follows almost by construction: in recessions by definition a lot of people are unemployed and hence more unemployment compensation (and once this runs out, welfare) is paid out. These welfare programs are sometimes called automatic stabilizers, as these programs provide more transfers in situations where incomes of households tend to be low on average, hence softening the decline in consumption expenditures and therefore the recession. In Figure 1.4 we plot the unemployment rate and government tax receipts as a fraction of GDP against time. We see that tax receipts are strongly procyclical, they increase in booms (low unemployment) and decline during recessions (high unemployment). In this sense taxes act as automatic stabilizers, too, since, due to the progressivity of the tax code, in good times households on average are taxed at a higher rate than in bad times. In this sense the tax system stabilizes after-tax incomes and hence spending. A second reason for declines of taxes in recessions is discretionary tax policy: cutting taxes may provide a stimulus for private consumption and hence may help to lead the economy out of a recession (we will later study a theorem that argues, however, that the timing of taxes is irrelevant for the real economy). For example, the tax cuts in the early 60's under President Kennedy were designed for this purpose; the recent Bush tax cuts were in part motivated by the same reason. So rather than being automatic stabilizers, taxes may be used deliberately in an attempt to fine-tune the business cycle.6 Now let us look at the government deficit over the business cycle. Figure 1.5 plots the federal budget deficit as a fraction of GDP and the unemployment are the covariances between the two variables and the standard deviations of the two variables, respectively. A positive correlation coefficient indicates that, on average, the variable x is high at the same time the variable y is high. 5Surprisingly, when one computes the coefficient of correlation between the unemployment rate and the share of government spending in GDP one obtains a slightly negative number, — 0.285, suggesting that that government spending is relatively low when the unemployment rate is high. This statistic, however, is driven entirely by the period from 1967 to 1972, which featured both a strong increase in the unemployment rate and a strong decline in the government expenditure share. Excluding this period one obtains a positive correlation of 0.43, suggesting that government spending was in fact anticyclical from the early 70's onward. 6The correlation between taxes and the unemployment rate is —0.28, significantly negative. Remember that a high unemployment rate means bad economic times, with low GDP. 1.3. FISCAL VARIABLES AND THE BUSINESS CYCLE 15 Figure 1.3: Unemployment Rate and Government Spending rate over time. The first observation is (see the right scale) that the federal budget had small surpluses in the late 60's, then went into (heavy) deficit for the next 35 years or so and only in the late 90's showed surpluses again, which disappeared in the year 2002. One clearly sees the large deficits during the oil price shock recession in 1974-75 and the large deficit during the early Reagan years, due to large increases of defense spending. Overall one observes that the budget deficit is clearly countercyclical: the deficit is large in recessions (as tax revenues decline and government outlays tend to increase) and is small in booms. In fact the extremely long and powerful expansion during the 90's resulted, in combination with federal government spending cuts, in the budget surpluses of the late 1990's.7 7 Again, the correlation of unemployment rate and the federal government deficit is strongly negative at —0.79: high unemployment rates go hand in hand with large deficits (remember that a deficit is a negative number). 16 CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY Taxes and Unemployment Rate, 1967-99 15 a) *= io (0 DC c a) E > o Q. E a) 5 5 Unemployment Rate 1965 1970 1975 1980 1985 Year 1990 1995 i0.35 0.3 a. a ü in a) x re > 0.25 O Ü J0.2 2000 Figure 1.4: Unemployment Rate and Tax Receipts How does one determine whether the federal government is loose or tight on fiscal policy. Just looking at the budget deficit may obscure matters, since the current government may either have generated a large deficit because of loose fiscal policy or because the economy is in a recession where taxes are typically low and transfer payments high, so that the large deficit was beyond the control of the government. Hence economists have developed the notion of the structural government deficit: it is the government deficit that would arise if the economy's current GDP equals its potential (or long run trend) GDP. The structural part of the deficit is not due to the business cycle, it is the deficit that on average arises given the current structure of taxes and expenditures. The cyclical government deficit is the difference between the actual and the structural deficit: it is that part of the deficit that is due to the business cycle. How loose or restrictive monetary policy is can then be determined by looking at the structural (rather than the actual) deficit. 1.4. GOVERNMENT DEFICITS AND GOVERNMENT DEBT 17 Federal Deficit and Unemployment Rate, 1967-99 12 |-1-1-1-1-1-1-10 1965 1970 1975 1980 1985 1990 1995 Year Figure 1.5: Unemployment Rate and Government Deficit 1.4 Government Deficits and Government Debt We previous defined the government budget deficit and related it to the change in the outstanding government debt. In table 1.4 we provide government deficit numbers for a cross-section of industrialized countries. We observe that, within the Euro area, there is substantial variation in the deficit-GDP ratio, ranging from a significant surplus in Finland to a substantial deficit of over 4% in France. However, comparing the Euro numbers to the US or Japan (or some countries in Europe not (yet) in the Euro area) we observe that deficit figures are not outrageous by international standards. However, note that the budget deficits of the US and Japan are the source of significant concern by policy makers and economists in the respective countries, so the fact the some European counties' substantial deficits are passed by other countries still should not be a sign of comfort. 18CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY International Deficit to GDP Ratios Country Def./GDP in 2003 Belgium 0.3 Germany -3.9 Greece -3.2 Spain 0.3 France -4.1 Ireland 0.2 Italy -2.4 Luxembourg -0.1 Netherlands -3.2 Austria -1.3 Portugal -2.8 Finland 2.3 Euro Area -2.7 Czech Republic -12.9 Denmark 1.5 Estonia 2.6 Cyprus -6.3 Latvia -1.8 Lithuania -1.7 Hungary -5.9 Malta -9.7 Poland -4.1 Slovenia -1.8 Slovakia -3.6 Sweden 0.7 UK -3.2 US -4.6 Japan -7.9 Table 1.4: Federal Government Deficits as fraction of GDP, 2003 1.4. GOVERNMENT DEFICITS AND GOVERNMENT DEBT 19 We now want to take a quick look at the stock of outstanding government debt, both in international comparison as well as over time for the US. For the US the outstanding government debt at the end of 2005 was $7, 932 billion, or about 64% of GDP (see above). The ratio between total government debt (which, roughly, equals federal government debt) and GDP is called the (government) debt-GDP ratio, and is the most commonly reported statistics (apart from the budget deficit as a fraction of GDP) measuring the indebtedness of the federal government. It makes sense to report the debt-GDP ratio instead of the absolute level of the debt because the ratio relates the amount of outstanding debt to the governments' tax base and thus ability to generate revenue, namely GDP. Lets have a look at some data the government debt, the accumulated deficits of the government. Figure 1.6 shows the explosion of the government debt outstanding in the last 70 years. The picture is obviously somewhat misleading, since it does not take care of inflation (inflation numbers before the turn of the century are somewhat hard to come by). But clearly visible is the sharp increase during World War II. Somewhat more informative is a plot of the debt-GDP ratio in figure 1.7. The main facts are that during the 60's the US continued to repay part of its WWII debt as debt grows slower than GDP, then, starting in the 70's and more pronounced in the 80's large budget deficits led to a rapid increase in the debt-GDP ratio, a trend that stopped and reversed in the late 1990's, but is expected to re-surface, due to the large tax cuts enacted by the Bush administration. Recent forecasts indicate (which we will look into in detail later in this course) that, to the very least until 2010, renewed and substantial federal budget deficits are to be expected, unless further drastic changes in fiscal policy are enacted in the near future. In order to gain some international comparisons, in table 1.5 we display debt-GDP ratios for various industrialized countries. Debt refers to the entire government sector, including the social insurance sector (which explains the different numbers for the US in the table and the figures above). Again, the variance of debt-GDP ratios within Europe is remarkable, with Belgium and Italy having government debt more than one years' GDP worth, whereas Luxembourg has hardly any government debt. Also observe that the former Communist east European countries tend to have low debt-GDP ratios, basically because they started with a blank slate at the collapse of the old regime at the end of the 1980's. Finally, Japan displays the largest debt to GDP ratio of the entire industrialized world, which may help explain the high private sector savings rate in Japan (somebody has to pay that debt, or at least the interest on that debt, with higher taxes sometime in the future). Note that a substantial fraction of this debt was accumulated during the 1990's, when various government spending and tax cut programs where enacted to try to bring Japan out of its decade-long recession. This concludes our brief overview over government spending, taxes, deficits and debt in industrialized countries. Once we have constructed, in the next chapters, our theoretical model that we will use to analyze the effects of fiscal 20 CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY International Debt to GDP Ratios Country Debt/GDP in 2003 Belgium 100.5 Germany 64.2 Greece 103.0 Spain 50.8 France 63.7 Ireland 32.0 Italy 106.2 Luxembourg 4.9 Netherlands 54.8 Austria 65.0 Portugal 59.4 Finland 45.3 Euro Area 70.6 Czech Republic 37.6 Denmark 45.0 Estonia 5.8 Cyprus 72.2 Latvia 15.6 Lithuania 21.9 Hungary 59.0 Malta 72.0 Poland 45.4 Slovenia 27.1 Slovakia 42.8 Sweden 51.8 UK 39.8 US 47.9 Japan 141.3 Table 1.5: Government Debt as Fraction of GDP, 2003 1.4. GOVERNMENT DEFICITS AND GOVERNMENT DEBT 21 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 Year Figure 1.6: US Government Debt policy, we will combine theoretical analysis with further empirical observations to arrive at a (hopefully) somewhat coherent and complete view of what a modern government does and should do in the economy. 22 CHAPTER 1. EMPIRICAL FACTS OF GOVERNMENT ECONOMIC ACTIVITY Government Debt as Fraction of GDP 2Q i_I_I_I_I_I_ 1950 1960 1970 1980 1990 2000 Year Figure 1.7: Government Debt as a Fraction of GDP, 1950-2005 Chapter 2 A Two Period Benchmark Model In this section we will develop a simple two-period model of consumption and saving that we will then use to study the impact of government policies on an individual households' consumption and saving decisions (in particular social security, income taxation and government debt). We will then generalize this model to more than two periods and study the empirical predictions of the model with respect to consumption and saving over the life cycle of a typical household. The simple model we present is due to Irving Fisher (1867-1947), and the extension to many periods is due to Albert Ando (1929-2003) and Franco Modigliani (1919-2003) (and, in a slightly different form, to Milton Friedman (1912-present)). 2.1 The Model Consider a single individual, for concreteness call this guy Hardy Krueger. Hardy lives for two periods (you may think of the length of one period as 30 years, so the model is not all that unrealistic). He cares about consumption in the first period of his life, c\ and consumption in the second period of his life, c2. His utility function takes the simple form U{c1,c2)=u{c1) + (3u{c2) (2.1) where the parameter [3 is between zero and one and measures Hardy's degree of impatience. A high [3 indicates that consumption in the second period of his life is really important to Hardy, so he is patient. On the other hand, a low [3 makes Hardy really impatient. In the extreme case of [3 = 0 Hardy only cares about his consumption in the current period, but not at all about consumption when he is old. The period utility function u is assumed to be at least twice differentiable, strictly increasing and strictly concave. This means that we can 23 24 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL take at least two derivatives of u, that u'(c) > 0 (more consumption increases utility) and u"(c) < 0 (an additional unit of consumption increases utility at a decreasing rate). Hardy has income 2/1 > 0 in the first period of his life and 2/2 > 0 in the second period of his life (we want to allow 2/2 = 0 in order to model that Hardy is retired in the second period of his life and therefore, absent any social security system or private saving, has no income in the second period). Income is measured in units of the consumption good, not in terms of money. Hardy starts his life with some initial wealth A > 0, due to bequests that he received from his parents. Again A is measured in terms of the consumption good, not in terms of money. Hardy can save some of his income in the first period or some of his initial wealth, or he can borrow against his future income 2/2- We assume that the interest rate on both savings and on loans is equal to r, and we denote by s the saving (borrowing if s < 0) that Hardy does. Hence his budget constraint in the first period of his life is ci + s = yi + A (2.2) Hardy can use his total income in period 1, 2/1 + A either for eating today c\ or for saving for tomorrow, s. In the second period of his life he faces the budget constraint c2 = y2 + (1 + r)s (2.3) i.e. he can eat whatever his income is and whatever he saved from the first period. The problem that Hardy faces is quite simple: given his income and wealth he has to decide how much to eat in period 1 and how much to save for the second period of his life. The is a very standard decision problem as you have studied left and right in microeconomics, with the only difference that the goods that Hardy chooses are not apples and bananas, but consumption today and consumption tomorrow. In micro our people usually only have one budget constraint, so let us combine (2.2) and (2.3) to derive this one budget constraint, a so-called intertemporal budget constraint, because it combines income and consumption in both periods. Solving (2.3) for s yields c2 - 2/2 s = - 1 + r and substituting this into (2.2) yields ci + —-=y!+A 1 + r or ci + t^=2/i + t^ + ^ (2.4) 1 + r 1 + r Let us interpret this budget constraint. We have normalized the price of the consumption good in the first period to 1 (remember from micro that we could multiply all prices by a constant and the problem of Hardy would not change). The price of the consumption good in period 2 is j^pp, which is also the relative 2.2. SOLUTION OF THE MODEL 25 price of consumption in period 2, relative to consumption in period 1. Hence the gross real interest rate 1 + r is really a price: it is the relative price of consumption goods today to consumption goods tomorrow (note that this is a definition).1 So the intertemporal budget constraint says that total expenditures on consumption goods c\ + j+f, measured in prices of the period 1 consumption good, have to equal total income y\ + measured in units of the period 1 consumption good, plus the initial wealth of Hardy. The sum of all labor income 2/1 + 1+^ is sometimes referred to as human capital. Let us by / = yi + + A denote Hardy's total income, consisting of human capital and initial wealth. 2.2 Solution of the Model Now we can analyze Hardy's consumption decision. He wants to maximize his utility (2.1), but is constrained by the intertemporal budget constraint (2.4). To let us solve max{«(ci) + ßu(c2)} ci,c2 S.t. C\ + 1 + r One option is to use the Lagrangian method, which you should have seen in microeconomics, and you should try it out for yourself. The second option is to substitute into the objective function for c\ = I — to get max \ u\ I---— ] + f3u(c2) C2 [ \ 1 + r J This is an unconstrained maximization problem. Let us take first order conditions with respect to c2 1 -u'(l--^-\ +f3u'(c2)=0 1 + r \ 1 + r u'[l--^-r)={l + r)ßu'{c2) (2.5) Using the fact that c\ = I — we have u'(Cl) = ß(l + r)u'(c2) 1The real interest rate r, the nominal interest rate i and the inflation rate are related by the equation 1 + i 1 + r = -. 1 + 7T Thus i C± r + 7T, which is a good approximation as long as m is small relative to i, r and 7r. 26 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL /Mc2) = _J_ , 6) m'(ci) 1 + r' ' ' This condition simply states that the consumer maximizes her utility by equalizing the marginal rate of substitution between consumption tomorrow and consumption today, , with relative price of consumption tomorrow to i consumption today, -^-p- = j^pp- Condition (2.6), together with the budget constraint (2.4), uniquely determines the optimal consumption choices (ci,c2), as a function of incomes (2/1,2/2), initial wealth A and the interest rate r? One can solve explicitly for (ci, c2) in a number of ways, either algebraically or diagrammatically. We will do both below. We will then document how the optimal solution (ci, c2) changes as one changes incomes (2/1,2/2), bequests A or the interest rate r. Example 1 Suppose that the period utility function is logarithmic, that is u(c) log(c). The equation (2.6) becomes P*- 1 < c2 _ x 1 + r ci pCl 1 C'2 1 + r c2 = p(l + r)Cl (2.7) 2 Strictly speaking, for a unique solution we require another assumption on the utility function, the so-called Inada condition lim u '(c) = 00. c—*0 There is another Inada condition that is sometimes useful: lim u'(c) = 0, c—»00 but this condition is not needed to prove existence and uniqueness of an optimal solution. With the first Inada condition it is straightforward to show the existence of a unique solution to (2.5). Either we plot both sides of (2.5) and argue graphically that there exists a unique intersection, or we use some math. The function /(c2) = «'(/- - (1 + r)/9u'(c2) is continuous on C2 £ (0, (1 + r) J), strictly increasing (since u is concave) and satisfies (due to the Inada conditions) lim f(c2) < 0 lim f(c2) > 0. c2 —(l+r)J Thus by the Intermediate Value Theorem there exists a (unique, since / is stictly increasing) c~2 such that f{c2) = 0, and thus a unique solution c^ to (2.5). 2.2. SOLUTION OF THE MODEL 27 Inserting equation (2.7) into equation (2.4) yields ^(1+^ = 1 + r Cl(l + /3) = J J ci = 1 + /3 1 1 + /3 \a± ' 1 + r ci(l/i,&i,r) = T-J-j fyi + 7^7- + A ) (2.8) Since c2 = /3(1 + r)ci we find ßO + r)T Co = -_/ 1 + /3 1 + /3 Finally, since savings s = yi + A — Ci ^+Ä-TTß(yi + Tf-r+Ä. l + /3vy± ; (l + r)(l + /3) which may be positive or negative, depending on how high first period income and initial wealth is compared to second period income. So Hardy's optimal consumption choice today is quite simple: eat a fraction of total lifetime income I today and save the rest for the second period of your life. Note that the higher is income y\ in the first period of Hardy's life, relative to his second period income, y2, the higher is saving s. For general utility functions u(.) we can in general not solve for the optimal consumption and savings choices analytically. But for the general case we can represent the optimal consumption choice graphically, using the standard microeconomic tools of budget lines and indifference curves. First we plot the budget line (2.4). This is the combination of all (ci,c2) Hardy can afford. We draw ci on the x-axis and c2 on the y-axis. Looking at the left hand side of (2.4) we realize that the budget line is in fact a straight line. Now let us find two points on the line. Suppose c2 = 0, i.e. Hardy does not eat in the second period. Then he can afford c\ = y\ + A + is the first period, so one point on the budget line is (c^cf) = (yi + A + j^:,0). Now suppose c\ = 0. Then Hardy can afford to eat c2 = (1 + r)(yi + A) + yi in the second period, so a second point on the budget line is (c^, c|) = (0, (l + r)(y1 + A) +2/2)- Connecting these two points with a straight line yields the entire budget line. We can also 28 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL compute the slope of the budget line as slope = - (l + r)(yi + A) + y2 = -(l + r) Hence the budget line is downward sloping with slope (1 + r). Now let's try to remember some microeconomics. The budget line just tells us what Hardy can afford. The utility function (2.1) tells us how Hardy values consumption today and consumption tomorrow. Remember that an indifference curve is a collection of bundles (ci,c2) that yield the same utility, i.e. between which Hardy is indifferent. Let us fix a particular level of utility, say v (which is just a number). Then an indifference curve consists of all (ci, c2) such that v = m(ci) + /3m(c2) (2.10) In order to determine the slope of this indifference curve we either find a micro book and look it up, or alternatively totally differentiate (2.10) with respect to (ci,c2). To totally differentiate an equation with respect to all its variables (in this case (01,02)) amounts to the following. Suppose we change c\ by a small (infinitesimal) amount dc\. Then the right hand side of (2.10) changes by dci *u'(ci). Similarly, changing c2 marginally changes (2.10) by dc2 *ftu'ipz). If these changes leave us at the same indifference curve (i.e. no change in overall utility), then it must be the case that dci * u'(ci) + dc2 * [3u'(c2) = 0 or dc2 u'(ci) dci [3u'{c2) which is nothing else than the slope of the indifference curve, or, in technical terms, the (negative of the) marginal rate of substitution between consumption in the second and the first period of Hardy's life.3 For the example above with u(c) = log(c), this becomes dc2 _ c2 >From (2.6) we see that at the optimal consumption choice the slope of the indifference curve and the budget line are equal or w'(ci) pu'{c2) -(1 + r) = slope 3 The marginal rate of substitution between consumption in the first and second period is f3u'(c2) MRS = ' y ' u'(ci) and thus the inverse of the MRS between consumption in the second and first period. 2.2. SOLUTION OF THE MODEL 29 MRS ßu'{c2) w'(ci) 1 (2.11) This equation has a nice interpretation. At the optimal consumption choice the cost, in terms of utility, of saving one more unit should be equal to the benefit of saving one more unit (if not, Hardy should either save more or less). But the cost of saving one more unit, and hence one unit lower consumption in the first period, in terms of utility equals u'(ci). Saving one more unit yields (1 + r) more units of consumption tomorrow. In terms of utility, this is worth (1 + r)[3u\c'2). Equality of cost and benefit implies (2.11), which together with the intertemporal budget constraint (2.4) can be solved for the optimal consumption choices. Figure 2.1 shows the optimal consumption (and thus saving choices) diagrammatically (1+r)(y1+A)+y2 /Budget Line Slope: -(1+r) y2 Optimal Consumption Choice, satisfies u'(c1)/ßu'(c2)=1+r Income Point Indifference Curve Slope: u'CcJ/pu'Co,) c*, y^A y^A+y^l+r) Figure 2.1: Optimal Consumption Choice 30 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL 2.3 Comparative Statics Government policies, in particular fiscal policy (such as social security and income taxation) affects individual households by changing the level and timing of after-tax income. We will argue below that an expansion of the government deficit (and hence its outstanding debt) may also change real interest rates. In order to study the effect of these policies on the economy it is therefore important to analyze the changes in household behavior induced by changes in after-tax incomes and real interest rates. 2.3.1 Income Changes First we investigate how changes in today's income yi, next period's income y2 and initial wealth A change the optimal consumption choice. First we do the analysis for our particular example 1, then for an arbitrary utility function u(c), using our diagram developed above. For the example, from (2.8) and (2.9) we see that both c\ and c2 increase with increases in either y\, y2 or A. In particular, remembering that i = yi + + A we have that = -^>0 dl 1 + /3 dcx 13(1 + r) dl 1 + /3 > 0 and thus dci dci 1 dci 1 —r = — =- > 0 and — =----- > 0 dA dVl l + (3 dy2 (l + /3)(l + r) dc2 dc2 (3(1 + r) dc2 (3 ■77 = -— = —-— > 0 and -— =--- > 0 dA dyx 1 +13 dy2 1 + f3 ds ds [3 ds 1 dA = d^1=TTp> an dy72=~(l + p)(l + r) < The change in consumption in response to a (small) change in income is often referred to as marginal propensity to consume. From the formulas above we see that current consumption c\ increases not only when current income and inherited wealth goes up, but also with an increase in (expected) income tomorrow. Standard Keynesian consumption functions typically ignore this later impact on consumption. Similarly consumption in the second period of Hardy's life increases not only with second period income, but also with income today. Finally, an increase in current income increases savings, whereas an increase in expected income tomorrow decreases saving, since Hardy finds it optimal to 2.3. COMPARATIVE STATICS 31 consume part of the higher lifetime income already today, and bringing some of the higher income tomorrow into today requires a decline in saving. For our example we could solve for the changes in consumption behavior induced by income changes directly. In general this is impossible, but we still can carry out a graphical analysis for the general case, in order to trace out the qualitative changes on consumption and saving. In figure 2.2 we show what happens when income in the first period y\ increases to y[ > y\. As a consequence the budget line shifts out in a parallel fashion (since the interest rate, which dictates the slope of this line does not change). At the new optimum both c\ and c2 are higher than before, just as in the example. The increase in consumption due to an income increase (in either period) is referred to as an income effect. If A increases (which works just as an increase in yi) it sometimes is also called a wealth effect. The income and wealth effects are positive for consumption in both periods for the (separable) utility functions that we will consider in this class, but you should remember from standard micro books that this need not always be the case (remember the infamous inferior goods). 2.3.2 Interest Rate Changes More complicated to analyze than income changes are changes in the interest rate, since a change in the interest rate will entail three effects. Looking back to the maximization problem of the consumer, the interest rate enters at two separate places. First, on the left hand side of the budget constraint ci + 7— = Vi + T— +A = I(r) 1 + r 1 + r as relative price of the second period consumption, and second as discount factor for second period income y2. Now for concreteness, suppose the real interest rate r goes up, say to r' > r. The first effect comes from the fact that a higher interest rate reduces the present discounted value of second period income, This is often called a (human capital) wealth effect, as it reduces total resources available for consumption, since I(r') < I(r). The name human capital wealth effect comes from the fact that income y2 is usually derived from working, that is, from applying Hardy's "human capital". Note that this effect is absent if Hardy does not earn income in the second period of his life, that is, if y2 = 0. The remaining two effects stem from the term -rr-- An increase in r reduces the price of second period consumption, j^pp, which has two effects. First, since the price of one of the two goods has declined, households can now afford more; a price decline is like an increase in real income, and thus the change in the optimal consumption choices as result of this price decline is called an income effect. Finally, a decline in -j-j^ not only reduces the absolute price of second period consumption, it also makes second period consumption cheaper, relative to first period consumption (whose price has remained the same). Since second period consumption has become relatively cheaper and first period consumption relatively more expensive, one would expect that Hardy substitutes second 32 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL c\ y1+A y'!+A Income Increase Figure 2.2: A Change in Income period consumption for first period consumption. This effect from a change in the relative price of the two goods is called a substitution effect. Table 2.1 summarizes these three effects on consumption in both periods. As before, let us first analyze the simple example 1. Repeating the optimal choices from (2.8) and (2.9) First, an increase in r reduces lifetime income I(r), unless yi = 0. This is the negative wealth effect, reducing consumption in both periods, ceteris paribus. Second, we observe that for consumption c\ in the first period this is the only 2.3. COMPARATIVE STATICS 33 Incr. in r Deer, in r Effect on Cl c2 Cl c2 Wealth Effect — — + + Income Effect + + — — Substitution Effect + + Table 2.1: Effects of Interest Rate Changes on Consumption effect: absent a change in I(r), C\ does not change. For this special example in which the utility function is u(c) = log(c), the income and substitution effect exactly cancel out, leaving only the negative wealth effect. In general, as indicated in Table 2.1, the two effects go in opposite direction, but that they exactly cancel out is indeed very special to log-utility. Finally, for c2 we know from the above discussion and Table 2.1 that both income and substitution effect are positive. The term —wmch depends positively on the interest rate r reflects this. However, as discussed before the wealth effect is negative, leaving the overall response of consumption c2 in the second period to an interest rate increase ambiguous. However, remembering that I(r) = A + y\ + -^p, we see that which is increasing in r. Thus for our example the wealth effect is dominated by the income and substitution effect and second period consumption increases with the interest rate. However, for general utility functions is need not be true. Let us now analyze the general case graphically. Again we consider an increase in the interest rate from r to r' > r; evidently a decline in the interest rate can be studied in exactly the same form. What happens to the curves in Figure 2.3 as the interest rate increases? The indifference curves do not change, as they do not involve the interest rate. But the budget line changes. Since we assume that the interest rate increases, the budget line gets steeper. And it is straightforward to find a point on the budget line that is affordable with old and new interest rate. Suppose Hardy eats all his first period income and wealth in the first period, c\ = y± + A and all his income in the second period c2 = y2, in other words, he doesn't save or borrow. This consumption profile is affordable no matter what the interest rate (as the interest rate does not affect Hardy as he neither borrows nor saves). This consumption profile is sometimes called the autarkic consumption profile, as Hardy needs no markets to implement it: he just eats whatever he has in each period. Hence the budget line tilts around the autarky point and gets steeper, as shown in Figure 2.3. In the figure consumption in period 2 increases, consumption in period 1 decreases and saving increases, just as for the simple example. Note, however, that we could have drawn this picture in such a way that both (ci, c2) decline or that ci increases and c2 decreases (see again Table 2.1). So for general utility functions it is hard to make firm predictions about the consequences of Figure 2.3: An Increase in the Interest Rate an interest change. If we know, however, that Hardy is either a borrower or a saver before the interest rate change, then we have some strong results. Proposition 2 Let (c*, c|, s*) denote the optimal consumption and saving choices associated with interest rate r. Furthermore denote by (c^c^s*) the optimal consumption-savings choice associated with interest r' > r 1. If s* > 0 (that is c* < A + yi and Hardy is a saver at interest rate r), then [/(c*,^) < [/(c*,^) and either c*{ < &{ or c?2 < c?2 (or both). 2. Conversely, if s* < 0 (that is c* > A + y\ and Hardy is a borrower at interest rate r'), then U{c\,<^) > U{c\,^) and either c\ > c\ or c| > c| (or both). Proof. We only prove the first part of the proposition; the proof of the second part is identical. Remember that, before combining the two budget 2.4. BORROWING CONSTRAINTS 35 constraints (2.2) and (2.3) into one intertemporal budget constraint they read as ci + s = y1 + A C'l = 2/2 + (1 + r)s Now consider Hardy's optimal choice (c*, c|, s*) for an interest rate r. Now the interest rate increases to r' > r. What Hardy can do (of course it may not be optimal) at this new interest rate is to choose the allocation (2i, £2, s) given by ci = c* > 0 S = s* > 0 and C2 = 2/2 + (1 + r')s > 2/2 + (l + r)s* =4 This choice (ci,C2,s) is definitely feasible for Hardy at the interest rate r' and satisfies c\ > c\ and C2 > c| and thus ^(cL^) < ^(ci,c2) But the optimal choice at r' is obviously no worse, and thus U(c*1,c*2) 0. (2.12) Let by (c^,C2,s*) denote the optimal consumption choice that Hardy would choose in the absence of the constraint (2.12). There are two possibilities. 36 CHAPTER 2. A TWO PERIOD BENCHMARK MODEL 1. If Hardy's optimal unconstrained choice satisfies s* > 0, then it remains the optimal choice even after the constraint has been added.4 In other words, households that want to save are not hurt by their inability to borrow. 2. If Hardy's optimal unconstrained choice satisfies s* < 0 (he would like to borrow), then it violates (2.12) and thus is not admissible. Now with the borrowing constraint, the best he can do is set ci = 2/i + A c-2 = 2/2 s = 0 He would like to have even bigger ci, but since he is borrowing constrained he can't bring any of his second period income forward by taking out a loan. Also note that in this case the inability of Hardy to borrow leads to a loss in welfare, compared to the situation in which he has access to loans. This is shown in Figure 2.4 which shows the unconstrained optimum {c\,c2) and the constrained optimum (ci = y\ + A, c2 = y2). Since the indifference curve through the latter point lies to the left of the indifference curve through the former point, the presence of borrowing constraints leads to a loss in lifetime utility. Note that the budget line, in the presence of borrowing constraints has a kink at (yi + A,y2). For c\ < y\ + A we have the usual budget constraint, as here s > 0 and the borrowing constraint is not binding. But with the borrowing constraint Hardy cannot afford any consumption c\ > yi + A, so the budget constraint has a vertical segment at y\ + A, because regardless of what c2, the most Hardy can afford in period 1 is yi + A. What the figure shows is that, if Hardy was a borrower without the borrowing constraint, then his optimal consumption is at the kink. Finally, the effects of income changes on optimal consumption choices are potentially more extreme in the presence of borrowing constraints, which may give the government's fiscal policy extra power. First consider a change in second period income y2. In the absence of borrowing constraints we have already analyzed this above. Now suppose Hardy is borrowing-constrained in that his optimal choice satisfies ci = 2/1 + A c2 = 2/2 s = 0 4Note that this is a very general property of maximization problems: adding constraints to a maximization problem weakly decreases the maximized value of the objective function and if a maximizer of the unconstrained problem satisfies the additional constraints, it is necessarily a maximizer of the constrained problem. The reverse is evidently not true: an optimal choice of a constrained maximization problem may, but need not remain optimal once the constraints have been lifted. 2.4. BORROWING CONSTRAINTS 37 Figure 2.4: Borrowing Constraints We see that an increase in y2 does not affect consumption in the first period of his life and increases consumption in the second period of his life one-for-one with income. Why is this? Hardy is borrowing constrained, that is, he would like to take out loans against his second period income even before the increase in y2. Now, with the increase in y2 he would like to borrow even more, but still can't. Thus c\ = y\ + A and s = 0 remains optimal. An increase in y\ on the other hand, has strong effects on c\. If, after the increase Hardy still finds it optimal to set s = 0 (which will be the case if the increase in y\ is sufficiently small, abstracting from some pathological cases), then consumption in period 1 increases one-for-one with the increase in current income and consumption c2 remains unchanged. Thus, if a government cuts taxes temporarily in period 1, this may have the strongest effects on those individual households that are borrowing-constrained. CHAPTER 2. A TWO PERIOD BENCHMARK MODEL Chapter 3 The Life Cycle Model The assumption that households like Hardy live only for two periods is of course a strong one. The generalization of the analysis above was pioneered in the 1950's independently by Franco Modigliani and Albert Ando, and by Milton Friedman, with slightly different focus. Whereas Modigliani-Ando's life cycle hypothesis stressed the implications of intertemporal consumption choice models for consumption and savings profiles as well as wealth accumulation over a households lifetime, Friedman's permanent income hypothesis focused more on the impact of the timing and the characteristics of uncertain income on individual consumption choices. For the purpose of our treatment we will not distinguish between the two hypotheses, but rather see that they will come out of the same theoretical model.1 We envision a household that lives for T periods. We allow that T = oo, in which case the household lives forever. In each period t of its life the household earns after-tax income yt and consumes ct. In addition the household may have initial wealth A > 0 from bequests. In each period the household faces the budget constraint ct + st = yt + (1 + r)st-i (3.1) Here r denotes the constant exogenously given interest rate, st denotes financial assets carried over from period t to period t + 1 and st-i denotes assets from period t — 1 carried to period t. In the simple model savings and assets were the same thing, now we have to distinguish between them. Savings in period t are defined as the difference between total income yt + rst-i (labor income and interest earned) and consumption ct. Thus savings are defined as savt = yt + rst-i - ct = st- st_i (3.2) where the last inequality comes from (3.1). Thus savings today are nothing else lrrhis chapter is a bit more technical. The economic intuition however, is hopefully clear even to those who are not familiar with the mathematical tools used in this section. 39 40 CHAPTER 3. THE LIFE CYCLE MODEL but the change in the asset position of a household between the beginning of the current period and the end of the current period. In period 1 the budget constraint reads as ci + si = A + It is the goal of the household to maximize its lifetime utility U(c1,c2,..., cT) = m(ci) + /3u(c2) + l32u(c3) + ... + /3T_1«(cT) (3.3) We will often write this more compactly as T U(c) = Y,Pt-Mct) (3.4) t=i where c = (ci, c2,..., cy) denotes the lifetime consumption profile and the symbol Ylt=i stands for the sum, from t = 1 to t = T. If expression (3.4) looks intimidating, you should always remember that it is just another way of writing (3-3). As above with the simple, two period model we can rewrite the period-by-period budget constraints as a single intertemporal budget constraint. To see this, take the first- and second period budget constraint ci + si = A + yi C2 + s2 = y2 + (l + r)s1 Now solve the second equation for si c2 + s2- 2/2 Sl= 1 + r and plug into the first equation, to obtain , c2 + s2 ~ V2 . , ci +-—-= A + 2/i 1 + r which can be rewritten as ci+ —— + ——= A +2/i +—— (3.5) 1 + r 1 + r 1 + r We now can repeat this procedure: from the third period budget constraint C3 + s3 = 2/3 + (1 + r)s2 we can solve for C3 + s3 - 2/3 S2= 1 + r and plug this into (3.5) to obtain (after some rearrangements) , c2 , c3 , s3 2/2 2/3 ci + "i—TT + TT~—3 + 7-75 = A + y! + 1 + r (1 + r)2 (1 + r)2 1 + r (1 + r)2 3.1. SOLUTION OF THE GENERAL PROBLEM 41 We can continue this process T times, to finally arrive at a single intertemporal budget constraint of the form . c2 c3 cT sT ,..2/2.2/3 . 2/T ci + ---1--o+- • -H--^~rH--= ^4+2/1 + ";--1--o ■ ■ -- 1 + r (1 + r)2 (l + rf-1 (1 + r)T_1 1 + r (1 + r)2 (1 + r)T~ .(3-6) Finally we observe the following. Since denotes the saving from period T to T + 1, but the household lives only for T periods, she has no use for saving in period T + 1 (unless she values her children and wants to leave bequests, a possibility that is ruled out for now by specifying a utility function that only depends on one's own consumption, as in (3.3)). On the other hand, we do not allow the household to die in debt (what would happen if we did?) Thus it is always optimal to set = 0 and we will do so until further notice. Then (3.6) reads as C2 , c3 cT 2/2 , 2/3 , 2/T --1--Ö + . . .H--Tf^—r = A+y1 + ---1--=■ . . .H--Tf^—r + r (l + rf (l + rf'1 1 + r (l + rf (l + rf'1 (3.7) or more compactly, as E-C±^~i=A+f\ Vtti (3-8) which simply states that the present discounted value of lifetime consumption (ci,..., ct) equals the present discounted value of lifetime income (2/1,..., 2/t) plus initial bequests. As in the simple two period model, it is the goal of the household to maximize its lifetime utility (3.3), subject to the lifetime budget constraint (3.7). The choice variables are all consumption levels (ci,..., cy). Now the use of graphical analysis is not helpful anymore, since one would have to draw a picture in as many dimensions as there are time periods T (you may want to try for T = 3). Thus the only thing we can do is to solve this constrained maximization problem mathematically. We will first do so for the general case, and then consider several important examples. 3.1 Solution of the General Problem In order to maximize the lifetime utility (3.3), subject to the lifetime budget constraint (3.7) we need to make use of the theory of constrained optimization. Rather than to give a general treatment of this important subject from applied mathematics, I will simply give a cookbook version of how to do this.2 The recipe works as follows: 1. First rewrite all constraints of the problem in the form stuff = 0 21 will only deal with equality constraints here. Inequality constraints can be treated in a similar fashion. 42 CHAPTER 3. THE LIFE CYCLE MODEL For our example there is only one constraint, (3.7), so rewrite it as , , , 2/2 , 2/3 2/T c2 c3 cT A+yi + —--h---o • • •+--7rTT-ci- 1 + r (l + ry (1 + ry-1 1 + r (1 + rf (1 + r)1 ~ 2. Write down the "Lagrangian"3: take the objective function (3.3), and add all constraints, each pre-multiplied by a so-called Lagrange multiplier. This mysterious entity, usually denoted by a Greek letter, say A (read lambda), can be treated, for our purposes, as a constant number. For our example the Lagrangian then becomes £(ci,...,cT) = m(ci) + /3u(c2) + l32u(c3) + ... + /3T_1«(cT) + , / , . . 2/2 , 2/3 2/T c2 c3 A A + 2/1 +-----o • • • H--— ci — i---^ \ 1 + r (1 + r)2 (1 + rf-1 1 + r (1 + r)" T / t t \ 2/t c« \ ^ V fe? + ^ (1 + r) 3. Do what you would usually would do when solving a standard maximization problem: take first order conditions with respect to all choice variables and set them equal to 0. These conditions, together with the constraints, then determine the optimal solution to the constrained maximization problem.4 For our example the choice variables are the consumption levels (ci,..., ct) in each period of the consumers' lifetime. Taking first order conditions with respect to c\ and setting it equal to zero yields u'(Cl) — A = 0 or m'(ci) = A. (3.9) Doing the same for c2 yields pu\c2) - A—j— = 0 1 + r or (1 + r)l3u'(c2) = A (3.10) and for an arbitrary ct we find (l + r)t-1f3t-1u'(ct) = X (3.11) 3 Named after the French mathematician Joseph Lagrange (1736-1813) who pioneered the mathematics of constrained optimization 4 Those of you with advanced knowledge in mathematics may ask whether we need to check second order conditions. We will only work on problems in this course in which the first order conditions are necessary and sufficient (that is, finite dimensional convex maximization problems). 3.2. IMPORTANT SPECIAL CASES 43 Therefore, using (3.9) to (3.11) we have u'(Cl) = (l+r)pu'(c2) = ... = [(1 + r)^-1 u'(ct) = [(1 + r)pf u'(ct+1) = ... = [(1 + r)/3]T"1 u'(cT) (3-12) These equations determine the relative consumption levels across periods, that is, the ratios —, — and so forth.5 In order to determine the absolute consumption levels we have to use the budget constraint (3.7). Without further assumptions on the interest rate r, the time discount factor [3 and income (yi,... ,yx) no progress can be made, and we will soon do so. Before jumping into specific examples let us carefully interpret conditions 3.12. These conditions that determine optimal consumption choices are often called Euler equations, after Swiss mathematician Leonard Euler (1707-1783) who first derived them. Let us pick a particular time period, say t = 1. Then the equation reads as u'(c1) = {\ + r)pu'(c2) (3.13) Remember that this is a condition the optimal consumption choices (ci,c2) have to satisfy. Thus the household should not be able to improve his utility by consuming a little less in period 1, save the amount and consume a bit extra in the second period. The cost, in terms of utility, of consuming a small unit less in period 1 is —u'(ci) and the benefit is computed as follows. Saving an extra unit to period 2 yields 1 + r extra units of consumption tomorrow. The extra utility from another consumption unit tomorrow is f3u'(c2), so the total utility consequences tomorrow are (1 + r)[3v!(c2). Thus the entire consequences from saving a little more today and eating it tomorrow are -u'(ci) + (1 + r)(3u'(c2) < 0 (3.14) because the household should not be able to improve his lifetime utility from doing so. Similarly, consuming one unit more today and saving one unit less for tomorrow should also not make the household better off, which leads to -«/(c1) + (l + r)/3«/(c2) >0. (3.15) Combining the two equations (3.14) and (3.15) yields back (3.13), which simply states that at the optimal consumption choice (ci,c2) it cannot improve utility to save either more or less between period 1 and 2. 3.2 Important Special Cases 3.2.1 Equality of [5 = ^ In this case the market discounts income tomorrow, versus income today, at the same rate as the household discounts utility today versus tomorrow, j3. In 5 Also note from equation (3.9) that the Lagrange multiplier can be interpreted as the "shadow cost" of the resource constraint: if we had one more unit of income (in period 1), we could buy one more unit of consumption in period 1, with associated utility consequences u'(ci) = A. Thus A measures the marginal benefit from relaxing the intertemporal budget constraint by one unit. 44 CHAPTER 3. THE LIFE CYCLE MODEL this case, since /3(1 + r) = 1, from (3.12) we find u'(ci) = u'(c2) = ... = u'(ct) = ... = u'(cT) But now we remember that we assumed that the utility function is strictly concave (i.e. u"(c) < 0), which means that the function u'(c) is strictly decreasing in c. We therefore immediately6 have that Cl = C2 = . . . = Cf = ... = cT and consumption is constant over a households' lifetime. Households find it optimal to choose a perfectly smooth consumption profile, independent of the timing of income. The level of consumption depends solely on the present discounted value of income, plus initial bequests, but the timing of income and consumption is completely de-coupled. The smoothness of consumption over the life cycle and the fact that the timing of consumption and income are completely unrelated are the main predictions of this model and the main implications of what is commonly dubbed the life cycle hypothesis. We will now derive its implication for life cycle savings and asset accumulation. Example 3 Suppose a household lives 60 years, from age 1 to age 60 (in real life this corresponds to age 21 to age 80; before the age of 21 the household is not economically active in that his consumption is dictated by her parents). Also suppose the household inherits nothing, i.e. A = 0. Finally assume that in the first 45 years of her life, the household works and makes a constant annual income of $40, 000 per year. For the last 15 years of her life the household is retired and earns nothing; for the time being we ignore social security. Finally we make the simplifying assumption that the interest rate is r = 0; since in this subsection we assume j3 = j^pp, this implies (3=1. We want to figure out the life cycle profile of consumption, saving and asset accumulation. From the previous discussion we already know that consumption over the households' lifetime is constant, that is c\ = c2 = ... = cqo = c. What we don't know is the level of consumption. But we know that the discounted value of lifetime consumption equals the discounted value of lifetime income. So let us first compute the lifetime value of lifetime income. Here the assumption r = 0 simplifies matters, because , 2/2,2/3 , 2/60 2/1 (1 + r)2'" (l + rf-1 1 + r = 2/1 + 2/2 + 2/3 •••+ 2/60 = 2/1 + 2/2 + 2/3 •••+ 2/45 = 45* $40,000 (3.16) where we used the fact that for the last 15 years the household does not earn anything. The total discounted lifetime cost of consumption, using the fact that 6If ci > C2 we have that u1 (c\) < u'(c2), since u'(c) is by assumption strictly decreasing. Reversely, if ci < C2 then u'(ci) > uf(c2). Thus the only possible way to get uf(c\) = uf(02) is to have c\ = C2. 3.2. IMPORTANT SPECIAL CASES 45 consumption is constant at c and that the interest rate is r = 0 is Cl C2 1 + r ci + c2 + .. 60 * c C3 (1 + rf + c60 c60 (1 + r) 59 (3.17) Equating (3.16) and (3.17) yields — * $40, 000 60 $30, 000 That is, in all his working years the household consumes $10,000 less than her income and puts the money aside for consumption in retirement. With a zero interest rate, r = 0, it is also easy to compute savings in each period. For all working periods, by definition savt = yt + rst-i - ct = Vt-ct = $40,000 - $30,000 = $10,000 whereas for all retirement periods savt = yt + rst-i - ct = -ct = -$30,000 Finally we can compute the asset position of the household. Remember from (3.2) that savt = st - st-i or st = st-i + savt That is, assets at the end of period t equal assets at the beginning of period t (that is, the end of period t—1) plus the saving in period t. Since the household 46 CHAPTER 3. THE LIFE CYCLE MODEL starts with 0 bequests, so = 0. Thus si = so + savi = $0 +$10,000 = $10,000 s2 = si + saw2 = $10,000 + $10,000 = $20,000 s3 = S2 + Sav3 = $20, 000 + $10,000 = $30, 000 545 = S44 + SGW45 546 = S45 + sav46 547 = S46 + Sftf47 s60 = S59 + saw60 = $30, 000 - $30,000 = $0 The household accumulates substantial assets for retirement and then runs them down completely in order to finance consumption in old age until death. Note that this household knows exactly when she is going to die and does not value the utility of her children (or has none), so there is no point for her saving beyond her age of sure death. The life cycle profiles of income, consumption, savings and assets are depicted in Figure 3.1. Note that the y-axis is not drawn to scale, in order to be able to draw all four variables on the same graph. Also remember that age 1 in our model corresponds to age 21 in the real world, age 45 to age 66 and age 60 to age SO. Again, the crucial features of the model, and thus the diagram, are the facts that consumption is constant over the life time, de-coupled from the timing of income and that the household accumulates assets until retirement and then de-saves until her death. The previous example was based on several simplifying assumptions. In exercises you will see that the assumption r = 0, while making our life easier, is not essential for the main results. The assumption that /3(1 + r) (that is, equality of subjective discount factor and market discount factor) however, is crucial, because otherwise consumption is not constant over the households' life time. 3.2.2 Two Periods and log-Utility In the case that f3 7^ j^pp, without making stronger assumptions on the utility function we usually cannot make much progress. So now suppose that the household only lives for two periods (that is, T = 2) and has period utility u(c) = log(c). Note that we have solved this problem already; here we merely want to check that our new method yields the same result. Remembering that for log-utility u'(c) = ^, equation (3.13) yields l_ _ (l + r)(3 ci c2 = $440,000 + $10,000 = $450,000 = $450,000 - $30,000 = $420,000 = $420,000 - $30,000 = $390,000 3.2. IMPORTANT SPECIAL CASES 47 $450,000 $40,000 $30,000 $10,000 -$30,000 Consumption ct 80 Age Figure 3.1: Life Cycle Profiles, Model c2 = (1 + r)pCl Combining this with the intertemporal budget constraint Cl C2 A + V! + 2/2 1 + r 1 + r yields back the optimal solution (2.8) and (2.9). 3.2.3 The Relation between (3 and and Consumption Growth We saw in subsection 3.2.1 that if f3 = j^pp, consumption over the life cycle is constant. In this section we will show that if interest rates are high and households are patient (i.e. have a high /3) then they will choose consumption 48 CHAPTER 3. THE LIFE CYCLE MODEL to grow over the life cycle, whereas if interest rates are low and households are impatient, then they will opt for consumption to decline over the life cycle. The Case f3 > j^p Now households are patient and the interest rate is high, so that j3 > or /3(1 + r) > 1. Intuitively, in this case we would expect that households find it optimal to have consumption grow over time. Since they are patient, they don't mind that much postponing consumption to tomorrow, and since the interest rate is high, saving an extra dollar looks really attractive. So one would expect ci < c2 < ... < ct < ct+1 < ... 1, and therefore m'(ci) > 1 u'(c2) u'(ci) > u'(c2) (3.19) But again remember that u'(c) is a strictly decreasing function, so the only way that (3.19) can be true is to have c\ < c2. Thus, consumption is higher in the second than in the first period of a households' life. For an arbitrary age t equation (3.12) implies [(1 + r)^-1 u'{ct) = [(1 + r)/3] V(ct+1) u'{ct) [{X + r)p] u'{ct+1) [(l + r)ß] - = (l + r)ß>l so that the same argument as for age 1 implies ct+1 > ct for an arbitrary age t. Thus consumption continues to rise throughout a households' life time, as proposed in (3.18). The exact growth rate and level of consumption, of course, can only be determined with knowledge of the form of the utility function u and the concrete values for income. The bottom line from this subsection: high interest rates and patience of households makes for little consumption expenditures today, relative to tomorrow. 3.3. EMPIRICAL EVIDENCE 49 The Case P < jjpp Now households are impatient and the interest rate is low, so that j3 < or [3(1 + r) < 1. Intuitively, we should obtain exactly the reverse result from the last subsection: we now would expect that households find it optimal to have consumption decline over time. Since they are impatient, they don't want to eat now rather than tomorrow, and since the interest rate is low, saving an extra dollar for tomorrow only brings a low return. An identical argument to the above easily shows that now ci > c2 > ... > ct > ... > cT. (3.20) Therefore low interest rates are conducive to high consumption today, relative to tomorrow, even more so if households are very impatient. This discussion concludes our treatment of the basic model which we will use in order to study the effects of fiscal policies. So far our households lived in isolation, unaffected by any government policy. The only interaction with the rest of the economy came through financial markets, on which the household was assumed to be able to borrow and lend at the market interest rate r. We will now introduce a government into our simple model and study how simple tax and transfer policies affect the private decisions of households. Before that we have a quick look at consumption over the life cycle from the data. 3.3 Empirical Evidence If one follows an average household over its life cycle, two main stylized facts emerge. First, disposable income follows a hump over the life cycle, with a peak around the age of 45 (the age of the household is defined by the age of the household head). This finding is hardly surprising, given that at young ages households tend to obtain formal education or training on the job and labor force participation of women is low because of child bearing and rearing. As more and more agents finish their education and learn on the job as well as promotions occur, average wages within the cohort increase. Average disposable income at age 45 is almost 2.5 times as high as average personal income at age 25. After the age of 45 disposable income first slowly, then more rapidly declines as more and more people retire and labor productivity (and thus often wages) fall. The average household at age 65 has only 60% of the personal income that the average household at age 45 obtains. The second main finding is the surprising finding. Not only personal income, but also consumption follows a hump over the life cycle. In other words, consumption seems to track income over the life cycle fairly closely, rather than be completely decoupled from it, as our model predicts. Figure 3.2 (taken from Krueger and Fernandez-Villaverde, 2003) documents the life cycle profile of consumption, with and without adjustment for family size. The key observation from this figure is that consumption displays a hump over the life cycle, and 50 CHAPTER 3. THE LIFE CYCLE MODEL that this hump persists, even after controlling for family size. The figure is constructed using semi-parametric econometric techniques, but the same picture emerges if one uses more standard techniques that control for household age with age dummies. Expenditures, Total and Adult Equivalent 4500 4000 3500 3000 2500 2000 1500 Total Adult Equivalent 20 30 40 50 60 Age 70 90 Figure 3.2: Consumption over the Life Cycle 3.4 Potential Explanations There are a number of potential extensions of the basic life cycle model that can rationalize a hump-shaped consumption. So far, the prediction of the model is that consumption is either monotonically upward trending, monotonically downward trending or perfectly fiat over the life cycle. So the basic theory can account for at most one side of the empirical hump in life cycle consumption. Here are several other factors that, once appropriately added to the basic model, may account for (part of) the data: 3.4. POTENTIAL EXPLANATIONS 51 • Changes in household size and household composition: Not only income and consumption follow a hump over the life cycle in the data, but also family size. Our simple model envisioned a single individual composing a household. But if household size changes over the life cycle (people move in together, get married, have children which grow and finally leave the household, then one of the spouses dies), it may be optimal to have consumption follow household size. The life cycle model only asserts that marginal utility of consumption should be smooth over the life cycle, not necessarily consumption expenditures themselves. However, in the previous figure we presented one line that adjusts the consumption data for household size, using so-called household equivalence scales. These scales try to answer the simple question as to how much more consumption expenditures as household have to have in order to obtain the same level of per capita utility, as the size of the household changes. Concretely, suppose that you move in with your boyfriend or girlfriend, the equivalence scale asks: how much more do you have to spend for consumption to be as happy off materially (that is, not counting the joy of living together) as before when you were living by yourself. The number researchers come up with usually is somewhere between 1 and 2, because it requires some additional spending to make you as happy as before (two people eat more than one), but it may not require double the amount (it takes about as much electricity to cook for two people than for one). Technically, this last consideration is called economies of scale in household production. So if one applies household equivalence scales to the data, the size of the hump in lifetime consumption is reduced by about 50%. That is, changes in household size and composition can account for half of the hump, with the remaining part being left unexplained by the life cycle model augmented by changes in family size.7 • The life cycle model was presented with exogenous income falling from the sky. If households have to work to earn their income and dislike work, that is, have the amount of leisure in the utility function, then things get more complicated. Suppose that consumption and leisure are separable in the utility function, that is, suppose that the utility function takes the form t U(c,l) = ^^-^(ct./t) t=i t t=l 7The exact fraction demographics can account for is still debated. See Fernandez-Villaverde and Krueger (2004) for a discussion. On a technical note, since there is no data set that follows individuals over their entire life time and collects consumption data, one has to construct these profiles using the synthetic cohort technique, pioneered by Deaton (1985). Again see Fernandez-Villaverde and Krueger (2004) for the details. CHAPTER 3. THE LIFE CYCLE MODEL where lt is leisure at age t and v is an increasing and strictly concave function. Then our theory above goes through unchanged and the predictions remain the same. But if consumption and leisure are substitutes (if you work a lot, the marginal utility from your consumption is high), then if labor supply is hump-shaped over the live cycle (because labor productivity is), then households may find it optimal to have a hump-shaped labor supply and consumption profile over the life cycle. This important point was made by Nobel laureate James Heckmen in his dissertation (1974). But Fernandez-Villaverde and Krueger (2004) provide some suggestive evidence that this channel is likely to explain only a small fraction of the consumption hump. We saw that the model can predict a declining consumption profile over the life cycle if /3(1 + r) < 1. Now suppose that young households can't borrow against their future labor income. Thus the best thing they can do is to consume whatever income whey have when young. Since income is increasing in young ages, so is consumption. As households age, at some point they want to start saving (rather than borrowing), and no constraint prevents them from doing so. But now the fact that /3(1 + r) < 1 kicks in and induces consumption to fall. Thus the combination of high impatience and borrowing constraints induces a hump-shaped consumption profile. Empirically, one problem of this explanation is that the peak of the hump in consumption does not occur until about age 45, a point in life where the median household already has accumulated sizeable financial assets, rather than still being borrowing-constrained. Finally, we may want to relax the assumption about certain incomes and certain lifetime. If an individual thinks that he will only survive until 100 with certain probability less than one, at age 20 he will plan to save less for age 100 than if she knows for sure she'll get that old. Thus realized consumption at age 100 will be smaller with lifetime uncertainty as without. Since death probabilities increase with age, this induces a decline in optimal consumption as the household ages. The death probabilities act like an additional discount factor in the household's maximization problem. On the other hand, suppose you are 25, with decent income, and you expect your income to increase, but be quite risky. Under the assumption that people have a precautionary savings motive (we will see below that this requires the assumption u"'(c) > 0), households will save for precautionary reasons and consume less when young than under certainty, even if income is expected to rise over their lifetime. Then, as the household ages and more and more uncertainty is resolved, the precautionary savings motive loses in importance, households start to consume more, and thus consumption rises over the life cycle, until death probabilities start to become important and consumption starts to fall again, rationalizing the hump in life cycle consumption in the data. Attanasio et al. (1999) show that a standard life cycle model, enriched by changes in household size and uncertainty about income and lifetime is capable of generating a 3.4. POTENTIAL EXPLANATIONS 53 hump in consumption over the life cycle of similar magnitude and timing as in the data. Rather than discussing these extensions of the model in detail we will now turn to the use of the life cycle model for the analysis of fiscal policy. At the appropriate points we will discuss how the conclusions derived with the simple model change once the model is enriched by some of the elements discussed above. CHAPTER 3. THE LIFE CYCLE MODEL Part II Positive Theory of Government Activity 55 Chapter 4 Dynamic Theory of Taxation In this chapter we want to study how government tax and transfer programs that change the size timing of after-tax income streams affect individual consumption and savings choices. We first discuss the government budget constraint, and then establish an important benchmark result that suggests that, under certain conditions, the timing of government taxes, does not affect the consumption choices of individual households. This result, first put forward by David Ricardo (1772-1823), is therefore often called Ricardian Equivalence. After analyzing the most important assumptions for the Ricardian Equivalence theorem to hold, we finally study the impact of consumption taxes, labor income taxes and capital income taxes on individual household decisions, provided that these taxes are not of lump-sum nature. In chapter 1 we presented data for the govvernment budget. For completeness, we here repeat the federal government budget for the U.S. for the year 2005. We now want to group the receipts and outlays of the government into three broad categories, in order to map our data into the theoretical analysis to follow. Let government expenditures Gt be comprised of1 Gt = Defense + International Affairs + Health + Other Outlays and net taxes Tt be comprised of Tt = Taxes + Social Insurance Receipts + Other Receipts - Medicare - Social Security - Income Security 1 There are small differences between government expenditures Gt as denned in this section and government consumption as measured in NIPA, but this fine distinction is inconsequential for our purposes. 57 58 CHAPTER 4. DYNAMIC THEORY OF TAXATION 2005 Federal Budget (in billion $) Receipts 2153.9 Individual Income Taxes 927.2 Corporate Income Taxes 278.3 Social Insurance Receipts 794.1 Other 154.2 Outlays 2472.2 National Defense 495.3 International Affairs 34.6 Health 250.6 Medicare 298.6 Income Security 345.8 Social Security 523.3 Net Interest 184.0 Other 339.9 Surplus -318.3 Table 4.1: Federal Government Budget, 2005 that is, Tt is all tax receipts from the private sector minus all transfers given back to the private sector. Finally let r denote the interest rate and Bt_i (for bonds) denote the outstanding government debt. Then rBt-i = Net Interest We now will discuss the government budget constraint, using only these symbols (Gt,Tt, Bt-i,r). The previous discussion should allow you to always go back from our theory to entities that you see in the data. 4.1 The Government Budget Constraint Like private households the government cannot simply spend money without having revenues. In developed countries the two main sources through which the government can generate revenues is to levy taxes on private households (e.g. via income taxes) and to issue government bonds (i.e. government debt).2 The main uses of funds are to finance government consumption (e.g. buying tanks), government transfers to private households (e.g. unemployment benefits) and the repayment of outstanding government debt. Let us formalize the government budget constraint. First assume that when the country was formed, the first government does not inherit any debt from the past. Denote by t = 1 the first period a country exists with its own government 2 In addition, the government usually can print fiat currency; the revenue from doing do, called "seigneurage" it a small fraction of total government revenues. It will be ignored from now on. 4.1. THE GOVERNMENT BUDGET CONSTRAINT 59 budget (for the purposes of the US, period 1 corresponds to the year 1776). At time 1 the budget constraint of the government reads as G1=T1+B1 (4.1) where Gi is government expenditures in period 1, T\ are total taxes taken in by the government (including payroll taxes for social security) minus transfers to households (e.g. social security payments, unemployment compensation etc.), and B\ are government bonds issued in period 1, corresponding to the outstanding government debt. For an arbitrary period t, the government budget constraint reads as Gt + (1 + r)Bt_! =Tt + Bt (4.2) where Bt_i are the government bonds issued yesterday that come due and need to be repaid, including interest, today. For simplicity we assume that all government bonds have a maturity of one period. First, we can rewrite (4.2) as Gt-Tt + rBt-i=Bt-Bt-i. (4.3) The quantity Gt —Tt, the difference between current government spending and tax receipts (net of transfers) is often referred to as the primary government deficit; it is the government deficit that ignores interest payments on past debt. This number is often used as a measure of current fiscal responsibility, since interest payments for past debt are inherited from past years (and thus past governments). The current total government deficit is given by the sum of the primary deficit and interest payments on past debt, or deft = Gt-Tt + rBt-i. (4.4) Equation (4.3) simply states that a government deficit (i.e. deft > 0) results in an increase of the government debt, since Bt — Bt-i > 0 and thus Bt > Bt-i-That is, the number of outstanding bonds at the end of period t is bigger than at the end of the previous period, and government debt grows. Obviously, if the government manages to run a surplus (i.e. deft < 0), then it can repay part of its debt. We now can do with the government budget constraint exactly what we did before for the budget constraint of private households. Equation (4.2), for t = 2, reads as G2 + (1 + r)B1 =T2 + B2 or T2 + B2 — G2 i3i = - 1 + r Plug this into equation (4.1) to obtain T2 + B2 — G2 1 + r G1 + -^- = T1 + -^- + -^_ 1+r 1+r 1+r 60 CHAPTER 4. DYNAMIC THEORY OF TAXATION We can continue this process further by substituting out for B2, again using (4.2), for t = 3 and so forth. At the end of this we obtain the intertemporal government budget constraint G2 G*3 Ct m T2 T3 Tj* Bj* G\ + ---1--o+- • -H--Tp-t = Tl~\----1--o+- • -H--Tp---t!" 1 + r (1 + r)2 (l + rf-1 1 + r (1 + rf (l + rf'1 (1 + r)T~ We will assume that even the government cannot die in debt and will not find it optimal to leave positive assets, so that B? = 0.3 Thus the intertemporal government budget constraint reads as G2 Gs Gt „ T2 Ts Tt Gi + ---1--0+. • -H--Tp—t = 11 + --1--o • -~l--?p—t 1 + r (1 + r)2 (l + rf'1 1 + r (1 + r)2 (l + rf'1 or more compactly, as t t Gt ^ Tt \ ^ "j-t _ \ ^ J-t If the country is assumed to live forever, we write the government constraint as Gt A Tt 5 (1 + r)*"1 5 (1 + r)*-1 t In short, the government is constrained in its tax and spending policy by a condition that states that the present discounted value of total government expenditures ought to equal the present discounted value of total taxes, just as for private households. The only real difference is that the government may live much longer than private households, but other than that the principle is the 4.2 The Timing of Taxes: Ricardian Equivalence 4.2.1 Historical Origin How should the government finance a given stream of government expenditures, say, for a war? There are two principal ways to levy revenues for a government, namely to tax in the current period or to issue government debt in the form of government bonds the interest and principal of which has to be paid via taxes in the future. The question then arise what the macroeconomic consequences of using these different instruments are, and which instrument is to be preferred from a normative point of view. The Ricardian Equivalence Hypothesis claims that it makes no difference, that a switch from taxing today to issuing debt and taxing tomorrow does not change real allocations and prices in the economy. It's origin dates back to the classical economist David Ricardo (1772-1823). He 3 We could do better than simply assuming this, but this would lead us too far astray. 4.2. THE TIMING OF TAXES: RIGARDIAN EQUIVALENCE 61 wrote about how to finance a war with annual expenditures of £20 millions and asked whether it makes a difference to finance the £20 millions via current taxes or to issue government bonds with infinite maturity (so-called consols) and finance the annual interest payments of £1 million in all future years by future taxes (at an assumed interest rate of 5%). His conclusion was (in "Funding System") that in the point of the economy, there is no real difference in either of the modes; for twenty millions in one payment [or] one million per annum for ever ... are precisely of the same value Here Ricardo formulates and explains the equivalence hypothesis, but immediately makes clear that he is sceptical about its empirical validity ...but the people who pay the taxes never so estimate them, and therefore do not manage their affairs accordingly. We are too apt to think, that the war is burdensome only in proportion to what we are at the moment called to pay for it in taxes, without reflecting on the probable duration of such taxes. It would be difficult to convince a man possessed of £20, 000, or any other sum, that a perpetual payment of £50 per annum was equally burdensome with a single tax of £1,000. Ricardo doubts that agents are as rational as they should, according to "in the point of the economy", or that they rationally believe not to live forever and hence do not have to bear part of the burden of the debt. Since Ricardo didn't believe in the empirical validity of the theorem, he has a strong opinion about which financing instrument ought to be used to finance the war war-taxes, then, are more economical; for when they are paid, an effort is made to save to the amount of the whole expenditure of the war; in the other case, an effort is only made to save to the amount of the interest of such expenditure. Ricardo thought of government debt as one of the prime tortures of mankind. Not surprisingly he strongly advocates the use of current taxes. Now we want to use our simple two-period model to demonstrate the Ricardian Equivalence result and then investigate the assumptions on which it relies. 4.2.2 Derivation of Ricardian Equivalence Suppose the world only lasts for two periods, and the government has to finance a war in the first period. The war costs G\ dollars. For simplicity assume that the government does not do any spending in the second period, so that G2 = 0. We want to ask whether it makes a difference whether the government collects taxes for the war in period 1 or issues debt and repays the debt in period 2. 62 CHAPTER 4. DYNAMIC THEORY OF TAXATION The budget constraints for the government read as Gi = T1 + B1 (l + rOBi = T2 where we used the fact that G2 = 0 and B2 = 0 (since the economy only lasts for 2 periods). The two policies are • Immediate taxation: T\ = G\ and B\ = T2 = 0 • Debt issue, to be repaid tomorrow: T\ = 0 and B\ = G\, T2 = (l + r)Bi = (l + rJGj. Note that both policies satisfy the intertemporal government budget constraint T2 1 + r Now consider how individual private behavior changes between the two policies. Remember that the typical household maximizes utility u(ci) + f3u(c2) subject to the lifetime budget constraint c1 + -^=y1 + -^ + A (4.5) 1 + r 1 + r where yi and y2 are the after-tax incomes in the first and second period of the households' life. Write Vi = ei - Ti (4.6) 2/2 = e2 - T2 (4.7) where e\, e2 are the pre-tax earnings of the household and T\, T2 are taxes paid by the household. The only thing that the government policies affect are the after tax incomes of the household. Substitute (4.6) and (4.6) into (4.5) to obtain . c2 e2 - T2 ci + —— = e1-T1 + ——--h A 1 + r 1 + r or ci + —— + Ti + —— = ei + —— + A 1 + r 1 + r 1 + r In other words, the household spends the present discounted value of pre-tax income, including initial wealth, e\ + _|_ A Qn the present discounted value of consumption expenses Cx +ancj the present discounted value of income taxes. Two tax-debt policies that imply exactly the same present discounted value of lifetime taxes therefore lead to exactly the same lifetime budget constraint and thus exactly the same individual consumption choices. This is the essence of the Ricardian Equivalence theorem, which we shall state in its general form below. Before that let us check the present discounted value of taxes under the two policy options discussed above 4.2. THE TIMING OF TAXES: RIGARDIAN EQUIVALENCE 63 • For immediate taxation we have T\ = G± and T2 = 0, and thus T\ + = • For debt issue we have T\ = 0 and T2 = (l + r)Gi, and thus T\ + j^p = G\ Therefore both policies imply the same present discounted value of lifetime taxes for the household; that is, the household perfectly rationally sees that, for the second policy, she will be taxed tomorrow because the government debt has to be repaid, and therefore prepares herself correspondingly. The timing of taxes does not matter, as long its lifetime present discounted value is not changed. Consumption choices of the household do not change, but savings choices do. This cannot be seen from the intertemporal household budget constraint (because this constraint was obtained substituting out savings), so let us go back to the period by period budget constraints ci + s = ei—Ti C'2 = e2 - T2 + (1 + r)s Let denote (c*, c?2) the optimal consumption choices in the two periods; we have already argued that these optimal choices are the same under both policies. Also let s* denote the optimal saving (or borrowing, if negative) choice under the first policy of immediate taxation. How does the household change its saving choice if we switch to the second policy, debt issue and taxation tomorrow. Let s denote the new saving policy. Again since the optimal consumption choice is the same between the two policies we have (remember T\ = 0 under the second policy) c* = ei — Ti — s* = e\ — s so that ei — Ti — s* = e\ — s S = s* + Ti. That is, under the second policy the household saves exactly T\ more than under the first policy, the full extent of the tax reduction from the second policy. This extra saving T\ yields (1 + r)T\ extra income in the second period, exactly enough to pay the taxes levied in the second period by the government to repay its debt. To put it another way, private households under policy 2 know that there will be higher taxes in the future and they adjust their private savings so to exactly be able to offset them with higher saving. Obviously the same argument can be done in a model where households and the government live for more than two periods, and for all kinds of changes in the timing of taxes. Let us now state Ricardian Equivalence in its general form. Theorem 4 (Ricardian Equivalence) A policy reform that does not change government spending (Gi,..., Gt), and only changes the timing of taxes, but leaves 64 CHAPTER 4. DYNAMIC THEORY OF TAXATION the present discounted value of taxes paid by each household in the economy has no effect on aggregate consumption in any time period. We could in fact have stated a much more general theorem, asserting that interest rates, GDP, investment and national saving (the sum of private and public saving) are unaffected by a change in the timing of taxes, but for this to be meaningful we would need a model in which interest rates, investment and GDP are determined endogenously within the model, which we have not yet constructed. Also, this theorem relies on several assumptions, which we have not made very explicit so far, but will do so in the next section. What does this discussion imply for the current government deficit? The theorem says that the timing of taxes (i.e. running a deficit today and repaying it with higher taxes tomorrow) should not matter for individual decisions and the macro economy, so long as government spending is left unchanged. This sounds good news, but one should not forget why the theorem is true: households foresee that taxes will increase in the future and adjust their savings correspondingly; after all, there is a government budget constraint that needs to be obeyed. In addition, the theorem requires a series of important assumptions, as we will now demonstrate. 4.2.3 Discussion of the Crucial Assumptions Absence of Binding Borrowing Constraint You already saw in chapter 1 and homework 1 that binding borrowing constraints can lead a household to change her consumption choices, even if a change in the timing of taxes does not change her discounted lifetime income. In the thought experiment above, if households are borrowing constrained then the first policy (taxation in period 1) leads to a decline in first period consumption by the full amount of the tax. Second period consumption, on the other hand, remains completely unchanged. With government debt finance of the reform, consumption in both periods may go down, since households rationally forecast the tax increase in the second period to pay off the government debt. Example 5 Suppose the Franch-British war in the U.S. costs £100 per person. Households live for two periods, have utility function log(ci) + log(c2) and pre-tax income of £1,000 in both periods of their life. The war occurs in the first period of these households' lives. For simplicity assume that the interest rate is r = 0. As before, the two policy options are to tax £100 in the first period or to incur £100 in government debt, to be repaid in the second period. Since the interest rate is 0, the government has to repay £100 in the second period (when the war is over). Without borrowing constraints we know from the general theorem above that the two policies have identical consequences. In particular, 4.2. THE TIMING OF TAXES: RIGARDIAN EQUIVALENCE 65 under both policies discounted lifetime income is £1, 900 and 1,900 ci = c2 = -L^— = 950 Now suppose there are borrowing constraints. The optimal decision with borrowing constraint, under the first policy is c\ = y\ = 900 and c2 = y2 = 1000, whereas under the second policy we have, under borrowing constraints, that c\ = c2 = 950 (since the optimal choice is to consume 950 in each period, and first period income is 1000, the borrowing constraint is not binding and the unconstrained optimal choice is still feasible, and hence optimal). This counter example shows that, if households are borrowing constrained, the timing of taxes may affect private consumption of households and the Ricar-dian equivalence theorem fails to apply. Current taxes have stronger effects on current consumption than the issuing of debt and implied future taxation, since postponing taxes to the future relaxes borrowing constraints and my increase current consumption. No Redistribution of the Tax Burden Across Generations If the change in the timing of taxes involves redistribution of the tax burden across generations, then, unless these generations are linked together by operative, altruistically motivated bequest motives (we will explain below what exactly we mean by that) Ricardian equivalence fails. This is very easy to see in another simple example. Example 6 Return to the Franch British war in the previous example, but now consider the two policies originally envisioned by David Ricardo. Policy 1 is to levy the £100 cost per person by taxing everybody £100 at the time of the war. Policy 2 is to issue government debt of £100 and to repay simply the interest on that debt (without ever retiring the debt itself). Let us assume an interest rate of 5%. Thus under policy 2 households face taxes of T2 = £5, T3 = £5 and so forth. Now consider a household born at the time of the French British war. Pre-tax income and utility function are identical to that of the previous example. Thus, under policy 1, his present discounted value of lifetime income is I = £1000 - £100 + £100° = 1852.38 1.05 and under policy 2 it is £950 £1000+-= 1904.76 1.05 Since with the utility function given above we easily see that under policy 1 consumption equals ci = 926.19 c2 = 972.50 66 CHAPTER 4. DYNAMIC THEORY OF TAXATION and under policy 2 it equals ci = 952.38 c2 = 1000.00 Evidently, because lifetime income is higher under policy 2, the household consumes more in both periods (without borrowing constraints) and strictly prefers policy 2. What happens is that under policy 2, part of the cost of the war is borne by future generations that inherit the debt from the war, at least the interest on which has to be financed via taxations The point that changes in the timing of taxes may, and in most instances will, shift the burden of taxes across generations, was so obvious that for the longest time Ricardian equivalence was thought to be an empirically irrelevant theorem (as a mathematical result it is obviously true, but it was thought to be irrelevant for the real world). Then, in 1974 Robert Barro (then at the University of Chicago, now a professor at Harvard University) wrote a celebrated article arguing that Ricardian equivalence may not be that irrelevant after all. While the technical details are somewhat involved, the basic idea is simple. First, let us suppose that households live forever (or at least as long as the government). Consider two arbitrary government tax policies. Since we keep the amount of government spending Gt fixed in every period, the intertemporal budget constraint requires that the two tax policies have the same present discounted value. But without borrowing constraints only the present discounted value of lifetime aftertax income matters for a household's consumption choice. But since the present discounted value of taxes is the same under the two policies it follows that (of course keeping pre-tax income the same) the present discounted value of aftertax income is unaffected by the switch from one tax policy to the other. Private decisions thus remain unaffected, therefore all other economic variables in the economy remain unchanged by the tax change. Ricardian equivalence holds. But how was Barro able to argue that households live forever, when in the real world they clearly do not. The key to his arguments are bequests. Suppose that people live for one period and have utility functions of the form where V is the maximal lifetime utility your children can achieve in their life if you give them bequests b. As before, c\ is consumption of the person currently alive. Now the parameter [3 measures intergenerational altruism (how much you love your children). A value of [3 > 0 indicates that you are altruistic, a value U(Cl) + ßV(h) 4 Note that even a positive probability of dying before the entire debt from the war is repaid is sufficient to invalidate Ricardian equivalence. 4.2. THE TIMING OF TAXES: RIGARDIAN EQUIVALENCE 67 of [3 < 1 indicates that you love your children, but not quite as much as you love yourself. The budget constraint is ci + b1 = y1 where y\ is income after taxes of the person currently alive. Bequests are constrained to be non-negative, that is b\ > 0. The utility function of the child is given by U(c2) + (3V(b2) and the budget constraint is c2 + b2 = y2 + (1 + r)bi By noting that V(pi) is nothing else but the maximized value of U(c2) + (3V(b2) one can now easily show that this economy with one-period lived people that are linked by altruism and bequests (so-called dynasties) is exactly identical to an economy with people that live forever and face borrowing constraints (since we have the restriction that bequests b\ > 0, b2 > 0 and so forth). Now from our previous discussion of borrowing constraints we know that binding borrowing constraints invalidate Ricardian equivalence, which leads us to the following Conclusion 7 In the Barro model with one-period lived individuals Ricardian equivalence holds if (and only if) a) individuals are altruistic ([3 > 0) and bequest motives are operative (that is, the constraint on bequests bt > 0 is never binding in that people find it optimal to always leave positive bequests). The key question for the validity of the Barro model (and thus Ricardian equivalence) is then whether the real world is well-approximated with all people leaving positive bequests for altruistic reasons.5 Thus a big body of empirical literature investigated whether most people, or at least those people that pay the majority of taxes, leave positive bequests. In class I will discuss some of the findings briefly, but the evidence is mixed, with slight favor towards the hypothesis that not enough households leave significant bequests for the infinitely lived household assumption to be justified on empirical grounds. Lump-Sum Taxation A lump-sum tax is a tax that does not change the relative price between two goods that are chosen by private households. These two goods could be consumption at two different periods, consumption and leisure in a given period, or leisure in two different periods. In section 4.4 we will discuss in detail how non-lump sum taxes (often call distortionary taxes, because they distort private decisions) impact optimal consumption, savings and labor supply decisions. Here we 5 One can show that if parents leave bequests to children for strategic reasons (i.e. threaten not to leave bequests if the children do not care for them when they are old), then again Ricardian equivalence breaks down, because a change in the timing of taxes changes the severity of the threat of parents (it's worse to be left without bequests if, in addition, the government levies a heavy tax bill on you). 68 CHAPTER 4. DYNAMIC THEORY OF TAXATION simply demonstrate that the timing of taxes is not irrelevant if the government does not have access to lump-sum taxes. Example 8 This example is similar in spirit to the last question of your first homework, but attempts to make the source of failure of Ricardian equivalence even clearer. Back again to our simple war finance example. Households have utility of log(ci) + log(c2) income before taxes of £1000 in each period and the interest rate is equal to 0. The war costs £100. The first policy is to levy a £100 tax on first period labor income. The second policy is to issue £100 in debt, repaid in the second period with proportional consumption taxes at rate r. As before, under the first policy the optimal consumption choice is ci = C2 = £950 s = £900 - £950 = -£50 The second policy is more tricky, because we don't know how high the tax rate has to be to finance the repayment of the £100 in debt in the second period. The two budget constraints under policy 2 read as ci + s = £1000 c2(l + r) = £1000 + s which can be consolidated to ci + (1 + t)c2 = £2000 Maximizing utility subject to the lifetime budget constraint yields ci = £1000 £1000 C2 = TT7 We could stop here already, since we see that under the second policy the households consumes strictly more than under the first policy. The reason behind this is that a tax on second period consumption only makes consumption in the second period more expensive, relative to consumption in the first period, and thus households substitute away from the now more expensive to the now cheaper good. The fact that the tax changes the effective relative price between the two goods qualifies this tax as a non-lump-sum tax. For completeness we solve for second period consumption and saving. The government must levy £100 in taxes. But tax revenues are given by r£1000 4.2. THE TIMING OF TAXES: RIGARDIAN EQUIVALENCE Setting this equal to 100 yields 69 100 = 1000* T 0.1 1 + t t 1 + t H= 0.1111 0.9 Thus c2 = 900 s = 0 Finally we can easily show that households prefer the lump-sum way of financing the war (policy 1) than the distortionary way (policy 2), since log(950) + log(950) > log(1000) + log(900). Even though this is just a simple example, it tells a general lesson: with distortionary taxes Ricardian equivalence does not hold and households prefer lump sum taxation for a given amount of expenditures to distortionary taxation. 70 CHAPTER 4. DYNAMIC THEORY OF TAXATION 4.3 An Excursion into the Fiscal Situation of the US In principal, the projection of the long-run fiscal situation of the government is straightforward. Start with the total debt the U.S. government owes in 2006, project both outlays and receipts into the future and thus arrive at the level of government debt at any time in the future. Obviously, the forecast of future outlays and revenues is far from a trivial task. It is as hard, and most likely quite harder, to forecast the level of government debt in 2050 than to forecast the weather in the same year. Fortunately, the report "Fiscal and Generational Imbalances" by Jagadeesh Gokhale and Kent Smetters finds a concise way to summarize the current long-run fiscal situation of the U.S. government. Before going into the details, a word about the authors. Gokhale was a consultant to the Department of the Treasury from July to December 2002 and is now a senior fellow at the Cato Institute. Smetters was assistant secretary at the U.S. Treasury from 2001 to 2002 and a consultant from August 2002 to February 2003 (before coming back to his regular job as an Associate Professor of risk and insurance at the Wharton School). 4.3.1 Two Measures of the Fiscal Situation Gokhale and Smetters define two crucial measures of the fiscal situation FL = PVEt - PVRt - A, where PVEt is the present discounted value of projected expenditures under current fiscal policy at the end of period t, PV Rt stands for the present discounted value of all projected receipts and At stands for assets (debt, if negative) that the government owns at the end of period t. Thus the measure Fit, which the authors call fiscal imbalance, measures the aggregate shortfall in the governments' finances, due to past behavior as captured in At, and current and future behavior, as captured in PVEt — PV Rt. In terms of our previous notation PVEt and PVRt T=t + 1 (1 + r)7 as well as t rp * s~i At = y—-^-Y t 4.3. AN EXCURSION INTO THE FISCAL SITUATION OF THE US 71 Thus our intertemporal budget constraint suggests that a fiscal policy that is feasible must necessarily have FIt = 0; in other words, if one computes a FIt > 0 under current and projected future policy, then fiscal policy has to change. Also note that one can compute the measure FIt for the entire federal government, or for selected programs (such as social security) separately. In order to assess which generations bear what burden of the total fiscal imbalance, an additional concept is needed. For example, FIt would not show a change if social security benefits today would be increased, to be financed with future increases in payroll taxes. In order to capture the effects of such policy changes Gokhale and Smetters define as GIt = PVE^ - PVRf - At where GIt is the generational imbalance at the end of period t, PVE\ is the present discounted value of outlays paid to generations currently alive in period t, with PVRt defined correspondingly. Thus GIt is that part of the fiscal imbalance FIt that results from transactions of the government with past (through At) and living generations; the difference Fit - GIt then denotes the projected part of fiscal imbalance due to future generations. 4.3.2 Main Assumptions In this subsection we collect the main assumption underlying Gokhale and Smetters' results. • Real interest rate (discount rate for the present value calculations) of 3.65% per annum (average yield on a 30 year Treasury bond in recent years). Note: using a larger discount factor would, other things equal, reduce the importance of future polices on the FIt measure. • An annual growth rate of labor productivity of 2.0%. • As we will see, the crucial assumptions for the numbers are related to health care costs. Here the authors use the assumptions of the Office of Management and Budget (OMB) of the White House for the annual increases in health care costs for the near future. These costs are assumed to grow substantially faster than GDP per capita until 2025, then growth of health care costs is assumed to fall, but still be 1.5% higher than growth of GDP per capita. For the long run Gokhale and Smetters assume that growth of health care costs slows down to the extent that health care outlays as a fraction of GDP stabilize. Overall the authors argue (and I would agree) that these are quite optimistic assumptions. • Social security benefits remain as mandated by current law. 72 CHAPTER 4. DYNAMIC THEORY OF TAXATION Fiscal Imbalance (Billion of Constant 2006 Dollars) Part of the Budget 2006 2009 2012 FI in Social Security 7, 684 8,672 9,737 FI in Medicare 65,181 72,291 79, 859 FI in Rest of Federal Government -9,190 -9,987 -10,762 Total FI 63, 675 70, 976 78, 834 Table 4.2: Fiscal and Generational Imbalance Fiscal Imbalance (as Percent of the Present Value of GDP) Part of the Budget 2006 2009 2012 FI in Social Security 0.8% 0.8% 0.9% FI in Medicare 6.7% 6.9% 7.1% FI in Rest of Federal Government -0.9% -1.0% -1.0% Total FI 6.6% 6.8% 7.0% Table 4.3: Fiscal and Generational Imbalance • In order to compute GI, one needs to break down taxes paid and outlays received by generations, which requires a host of ancillary assumptions, in particular on demographic trends. 4.3.3 Main Results Table 4.2 presents measures of the fiscal imbalance for the U.S. federal government. Table 4.3 relates the fiscal imbalance to the present value of GDP. It shows what fraction of GDP the government would have to confiscate, starting today and into the indefinite future, to pay for the entire fiscal imbalance 4.3.4 Interpretation The key findings from the preceding table can be summarized as follows: 1. The total fiscal imbalance of the government is huge; it requires the confiscation of 7% of GDP in perpetuity to close this imbalance. Expressed in terms of a required increase in labor income taxes, this would come to about a 15% increase (over and above the taxes already in place, and assuming no negative effects on labor supply induced by the tax hike). 2. If no policy changes are taken, the measure FI grows over time at a gross rate of (1 + r) = 1.0365 per year. It is a debt that, without any action of repayment, simply accumulates at the gross interest rate. 4.3. AN EXCURSION INTO THE FISCAL SITUATION OF THE US 73 Generational Imbalance (Billion of Constant 2006 Dollars) Part of the Budget 2006 2009 2012 FI in Social Security 7, 684 8,672 9,737 GI in Social Security (incl. Trust Fund) 11,019 12,157 13,372 FI — GI in Social Security -3,335 -3, 485 -3,635 FI in Medicare GI in Medicare FI - GI in Medicare 65,181 26,496 38,685 72, 291 30,417 41,874 79, 859 34, 747 45, 111 Table 4.4: Fiscal and Generational Imbalance 3. By far the largest part of the fiscal imbalance is due to medicare. The fiscal imbalance form this government program is about nine times as large as the fiscal imbalance arising from social security (surprisingly there seems to be much more discussion about social security than medicare reform). The large imbalance in this program stems from two elements: per-capita expenditures grow faster than GDP per capita and the population rapidly ages, and thus the number of people eligible for medicare increases. 4. The rest of the government programs contribute only marginally to total fiscal imbalance. This result, however, is subject to quite conservative estimates of the increase in government discretionary spending and optimistic estimates of government revenues (it, for example, does not include recent tax cut proposals and assumes that the tax cuts already enacted expire, as under current law). 5. The total fiscal imbalance measured by Gokhale and Smetters dwarfs the official most commonly reported measure of government indebtedness, namely the outstanding government debt, roughly by a factor of 8. Now we turn to the generational imbalance, splitting the fiscal imbalance between generations living today and future generations. Future generations means being born 15 years ago or later; for example, for 2006 these are all generations born in 1992 or later. Of particular interest is the division of the entire imbalance for the two programs that transfers across generations, namely social security and medicare. 1. The table shows that the majority of the fiscal imbalance in medicare is due to future generations (FI — GI). The implication of these numbers is clear. Current or future generations have to pay more for or receive less in health care when old. 2. In contrast, the fiscal imbalance in social security is due entirely to past and current generations. In fact, the contribution of future generations FI — GI is negative, indicating that future generations, under current law, are set receive less in present value than they pay in the form of payroll 74 CHAPTER 4. DYNAMIC THEORY OF TAXATION taxes. As mentioned above, the total fiscal imbalance from social security is only a small fraction of that due to medicare, but its absolute size is still comparable to that of the total outstanding government debt. 3. The results presented in the last table are somewhat sensitive to assumptions regarding the discount rate, the assumed growth of GDP per capita and the growth rate of labor productivity as well as the growth differential between health care expenditures per capita and GDP per capita. In table 5 of their paper Gokhale and Smetters present a number of sensitivity analyses. Despite the fact that the exact numbers of fiscal imbalance are somewhat sensitive to the exact assumptions made, the general conclusion from Gokhale and Smetters' report persist: large spending cuts in government programs or substantial tax increases are required to restore fiscal balance. What are the effects on the economy? In order to answer that question we first need to study what effects tax increases have, if these taxes are distortionary, as all real world taxes are. 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 75 4.4 Consumption, Labor and Capital Income Taxation 4.4.1 The U.S. Federal Personal Income Tax Brief History6 In early U.S. history the country relied on few taxes, on alcohol, tobacco and snuff, real estate sold at auctions, corporate bonds and slaves. To finance the war in 1812 sales taxes on gold, silverware and other jewelry were added. All internal taxes (that is taxes on residents of the U.S.) were abolished in 1817, with the government relying exclusively on tariffs on imported goods to fund its operations. The civil war from 1861-1865 demanded increased funds for the federal government. In 1862 the office of Commissioner of Internal Revenue (the predecessor of the modern IRS) was established, with the rights to assess, levy and collect taxes, and the right to enforce the tax laws though seizure of property and income and through prosecution. During the civil war individuals earning between $600 — $10000 had to pay an income tax of 3%, with higher rates for people with income above $10000. Note however, that $600 was well beyond the average income of a person at that time, so that a majority of the U.S. population was exempted from the income tax. In addition to income taxes, additional sales and excise taxes were introduced (an excise tax is a sales tax levied on a particular set of commodities; alcohol, tobacco and gambling are the most common goods to which excise taxes are applied). Furthermore, for the first time an inheritance tax was introduced. Total tax collections reached $310 million in 1866, the highest amount in U.S. history to that point, and an amount not reached again until 1911. The general income tax was scrapped again in 1872, alongside other taxes besides excise taxes on alcohol and tobacco. It was briefly re-introduced in 1894, but challenged in court and declared unconstitutional in 1895, because it did not levy taxes and distribute the funds among states in accordance with the constitution. The modern federal income tax was permanently introduced in the U.S. in 1913 through the 16-th Amendment to the Constitution. The amendment gave Congress legal authority to tax income and pathed the way for a revenue law that taxed incomes of both individuals and corporations. By 1920 IRS revenue collections totaled $5.4 billion dollars, rising to $7.3 billion dollars at the eve of WWII. However, the income tax was still largely a tax on corporations and very high income individuals, since exemption levels were high. Consequently the majority of U.S. citizens were not subject to federal income taxes. However, in 1943 the government introduced a withholding tax on wages (before taxes were paid once a year, in one installment) that covered most working Americans. Consequently, by 1945 the number of income taxpayers increased to 60 million (out of a population of 140 million) and tax revenues The subsection is based on "History of the Income Tax in the United States" by Scott Moody, senior economist at the Tax Foundation. 76 CHAPTER 4. DYNAMIC THEORY OF TAXATION from the federal income tax increased to $43 billion, a six-fold increase from the revenues in 1939. The universal federal income tax in the U.S. was born. The post-war period saw a series of changes in tax laws. The most far-reaching tax reforms in recent history have been the tax reforms by President Reagan of 1981 and 1986,President Clinton's tax reform of 1993 and the recent tax reforms of President George W. Bush in 2001-2003. The Reagan tax reforms reduced income tax rates by individuals drastically (with a total reduction amounting to the order of $500 — 600 billion), partially offset by an increase in tax rates for corporations and moderate increases of taxes for the very wealthy.. Under the pressure of mounting budget deficits President Clinton partially reversed Reagan's tax cuts in 1993, in order to avoid extensive budget deficits and thus the expansion of government debt in the future. Further tax reforms under the Clinton presidency included tax cuts for capital gains, the introduction of a $500 tax credit per child and tax incentives for education expenses. Finally, the recent large tax cuts in 2003 by President Bush temporarily (the package is scheduled to expire in 2012) reduce dividend and capital gains taxes as well as marginal income taxes, and increased child tax credits for most American tax payers. In order to analyze the economic impacts of these different changes in the tax code on private decisions, the distribution of income, wealth and welfare as well as to discuss the normative rationale (are these reforms "good") we need to go beyond simply describing the reforms and return to our theoretical analysis of what taxes and tax reforms do to private households. Concepts Let by y denote taxable income, that is, income from all sources excluding deductions. A tax code is defined by a tax function T(y), which for each possible taxable income gives the amount of taxes that are due to be paid. In both the political as well as the academic discussion two important concepts of tax rates emerge. Definition 9 For a given tax code T we define as 1. the average tax rate of an individual with taxable income y as y for all y > 0. 2. the marginal tax rate of an individual with taxable income y as r(y) = T'(y) whenever T''(y) is well-defined (that is, whenever T'(y) is differentiable). 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 77 The average tax rate t(y) indicates what fraction of her taxable income a person with income y has to deliver to the government as tax. The marginal tax rate r(y) measures how high the tax rate is on the last dollar earned, for a total taxable income of y. It also answers the question how many cents for an additional dollar of income a person that already has income y needs to pay in taxes. Evidently one can also define a tax code by the average tax rate schedule, since where the equality follows from the fundamental theorem of calculus. In fact, the current U.S. federal personal income tax code is defined by a collection of marginal tax rates; the tax code T(y) can be recovered using (4.8). So far we have made no assumption on how the tax code looks like. It turns out that tax codes can be broadly classified into three categories. Definition 10 A tax code is called progressive if the function t(y) is strictly increasing in y for all income levels y, that is, if the share of income due to be paid in taxes strictly increases with the level of income. A tax system is called progressive over an income interval (yi,yh) if t(y) is strictly increasing for all income levels y £ (yi,yh)- Definition 11 A tax code is called regressive if the function t(y) is strictly decreasing in y for all income levels y, that is, if the share of income due to be paid in taxes strictly decreases with the level of income. A tax system is called regressive over an income interval (yi,yh) if t(y) is strictly decreasing for all income levels y £ (yi,yh)- Definition 12 A tax code is called proportional if the function t(y) is constant y for all income levels y, that is, if the share of income due to be paid in taxes is constant in the level of income. A tax system is called proportional over an income interval (yi,yh) ift(y) is constant for all income levels y £ (yi,yh)- Let us look first at several examples, and then at some general results concerning tax codes. Example 13 A head tax or poll tax where T > 0 is a number. That is, all people pay the tax T, independent of their income. Obviously this tax is regressive since T(y) = y* t(y) or by the marginal tax rate schedule, since (4.8) T(y) = T 78 CHAPTER 4. DYNAMIC THEORY OF TAXATION is a strictly decreasing function ofy. Also note that the marginal tax is r(y) = 0 for all income levels, since the tax that a person pays is independent of her income Example 14 A flat tax or proportional tax that is, average and marginal tax rates are constant in income and equal to the tax rate r. Clearly this tax system is proportional. Example 15 A flat tax with deduction where d,t > 0 are parameters. Here the household pays no taxes if her income does not exceed the exemption level d, and then pays a fraction r in taxes on every dollar earned above d. One can compute average and marginal tax rates to be Thus this tax system is progressive for all income levels above d; for all income levels below it is trivially proportional. Example 16 A tax code with step-wise increasing marginal tax rates. Such a tax code is defined by its marginal tax rates and the income brackets for which these taxes apply. I constrain myself to three brackets, but one could consider as many brackets as you wish. The tax code is characterized by the three marginal rates (ti, t2, T3) and income cutoffs (&i,&2) that define the income tax brackets. It is somewhat burdensome7 to derive the tax function T(y) and the average tax t(y); here we simply state without proof that if t\ < r2 < T3 then this tax system is proportional for y £ [0, 61] and progressive for y > b\. Obviously, with just two brackets we get back a flat tax with deduction, if t\ = 0. 7 In fact, it is not so hard if you know how to integrate a function. For 0 < y < b\ we have T(y) =r*y where r £ [0,1) is a parameter. In particular, t(y) = r(y) = t and 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 79 The reason we looked at the last example is that the current U.S. tax code resembles the example closely, but consists of six marginal tax rates and five income cut-offs that define the income tax brackets. The income cut-offs vary with family structure, that is, depend on whether an individual is filing a tax return as single, as married filing jointly with her spouse or as married but filing separately. Now let us briefly derive an important result for progressive tax systems. Since it is easiest to do the proof of the result if the tax schedule is differentiable (that is T'(y) is well-defined for all income levels), we will assume this here. Theorem 17 A tax system characterized by the tax code T(y) is progressive, that is, t(y) is strictly increasing in y (i.e. t'(y) > 0 for all y) if and only if the marginal tax rate T'(y) is higher than the average tax rate t(y) for all income levels y > 0, that is T'(y) > t(y) Proof. Average taxes are defined as But using the rule for differentiating a ratio of two functions we obtain yT'(y) - T(y) t'(v) y2 But this expression is positive if and only if yT'(y)-T(y)>0 T'(y)>^l=t(y) for bi < y < b2 we have f-y pbi ry lb! and finally for b\ < y < 62 we have rbi rb2 ry ry rb\ ry T{v) = I T~{y)dy = I Tidy + / T2dy = Tibi + T2(y — 61) JO JO J61 rb\ ro2 ry T{v) = / Tidy + I T2dy + / T3dy = Tibi + r2&2 + r3(y - b2) JO Jbx Jb2 Consequently average tax rates are given by Tl if 0 < y < bi ti b t(y) = { if+-2(l-^) if6i bi. 80 CHAPTER 4. DYNAMIC THEORY OF TAXATION Intuitively, for average tax rates to increase with income requires that the tax rate you pay on the last dollar earned is higher than the average tax rate you paid on all previous dollars. Another way of saying this: one can only increase the average of a bunch of numbers if one adds a number that is bigger than the average. This result provides us with another, completely equivalent, way to characterize a progressive tax system. Obviously a similar result can be stated and proved for a regressive or proportional tax system. The Current Tax Code Now let us look at the current U.S. federal personal income tax code. The tax rates an individual faces depends on whether the individual is single or married, and if married, if she files a tax return jointly with her spouse or not. Before looking at the tax rates applying to different levels of income, we first have to discuss what income is subject to income taxes. The income that is subject to income taxes is called taxable income; this is the entity we have previously denoted by y. Two steps have to be taken to obtain taxable income from gross income. This gross income consists of Gross Income = Wages and Salaries +Interest Income and Dividends +Net Business Income +Net Rental Income +Other Income Most of these categories are self-explanatory; the "net" in net business income and net rental income refers to income net of business expenses or expenses for the rental property on which income is earned. Other income includes unemployment insurance benefits, alimony, income from gambling, income from illegal activities (which is evidently often not reported). There are, however, important sources of income that are not part of Gross Income and thus not taxable. Examples include child support, gifts below a certain threshold, interest income from state and local bonds (so-called Muni's), welfare and veterans benefits. Also, certain parts of employee compensation, such as employer contributions for health insurance and retirement accounts, are not part of Gross Income and thus not taxable. From Gross Income one arrives at Adjusted Gross Income (AGI) by subtracting contributions to Individual Retirement Accounts (IRA's), alimony, and health insurance payments by self-employed for themselves and their families. Finally, taxable income is derived from AGI by subtracting deductions and exemptions. Personal exemptions are amounts by which AGI is reduced that depend on the number of family members the tax payer supports. Everybody is entitled to claim him- or herself as an exemption; in addition, one can claim his/her spouse and children, unless the spouse files for taxes separately. In 2006, per exemption, the tax payer is entitled to deduct $3,300 from AGI. With respect to deductions, each tax payer has the choice to claim the standard deduction, 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 81 Tax Rates for 2006, Singles Income T'(y) T(y) 0 < y < $7, 550 10% 0.12/ $7, 550 < y < $30, 650 15% $755 + 0.15(2/ - 7, 550) $30, 650 < y < $74, 200 25% $4, 220 - 1- 0.25(2/ - - 30, 650) $74, 200 < y < $154, 800 28% $15,107.5- 1- 0.28(2/ - - 74, 200) $154, 800 < y < $336,550 33% $37, 675.5 + 0.33(y - 154, 800) $336, 550 < y < oo 35% $97, 653 + 0.35(2/ - 336, 550) Table 4.5: Marginal Tax Rates in 2003, Households Filing Single or to claim itemized deductions. The standard deduction for 2006 amounts to $10,300 for married households filing jointly and $5,150 for single households. If one opts to use itemized deductions, the include mortgage interest payments state and local income and property taxes, medical expenses in excess of 7.5% of AGI, charitable contributions, moving expenses related to relocation for employment and other small items. Whether to claim the standardized deduction or to use itemized deductions obviously depends on the amount the itemized deductions add up to for a given tax payer. In general, households with large mortgages or huge medical bills tend to opt for the itemized deduction option. After these adjustments to AGI one finally arrives at taxable income y, on which the tax code is applied to figure the tax liability of a household. Comparing this liability to the withholdings of the tax year, and subtracting tax credits (such as child care expenses, taxes paid in foreign countries, the earned income tax credit and tax credits for college tuition) finally yields the taxes that are due upon filing your income tax on April 15 (or the rebate owed to the tax payer should tax liabilities minus credits fall short of withholdings). Now we want to discuss the current U.S. tax code, that is, the schedule that for each taxable income determines how large the tax payers' liabilities are. Table 4.5 summarizes marginal tax rates and the total tax code for individuals filing single. As we can see from the tax schedule, for the first $7, 550 the tax code is proportional, with a marginal and average tax rate of 10%. After that, the tax code becomes progressive, since marginal tax rates are increasing (strictly so at the income bracket points) in income y. The last column shows total taxes owed; the function T(y) is derived by using T(0) = 0 and integrating the marginal tax schedule with respect to income. The tax code is depicted graphically in figures 4.1 and 4.2. We see that the tax code is defined by six marginal tax rates and six income brackets for which the marginal tax rates apply. Since marginal tax rates are increasing with income, average tax rates are increasing with income as well, strictly so after the first income bracket. This is exactly what figure 4.2 documents. Average taxes are fiat at 10% for the first $7, 550 and then strictly increasing. A comparison of the two graphs also shows that average taxes are always lower than marginal taxes, strictly so 82 CHAPTER 4. DYNAMIC THEORY OF TAXATION 0.4 0.35 0.3 0) T5 0.25 cc x CO t 0.2 CO _c JS 0.15 0.1 J 0.05 Marginal Income Tax Rates for the US, Singles: 2006 0.5 1.5 2 2.5 Income x 10 3.5 5 Figure 4.1: U.S. Marginal Income Taxes, Individuals Filing Single for all income levels above $7, 500. Note that if we were to continue the average tax plot for higher and higher income, average taxes would approach the 35% mark, the highest marginal tax rate, as income becomes large. In Table 4.5 we document the tax code applying for a married couple that files a joint tax return. Encoded in these two tax schedules is the so-called marriage penalty. Consider the following hypothetical situation: Angelina and Brad are madly in love and think about getting married. Each of them is making $100,000 as taxable income. Simply living together without being married, Angelina pays taxes TA = T(100, 000) = $15,107.5 + 0.28(100,000 - 74, 200) = $22, 331.50 = TB and Brad pays the same amount. So the joint tax liability of the couple is $44, 663. If they marry (even ignoring the $100, 000 cost for the wedding), they now pay taxes of r = T(200,000) = $42,170 + 0.33(200,000 - 188,450) = $45, 981.50 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 83 Average Income Tax Rates for the US, Singles: 2006 x 10 Figure 4.2: U.S. Average Tax Rate for Individuals Filing Single that is, the mere act of marriage increases their joint tax liability by about $1,300. Note that one can also construct a reverse example. Now suppose that Angelina makes $150, 000 and Brad makes $30, 650. Getting married and filing a single tax return yields taxes for the family of TA+D = T(180; 650) = $39j 9g6 whereas pre-marriage taxes are given by TA = T(150, 000) = $15,107.5 + 0.28(150,000- 74,200) = 36,331.50 TB = T(30, 650) = $4, 220 and thus total tax liabilities without getting married, total $40, 551.50. Thus, whether it pays to get married for tax reasons depends on how incomes within the couple are distributed. In general, with fairly equal incomes it does not pay, 84 CHAPTER 4. DYNAMIC THEORY OF TAXATION Tax Rates for 2006, Married Filing Jointly Income T'(y) T(y) 0 < y < $15, 100 10% 0.12/ $15,100 < y < $61, 300 15% $1510 4- 0.15(2/ - - 15,100) $61,300 < y < $123, 700 25% $8,440 4 0.25(2/- - 61, 300) $123,700 < y < $188, 450 28% $24,040 4 0.28(2/ - 123, 700) $188,450 < y < $336, 550 33% $42,170 4 0.33(y - 188, 450) $336, 550 < y < oo 35% $91,043 4 0.35(2/ - 336, 550) Table 4.6: Marginal Tax Rates in 2003, Married Households Filing Jointly whereas with incomes substantial different between the two partners it pays to get married and file taxes jointly. Normative Arguments for Progressive Taxation For simplicity assume that there are only two households in the economy, household 1 with taxable income of $100,000 and household 2 with taxable income of $20, 000. Again for simplicity assume that their lifetime utility u(c) only depends on their current after-tax income c = y — T(y), which we assume to be equal to consumption (implicitly we assume that households only live for one period). Finally assume that the lifetime utility function u(c) is of log-form.8 We want to compare social welfare under two tax systems, a hypothetical proportional tax system and a system of the form in the last example. For concreteness, let the second tax system be given by ( 0% if 0 < y < 15000 T(y) = ! 10% if 15000 < y < 50000 [ 20% if 50000 < y < oo Under this tax system total tax revenues from the two agents are T(15,000)4T(100,000) = 0.1* (20000- 15000) 40.1 * 35000 4 0.2(100000 - 50000) = $5004$13500 = $14000 8 For the argument to follow it is only important that u is strictly concave. The log-formulation is chosen for simplicity. Also, as long as current high income makes future high income more likely, the restriction to lifetime utility being denned over current after-tax income does not distort our argument. If we define the function V(c) as the lifetime utility of a person with current after tax labor income c, as long as this function is increasing and strictly concave in y (which it will be if after-tax income is positively correlated over time and the period utility function is strictly concave), the argument below gues through unchanged. 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 85 and consumption for the households are ci = 20000 - 500 = 19500 c2 = 100000 - 13500 = 86500 In order to enable the appropriate comparison, we first have to determine the proportional tax rate r such that total tax revenues are the same under the hypothetical proportional tax system and the progressive tax system above system. We target total tax revenues of $14000. But then 14000 = t * 20, 000 + t * 100,000 = r * 120, 000 14,000 120,000 11.67% is the proportional tax rate required to collect the same revenues as under our progressive tax system. Under the proportional tax system consumption of both households equals Cl = (1 - 0.1167) * 20000 = 17667 c2 = (1 - 0.1167) * 100000 = 88333 Which tax system is better? This is a hard question to answer in general, because under the progressive tax system the person with 20, 000 of taxable income is better off, whereas the person with 100, 000 is worse off than under a pure proportional system. So without an ethical judgement about how important the well-being of both households is we cannot determine which tax system is to be preferred. Such judgements are often made in the form of a social welfare function W{u(cij,...,u{cN)) where N is the number of households in the society and W is an arbitrary function, that tells us, given the lifetime utilities of all households, u{ci),..., m(cjv), how happy the society as a whole is. So far we have not made any progress, since we have not said anything about how the social welfare function W looks like. Here are some examples: Example 18 Household i is a "dictator" W{u{cij,u(cN)) = u{ci) This means that only household i counts when calculating how well-off a society is. Obviously, under such a social welfare function the best thing a society can do is to maximize household i's lifetime utility. For the example above, if the dictator is household 1, then the progressive tax system is preferred by society to the proportional tax system, and if household 2 is the dictator, the proportional tax system beats the progressive system. Note that even though dictatorial social welfare functions seem somehow undesirable, there are plenty 86 CHAPTER 4. DYNAMIC THEORY OF TAXATION of examples in history in which such a social welfare function was implemented (you pick your favorite dictator). Clearly the previous social welfare functions seem unfair or undesirable (although there is nothing logically wrong with them). Two other types of social welfare functions have enjoyed popularity among philosophers, sociologists and economists: Example 19 Utilitarian social welfare function W(u(cij,..., u(cN)) = u(ci) + ... + u(cN) that is, all household's lifetime utilities are weighted equally. This social welfare function posits that everybody's utility should be counted equally. The intellectual basis for this function is found in John Stuart Mill's (1806-1873) important work "Utilitarism" (published in 1863). In the book he states as highest normative principle Actions are right in proportion as they tend to promote happiness; wrong as they tend to produce the reverse of happiness He refers to this as the "Principle of Utility". Since everybody is equal according to his views, society should then adopt policies that maximize the sum of utility of all citizens. For our simple example the Utilitarian social welfare function would rank the progressive tax code and the proportional tax code as follows and thus the progressive tax code dominates a purely proportional tax code, according to the Utilitarian social welfare function. Example 20 Rawlsian social welfare function that is, social welfare equals to the lifetime utility of that member of society that is worst off. The idea behind this function is some kind of veil of ignorance. Suppose you don't know whether you are going to be born as a household that will have low or high income. Then, if, pre-natally, you are risk-averse you would like to live in a society that makes you live a decent life even in the worst possible realization of your income prospects. That is exactly what the Rawlsian social welfare function posits. For our simple example it is easy to see that the progressive tax system is preferred to a proportional tax system since TUpros(«(Cl), u{c2)) = min{log(Cl), log(c2)} = log(ci) = log(19500) TUproP(«(Cl),«(c2)) = min{log(Cl),log(c2)} = log(ci) = log(17667) < TUpros(«(Cl),u{c2)) W^(u(Cl),u(c2)) W^(u(Cl),u(c2)) log(19500) + log(86500) = 21.2461 log(17667) + log(88333) = 21.1683 W{u{cij,.. .,u(cN)) min{«(ci),..., m(cjv)} 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 87 In fact, under the assumption that taxable incomes are not affected by the tax code (i.e. people work and save the same amount regardless of the tax code - it may still differ across people, though-) then one can establish a very strong result. Theorem 21 Suppose that u is strictly concave and the same for every household. Then under both the Rawlsian and the Utilitarian social welfare function it is optimal to have complete income redistribution, that is Vi + V2 + ■ ■ ■ + VN - G Y-G C1 = C2 = ... = CN =-_-= __ where G is the total required tax revenue and Y = y\ + y2 + ■ ■ ■ + yN is total income (GDP) in the economy. The tax code that achieves this is given by . . Y-G T{yi) =yi--— i.e. to tax income at a 100% and then rebate Y~^G back to everybody. We will omit the proof of this result here (and come back to it once we talk about social insurance). But the intuition is simple: suppose tax policy leaves different consumption to different households, for concreteness suppose that N = 2 and c2 > c\. Now consider taking way a little from household 2 and giving it to household 1 (but not too much, so that afterwards still household 2 has weakly more consumption than household 1). Obviously under the Rawlsian social welfare function this improves societal welfare since the poorest person has been made better off. Under the Utilitarian social welfare function, since the utility function of each agent is concave and the same for every household, the loss of agent 2, u'(c2) is smaller than the gain of agent 1, u'{ci), since by concavity c2 > c\ implies u'(ci) > u'(c2). Evidently the assumption that changes in the tax system do not change a households' incentive to work, save and thus generate income is a strong one. Just imagine what household would do under the optimal policy of complete income redistribution (or take your favorite ex-Communist country and read a history book of that country). Therefore we now want to analyze how income and consumption taxes change the economic incentives of households to work, consume and save. 4.4.2 Theoretical Analysis of Consumption Taxes, Labor Income Taxes and Capital Income Taxes In order to meaningfully talk about the trade-offs between consumption taxes, labor income taxes and capital income taxes we need a model in which households decide on consumption, labor supply and saving. We therefore extend our simple model and allow households to choose how much to work. Let I denote 88 CHAPTER 4. DYNAMIC THEORY OF TAXATION the total fraction of time devoted to work in the first period of a household's life; consequently 1 — Z is the fraction of total time in the first period devoted to leisure. Furthermore let by w denote the real wage. We assume that in the second period of a person's life the household retires and doesn't work. Also, we will save our discussion of a social security system for the next chapter and abstract from it here. Finally we assume that households may receive social security benefits b > 0 in the second period of life. The household maximization problem becomes max log(ci) + 01og(l - I) + /31og(c2) (4.9) s.t. (l + rCl)Cl + s = (l-ri)wl (4.10) (l + rC2)c2 = (l + r(l-ts))s + b (4.11) where 6 and [3 are preference parameters, rCl, rC2 are proportional tax rates on consumption, t; is the tax rate on labor income, r is the return on saving, and ts is the tax rate on that return. The parameter [3 has the usual interpretation, and the parameter 6 measures how much households value leisure, relative to consumption. Obviously there are a lot of different tax rates in this household's problem, but then there are a lot of different taxes actual U.S. households are subject to. To solve this household problem we first consolidate the budget constraints into a single, intertemporal budget constraint. Solving equation (4.11) for s yields _ (1 + tc2)c2 - b S~ (l + r(l-ra)) and thus the intertemporal budget constraint (by substituting for s in (4.10)) (x + T-)Cl + nl+nC2>2^ = t1" Tl>1 + 77X7?-^ (l + r(l-tsj) (l + r(l-tsj) In order to solve this problem, as always, we write down the Lagrangian, take first order conditions and set them to zero. Before doing so let us rewrite the budget constraint a little bit, in order to provide a better interpretation of it. Since I = 1 — (1 — I) the budget constraint can be written as <1 + ^ + (i(Jtd-?,)) = a-^.(i-d-o)+(1 + r(;_Tj)) '^^■V^-V'1-'''1-^' = (1-Tl)"+(l + ^-r.)) The interpretation is as follows: the household has potential income from social security ^1+r^_T ^ and from supplying all her time to the labor market. With this she buys three goods: consumption c\ in the first period, at an effective 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 89 (including taxes) price (1 + tci), consumption c2 in the second period, at an effective price ^^^J^! ^ and leisure 1 — I at an effective price (1 — ti)w, equal to the opportunity cost of not working, which is equal to the after-tax wage. The Lagrangian reads as L = log(ci) + 01og(l-O + /?log(c2) +A O1 - ^ + (l + r(Lr,)) " (1 + ^ ~ (l + rd-?,)) " (1 " " ^ and we have to take first order conditions with respect to the three choice variables ci,c2 and I (or 1 — I, which would give exactly the same results). These first order conditions, equated to 0, are --A(l + rCl) = 0 c2 (l + r(l-ra)) ^ + A(l - n)w = 0 1 Cl A(l + rCl) (4.12) 1 - I I = A ^ + ^) , (4.13) c2 (l + r(l-ra)) V 7 A(l-r;)w (4.14) Now we can, as always, substitute out the Lagrange multiplier A. Dividing equation (4.13) by equation (4.12) one obtains the standard intertemporal Euler equation, now including taxes: ^=Si+^;*7_i__ (4.i5) c2 (l + rCl) (l + r(l-ra)) V ' and dividing equation (4.14) by equation (4.12) yields the crucial intra-temporal optimality condition of how to choose consumption, relative to leisure, in the first period: 6>Cl (1-ti)w I- I (1 + tc1) (4.16) These two equations, together with the intertemporal budget constraint, can be used to solve explicitly for the optimal consumption and labor (leisure) choices ci,c2,l (and, of course, equation (4.10) can be used to determine the optimal savings choice s). Before doing this we want to interpret the optimality conditions (4.15) and (4.16) further. Equation ((4.15)) is familiar: if consumption 90 CHAPTER 4. DYNAMIC THEORY OF TAXATION taxes are uniform across periods (that is, rCl = rC2) then it says that the marginal rate of substitution between consumption in the second and consumption in the first period (iu'(c2) _ (ic1 w'(ci) c2 should equals to the relative price between consumption in the second to consumption in the first period, ^1+r^_T )); the inverse of the gross after tax interest rate. With differential consumption taxes, the relative price has to be adjusted by relative taxes I^+t"2) • intertemporal optimality condition has the following intuitive comparative statics properties Proposition 22 1. An increase in the capital income tax rate ts reduces the after-tax interest rate l + r(l — ts) and induces households to consume more in the first period, relative to the second period (that is, the ratio ^ increases). 2. An increase in consumption taxes in the first period tCi induces households to consume less in the first period, relative to consumption in the second period (that is, the ratio ^ decreases). 3. An increase in consumption taxes in the second period rC2 induces households to consume more in the first period, relative to consumption in the second period (that is, the ratio ^ increases). Proof. Obvious, simply look at the intertemporal optimality condition. ■ The intra-temporal optimality condition is new, but equally intuitive. It says that the marginal rate of substitution between current period leisure and current period consumption, 6u'(l - I) 0ci m'(ci) 1 - I should equal to the after-tax wage, adjusted by first period consumption taxes (1-T,)W (that is, the relative price between the two goods) Q Tl'>w]. Again we obtain the following comparative statics results Proposition 23 1. An increase in labor income taxes t% reduces the aftertax wage and reduces consumption, relative to leisure, that is -^j falls. This substitution effect suggests (we still have to worry about the income effect) that an increase in t% reduces both current period consumption and current period 2. An increase in consumption taxes tCi reduces consumption, relative to leisure, that is jzrt falls. Again, this substitution effect suggests that an increase in tCi reduces both current period consumption and current period labor supply. 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 91 Proof. Obvious, again simply look at the intratemporal optimality condition. ■ According to Edward Prescott, this years' Nobel price winner in economics (and, incidentally, my advisor) this proposition is the key to understanding recent cross-country differences in the amount of hours worked per person.9 Before developing his arguments and the empirical facts that support them in more detail we state and prove the important, useful and surprising result that uniform proportional consumption taxes are equivalent to a proportional labor income tax. Proposition 24 Suppose we start with a tax system with no labor income taxes, Ti = 0 and uniform consumption taxes tci = rC2 = rc (the level of capital income taxes is irrelevant for this result). Denote by ci, c2, /, s the optimal consumption, savings and labor supply decision. Then there exists a labor income tax Ti and a lump sum tax T such that for rc = 0 households find it optimal to make exactly the same consumption choices as before. Proof. Under the assumption that the consumption tax is uniform, it drops out of the intertemporal optimality condition (4.15) and only enters the optimality condition (4.15). Rewrite that optimality condition as Oci = (1-tQ (l-l)W (1 + tc) The right hand side, for t% = 0, is equal to 1 0- + tc) But if we set f% = and fc = 0, then (1~^)_1 rc _ 1 (l + fc) 1 + TC (1 + tc)' that is, the household faces the same intratemporal optimality condition as before. This, together with the unchanged intertemporal optimality condition, leads to the same consumption, savings and labor supply choices, if the budget constraint remains the same. But this is easy to guarantee with the lump-sum tax T, which is set exactly to the difference of tax receipts under consumption and under labor taxes. ■ Before making good use of this proposition in explaining cross-country differences in hours worked we now want to give the explicit solution of the household decision problem. From the intratemporal optimality condition we obtain 9 See Edward C. Prescott (2004), "Why do Americans Work so much more than Europeans?," NBER Working Paper 10316. 92 CHAPTER 4. DYNAMIC THEORY OF TAXATION intertemporal optimality condition we obtain c2 = ßCl(l + r(l - ts)) (l + tCl)0 (1 + Tc2) (l-Tl)(l-l)wp(l + r(l-Ts)) e (i + tc2) Plugging all this mess into the budget constraint (i-^)(i-0»>+^(i-^Ki-O^ = (1_tjH + (4.18) (l + r(l-ra)) N(1-t;)(1-/)w , n & (l + r(l-ra)) which is one equation in the unknown I. Sparing you the details of the algebra, the optimal solution for labor supply I is r = -l±l___*_ (419) In particular, if there are no social security benefits (i.e. 6 = 0), then the optimal labor supply is given by l* = TTpTo^0^ The more the household values leisure (that is, the higher is 6), the less she finds it optimal to work. In this case labor supply is independent of the after-tax wage (and thus the labor tax rate), since with log-utility income and substitution effect cancel each other out. With b > 0, note that higher social security benefits in retirement reduce labor supply in the working period (partly we have implicitly assumed that current labor income does not determine future retirement benefits). Finally note that if b gets really big, then the optimal I* = 0 (the solution in (4.19) does not apply anymore). Obviously one can now compute optimal consumption and savings choices. Here we only give the solution for b = 0; it is not particularly hard, but algebraically messy to give the solution for b > 0. From (4.17) we have Cl (i-n)(i-i*)w (1 + tC1)0 (l + tCl)(l + /?+l and from (4.18) we have _ ß(l-Tl)(l + r(l-Ts)) 02 (i + ß + e)(i + rC2) w 4.4. CONSUMPTION, LABOR AND CAPITAL INCOME TAXATION 93 and finally from the first budget constraint (4.10) we find S = (1 - Tt)wl - (1 + TCl )ci (1 + p) (1 - rt)w (1-tAw PJX-ri)w 1 + /3 + 6* International Differences in Labor Income Taxation and Hours Worked The last proposition in the previous section shows that what really matters for household consumption and labor supply decisions is the tax wedge |^+t') the intratemporal optimality condition 0C (1-Tj) I- I (1 + TC) w (4.20) where we have dropped the period subscript on consumption. Clearly both labor and consumption taxes are crucial determinants of labor supply. In order to make this equation useful for data work we need to specify wages. For this we consider a typical firm in the economy. This firm uses labor and physical capital to produce output. Thus the production technology is given by V AkPl1- and the firm solves the maximization problem m&xAkan1~a — wn — rk k,n where k is the capital stock used by the firm, r is the rental rate of capital (equal to the interest rate), and n is the amount of labor hired at wages w. The parameter a is telling us how important capital is, relative to labor, in the production of output. It also turns out to be equal to the capital share (the fraction of income accruing to capital income). Taking the first order condition with respect to n and setting it equal to 0 yields (1 - a)Akan~a = w (1 - a)Akan1-a ---- = w n (1 — a)— = w n But this firm is representative of the entire economy, and our household is representative of the entire population. Thus we can interpret y as total output (or GDP) of a country and we need that the amount of labor hired by the 94 CHAPTER 4. DYNAMIC THEORY OF TAXATION firm equals the labor supplied by the household, or I = n. Then this equation becomes (l-a)y = w (4.21) (1 — a)y = wl The last expression demonstrates that the labor share Lab°GDpC°me = ^ equals 1 — a, so that the capital share equals a. Now we use equation (4.21) to substitute out the wage w in equation (4.20) to obtain 9c -^l-Ti\i-a)y 1-1 (1 + tc)v 'I Solving this equation for labor supply I yields, after some tedious algebra '=1_ar^6(M) 1- Since real wages grow over time, and thus the tax base for social security payroll taxes, the current system lets retired households benefit from this growth in real wages. In addition to inflation earnings in early years of our person's life are therefore adjusted in the following fashion Yt = yt * Ct,2007 where Yt is called the indexed earnings from year t. (d) After this ordeal we arrive at 45 numbers, {Y1963, ii96ij • • • j ^2007}-We compute average indexed monthly earnings by selecting the 35 highest entries from the list {Y1963, ii96ij • • • j ^2007}, summing them up and dividing by 35 (that is, taking the average of the 35 best earnings years. This yields the person's AIME. 5.3. THE CURRENT US SYSTEM 105 2. The second step of computing social security benefits is considerably easier. We simply insert AIME into the following benefit formula4 ( 0.9AIME if AIME < $680 b = < 612 + 0.32(AIME - 680) if $680 < AIME < $4,100 [ 1706.4 + 0.15(^^1^-4,100) if $4,100 < AIME (5.1) This looks messy, but simply states that for each of the first 680 dollars 90 cents of benefits are earned. For each additional dollar earned between $606 and $4,100 an additional 32 cents in benefits are obtained, and for each dollar above $4,100 another 15 cents are added to benefits. Equation (6.4) gives our person's benefits in 2007. For that point on every year his so-computed benefits are simply indexed by inflation, that is, if the inflation rate between 2007 and 2008 is 3%, then his benefits increase by 3% between 2007 and 2007. Benefits are paid until our person dies. From the previous discussion we see that social security benefits are perfectly determined by average indexed monthly earnings, that is, by the best 35 working years. Since benefits depend positively on AIME, rational forward-looking household understand that working more today will increase social security benefits, although the link becomes weaker the higher is income. Define the replacement rate as rr(AIME) = b(?ME> y ' AIME that is, as the ratio between social security benefits and AIME. Obviously the replacement rate depends on AIME. Figure 5.1 plots the replacement rate, as a function of AIME. It is first constant at 90% and then strictly declining. This means that the higher your average indexed monthly earnings are, the lower is the fraction of these earnings that you receive as benefits. Remember that payroll taxes are proportional to earnings. Thus the social security system contains a redistributive component: households with low earnings receive more in benefits than they contributed in taxes, whereas for high income earners the situation is reversed. One can also interpret this redistribution as insurance: if you don't know whether you are born as a person with high abilities and thus high income or a person with low earnings abilities, then ex ante (pre-birth) you like a system that redistributes between low-and high income earners, if you are risk-averse. So what is redistribution ex post is insurance ex ante. But also note that the extent of redistribution is limited since there is a cap on the social security taxes, as discussed before. In addition capital income, by construction, is not subject to social security taxes, so no redistribution between workers and capitalists takes place. 4This is the benefit formula for 2006. The formula for 2007 will slightly differ from the one given in the text. 106 CHAPTER 5. UNFUNDED SOCIAL SECURITY SYSTEMS Marginal Benefits and Replacement Rate, 2007 Marginal Benefit Replacement Rate 1000 2000 3000 4000 5000 6000 7000 8000 Average Indexed Monthly Earnings Figure 5.1: Social Security Replacement Rate and Marginal Benefits After this discussion of the actual system we will now use our theoretical model to analyze the positive and normative effects of a pay-as-you go social security system. We will first show that such a system decreases private savings rates, and then discuss under what condition the introduction of a social security system is, in fact, a good idea. 5.4 Theoretical Analysis 5.4.1 Pay-As-You-Go Social Security and Savings Rates Now we use the model to analyze a policy issue that has drawn large attention in the public debate. The personal saving rate -the fraction of disposable income that private households save- has declined from about 7-10% in the 60's and 70's to close to 0 right now. Since saving provides the funds for investment a lower 5.4. THEORETICAL ANALYSIS 107 saving rate, so a lot of people argue, harms growth be reducing investment.5 Some economists argue that the expansion of the social security system has led to a decline in personal saving. We want to analyze this claim using our simple model. We look at a pay-as-you go social security system, in which the currently working generation pays payroll taxes, whose proceeds are used to pay the pensions of the currently retired generation. The key is that current taxes are paid out immediately, and not invested. We make the following simplifications to our model. We interpret the second period of a person's life as his retirement, so in the absence of social security he has no income apart from his savings, i.e. 2/2 = 0. Let y denote the income in the first period. The household maximizes max log(ci) + /31og(c2) (5.2) ci,c2,s S.t. ci + s = (1 - r)y c2 = (l + r)s+6 Let us assume that the population grows at rate n, so when the household is old there are (1 + n) as many young guys around compared when he was young. Also assume that incomes grow at rate g (because of technical progress) making younger generations having higher incomes. Finally assume that the social security system balances its budget, so that total social security payments equal total payroll taxes. This implies that b=(l + n)(l + g)Ty (5.3) The household benefits from the fact that population grows over time since when he is old there are more people around to pay his pension. In addition these people paying for pensions have higher incomes because of technical progress. Using the social security budget constraint (5.3) we can rewrite the budget constraints of the household as ci + s = (1 - r)y c2 = (1 + r)s + (1 + n)(l + g)TVl Again we can write this as a single intertemporal budget constraint c2 /-, \ (1 + n)(l + g)ry Cl + TT7 = (1"r)2/+^-l + r =/(T) where we emphasize that now discounted lifetime income depends on the size of the social security system, as measured by the tax rate r. Maximizing (2.1) 5This argument obviously ignores the increased inflow of foreign funds into the US. 108 CHAPTER 5. UNFUNDED SOCIAL SECURITY SYSTEMS subject to (5.4) yields, as always ci 1 + ß c2 = T^d + O/ * = a-^y-Y^g (5-5) So what does pay-as-you go social security do to saving? Using the definition of I (t) in (5.5) we find T s = (l-r)y = (l-r)y 1 + ß (1-t). (l + n)(l + g)ry 1 + ß (l + r)(l + ß) ß(l-T)y _ (l + n)(l + g)Ty 1 + ß (l + r)(l + ß) ßy (1 + n)(l + g)ry + ßTy(l + r) 1 + ß (l + r)(l + ß) ßy (l + n)(l + g) + ß(l + r) 1 + ß (l + r)(l + ß) * ry which is obviously decreasing in r. So indeed the bigger the public pay-as-you-go system, the smaller are private savings. Note that due to the pay-as-you go nature of the system the social security system itself does not save, so total savings in the economy unambiguously decline with an increase in the size of the system as measured by r. To the extent that this harms investment, capital accumulation and growth the pay-as-you-go social security system may have substantial negative long-run effects. 5.4.2 Welfare Consequences of Social Security Second, we use the model to analyze a policy issue that has drawn large attention in the public debate. From a normative perspective, should the government run a pay-as-you go social security system or should it leave the financing of old-age consumption to private households (which is equivalent, under fairly weak conditions, to a fully funded government-run pension system). In a pure pay-as-you go social security system currently working generation pays payroll taxes, whose proceeds are used to pay the pensions of the currently retired generation. The key is that current taxes are paid out immediately, and not invested. In a fully funded system the contributions of the current young are saved (either by the households themselves in private accounts akin to the Riester Rente, or by the government). Future pension benefits are then financed by these savings, including the accumulated interest. The key difference is that with in a pay-as-you go system current contributions are used for current consumption of the old 5.4. THEORETICAL ANALYSIS 109 (as long as these generations do not save), whereas with a funded system these contributions augment savings (equal to investment in a closed economy). One can show that under fairly general conditions the physical capital stock in an economy with pay-as-you go social security system is lower than in an identical economy with a fully funded system. But rather than studying capital accumulation directly, we restrict our analysis to a partial equilibrium analysis, asking whether individual households are better off in a pay-as-you-go system relative to a fully funded system, keeping the interest rate fixed (in a closed production economy the interest rate equals the marginal product of capital and thus is lower in an economy with more capital). So under which condition is the introduction of social security good for the welfare of the household in the model? This has a simple and intuitive answer in the current model. When maximizing (5.2), subject to (5.4), we see that the social security tax rate only appears in I(r), which is given as I(f) = (l-t)y+-7-j---. (5.6) So the question of whether social security is beneficial boils down to giving conditions under which I(r) is strictly increasing in r. Rewriting (5.6) yields I(t) = yi-ry+- y- 1 + r (l + g)(l + n) 1 1 + r ry and thus the pay-as-you go social security system is welfare improving if and only if (l + n)(l + g) > 1 + r. Since, empirically speaking, n*g is small relative to n,g or r (on an annual level g is somewhere between 1 — 2% for most industrialized countries, n is even smaller and in some countries, including Germany, negative), the condition is well approximated by n + g > r That is, if the population growth rate plus income growth exceeds the private returns on the households's saving, then a given household benefits from pay-as-you-go social security This condition makes perfect sense. If people save by themselves for their retirement, the return on their savings equals 1 + r. If they save via a social security system (are forced to do so), their return to this forced saving consists of (1 + n)(l + g) (more people with higher incomes will pay for the old guys). This result makes clear why a pay-as-you-go social security system may make sense in some countries (those with high population growth), but not in others, and that it may have made sense in Germany in the 60's and 70's, but not in the 90's. Just some numbers: the current population growth rate in Germany is, say about n = 0% (including immigration), productivity growth is about g = 1% and the average return on the stock market for the last 110 CHAPTER 5. UNFUNDED SOCIAL SECURITY SYSTEMS 100 years is about r = 7%. This is the basis for many economists to call for a reform of the social security system in many countries. Part of the debate is about how one could (partially) privatize the social security system, i.e. create individual retirement funds so that basically each individual would save for her own retirement, with return 1 + r > (l + n)(l + g). Abstracting from the fact that saving in the stock market is fairly risky even over longer time horizons (and the return on saver financial assets is not that much higher than n + g), the biggest problem for the transition is one missing generation. At the introduction of the system there was one old generation that received social security but never paid taxes for it. Now we face the dilemma: if we abolish the pay-as-you go system, either the currently young pay double, for the currently old and for themselves, or we just default on the promises for the old. Both alternatives seem to be difficult to implement politically and problematically from an ethical point of view. The government could pay out the old by increasing government debt, but this has to be financed by higher taxes in the future, i.e. by currently young and future generations. Hence this is problematic, too. The issue is very much open, and since I did research on this issue in my own dissertation I am happy to talk to whoever is interested in more details. 5.4.3 The Insurance Aspect of a Social Security System Modern social security systems provide some form of insurance to individuals, namely insurance against the risk of living longer than expected. In other words, social security benefits are paid as long as the person lives, so that people that live (unexpectedly) longer receive more over their lifetime than those that die prematurely. Note, however, that such insurance need not be provided by the government via social security, but could also be provided by private insurance contract. In fact, private annuities are designed to exactly provide the same insurance. We will briefly discuss below why the government may be in a better position to provide this insurance. Before doing so I first want to demonstrate that providing such insurance, privately or via the social security system is indeed beneficial for private households. First we consider a household in the absence of private or public insurance markets. The household lives up to two periods, but may die after the first period. Let p denote the probability of surviving. We normalize the utility of being dead to 0 (this is innocuous because our households can do nothing to affect the probability of dying) and for simplicity abstract from time discounting. The agent solves max log(ci) +plog(c2) ci ,c2 ,s S.t. ci + s = y c2 = (1 + r)s Note that we have implicitly assumed that the household is not altruistic, so that the savings of the household, should she die, are lost without generating 5.4. THEORETICAL ANALYSIS 111 any utility. As always, we can consolidate the budget constraint, to yield ci + —— = y 1 + r and the solution to the problem takes the familiar form 1 ci = -r—y l + p p(l + r) ci = —j——y l + p where p takes the place of the time discount factor [3. Now consider the same household with a social security system in place. The budget constraints reads as usual ci + s = (1 - r)y c2 = (1 + r)s + b But now the budget constraint of the social security administration becomes pb = (1 + n)(l + g)ry The new feature is that social security benefits only need to be paid to a fraction p of the old cohort (because the rest has died). Consolidating the budget constraints an substituting for b yields ci + —— = y + ry[---1 1 + r \ p(l + r) The household may benefit from a pay-as-you-go social security system for two reasons. First, as we saw above, if (1 + n){l + g) > 1 + r, the implicit return on social security is higher than the return on private assets. This argument had nothing to do with insurance at all. But now, as long as p < 1, even if (1 + n)(l + g) < 1 + r social security may be good, since the surviving individuals are implicitly insured by their dead brethren: the implicit return on social security is 1+g-* > (1 + n)(l + g). If you survive you get higher benefits, if you die you don't care about receiving nothing. Now suppose that (1 + n)(l + g) = 1 + r, that is, the first reason for social security is absent by assumption, because we want to focus on the insurance aspect. The implicit return on social security is then _ l+r Now consider the other alternative of providing insurance, via the purchase of private annuities. An annuity is a contract where the household pays 1 Euro today, for the promise of the insurance company to pay you l+ra Euros as long as you live, from tomorrow on (and in the simple model, you live only one more period). But what is the equilibrium return 1 + ra on this annuity. Suppose there is perfect competition among insurance companies, resulting in zero profits. The insurance company takes 1 Euro today (which it can invest at the market interest 112 CHAPTER 5. UNFUNDED SOCIAL SECURITY SYSTEMS rate 1 + r). Tomorrow it has to pay out with probability p (or, if the company has many customers, it has to pay out to a fraction p of its customers), and it has to pay out 1 + ra per Euro of insurance contract. Thus zero profits imply 1 + r = p(l + ra) or V This is the return on the annuity, conditional on surviving, which coincides exactly with the expected return via social security, as long as (1 + n)(l + g) = 1 + r. That is, insurance against longevity can equally be provided by a social security system or by private annuity markets. The only difference is that the size of the insurance is fixed by the government in the case of social security, and freely chosen in the case of private annuities. In practice in the majority of the countries it is the government, via some sort of social security, that provides this insurance. Private annuity markets do exist, but seem to be quite thin (that is, not many people purchase these private annuities). There are at least two reasons that I can think of • If there is already a public system in place (for whatever reason), there are no strong incentives to purchase additional private insurance, unless the public insurance does not extent to some members of society. • In the presence of adverse selection private insurance markets may not function well. If individuals have better information about their life expectancy than insurance companies, then insurance companies will offer rates that are favorable for households with high life expectancy and bad for people with low life expectancy. The latter group will not buy the insurance, leaving only the people with bad risk (for the insurance companies) in the markets. Rates have to go up further. In the end, the private market for annuities may break down (nobody but the very worst risks purchase the insurance, at very high premium). The government, on the other hand, can force all people into the insurance scheme, thus avoiding the adverse selection problem. Another problem with insurance, so called moral hazard, will emerge in the next section where we discuss social insurance, especially unemployment insurance. Chapter 6 Social Insurance The term "Social Insurance" stands for a variety of public insurance programs, all with the aim of insuring citizens of a rich, modern society against the major risks of life: unemployment (unemployment insurance), becoming poor at young and middle ages (welfare, food stamps), becoming poor in old age because of unexpected long life (social security), becoming sick in old age (medicare). These risks and policies to insure the risks vary in their details, but their basic features are similar. Therefore, rather than describing all of them in detail, we will focus on the main risk during a person's working life: unemployment. 6.1 International Comparisons of Unemployment Insurance Before providing a theoretical rationale for publicly provided unemployment insurance (and the limits thereof) we first want to document and discuss the astounding international differences in the generosity and length of unemployment insurance benefits. Before doing so, let us first briefly discuss when people tend to get unemployed in modern economies The unemployment rate is very counter-cyclical. It increases in recessions and increases in expansions. Figure 6.1 plots the unemployment rate for the U.S. for the last 35 years. We clearly see that the unemployment rate increased during all recessions in the last 35 years. Formally a recession may be defined as two consecutive quarters of declining real GDP, but since output is produced using capital and labor, a decline in output almost automatically means a reduction in labor being used in production. Ignoring the important possibility that laid-off workers leave the labor force, this means that the unemployment rate increases in economic downturns. But why exactly does the unemployment rate go up in recessions? Is it because more people than normal get fired, or less people than normal get hired. Here are the basic facts on job creation and job destruction for the U.S. 113 114 CHAPTER 6. SOCIAL INSURANCE Figure 6.1: The U.S. Unemployment Rate manufacturing sector:1 • Job turnover is large. In a typical year 1 out of every ten jobs in manufacturing is destroyed and a comparable number of jobs is created at different plants. • Most of the job creation and destruction over a twelve-month interval reflects highly persistent plant-level employment changes. This persistence implies that most jobs that vanish at a particular plant in a given twelvemonth period fail to reopen at the same location within the next two years. • Job creation and destruction are concentrated at plants that experience large percentage employment changes. Two-thirds of job creation and destruction takes place at plants that expand or contract by 25% or more 1 These facts come from the great book by Davis, Haltiwanger and Schuh Job Creation and Destruction which collects and describes these facts for the manufacturing sector. 6.1. INTERNATIONAL COMPARISONS OF UNEMPLOYMENT INSURANCE115 Unemployment Spell 1989 1992 < 5 weeks 49% 35% 5-14 weeks 30% 29% 15 - 26 weeks 11% 15% > 26 weeks 10% 21% Table 6.1: Length of Unemployment Spells within a twelve-month period. About one quarter of job destruction takes place at plants that shut down. • Job destruction exhibits greater cyclical variation than job creation. In particular, recessions are characterized by a sharp increase in job destruction accompanied by a mild slowdown in job creation. • Gross job creation is relatively stable over the business cycle, whereas gross job destruction moves strongly countercyclical: it is high in recessions and low in booms. • In severe recessions such as the 74-75 recession or the 80-82 back to back recessions up to 25% of all manufacturing jobs are destroyed within one year, whereas in booms the number is below 5%. • Time a worker spends being unemployed also varies over the business cycle, with unemployment spells being longer on average in recession years than in years before a recession. This last fact is made concise in table 6.1. It shows the average length of unemployment spells in two years, 1989, the last year of the expansion of the late 80's and 1992, the end of the recession of the early 90's We observe that in a recession year many more unemployment spells last for more than half a year than in an expansion, where most workers that are laid off find a new job within a matter of 5 weeks. By international standards the fraction of households in long term unemployment (longer than six months or longer than one year) in the U.S. is small, as the next table demonstrates. From table 6.2 we observe several things. First, unemployment rates in Europe were not always higher than in the U.S. In fact, in the 70's it was the U.S. that had higher unemployment rates than Europe, but then the situation reversed. Second, and crucially, from the data on long-term unemployment we see that the fraction of all unemployed that are long-term unemployed is quite low in the U.S. (less than 10% of one defines log-term unemployment to be longer than one year). In Europe, in contrast, in most countries a majority of all unemployed is without a job for more than half a year, at many are unemployed for longer than one year. The fraction of long-term unemployed have gone up over time as well, so that one can characterize the European unemployment dilemma as a dilemma of long-term unemployment. 116 CHAPTER 6. SOCIAL INSURANCE Unemployment (%) > 6 Months > 1 Year 74-9 80-9 95 79 89 95 79 89 95 Belgium 6.3 10.8 13.0 74.9 87.5 77.7 58.0 76.3 62.4 France 4.5 9.0 11.6 55.1 63.7 68.9 30.3 43.9 45.6 Germany 3.2 5.9 9.4 39.9 66.7 65.4 19.9 49.0 48.3 Netherlands 4.9 9.7 7.1 49.3 66.1 74.4 27.1 49.9 43.2 Spain 5.2 17.5 22.9 51.6 72.7 72.2 27.5 58.5 56.5 Sweden 1.9 2.5 7.7 19.6 18.4 35.2 6.8 6.5 15.7 UK 5.0 10.0 8.2 39.7 57.2 60.7 24.5 40.8 43.5 US 6.7 7.2 5.6 8.8 9.9 17.3 4.2 5.7 9.7 OECD Eur. 4.7 9.2 10.3 - - - 31.5 52.8 - Tot. OECD 4.9 7.3 7.6 - - - 26.6 33.7 - Table 6.2: Unemployment Rates, OECD Age Group 15-24 25-44 > 45 Belgium 17 62 20 France 13 63 23 Germany 8 43 48 Netherlands 13 64 23 Spain 34 38 28 Sweden 9 24 67 UK 18 43 39 US 14 53 33 Table 6.3: Long-Term Unemployment by Age, OECD Who are these long-term unemployed? Table 6.3 gives the fraction of all long-term unemployed (unemployed longer than one year) by age in 1990. Even though the number of long-term unemployed is much higher in Europe than in the U.S., its distribution is somewhat similar, with the bulk at prime ages 25 — 44 and a sizeable minority of old long-term unemployed. How can the dramatic differences in unemployment rates between the U.S. and Europe, and in particular the large difference in long-term unemployed, be explained. This is a very complex problem. In a very influential paper Lars Ljungqvist and Tom Sargent relate long-term unemployment rates to the generosity of the European unemployment benefits. Table 6.4 summarizes unemployment benefit replacement rates for various countries, as a function of the length of unemployment, for the mid-90's. The table has to be read as follows. A 79 for Belgium in year 1 means that a typical worker in Belgium that is unemployed for no more than one year receives 79% of her last wage as unemployment compensation. This table tells us the following. First, replacement rates are much lower 6.1. INTERNATIONAL COMPARISONS OF UNEMPLOYMENTINSURANCE117 Single With Dependent Spouse 1. Y. 2.-3. Y. 4.-5. Y. 1. Y. 2.-3. Y. 4.-5. Y. Belgium 79 55 55 70 64 64 France 79 63 61 80 62 60 Germany 66 63 63 74 72 72 Netherlands 79 78 73 90 88 85 Spain 69 54 32 70 55 39 Sweden 81 76 75 81 100 101 UK 64 64 64 75 74 74 US 34 9 9 38 14 14 Table 6.4: Unemployment Benefit Replacement Rates in the U.S. than in Europe. Second, and possibly more important, while in the U.S. benefits drop sharply after 13 weeks, in many European countries the replacement rate remains over 60% three years into an unemployment spell. Imagine what this may do to incentives to find a new job. As we will show below, publicly provided unemployment benefits may provide very valuable social insurance. On the other hand, it may reduce incentive to keep job or find new ones. What is puzzling, however, is why, basically with unchanged benefit schemes over time, Europe did very well in the 60's and 70's, but fell behind (in the performance of their labor markets) in the 80's and 90's. Prescott's taxation story, discussed, maybe part of the story. Ljungqvist offer the following explanation. The 60's and 70's were a period of tranquil economic times, in the sense that a laid-off worker did not suffer large skill losses when being laid off. In the 80's the situation changed and laid-off workers faced a higher risk of loosing their skills when becoming unemployed (they call this increased turbulence). Thus in earlier times the European benefit system was not too distortive; it provided insurance and didn't induce laid-off households not to look for new jobs (because they had good skills and thus could find new, well-paid jobs easily). In the 80's, with higher chances of skill losses upon lay-off the benefit system becomes problematic. A newly laid off worker in Europe has access to high and long-lasting unemployment compensation; on the other hand, he may have lost his skill and thus is not offered new jobs that are attractive enough. Now he decides to stay unemployed, rather than accept a bad job. Higher turbulence plus generous benefits create the European unemployment dilemma. After having discussed what all may be wrong with generous unemployment benefits, let us provide a theoretical rationale for its existence in the first place, before coming back to the incentive problems such a system may create. 118 CHAPTER 6. SOCIAL INSURANCE 6.2 Social Insurance: Theory In this section we will study a simple insurance problem, first in the absence, then in the presence of a government-run public insurance system. We will focus on unemployment as the risk the household faces and on unemployment insurance as the government policy enacted to deal with it. Exactly the same analysis can be carried out for health risk and public health insurance, and death risk and social security. 6.2.1 A Simple Intertemporal Insurance Model Our agent lives for two periods. In the first period he has a job for sure and earns a wage of 2/1 • In the second period he may have a job and earn a wage of j/2 or be unemployed and earn nothing. Let p denote the probability that he has a job and 1—p denote the probability that he is unemployed. For simplicity assume that the interest rate r = 0. The utility function is given by log(ci) + p log(c|) + (1 - p) log(c^) where c| is his consumption if he is employed in the second period and c2 is his consumption if he is unemployed in the second period. His budget constraints are 2/1 2/2 + s s 6.2.2 Solution without Government Policy Let us start solving the model without government intervention. For now there is no public unemployment insurance. For concreteness suppose that income in the first period is given by yi and income in the second period is y2. First let's assume that p = 1, i.e. the household has a job for sure in the second period, and 2/1=2/2=2/ (that is, he keeps his same job with same pay). Then the maximization problem reads as maxlog(ci) + log(c|) + 0 * log(c^) s.t. ci + s = y cf2 = V + s cl = s Obviously in this situation the household does not face any uncertainty, and his choice problem is the standard one studied many times before in this class. Its optimal solution is ci ci = c2 = y s = 0 6.2. SOCIAL INSURANCE: THEORY 119 Note that the choice c2 is irrelevant, since (1 — p) = 0. Now let us introduce uncertainty: let yi = y, and p = 0.5 and y2 = 1y\ = 2y. That is, the household's expected income in the second period is 0.5 * 2y + 0.5 * 0 = y as before. But now the household does face uncertainty and we are interested in how his behavior changes in the light of this uncertainty. His maximization problem now becomes maxlog(ci) + 0.51og(cf) + 0.5 log^) s.t. ci + s = y (6.1) 4 = 2y + S (6.2) cl = s (6.3) This is a somewhat more complicated problem, so let us tackle it carefully. There are 4 choice variables, (ci, c|, c%, s). One could get rid of one by consolidating two of the three budget constraints, but that makes the problem more complicated than easy. Let us simply write down the Lagrangian and take first order conditions. Since there are three constraints, we need three Lagrange multipliers, Ai, X2, A3. The Lagrangian reads as L = log(ci)+0.51og(c|)+0.51og(c2)+Ai (y - cx - s)+A2 (2y + s - c|)+A3 (s - c£) Taking first order conditions with respect to (ci, c|, c%, s) yields — - Ai = 0 ci 0.5 —-A2 = 0 c2 0.5 --A3 = 0 -Ai + A2 + A3 = 0 ci 0.5 0.5 A3 c2 A2 + A3 = Ai 120 CHAPTER 6. SOCIAL INSURANCE Substituting the first three equations into the last yields 0.5 0.5 1 , N — + — = — (6.4) c2 c2 c\ Now we use the three budget constraints (6.1)-(6.3) to express consumption in (6.4) in terms of saving: 0.5 0.5 1 2y + s s (y- s) which is one equation in one unknown, namely s. Unfortunately this equation is not linear in s, so it is a bit more difficult to solve than usual. Let us bring the equation to one common denominator, s * (2y + s) * (y — s), to obtain 0.5s(y - s) 0.5 (2y + s) (y - s) s (2y + s) s (2y + s)(y- s) s (2y + s) (y - s) s (2y + s) (y - s) or 0.5s(y -s) + 0.5 (2y + s) (y - s) - s (2y + s) _ s (2y + s)(y- s) But this can only be 0 if the numerator is 0, or 0.5s(y -s) + 0.5 (2y + s) (y - s) - s (2y + s) = 0 Multiplying things out and simplifying a bit yields s2 +ys- ^y2 = 0 This is a quadratic equation, which has in general two solutions.2 They are S1 = -i-/5=-^(i+v^)o J Remember that if you have an equation where a, b are parameters, then the two solutions are given by / a2 / a2 X2 = —2+\jT~b 2 For these solutions to be well-defined real numbers we require --b > 0 6.2. SOCIAL INSURANCE: THEORY 121 The first solution can be discarded on economic grounds, since it leads to negative consumption c_ = s = — (l + \/3) . Thus the optimal consumption and savings choices with uncertainty satisfy i.(vs-i) > 0 ci = y-\y(Vl-l) = \v(z- v^) y V3-11 We make the following important observation. Even though income in the first period and expected income in the second period has not changed at all, compared to the situation without uncertainty, now households increase their savings and reduce their first period consumption level: \v (3 " V3) ci = T,y ( 3 - V 3 ) < y = ci > 0 = s This effect of increasing savings in the light of increased uncertainty (again: expected income in the second period remains the same, but has become more risky) is called precautionary savings. Households, as precaution against income uncertainty in the second period, save more with increased uncertainty, in order to assure decent consumption even when times turn out to be bad. We assumed that households have log-utility. But our result that households increase savings in response to increased uncertainty holds for arbitrary strictly concave utility functions that have a positive third derivative, or u"'(c) > 0 (one can easily check that log-utility satisfies this). Note that strict concavity alone (that is, risk-aversion) is not enough for this result. In fact, if utility is m(c) = — ^(c — 100, 000)2 (with 100,000 being the bliss point of consumption) then the household would choose exactly the same first period consumption and savings choice with or without uncertainty. In this case the first order conditions become -id - 100, 000) = Ai -0.5(4 - 100, 000) = A2 -0.5(c_ - 100,000) = A3 A2 + A3 = Ai Inserting the first three equations into the fourth yields -(ci - 100,000) = -0.5(c| - 100,000) - 0.5(c_ - 100, 000) ci = 0.5(4 + CV) 122 CHAPTER 6. SOCIAL INSURANCE Now using the budget constraints one obtains y-s = 0.5(2y + s+ s) y-s = y+s 2s = 0 and thus the optimal savings choice with quadratic utility is s = 0, as in the case with no uncertainty. Economists often say that under quadratic utility optimal consumption choices exhibit "certainty equivalence", that is, even with risk households make exactly the same choices as without uncertainty. Note that obviously realized consumption in period differs with and without uncertainty. With uncertainty one consumes 2y with probability 0.5 and 0 with probability 0.5, whereas under certainty one consumes y for sure. So while expected consumption remains the same, realized consumption (and thus welfare) does not. Finally note that with quadratic utility households are risk-averse and thus dislike risk, but they optimally don't change their saving behavior to hedge against it. It is easy to verify that with quadratic utility u'" = 0, thus providing no contradiction to our previous claim about precautionary savings. 6.2.3 Public Unemployment Insurance Rather than to dwell on this point, let us introduce a public unemployment insurance program and determine how it changes household decisions and individual welfare. The government levies unemployment insurance taxes on employed people in the second period at rate r and pays benefits b to unemployed people, so that the budget of the unemployment insurance system is balanced. There are many people in the economy, so that the fraction of employed in the second period is p = 0.5 and the fraction of unemployed is 1 — p = 0.5 Thus the budget constraint of the unemployment administration reads as ry2 = b and the budget constraints in the second period become c2 = (i - T)y-2 +s c% = b + s = ry2 + s For concreteness suppose that r = 0.5 and y2 = 2yi = 2y as before, so that 0.5ry2 = 0.5b or c2 = y + s c2 = y + s (6.5) (6.6) That is, the unemployment system perfectly insures the unemployed: unemployment benefits are exactly as large as after tax income when being employed. We 6.2. SOCIAL INSURANCE: THEORY 123 can again solve for optimal consumption and savings choices. One could set up a Lagrangian and proceed as always, but in this case a little bit of clever thinking gives us the solution much easier. From (6.5) and (6.6) it immediately follows that c| = eg = c2 no matter what s is. But then the maximization problem of the household boils down to maxlog(c!) + 0.51og(c2) + 0.51og(c2) maxlog(ci) + log(c2) s.t. y y + s with obvious solution Cl + s = C'2 = ci = c2 = y s = 0 exactly as in the case without income uncertainty. That is, when the government completely insures unemployment risk, private households make exactly the same choices as if there was no income uncertainty. Three final remarks: 1. In terms on welfare, would individuals rather live in a world with or without unemployment insurance? With perfect unemployment insurance their lifetime utility equals Vms = log(y) + log(y) which exactly equals the lifetime utility without income uncertainty. Without unemployment insurance lifetime utility is Vno = log (3 - V3)) +0.5 log (3 + v^)) +0.5 log Qy (y/3 - l) and it is easy to calculate that Vms > Vno. 2. Even if the unemployment insurance would only provide partial insurance, that is 0 < r < 0.5, the household would still be better of with that partial insurance than without any insurance (although it becomes more messy to show this). Risk-averse individuals always benefit from public (or private) provision of actuarially fair insurance; but they prefer more insurance to less, absent any adverse selection or moral hazard problem. 124 CHAPTER 6. SOCIAL INSURANCE 3. Above we have made a strong case for the public provision of complete unemployment insurance. No country provides full insurance against being unemployed, not even the European welfare states. Why not? In contrast to the model, where getting unemployed is nothing households can do something about, in the real world with perfect insurance a strong moral hazard problem arises. Why work if one get's the same money by not working. As always, the policy maker faces an important and difficult trade-off between insurance and economic incentives. If the government could perfectly monitor individuals and thus observe whether they became unemployed because of bad luck or own fault and also monitor their intensity in looking for a new job, then things would be easy: simply condition payment of benefits on good behavior. But if these things are private information of the households, then the complicated trade-off between efficiency and insurance arises, and the optimal design on an optimal unemployment insurance system becomes a difficult theoretical problem, one that has seen very many interesting research papers in the last 5 years. These, however, are well beyond the scope of this class. Part III Optimal Fiscal Policy 125 Chapter 7 Optimal Fiscal Policy with Commitment 7.1 The Ramsey Problem 7.2 Main Results in Optimal Taxation 127 128 CHAPTER 7. OPTIMAL FISCAL POLICY WITH COMMITMENT Chapter 8 The Time Consistency Problem 129 CHAPTER 8. THE TIME CONSISTENCY PROBLEM Chapter 9 Optimal Fiscal Policy without Commitment 131 132CHAPTER 9. OPTIMAL FISCAL POLICY WITHOUT COMMITMENT Part IV The Political Economics of Fiscal Policy 133 Chapter 10 Intergenerational Conflict: The Case of Social Security 135 136CHAPTER 10. INTERGENERATIONAL CONFLICT: THE CASE OF SOCIAL SECURITY Chapter 11 Intragenerational Conflict: The Mix of Capital and Labor Income Taxes 137 138CHAPTER 11. INTRAGENERATIONAL CONFLICT: THE MIX OF CAPITAL AND LABOR INC Bibliography [1] Attanasio, O., J. Banks, C. Meghir and G. Weber (1999), "Humps and Bumps in Lifetime Consumption". Journal of Business and Economics Statistics 17, 22-35. [2] Barro, R. (1974), "Are Government Bonds Net Wealth?," Journal of Political Economy, 82, 1095-1117. [3] Deaton, A. (1985), "Panel Data from Time Series of Cross-Sections". Journal of Econometrics 30, 109-126. 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