Overlapping Generations Models Kaiji Chen University of Oslo October 29, 2008 1 Review of Last Class • Definition of Great Depression — A large negative deviation from trend (balanced) growth path. — On balanced growth path, capital output ratio is constant, and all per capita variables grow at constant rate except hours per working age person. — episode including not only sharp decline but also probably slow recovery. 2 • Great Depression Methodology — Growth accounting: various shocks affect aggregate output during depression through three channels: efficiency that inputs are combined together for production, capital input, and labor input (from both supply and demand sides). — identify the quantitative importance of these channels through dynamic general equilibrium model. — Simulation tells us sharp declines in TFP are important for the output drop during U.S. Great Depression, but not the slow recovery. — direction for future research: what are the shocks that cause the decline in TFP? 3 Motivation • Life cycle profile of consumption and asset accumulation are hump-shaped. • Policy analysis — social security — effects of taxes on retirement decisions — distribution effects of taxes — effects of life time saving on capital accumulation — demographic transition 1 • Need a model in which agents experience a life cycle and people of different ages live at the same time. The OLG models are very useful tools for policy analysis. • Also, OLG models have some different theoretical implications than infinite-horizon models. 2 Road map • OLG setup • Equilibrium • dynamics and the balanced-growth path 3 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION 1 Overlapping Generation model with Production 1.1 Environment • Discrete time: t = 1, 2, 3... • Demography: — A new generation is born into the economy at the beginning of each period t > 1. Agents within each generations are homogeneous. Each generation is indexed according to his date of birth (e.g. agents who were born at date 1 are called cohort 1). 4 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION — Agents live for two periods. — At t = 1, there are some old agents who were born at t = 0. Normalize the size of cohort 0 to 1. — Population grows at a constant rate: Lt = Lt_ 1 (1 + n). • Preference — newborn at t > 1 : U (c1t,C2t+1) = u (c1t) + fiu (c2t+1),where u1 (•) > 0,u" (•) < 0. — the old at t = 1 : U (C21) = u (C21) 5 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • Endowment — cohorts t > 1 have one unit of time to work when young, no endowment when old. — initial old is endowed with capital ki > 0, which is given exoge-nously. • Technology — A representative firm owns technology Yt = F (Kt,Lt), renting labor Lt and capital Kt from agents alive at time t. For simplicity, no capital depreciation. — Assuming constant return to scale to production (CRS) ) yt = f (kt); where the lower case variables denote the corresponding variables in per capita term. 6 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION_ — f (•) > 0, /" (•) < 0. In addition, / (0) = 0, f (0) = 1, f0 (1) = 0 (Inada conditions). 7 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • Timing of events at a certain period — At the beginning of period t, young agents supply labor and old agents supply capital (saved at the end of t — 1) to the firm, who produces goods Yt during period t. — At the end of period t, young agents get their labor income and decide how much to consume (cit) and how much to save (sit) . saving occurs in the form of physical capital, the only asset in the economy. — At the beginning of t +1, production takes place with labor from cohort (born at) t +1, and capital from cohort (born at) t. Net rate of return to savings equals rt+i. — At the end of period t + 1, cohort t consumes what saved at the last period plus interest come, i.e. (1 + rt+i)sit, and dies. 8 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • All agents have perfect foresight. 9 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION 1.2 Equilibrium The problem for cohort t > 1 Ci+ ^ s1+U (c1t) + /u (c2t+1) (1) c1t,c2t+1 ,s1t subject to cit + sit < wt (2) c2t+1 < (1 + rt+1) s1t (3) 10 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION The problem for cohort 0 max™ (C21) (4) subject to c21 < (1 + r1) k1 The representative firm's problem max F (Kt, Lt) - wtLt - rtKt (5) Kt,Lt 11 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION Competitive Equilibrium Definition: A competitive equilibrium is a sequence of allocation for agents b21; {bit, b2t+i, bt}1=i, a sequence of allocation for the firm JKt, ^t]^ 1 , and prices {b^wDt}1^, such that • for all cohorts t > 1, given prices {bt, wbt}1=i, {bit, b2t+1, bt} solve the problem of cohort t, i.e. (1). • for initial old generation, given prices {bt,wbt}£=!, b2i solve the problem facing the initial old generation. • for all t > 1,given prices {bt,wbt}1=i, JKt,Ltj solve the represen- t=1 t=1 tative firm's problem. 12 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • {r^wDtg^i are such that all markets clear. — goods market clears Ltbit + Lt_iC2t + Kt+i = F (Ku Lt) + Kt (6) The left-hand side (LHS) of (6) is the consumption of the young, plus the consumption of the old alive at period t, plus the total saving of the young (resource put aside for tomorrow's production). The right-hand-side (RHS) of (6) is the quantity of goods available at period t. That is the consumption goods produced at time t, plus the capital that is left after production has taken place. — Labor market clears. 13 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION — Capital market clears Ltbit ~ Lt-i bit-i = Kt+i - Kt (7) The LHS of (7) is the aggregate savings of this economy at period t, denoted as St (i.e. the savings of the young, LtSit, minus the dissaving of the old, Lt_ibit—1), while the RHS of (7) is the aggregate investment at time t, It. In other words, capital market clearing condition implies St = It-Note if we denote the saving of the old at time t as b2t, then b2t = rtsit_i - C2t = rtsit-i - (1 + rt) «it-i = —sit—i- That is the old at time t simply dissave what they have saved at time t — i. i4 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION 1.3 Solving the model Cohort t > 1 • replace cit and C2t+i in the utility function with sit, using period budget constraints (2) and (3). • first order condition (Euler Equation) u (wt - sit) = /3 (1 + rt+i) u (sit(1 + rt+i)) (8) — LHS of Equation (8) is the marginal disutility of giving up one unit of consumption (or increasing one unit of saving) at time t; RHS of Equation (6) is the increase in utility (discounted back to time t) by the increase of one unit of saving at time t. 15 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • Equation (8) implies s1t = s (wt,rt+1) (9) 16 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • Assuming goods are normal, we have 0 <@s1t< 1 — if income today, wt, increases by one unit, agents will increase both periods' consumption. — Question: if cohort t also earn a wage income at time t +1, denoted as wt+1, what is the sign of @Wt+1 and why? • but @r 0,%f > 0. 22 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION Example • Constant Relative Risk Aversion (CRRA) Utility, 0 0 1—a i si—® 1 u = Cit " 1 + /C2t+1"1 1 - a 1 - a • Euler equation clta = ^ (1 + rt+1) c2t+1 or C2t±1 = [// (1 + rt+i)]1 (14) c1t — the more patient an agent is (the higher is //), the higher is the consumption growth rate. 23 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION 1 the cheaper is future consumption (the smaller is ), the higher is consumption growth. Question: how does the magnitude of intertemporal elasticity of substitution (^) affect the consumption growth? • substitute C2t+i into the intertemporal budget constraint, we get cit =-1-i-TTwt (i5) • Note a change of rt+i may affect the fraction of lifetime income to be consumed in the first period. 24 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION Effects of a change of rt+1 on s1t • Income effect: rt+i " =) Equation (10) indicates that the price of consumption at t + 1 # =) agents can afford more consumption given the same wage income as before =) both cit and C2t+i increase =) • Substitution effect: rt+i " =) the price of consumption at t relative to consumption at t +1 increases =) substitute C2t+i for cit =) sit " . 25 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • With CRRA utility, the relative magnitude of these two effects depend on ^ (intertemporal elasticity of substitution) — 1 > 1 (or a < 1), substitution effect dominates. (Mathematically, a determines the curvature of the utility function and the indifference curve.) — 1 < 1, income effect dominates — 1 = 1 (log utility), both effects cancel each other. As a result, a change of rt+1 has no impact on the fraction of lifetime resource that is consumed today. In this framework, C1t and thus saving rate (i.e. (wt — ct) /wot) are unchanged. 26 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION If agents also receive income wut+1 when old, how saving is affected by a change in rt+1? • Solving for the agent's problem, you will get _1_( , wt+1 c1t =-1-FT wt + 1 1 l t i 1 + /35 (1 + rt+1)5-H 1 + rt+1 ) • Still, our previous analysis regarding the income efect and substitution efect holds. • But now, agents' life time wealth ^w;t + 1+^++^ decreases due to an increase in the interest rate. 27 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • As a result, c1t decreases (also C2t+1 decreases). We call the impact of a change in rt+1 on S1t through its impact on lifetime wealth the wealth efects. 28 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION Special case: log utility and C-D production: a = 1, f (k) = kf c1t = (16) s1t = wt = (1 - a) kf (17) kt+1 = (1 + // + n)k (18) • Note, in this special case, rt+1 has no impact on S1t, and therefore, the saving rate. 29 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION At Balanced-Growth Path kt+i = kt = k*. Then (18) implies k* = —-±-1— (ig) Also, Kt+1 = (1 + n) Kt. Aggregate saving St = K+1 — Kt = nKt. Then aggregate saving rate, denoted as follows Sa = Yt = Yt = (1 + ß + n) ( ) where the last equality of Equation(20) derives from Equation (18) at the balanced-growth path. 30 1 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION Note increases in n. • When n becomes smaller (i.e. population ages), the size of young agents is smaller relative to the old at any period. Since the saving is done by the young, the saving rate becomes smaller (supplycurve shifts to the left.) • When n becomes smaller, the supply of next period labor is also smaller, which means labor becomes more scarce. The marginal productivity of capital thus is smaller. Hence, the demand curve also shifts to the left. 3i 1. OVERLAPPING GENERATION MODEL WITH PRODUCTION • In any case, the equilibrium saving rate is smaller. • But not necessary for r* (in this specific example, r* increases in n). 32