Budget Constraint, Preferences and Utility Varian: Intermediate Microeconomics, 8e, Chapters 2, 3 and 4 Consumer Theory Consumers choose the best bundles of goods they can afford. • This is virtually the entire theory in a nutshell. • But this theory has many surprising consequences. Two parts to consumer theory • "can afford" - budget constraint • "best" - according to consumers' preferences Consumer Theory (cont'd) What do we want to do with the theory? • Test it. See if it is adequate to describe consumer behavior. • Predict how behavior changes as economic environment changes. • Use observed behavior to estimate underlying values. These values can be used for • cost-benefit analysis, • predicting impact of some policy. Budget Constraint The first part of the lecture explains • what is the budget constraint and • how changes in income and prices affect the budget line, • how taxes, subsidies and rationing affect the budget line. Consumption Bundle For goods 1 and 2, the consumption bundle {x1,x2) shows how much of each good is consumed. Suppose that we can observe • the prices of the two goods (pi, p2) • and the amount of money the consumer has to spend m (income). The budget constraint can be written as pixi + p2x2 < m. The affordable consumption bundles are bundles that don't cost more than income. The set of affordable consumption bundles is budget set of the consumer. Two Goods Theory works with more than two goods, but can't draw pictures. We often think of good 2 (say) as a composite good, representing money to spend on other goods. Budget constraint becomes pxXx + x2 < m. Money spent on good 1 (pixi) plus the money spent on good 2 (x2) has to be less than or equal to the available income (m). Budget Line Budget line is piXi + p2x2 = m. It can be also written as m pi x2 =---Xi. P2 P2 Slope of budget line = opportunity cost of good 1. Horizontal intercept = m/p. Change in Income Increasing m makes parallel shift out. The vertical intercept increases and the slope remains the same. m./p m'/pi A', Change in One Price Increasing pi makes the budget line steeper. The vertical intercept remains the same and the slope changes. nv'pi m/pi x Changes in More Variables Multiplying all prices by t is just like dividing income by t: m tPiXi + tp2x2 = m <^=^ pxXx + P2X2 = j. Multiplying all prices and income by t doesn't change budget line: tpiXi + tp2X2 = tm piXi + P2X2 = m. A perfectly balanced inflation doesn't change consumption possibilities. Numeraire We can arbitrarily assign one price or income a value of 1 and adjust the other variables so as to describe the same budget set. Budget line: piXi + P2X2 = m The same budget line for p2 = 1: Pi , rn —xi +x2 = —. P2 P2 The same budget line for m = 1: Pi . P2 * —xi H--x2 = 1. m m The price adjusted to 1 is called the numeraire price. Useful when measuring relative prices; e.g. English pounds per dollar, 1987 dollars versus 1974 dollars, etc. Taxes Three types of taxes: • quantity tax - consumer pays amount t for each unit she purchases. —>■ Price of good 1 increases to px + t. • value tax (or ad valorem tax) - consumer pays a proportion of the price r. —>■ Price of good 1 increases to px + rpi = (1 + r)pi. • lump-sum tax - amount of tax is independent of the consumer's choices. —>■ The income of consumer decreases by the amount of the tax. Subsidies Subsidies - opposite effect than the taxes • quantity subsidy of s on good 1 —>■ Price price of good 1 decreases to px — s. • ad valorem subsidy at a rate of a on good 1 —>■ Price price of good 1 decreases to px — opt = (1 — cr)pi. • lump-sum subsidy —)- The income increases by the amount of the subsidy. Rationing Rationing - can't consume more than a certain amount of some good. Good 1 is rationed, no more than x units of good 1 can be consumed by any consumer. *2 Budget Taxing Consumption Greater than xi Taxed only consumption of good 1 in excess of x1: the budget line becomes steeper right of xi The Food Stamp Program Before 1979 was an ad valorem subsidy on food • paid a certain amount of money to get food stamps which were worth more than they cost • some rationing component — could only buy a maximum amount of food stamps After 1979 got a straight lump-sum grant of food coupons. Not the same as a pure lump-sum grant since could only spend the coupons on food. »Mt FOOD COUPONS VALUE $7.00 FOOD COUPONS VALUE «5.00 Budget line without food stamps Budget line with food stamps Budget \ I line without \ \ food K stamps $200 F(K)I) B FIGURE 2.6 Food stamps Intermediate Micmerrmomics. ävi Edition rnpyritjh! ö 2010 W. W. Nnrtm S, Company Summary The budget set consists of bundles of goods that the consumer can afford at given prices and income. Typically assume only 2 goods -one of the goods might be composite good. The budget line can be written as PlXl + P2*2 = m. Increasing income shifts the budget line outward. Increasing price of one good changes the slope of the budget line. Taxes, subsidies, and rationing change the position and slope of the budget line. Preferences The second part of the lecture explains • what are consumer's preferences, • what properties have well-behaved preferences, • what is marginal rate of substitution. Preferences - Introduction Economic model of consumer behavior -choose the best things they can afford • up to now, we clarified "can afford" • next, we deal with "best things" people Several observations about optimal choice from movements of budget lines • perfectly balanced inflation doesn't change anybody's optimal choice • after a rise of income, the same choices are available - consumer must be at least as well of as before °4i" 8 ' Preferences Preferences are relationships between bundles. • If a consumer chooses bundle (xltx2) when (yi,y2) is available, then it is natural to say that bundle (x1,x2) is preferred to (y!,y2) by this consumer. • Preferences have to do with the entire bundle of goods, not with individual goods. Notation • (xiix2) >~ (yiiy2) means the x-bundle is strictly preferred to the y-bundle. • (xiix2) ~ (yiiy2) means that the x-bundle is regarded as indifferent to the y-bundle. • {xi,X2) >z (yiiYi) means the x-bundle is at least as good as (or weakly preferred) the y-bundle. Assumptions about Preferences Assumptions about "consistency" of consumers' preferences: • Completeness — any two bundles can be compared: (xi,x2) h (yi,y2), or (xi,x2) ^ (yi,y2), or both • Reflexivity — any bundle is at least as good as itself: (*i,x2) h (xi, x2) • Transitivity — if the bundle X is at least as good as Y and Y at least as good as Z, then X is at least as good as Z: If (xi,x2) y (yi,y2), and (yi,y2) h (zi,z2), then (xi,x2) h {zi,z2) Transitivity necessary for theory of optimal choice. Otherwise, there could be a set of bundles for which there is no best choice. Scissors Indifference Curves Weakly preferred set are all consumption bundles that are weakly preffered to a bundle (xi,x2). Indifference curve is formed by all consumption bundles for which the consumer is indifferent to (xx,x2) - like contour lines on a map. Weekly preferred sel: bundles weakly preferred lo (x,, x2) Indifference curve: bundles indifferent to (x,,x2) Indifference Curves (cont'd) Note that indifference curves describing two distinct levels of preference cannot cross. Proof— we know that X ^ Z and Z ~ Y. Transitivity implies that X ~ Y. This contradicts the assumption that X >- Y. x2 M leged indifference Examples: Perfect Substitutes Perfect substitutes have constant rate of trade-off between the two goods; constant slope of the indifference curve (not necessarily —1). E.g. red pencils and blue pencils; pints and quarts. Examples: Perfect Complements Perfect complements are consumed in fixed proportion (not necessarily 1:1). E.g. right shoes and left shoes; coffee and cream. Indifference curves RIGHT SHOES Examples: Bad Good A bad is a commodity that the consumer doesn't like. Suppose consumer is doesn't like anchovies and likes pepperoni. Pri'PFROM Examples: Neutral Good Consumer doesn't care about the neutral good. Suppose consumer is neutral about anchovies and likes pepperoni. ANCHOVIES __ Indifference curves PHWKONI Examples: Satiation Point Satiation or bliss point is the most preferred bundle (xi,x2) • When consumer has too much of good, it becomes a bad -reducing consumption of the good makes consumer better off. • E.g. amount of chocolate cake and ice cream per week Indifference _ _ curves Satiation \s>______ point Examples: Discrete Good Discrete good is only available in integer amounts. • Indiference "curves" - sets of discrete points; weakly preferred set - line segments. • Important if consumer chooses only few units of the good per time period (e.g. cars). N j 1 2 I C(fD A Indifference "curves" Bundles 7 weakly preferred to ("I, x2) 12 3 GOOD Weakly preferred set Wei I-Behaved Preferences Monotonicity - more is better (we have only goods, not bads) => indifference curves have negative slope (see Figure 3.9): If (ylt y2) has at least as much of both goods as (x1,x2) and more of one, then (yi,y2) >- (xi,x2). Convexity - averages are preferred to extremes =>• slope gets flatter as you move further to right (see Figure 3.10): If (xi, x2) ~ (yi, y2), then (txi + (1 - t)yu tx2 + (1 - t)y2) h (xi, x2) for all 0 < t < 1 • non convex preferences - olives and ice cream • strict convexity - If the bundles (x!,x2) ~ (yi,y2), then (txi + (1 - t)yu tx2 + (1 - t)y2) > (xi, x2) for all 0 < t < 1 A Convex B Nonronvex C Concave preferences preferences preferences FIGURE 3.10 Various kinds of preferences Intermediate Microeconomics, 8th Edition Marginal Rate of Substitution Marginal rate of substitution (MRS) is the slope of the indifference curve: MRS = Ax2/Axx = dx2/dx1. Sign problem — natural sign is negative, since indifference curves will generally have negative slope. Marginal Rate of Substitution (cont'd) MRS measures how the consumer is willing to trade off consumption of good 1 for consumption of good 2 (see Figure 3.12). For strictly convex preferences, the indifference curves exhibit diminishing marginal rate of sustitution Other interpretation: marginal willingness to pay - how much of good 2 is one willing to pay for a extra consumption of good 1. If good 2 is a composite good, the willingness-to-pay interpretation is very natural. Not the same as how much you have to pay. FIGURE 3.12 Trading al an exchange rate Example: Slope of the Indifference Curve 1) Calculate the slope of the indifference curve x2 = 4/xi at the point (x1,x2) = (2,2). Slope of the indifference curve = MRS = —— = —= —1. dx2 _ -4 2) Calculate the slope of the indifference curve x2 = 10 — 6v/x[ at the point (xi,x2) = (4, 5). ďx2 —3 —3 Slope of the indifference curve = MRS = —— = —= = —. Summary Economists assume that a consumer can rank consumption bundles. The ranking describes the consumer's preferences. The preferences are assumed to be complete, reflexive and transitive. Well-behaved preferences are monotonie and convex. MRS measures the slope of the indifference curve. MRS can be interpreted as how much of good 2 is one willing to pay for an extra consumption of good 1. Utility The third part of the lecture explains • what is utility, • what is a utility function, • what is a monotonie tranformation of a utility function, • how can we use utility function to calculate MRS. Utility Two ways of viewing utility: Old way - measures how "satisfied" you are (cardinal utility) • not operational • many other problems New way - summarizes preferences, only the ordering of bundles counts (ordinal utility) • operational • gives a complete theory of demand 2 ......i7.«*,, two 1 ■ ■ ■ seven 3 L 871 eight »» ?JI«Hf!° Ordinal Utility A utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers. If (xi,x2) >- (yi,y2), then u(xi,x2) > u{y1,y2). Three ways to assign utility that represent the same preferences: Bundle u, U2 u3 A 3 17 -1 B 2 10 -2 C 1 .002 -3 Utility Function is Not Unique A positive monotonic transformation f(u) is any increasing function. Examples: f(u) = 3u, f(u) 3, f(u) If u(xltx2) is a utility function that represents some preferences, then f(u(xlt x2)) represents the same preferences. Why? Because u(xi,x2) > u(y1,y2) only if f{u(x1,x2)) > f{u(y1, y2)). Nondccrcaslng iffxl > 0 Constructing Utility Functions Mechanically using the indifference curves. *> Examples: Utility to Indifference Curves Easy —just plot all points where the utility is constant Utility function u(x1,x2) = x1x2; Indifference curves: k = xxx2 x2 = — Examples: Indifference Curves to Utility More difficult - given the preferences, what combination of goods describes the consumer's choices. Perfect substitutes • All that matters is total number of pencils, so u(xi, x2) = xi + x2 does the trick. • Can use any monotonie transformation of this as well, such as ln(xi + x2). Perfect complements • What matters is the minimum of the left and right shoes you have, so u(xi,X2) = min{xi,X2} works. • In general, if it not 1:1, the utility function is Lř(x!, x2) = min{ax!, bx2}, where a and b are positive numbers. Examples: Indifference Curves to Utility (cont'd) Quasilinear preferences • Indifference curves are vertically parallel (see Figure 4.4). Not particularly realistic, but easy to work with. • Utility function has form u(xlt x2) = v(x1) + x2 • Specific examples: u(xlt x2) = ^/xi + x2 or u(xi, x2) = In x1 + x2 Cobb-Douglas preferences • Simplest algebraic expression that generates well-behaved preferences. • Utility function has form u(x1,x2) = x*x| (See Figure 4.5). • Convenient to take transformation f(u) = u~< and write xf^xf1 or x{x\~3, where a = b/(b + c). A I = 1/2 it is not an operational concept. • However, ML) is closely related to MRS, which is an operational concept. Relationship between Mil and MRS An indifference curve u(x;l,x2) = k, where k is a constant. We want to measure slope of indifference curve, the MRS. So consider a change (Axx, Ax2) that keeps utility constant. Then, iWiAxi + MU2Ax2 = 0 du . du . —Axx + — Ax2 = 0. ox1 ox2 Hence, Ax2 _ MUi AxT ~ ~W2 So we can compute MRS from knowing the utility function. Example: Utility for Commuting Question: Take a bus or take a car to work? Each way of transport represents bundle of different characteristics: Let xx be the time of taking a car, yi be the time of taking a bus. Let x2 be cost of car, etc. Suppose utility function takes linear form (7(x1,...,xn) = /31x1 + ... + /3nxn. We can observe a number of choices and use statistical tech- niques to estimate the parameters /?,- that best describe choices. Example: Utility for Commuting (con't) Domenich a McFadden (1975) report a utility function U{ TE, TT, C) = -0.147 TW - 0.0411 TT - 2.24C, where TW = total walking time to and from bus or car in minutes TT = total time of trip in minutes C = total cost of trip in dollars. Once we have the utility function we can do many things with it: • Calculate the marginal rate of substitution between two characteristics. How much money would the average consumer give up in order to get a shorter travel time? • Forecast consumer response to proposed changes. • Estimate whether proposed change is worthwhile in a benefit-cost sense. Summary A utility function is a way to represent a preference ordering. The numbers assigned to different utility levels have no intrinsic meaning. Any monotonie transformation of a utility function will represent the same preferences. The marginal rate of substitution is equal to MRS = Ax2/Axi = -MU!/MU2.