Choice and Revealed Preference Varian, Intermediate Microeconomics, 8e, Ch. 5 and Sections 7.1–7.7 Choice In this lecture explains: • What will be the optimal choice of consumer. • How this choice depends on the consumer’s preferences. • How can we estimate utility function from consumption choices we observe. • Some of the implications of optimal choice. Optimal Choice The optimal choice is where the indifference curve is tangent to the budget line: MRS = −price ratio = − p1 p2 Exceptions: Kinky Tastes The IC doesn’t cross the budget line, but there is no unique tangent line at the optimal point. For tangency, we need smooth indifference curves. Exceptions: Boundary Optimum The IC doesn’t cross the budget line, but the slopes are different. For tangency, we need an interior optimum = both goods are consumed. Exceptions: More Than One Tangency Tangency condition is only a necessary condition for optimal choice. It is sufficient in the case of convex preferences. Consumer Demand Optimal choice is the consumer’s demanded bundle. As we vary prices and income, we get demand functions. Demand functions will depend on both prices and income: x1(p1, p2, m) and x2(p1, p2, m). Different preferences lead to different demand functions We want to study how the demanded bundle changes as price and income change. Examples: Perfect Substitutes If goods 1 and 2 are perfect substitutes (1:1 trade-off), the demand function for good 1 is x1 =    m/p1 when p1 < p2; any number between 0 and m/p1 when p1 = p2; 0 when p1 > p2. Examples: Perfect Complements If goods 1 and 2 are perfect complements and consumer purchases the amount x of both goods (left and right shoe), the demand function can be derived from the budget constrant as follows: p1x + p2x = m ⇐⇒ x1 = x2 = x = m p1 + p2 . Examples: Neutrals and Bads The consumer spends all money on the good and doesn’t purchase the neutral good or the bad. If commodity 1 is good and commodity 2 is neutral or bad, the demand function is x1 = m p1 and x2 = 0. Examples: Discrete Goods Good 1 is discrete good (integer units) and good 2 is money. Consumption bundles: (1, m − p1), (2, m − 2p1), (3, m − 3p1), . . . Examples: Concave Preferences Tangency doesn’t work – like in the case of perfect substitutes. E.g. olives and ice cream. Examples: Cobb-Douglas Preferences The Cobb-Douglas utility function is u(x1, x2) = xc 1 xd 2 . It is convenient to use logs of the Cobb-Douglas utility function u(x1, x2) = ln(xc 1 xd 2 ) = c ln x1 + d ln x2. The problem we want to solve is max c ln x1 + d ln x2 such that p1x1 + p2x2 = m. Using MRS = −p1/p2, we get two equations with two unknowns: cx2 dx1 = p1 p2 , and p1x1 + p2x2 = m. Examples: Cobb-Douglas Preferences (cont’d) The solution of the equations are the Cobb-Douglas demand functions x1 = c c + d m p1 , x2 = d c + d m p2 Convenient property: The Cobb-Douglas consumer spends a fixed fraction of his income on each good: p1x1 m = p1 m c c + d m p1 = c c + d p2x2 m = p2 m d c + d m p2 = d c + d . Convenient to use the Cobb-Douglas utility function u(x1, x2) = xa 1 x1−a 2 where the parameter a is the fraction of income spent on good 1. Estimating Utility Function Examine the consumption data and see if you can “fit” a utility function to it. The income shares (s1, s2) are more or less constant =⇒ Cobb-Douglas utility function u(x1, x2) = x 1/4 1 x 3/4 2 seems to fit these data well. Estimating Utility Function (cont’d) We can use the fitted utility function as guide to policy decisions. Suppose a system of taxes that would result in prices (2,3) and the income of 200. At these prices, the demanded bundle is x1 = 1 4 200 2 = 25 x2 = 3 4 200 3 = 50 The estimated utility of this bundle is u(x1, x2) = 251/4503/4 ≈ 42, which is more than in year 2 but less than in year 3. In real life more complicated forms are used, but basic idea is the same. Implications of MRS Condition Why do we care that MRS = −price ratio? If everyone faces the same prices, then everyone has the same local trade-off between the two goods. This is independent of income and tastes. Since everyone locally values the trade-off the same, we can make policy judgments. Is it worth sacrificing one good to get more of the other? Prices serve as a guide to relative marginal valuations. Application: Choosing a Tax Which is better, a quantity tax or an income tax? We can show an income tax is always better in the sense that given any quantity tax, there is an income tax that makes the consumer better off. Application: Choosing a Tax (Outline of Argument) Quantity tax • original budget constraint: p1x1 + p2x2 = m • budget constraint with tax: (p1 + t)x1 + p2x2 = m • optimal choice with tax: (p1 + t)x∗ 1 + p2x∗ 2 = m • tax revenue raised: tx∗ 1 . Income tax that raises same amount of revenue leads to budget constraint: p1x1 + p2x2 = m − tx∗ 1 . • This line has same slope as original budget line. • also passes through (x∗ 1 , x∗ 2 ) – proof: p1x∗ 1 + p2x∗ 2 = m − tx∗ 1 . • this means that (x∗ 1 , x∗ 2 ) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x∗ 1 , x∗ 2 ). Application: Choosing a Tax (Caveats) • Only applies for one consumer, no longer true if we have to have the same income tax for all consumers. A person that doesn’t consume any of good 1 will certainly prefer the quantity tax to a uniform income tax. • We have assumed that income is exogenous. If income responds to tax, e.g. tax dicourages people from earning income =⇒ problems. • We left out supply response. A complete analysis would have to take supply response into account. Application: The Cost of Christmas Joel Valdfogel, “The Deadweight Loss of Christmas” (AER, 1993): • „In the standard microeconomic framework of consumer choice, the best a gift-giver can do with, say, $10 is to duplicate the choice that the recipient would have made. (p. 1328) • In most situations, the the recipient is worse off.. • Estimates cost of gift-giving using surveys given to Yale undergraduates. • Gift-giving destroys between 10 % and 1/3 of the value of the gift: min. loss $4 billions. • The worst gifts by extended family – also give money most often. Summary • The optimal choice is the bundle in the consumer’s budget set that lies on the highest indifference curve. • In optimum, the MRS is typically equal to the slope of budget line. • We can estimate a utility function from consumption choices and use it to evaluate economic policies. • If everyone faces the same prices of two goods, then everyone will have the same MRS between the two goods. Revealed Preference The second part of the lecture explains • what does it mean if a bundle is revealed preferred to another, • how can we recover preferences from the observed choices, • what are the Weak and Strong Axioms of Revealed Preference. The Motivation of Revealed Preference Up until now we’ve started with preference and then described behavior. But in real life, preferences are not directly observable. Revealed preference is “working backwards” — start with behavior and describe preferences. If we want to “recover” preferences from behavior of people, we have to assume that preferences don’t change over time. In this lecture, we also assume that the preferences are strictly convex – we get a unique demanded bundle (not necessary for the theory of revealed preferences, simplifies the exposition). The Idea of Revealed Preference If (x1, x2) is chosen when (y1, y2) is affordable, then we know that (x1, x2) is better than (y1, y2). In equations: If (x1, x2) is chosen when prices are (p1, p2) and (y1, y2) is some other bundle such that p1x1 + p2x2 ≥ p1y1 + p2y2, then if the consumer is choosing the most preferred bundle she can afford, we must have (x1, x2) (y1, y2). The Idea of Revealed Preference (cont’d) If p1x1 + p2x2 ≥ p1y1 + p2y2, we say that (x1, x2) is directly revealed preferred to (y1, y2). If X is revealed preferred to Y , and Y is revealed preferred to Z, then transitivity implies that X is indirectly revealed preferred to Z. The Idea of Revealed Preference (cont’d) The more choices, the more information about the consumer’s preferences. The figure uses the demanded bundles for recovering preferences – the indifference curve lies between the two shaded areas. Choices Inconsistent with the Model of Consumer Choice In the figure, (x1, x2) is directly revealed preferred to (y1, y2) and (y1, y2) is directly revealed preferred to (x1, x2). Or for a bundle (x1, x2) purchased at prices (p1, p2) and a different bundle (y1, y2) purchased at prices (q1, q2), we have p1x1 + p2x2 > p1y1 + p2y2 and q1y1 + q2y2 > q1x1 + q2x2. Weak Axiom of Revealed Preference Weak Axiom of Revealed Preference (WARP): If (x1, x2) is directly revealed preferred to (y1, y2), then (y1, y2) cannot be directly revealed preferred to (x1, x2). In other words, for a bundle (x1, x2) purchased at prices (p1, p2) and a different bundle (y1, y2) purchased at prices (q1, q2), if p1x1 + p2x2 ≥ p1y1 + p2y2, then it must not be the case that q1y1 + q2y2 ≥ q1x1 + q2x2. How to Check WARP How to test systematically WARP? Consider the following consumption data: The table below shows cost of each bundle at each set of prices. The choices (e.g. bundle 1, prices 1) are directly revealed preferred to baskets in the same row with ∗ (e.g. bundle 2, prices 1). Strong Axiom of Revealed Preference WARP is only a necessary condition for behavior to be consistent with utility maximization. Strong Axiom of Revealed Preference (SARP): if (x1, x2) is directly or indirectly revealed preferred to (y1, y2), then (y1, y2) cannot be directly or indirectly revealed preferred to (x1, x2). If his observed behavior is consistent with SARP, then we can always find well-behaved preferences (utility function) that explain the behavior of the consumer as maximizing behavior. SARP is a necessary and sufficient condition for utility maximization. How to Check SARP The table below shows cost of each bundle at each set of prices. The choices are indirectly revealed preferred to baskets in the same row with (∗) (e.g. at prices 1, the bundle 1 is indirectly revealed preferred to bundle 3). This gives us a completely operational test for whether a particlular consumer’s choices are consistent with economic theory. We could use it also for units consisting of several people like households or universities. Summary • If one bundle is chosen when another bundle could have to be chosen, it is revealed prefered to the second. • The consumer chooses the the most preferred bundle she can afford. Therefore, the chosen bundles must be preferred to the affordable bundles that were not chosen. • We can estimate consumer’s preferences using observed choices. • The Strong Axiom of Revealed Preference (SARP) is a necessary and sufficient condition that consumer choices must obey if they are to be consistent with the economic model of optimizing choice.