Economics 2010a Fall 2003 Edward L. Glaeser Lecture 2 2. Choice and Utility Functions a. Choice in Consumer Demand Theory and Walrasian Demand b. Properties of demand from continuity and properties from WARP c. Representing Preferences with a Utility Function d. Demand as Derived from Utility Maximization e. Application: Fertility x i denotes commodities, continuous numbers x ? ?x 1 ,x 2 ,....x L ? vector of discrete commodities p ? ?p 1 ,p 2 ,....p L ? vector of prices w ? wealth available to be spent The budget constraint p ? x ? ? i?1 L p i x i ? w MWG Definition 2.D.1 The Walrasian Budget Set B p,w ? ?x ? ? ? L : p ? x ? w? is the set of all feasible consumption bundles for the consumer faces market prices p and has wealth w. Note: We will be treating all prices and consumption levels as being weakly positive. Prices are treated as exogenous– as they will be in the production case. While neither consumer nor producer chooses prices (generally) prices are the extra parameter in each side’s problem that ensures that demand and supply are equal. Non-linear prices are certainly possible (example 2.D.4). The Walrasian Demand Function is the set C?B p,w ? which is defined for all ?p,w?, or at least for a full dimensional subset ?p,w? ? ? ? L?1 We generally assume that C?B p,w ? has a single element (for convenience) but it doesn’t need to. We write C?B p,w ? ? x?p,w? ? ?x 1 ?p,w?,...x L ?p,w?? We will also generally assume that demand is continuous and differentiable. MWG Definition 2.E.1: The Walrasian Demand Function is homogeneous of degree zero if x??p,?w? ? x?p,w? for any p,w and ? ? 0. This property follows from the fact that choice is only a function of the budget set and B p,w ? ?x ? ? ? L : p ? x ? w? is the same set as B ?p,?w ? ?x ? ? ? L : ?p ? x ? ?w? This fairly obvious claim is in many ways the underlying intellectual basis for the economic bias that the price level doesn’t matter. Differentiating x??p,?w? ? x?p,w? totally with respect to ? gives us the following equation: ? i?1 L ?x k ?p,w? ?p i p i ? ?x k ?p,w? ?w w ? 0 ? i?1 L ?x k ?p,w? ?p i p i x k ? ?x k ?p,w? ?w w x k ? ? i?1 L ? p i k ?? w k ? 0 This tells you that for any commodity, the sum of own and cross price elasticities equals -1 times the income elasticity. MWG Definition 2.E.2 Walras’ Law: The Walrasian Demand correspondence x?p,w? satisfies Walras’ law if for every p ? 0 and w ? 0, we have p ? x ? w for all x ? x?p,w?. This just says that the consumer spends all of his wealth. Looking ahead, Walras’ law will come about as long as consumers are not “satiated” in at least one of the goods. Walras’ Law and differentiability give us two convenient equalities. Differentiating p ? x ? w with respect to w yields: ? i?1 L p i ?x i ?p,w? ?w ? 1 or manipulating this slightly yields: ? i?1 L ?x i ?p,w? ?w w x i p i x i w ? ? i?1 L ? w i ? i ? 0, where ? i ? p i x i w , the budget share of good i. This means that income elasticities (when weighted by budget shares) sum to one. All goods can’t be luxuries, etc. Sometimes this is known as Engel aggregation (income effects are after all drawn with Engel curves). Differentiating p ? x ? w with respect to p k yields: ? i?1 L p i ?x i ?p,w? ?p k ?x k ? 0 or multiplying the whole expression by p k w : ? i?1 L ?x i ?p,w? ?p k p k x i p i x i w ? p k x k w ? ? i?1 L ? p k i ? i ?? k ? 0, where again ? i ? p i x i w This means that cross price elasticities sum to -1 times the budget share of the relevant good. Overall, these elasticities have to sum to a negative number. MWG Definition 2.F.1 The Walrasian Demand function satisfies the weak axiom of revealed preference if the following property holds for any two price-wealth situations ?p,w? and ?p ? ,w ? ? : If p ? x?p ? ,w ? ? ? w and x?p ? ,w ? ? ? x?p,w? then p ? ? x?p,w? ? w ? . In words– if the goods that are chosen with budget set (a) are affordable at budget set (b), and not the same as the goods that are chosen at budget set (b), then the goods that are chosen at budget set (b) are not affordable at budget set (a). Just like in the last lecture, WARP means that if bundle (b) is preferred to bundle (a) in one setting, it will be preferred in all other settings. A property that follows from WARP: price changes that are fully income compensated make consumers weakly better off. Take any ?p,w? and let p ? ? p??p Compensate the consumer with an income change so that the old bundle is exactly affordable at the new prices, i.e.: w ? ? w??w ? x?p,w? ??p The consumer’s new consumption level at ?p ? ,w ? ? satisfies the WARP condition: p ? ? x?p,w? ? w ? , which then implies that p ? x?p ? ,w ? ? ? w, and this holds with strict equality when x?p ? ,w ? ? ? x?p,w? Given that the consumer could have chosen the old bundle, the consumer must have weakly preferred the new bundle. MWG Proposition 2.F.1: WARP implies the law of compensated demand. If the Walrasian demand function x?p,w? satisfies Walras’ Law and is homogeneous of degree zero, then x?p,w? satisfies the weak axiom if and only if the following property holds. For any compensated price change from an initial situation ?p,w? to a new price wealth pair ?p ? ,w ? ? ? ?p ? ,p ? ? x?p,x??, then ?p ? ? p? ? ?x?p ? ,w ? ? ? x?p,x?? ? 0, and this equality holds strictly when x?p ? ,w ? ? ? x?p,x?. We are now interested only in the weak axiom -? law of demand part of the proposition. First, note that if x?p ? ,w ? ? ? x?p,w? we’re done, so consider only the case where x?p ? ,w ? ? ? x?p,w? . In that case, Walras’ law gives us: p ? ? x?p ? ,w ? ? ? w ? , and we have defined w ? so that w ? ? p ? ? x?p,w? Together, this gives us that p ? ? ?x?p ? ,w ? ? ? x?p,w?? ? 0 But we also know that as ?p ? ,w ? ? satisfies the WARP condition: p ? ? x?p,w? ? w ? and x?p ? ,w ? ? ? x?p,w? it follows that p ? x?p ? ,w ? ? ? w . Using p ? x?p,w? ? w (again by Walras’ Law), this gives us that: p ? ?x?p ? ,w ? ? ? x?p,w?? ? 0 Subtraction then yields: ?p ? ? p? ? ?x?p ? ,w ? ? ? x?p,w?? ? 0 Change in prices times change in quantities is negative. Since ?p ??x ? 0 for all compensated price changes, this also holds in the limit for very small price changes and dp ? dx ? 0. We can use matrix notation and write: dx ? D p x?p,w?dp?D w x?p,w?dw which just means: dx ? ? i?1 L ?x 1 ?p i dp i ? ?x 1 ?w dw,.., ? i?1 L ?x L ?p i dp i ? ?x L ?w or dx ? D p x?p,w?dp?D w x?p,w??x?p,w? ? dp? or dx ? D p x?p,w? ?D w x?p,w?x?p,w? T dp The D p x?p,w? ?D w x?p,w?x?p,w? T term is a matrix where the element in row i, column j of the matrix is ?x i ?p k ? ?x i ?w x k ?x 1 ?p 1 ? ?x 1 ?w x 1 .... ?x 1 ?p L ? ?x 1 ?w x L .... .... ..... ?x L ?p 1 ? ?x L ?w x 1 .... ?x L ?p L ? ?x L ?w x L and then dp ? D p x?p,w? ?D w x?p,w?x?p,w? T dp ? 0 The term in brackets is the Slutsky matrix. MWG Proposition 2.F.2: If a differentiable Walrasian demand function x?p,w? satisfies Walras’ law, homogeneity of degree zero and WARP, then at any ?p,w? the Slutsky matrix is negative semi-definite, i.e. v ? Sv ? 0 for any v ? ? ? L . MWG Proposition 2.F.3: Suppose that the Walrasian demand function x?p,w? is differentiably homogeneous of degree zero, and satisfies Walras’ Law. Then p ? S?p,w? ? 0 and S?p,w?p ? 0 for any ?p,w? Utility Functions– Finally getting to the basic tool of 99% of economics. Back to preferences. We want four attributes: (1) completeness and (2) transitivity These were defined in lecture 1 and we refer to preferences with these attributes as being rational. Definition 3.B.2.: The preference relation ? on X is monotone if x,y ? X, and y ? x implies y ? x. It is strongly monotone if y ? x and y ? x implies y ? x. In many cases, we won’t naturally have monotonicity, but then a little redefinition of variables does the trick (turn a bad into a good by multiplying by -1). A slight twist (I mentioned this earlier): Definition 3.B.3: The preference relation ? on Xis locally nonsatiated if for every x,y ? X, and every ??0, there exists a y such that ?y ? x? ? ? and y ? x. This means that there exists a y vector that is arbitrarily close to x that is strictly preferred to x. One last property: Definition 3.C.1.: The preference relation ? on X is continuous if it is preserved under limits. That is, for any sequence of pairs ??x n ,y n ?? n?1 ? with x n ? y n for all n, x ? lim n?? x n and y ? lim n?? y n we have x ? y. The famous counterexample is lexicographic preferences, where more of good 2 is preferred to less, but unless the bundles have the same amount good 1, then the bundle with more good 1 is always preferred. Consider sequence 1, with no units of good 2 and 1/n units of good 1, and sequence 2 with no units of good 1, and 1 ? 1/n units of good 2. For all finite n, sequence 1 is preferred to sequence 1, but in the limit sequence 1 yields zero unit of either good and sequence 2 yields 1 unit of good 2 and is therefore preferred. Lexicographic preferences are a famous example, but hardly a mainstay of either theory or empirical work. MWG Definition 1.B.2: A function u : X ??is a utility function representing preference relation ? if for all x,y ? X, x ? y if and only if u?x? ? u?y? Proposition 3.C.1: If the rational preference relation ? on ? L is continuous , then there is a continuous utility function u?x? that represents ? . The proof in MWG requires monotonicity– a slight variant is to to take a probability measure on ? L that has positive density everywhere, and then let u?x? ? 1 ? prob??y ? y ? x??. By construction, if x ? y then u?x? ? u?y? and if u?x? ? u?y? then x ? y . Now we’ve gotten to a utility function and we know that it is continuous. MWG Definition 3.B.4: The preference relation ? is convex if for every x ? X the upper contour set ?y ? Y : y ? x? is convex, that is if y ? x and z ? x then ?y? ?1 ? ??z ? x for every ? ? ?0,1?. MWG Definition 3.B.5: The preference relation ? is strictly convex if for every x ? X if y ? x and z ? x and y ? z implies that ?y? ?1 ? ??z ? x for every ? ? ?0,1?. Convex preferences imply that u?.? is quasi-concave, i.e. the set ?y ? ? ? L : u?y? ? u?x?? is concave (equivalently u??y? ?1 ? ??x? ? Min?u?x?,u?y?? for any x,y and all ? ? ?0,1?. Strict convexity implies strict quasi-concavity. The utility maximization problem is to maximize u?x? subject to the constraint p ? x ? 0. MWG Proposition 3.D.1: If p ? 0 and u?.? is continuous then the utility maximization problem has a solution. The proof relies on the fact that a continuous function always has a maximum on a compact set, so you just need to show that the budget set is closed and bounded, i.e. compact. Proposition 3.D.2: Suppose that u?.? is a continuous utility function representing a locally non-satiated preference relation ? defined on the consumption set X ?? ? L . Then the Walrasian demand correspondence has the following properties: (1) x?p,w? is homogeneous of degree zero in prices and wealth, i.e. x?p,w? ? x??p,?w? for every ? ? 0 (2) Walras’ law p ? x?p,w? ? w (3) If ? is convex so that u?.? is quasi-concave, then x?p,w? is a convex set. Moreover if ? is strictly convex, so that u?.? is strictly quasi-concave, then x?p,w? consists of a single element. In practice, we write down a Langrangian max x,? U?x 1 ,x 2 ,...x L ? ?? w ? ? i?1 L p i x i which yields us a system of first order conditions: ?U?x 1 ,x 2 ,...x L ? ?x i ? ?p i These are L equations and we have L? 1 unknowns, so we need to use the budget set as well to solve the problem. An application: fertility decisions. Many empirical puzzles: (1) Why does fertility drop with income so substantially across countries? (2) Why is fertility below replacement in all of Europe, but not in the U.S.? How would we capture this: Begin with U?C,N?– utility over consumption and kids. Assume that kids have both a cash cost (k),andatimecost(t) Assume that you have a time budget that can either be used producing kids or making money. Write down the total budget constraint WT?Y ? C? ?k?tW?N Solving C from the budget set the agent’s maximization problem reduces to max N U?WT?Y ? ?k?tW?N,N? Which in turn produces a F.O.C. ? ?k?tW?U 1 ?WT?Y ? ?k?tW?N,N? ?U 2 ?WT?Y ? ?k?tW?N,N? ? 0 where U 1 is the partial w.r.t. the 1st argument of U and similarly for U 2 . The marginal utility of another kid is weighed off against the marginal cost in terms of time and money. Comparative statics can be derived by (1) Using the implicit function theorem to define N ? ?Z?, where Z is a vector representing all of the parameters in this equation (i.e., Z ? ?t,T,Y,W,k?) and (2) Totally differentiating the F.O.C. to find ?N ? ?Z . Let F?N,Z? ???k?tW?U 1 ?WT?Y ? ?k?tW?N,N? ?U 2 ?WT?Y ? ?k?tW?N,N? so that the F.O.C is F?N ? ?Z?,Z? ? 0. Differentiation w.r.t parameter Z then gives us: F 1 ?N ? ?Z?,Z? ?N ? ?Z ?F 2 ?N ? ?Z?,Z? ? 0 from which ?N ? ?Z ? ?F 2 ?N ? ?Z?,Z? F 1 ?N ? ?Z?,Z? . In this case F 1 ?N ? ,Z? ? ??k?tW? 2 U 11 ?C ? ,N ? ? ? U 22 ?C ? ,N ? ? ? U 22 ?C ? ,N ? ? ? 2?k?tW?U 12 ?C ? ,N ? ? and C ? ? WT?Y ? ?k?tW?N ? . We generally assume that terms like F 1 ?N ? ?Z?,Z? are negative (Why?) so the sign of ?N ? ?Z depends entirely on the sign of ?F 2 ?N ? ,Z? ? ?F?N ? ,Z? ?Z . In the case of Z ? Y for example: ? ?Y ??k?tW?U 1 ?WT?Y ? ?k?tW?N ? ,N ? ? ? U 2 ?WT?Y ? ?k?tW?N ? ,N ? ? ??k?tW?U 11 ?WT?Y ? ?k?tW?N ? ,N ? ? ? U 12 ?WT?Y ? ?k?tW?N ? ,N ? ? What can we say about this? Does this give us any intuition about anything? In the case of W: ? ?W ??k?tW?U 1 ?WT?Y ? ?k?tW?N ? ,N ? ? ? U 2 ?WT?Y ? ?k?tW?N ? ,N ? ? ??k?tW??T ? tN ? ?U 11 ?WT?Y ? ?k?tW?N ? ,N ? ? ? ?T ? tN ? ?U 12 ?WT?Y ? ?k?tW?N ? ,N ? ? ? tU 1 ?WT?Y ? ?k?tW?N ? ,N ? ? It’s the same as the unearned income effect expect for the third term. What does that third term represent? Now let’s get a little more fancy and assume that parents care about both quantity and quantity, i.e. U?C,Q,N?. For simplicity assume utility is a separable and quasi-linear in consumption, so U?C,Q,N? ? C?V?Q,Z Q ? ?W?N,Z N ?. Assume children cost WtQN. Then the first order conditions are: ?V?Q,Z Q ? ?Q ? WtN and ?W?N,Z N ? ?N ? WtQ If a parameter increases the marginal return to quality– this will have the following effect ? 2 V?Q,Z Q ? ?Q 2 ?Q ?Z Q ? ? 2 V?Q,Z Q ? ?Q?Z Q ? Wt ?N ?Z Q and ? 2 W?N,Z N ? ?N 2 ?N ?Z Q ? Wt ?Q ?Z Q or ?Q ?Z Q ? ? 2 V?Q,Z Q ? ?Q?Z Q ? ? 2 V?Q,Z Q ? ?Q 2 ? ?Wt? 2 ? 2 W?N,Z N ? ?N 2 and ?N ?Z Q ? ? 2 V?Q,Z Q ? ?Q?Z Q ? ? 2 V?Q,Z Q ? ?Q 2 ? 2 W?N,Z N ? ?N 2 ??Wt? 2 How can we sign these two things? Each equation also tells you that the price per child rises as the quality of each child increases, and the price of quality rises as the number of children rises. Through the budget set there is an inherent substitutability between quantity and quality of children.