Economics 2010a Fall 2003 Edward L. Glaeser Lecture 3 3. Comparative Statics a. Indirect Utility Functions b. Expenditure Functions and Duality c. Expenditure Function and Price Indices d. Slutsky via Utility Functions e. Slutsky via Preferences f. Composite Commodity Theorem g. Application: Labor Supply Indirect Utility Functions Indirect Utility functions represent the level of utility as a function of prices and wages, and we write v?p,w?. It is useful many times, to have utility solely as a function of "exogenous" parameters. Define the indirect utility function as v?p,w? ? u?x?p,w?? where x?p,w? is the Marshallian demand function, which solves the consumers problem to maximize u?x? subject to w ? p ? x. As an aside, remember that utility or indirect utility units have no meaning. The same preferences are u?x? and f?u?x?? if f?. ? is a strictly monotonic function. Properties of the indirect utility function. MWG Proposition 3.D.3: Suppose that u?. ? is a continuous utility function representing a locally non-satiated preference relation ? defined on the consumption set ? ? L . The indirect utility function v?p,w? is: (a) homogeneous of degree zero, (b) strictly increasing in w and non-increasing in p, (c) quasiconvex; that is, the set ??p,w? : v?p,w? ? v? is convex for any v, and (d) continuous in p and w. We have already proven that Marshallian demand is homogeneous of degree zero, this implies that indirect utility is homogeneous of degree zero. Nonincreasing in p follows from the fact that if p falls you can always buy the old bundle, and thus be no worse off. Increasing in w uses that fact plus local non-satiation: with an increase in w you can always buy the old bundle plus a little bit more of the good that you are not satiated with. Continuity I leave up to you. To show quasi-convexity, assume that v?p,w? ? v and v?p ? ,w ? ? ? v . For any ? ? ?0,1? consider the price wealth pair ?p ?? ,w ?? ? ? ??p ? ?1 ? ??p ? ,?w ? ?1 ? ??w ? ? Assume that v?p,w? is not quasi convex, i.e. there exists an x, such that ?p ? x ? ?1 ? ??p ? ? x ? ?w ? ?1 ? ??w ? , but u?x? ? v. If u?x? ? v, then x must not have been affordable at the old budget sets (otherwise it would have been chosen and would have yielded higher utility), which implies that p ? x ? w and p ? ? x ? w ? . But these together imply that ?p ? x ? ?1 ? ??p ? ? x ? ?w ? ?1 ? ??w ? which is a contradiction. Quasi-convexity gives us the idea that certain types of variation in prices are actually good, not bad for consumers. As an obvious example, consider the utility function x?y?z (three goods), where consumption is contrained to be non-negative. The price of x is always one. Income is always one. What is utility if the price of y and z are both one? What is utility if the price of y equals 1.5 and the price of z equals .5? What is utility if the price of y equals .5 and the price of z equals 1.5? Roy’s Identity: MWG Proposition 3.G.4: Suppose that u?. ? is a continous utility function representing a locally non-satiated and strictly convex preference relation ? defined on the consumption set X ? ? ? L . Suppose also that the indirect utility function is differentiable at ?p,w? ? 0. Then x?p,w? ? ? 1 ? w v?p,w? ? p v?p,w? that is for every j?1,2,...L, x j ?p,w? ? ? ?v?p,w? ?p j / ?v?p,w? ?w In words, marshallian demand for a good equals the ratio of the derivative of indirect utility with respect to the price of that good divided by the derivative of indirect utility with respect to wealth. Proof: My favorite is the envelope theorem argument: v?p,w? ? U?x ? ? ? ??w ? p ? x ? ? ?v?p,w? ?p j ? ??x j ? ? ?x j ? ?p j ?U ?x j ? ?p But the second term is zero (that’s the envelope argument) at the maximum. Likewise ?v?p,w? ?w ? ? and we’re done. The expenditure function. Previously, we have discussed the utility maximization problem (i.e. maximize utility subject to a fixed budget constraint). In many cases, it is valuable to discuss the expenditure minimization problem (i.e. minimize expenditure subject to a fixed utility level). Formally, this problem is Minimize p ? x subject to u?x? ? u We define the expenditure function e?p,u? as the lowest level of income needed to provide utility level u given prices p, The value of e?p,u? equals p ? x ? where x ? minimizes p ? x subject to u?x? ? u. There is a fundamental equivalence between utility maximization and expenditure minimization captured in: MWG Proposition 3.E.1: Suppose that u?. ? is a continuous utility function representing a locally non-satiated preference relation ?defined on the consumption set X ? ? ? L and that the price vector is p ? 0 . Then: (i) if x ? maximizes utility for w?0, then x ? minimizes expenditure when the required utility level is u?x ? ?. Moreover the minimized expenditure level is exactly w. (ii) if x ? minimizes expenditure then the required utility level is u then x ? maximzes utility when wealth equals p ? x ? . Moreover the maximized utility level is exactly u. Proof (i): Suppose that x ? maximizes utility and does not minimize expenditure, this implies that there exists an x ? such that u?x ? ? ? u?x ? ? and p ? x ? ? p ? x ? . But then local nonsatiation implies that by spending alittlemorethanp ? x ? on the nonsatiated good, we can find an x ?? such that u?x ?? ? ? u?x ? ? and p ? x ? ? p ? x ? ? w, and this contradicts maximization. The proof to the second part is quite similar. Properties of the Expenditure Function MWG Proposition 3.E.2: Suppose that u?. ? is a continuous utility function representing a locally non-satiated preference relation ?defined on the consumption set X ? ? ? L . The expenditure function e?p,u? is: (i) homogeneous of degree one in p. (ii) strictly increasing in u and nondecreasing in p j for any j. (iii) Concave in p. (iv) Continuous in p and u. Proof: (i) The problem Minimize p ? x subject to u?x? ? u and minimize ?p ? x subject to u?x? ? u , yields exactly the same optimal value for the x vector, denoted x ? . As the expenditure function equals prices times quantities, when prices are multiplied by ?, the expenditure function must be multiplied by the same amount. (ii) Assume that e?p,u? is not strictly increasing in u , and let x ? and x ?? denote optimal consumption bundles for utility levels u ? and u ?? respectively, where u ?? ? u ? and p ? x ? ? p ? x ?? ? 0. Continuity ensures that there exists a value of ? which is less than one but sufficiently close to one so that u??x ?? ? ? u?x ? ? and p ? ?x ?? ? p ? x ?? ? p ? x ? but then the bundle ?x ?? yields more utility at less cost than the bundle x ? so x ? is not expenditure minimizing and we have a contradiction. (iii) Fix a utility level u, and consider two price level p and p ? and let p ?? ? ?p ? ?1 ? ??p ? for ? ? ?0,1?. Let x ?? denote the bundle that minimizes expenditures and achieves utility level u, when prices are p ?? . If so then e?p ?? ,u? ? p ?? ? x ?? ? ?p ? x ?? ? ?1 ? ??p ? ? x ?? ? ?e?p,u? ? ?1 ? ??e?p ? ,u? Concavity of the expenditure function with respect to prices is the natural parallel of convexity of the indirect utility function with respect to prices. Under some circumstances, price variation is good, not bad. Again consider the utility function x?y?z, and think about the expenditures need to yield one unit of utility. In the case where the price of all three goods is one, the one unit of currency is needed. In the case where one of the goods costs 1.5 and the other .5, then only .5 units of currency are needed. Two useful equalities v?p,e?p,u?? ? u and e?p,v?p,w?? ? w The Hicksian demand function is defined as h?p,u? as a function of the price level and the utility level and Hicksian demand is defined as the values of x that minimize expenditures for utility level u. The following properties of Hicksian demand are useful: Proposition 3.E.3: Suppose that u?. ? is a continuous utility function representing a locally nonsatiated preference relation ? defined on consumption set X ? ? ? L . Then for any p ? 0 the Hicksian demand correspondence h?p,u? possesses the following properties: (i) homogeneity of degree zero in p, (ii) no excess utility: for any x ? h?p,u?, u?x? ? u (iii) Convexity, if ? is convex then h?p,u? is a convex set; and if ? is strictly convex, so that u?. ? is strictly quasi-concave then there is a unique element in h?p,u?. Proof of (i) h?p,u? solves the problem of minimizing p ? x such that u?x? ? u. The same values of x that minimize p ? x also minimize ?p ? x. Proof of (ii) Assume that there exists an x ? h?p,u?, u?x? ? u Continuity of the utility function implies that there exists a bundle ?x where ? is less than one, but sufficiently close to one, so that u??x? ? u and obviously since 1 ? ? , p ? x ? ?p ? x as a result, the vector x doesn’t minimize expenditures subject to u?x? ? u Implication of homogeneity of degree zero and differentiability: h??p,u? ? h?p,u?, so differentiate with respect to ? and evaluate at ? ? 1 to get: ? j?1 L ?h i ?p,u? ?p j p j ? 0,or,? p h i ?p,u? ? p ? 0 for all i ,whichcanbewrittenas D p h?p,u? ? p ? 0 . Another law of compensated demand: Proposition 3.E.4: Suppose that u?. ? is a continuous utility function representing a locally nonsatiated preference relation ? and that h?p,u? consists of a single element for all p ? 0. Then the Hicksian demand function h?p,u? satisfies the law of compensated demand, i.e. for all p ? and p ?? , we have ?p ?? ? p ? ? ? ?h?p ?? ,u? ? h?p ? ,u?? ? 0 Proof: Since hicksian demand minimizes expenditure for a given set of prices, p ?? ? h?p ?? ,u? ? p ?? ? h?p ? ,u? or p ?? ? ?h?p ?? ,u? ? h?p ? ,u?? ? 0 and p ? ? h?p ?? ,u? ? p ? ? h?p ? ,u? or p ? ? ?h?p ?? ,u? ? h?p ? ,u?? ? 0. Just subtract the two inequalities and you are done. The expenditure– hicksian demand link: Proposition 3.G.1: Suppose that u?. ? is a continuous utility function representing a locally nonsatiated and strictly convex preference relation ? defined on the consumption set X ? ? ? L . For all p and u , the Hicksian demand h?p,u? is the derivative of the expenditure function with respect to prices: h?p,u? ? ? p e?p,u? That is h i ?p,u? ? ?e?p,u? ?p i for all i?1, 2, ... L. Proof in the case where h?p,u? ? 0 and h?p,u? is differentiable: Start with the fact that e?p,u? ? ? j?1 L p j h j ?p,u? and differentiate this totally with respect to p i : ?e?p,u? ?p i ? h i ?p,u? ? ? j?1 L p j ?h j ?p,u? ?p i ? h i ?p,u? But what does the last term equal and why? So we’re done. Final Expenditure Function Properties: Proposition 3.G.1: Suppose that u?. ? is a continuous utility function representing a locally nonsatiated and strictly convex preference relation ? defined on the consumption set X ? ? ? L . Suppose also that h?.,u? is continuously differentiable at ?p,u? amd denote its L ? L derivative matrix by D p h?p,u?. Then: (i) D p h?p,u? ? D p 2 e?p,u?, (ii) D p h?p,u? is a negative semidefinite matrix, (iii) D p h?p,u? is a symmetric matrix, (iv) D p h?p,u?p ? 0 Proof: (i) follows from 3.G.1 and differentiation, (ii) and (iii) follow from (i) and the fact that e?p,u? is a twice continuously differentiable concave function which therefore has a symmetric and negative semi-definite matrix of second derivatives. The last property we already proved as part of homogeneity of degree zero and even used in the proof of 3.G.1. Expenditure Functions and Price Indices The price index problem– we are interested in knowing how much welfare has changed over time, given an increase in prices and wealth, or alternatively how much more needs to be spent to the same level of utility. This is exactly what the expenditure function is good at– after all it tells us the answer to exactly that question. In some cases, things are easy. For example if all prices change together and are multiplied by a constant ? then we know what to do. Second, for infintesimal changes we also know what the answer is: de?p,u? ? ? i?1 L ?e?p,u? ?p i dp i ? ? i?1 L h i ?p,u?dp i So for infinitesimal changes, you can weight by quantities– but we measure prices at more discrete intervals than that. So– some different indices have been proposed: Laspeyres The Laspeyres Price Index is p ? ?q p?q Paasche Price Index is p ? ?q ? p?q ? With w p ? ?q p?q can you do better or worse in period 2 than you did in period 1? If utility levels are constant across periods, can you do better with w p ? ?q ? p?q ? in period 2? So which index overstates increases in the cost of living, and which understates it? Deriving the Slutsky Equation the easy way: Start with the equality h?p,u? ? x?p,e?p,u?? Differentiate this totally with respect to some price p i , then we get: ?h i ?p,u? ?p j ? ?x i ?p,w? ?p j ? ?x i ?p,w? ?w ?e?p,u? ?p j ? ?x i ?p,w? ?p j ? ?x i ?p,w? ?w or in matrix notation D p h?p,u? ? D p x?p,w? ? D w x?p,w?x?p,w? T Deriving the Slutsky Equation via Preferences (i.e. no utility functions) You can define the expenditure function on a reference level of consumption, rather than a utility function, i.e. e?p,x 0 ? is the minimum amount of expenditure needed to generate a bundle that is weakly preferred to x 0 , or more formally: e?p,x 0 ? ? minp ? x, such that x ? x 0 Likewise, you can define the Hicksian demand such that h?p,x 0 ? ?arg min ?x|x?x 0 ? p ? x As before: ?e?p,x 0 ? ?p i ? h?p,x 0 ? And as before h?p,x 0 ? ? x?p,e?p,x 0 ?? Differentiating with respect to p j yields: ?h i ?p,x 0 ? ?p j ? ?x i ?p,w? ?p j ? ?x i ?p,w? ?w ?e?p,x 0 ? ?p j ? ?x i ?p,w? ?p j ? Composite Commodity Theorem (Hicks/Leontief) In many cases, we are only interested in the price movements of a subset of commodities, and basically want to agglomerate the other commodities into a single group. The composite commodity theorem allows us to do this. Suppose that u?. ? is a continuous utility function representing a locally nonsatiated preference relation ? defined on the consumption set X ? ? ? L , and that for a subset of commodities j where k ? j ? L, ?p k ,p k?1 ,...p L ? ? ??p k ,?p k?1 ,...?p L ?, where ?p k ,p k?1 ,...p L ? is constant, then exists a function ??x 1 ,x 2 ,...x k?1 ,z?, where z ? ? i?k L p i x i , such that values of ?x 1 ,x 2 ,...x k?1 ? which maximize ??x 1 ,x 2 ,...x k?1 ,z?, subject to the constraint: ? i?1 k?1 p i x i ? ?z ? w areexactlythesameas values ?x 1 ,x 2 ,...x k?1 ? which maximize u?x 1 ,x 2 ,...x L ? subject to ? i?1 L p i x i ? w Define a preference ordering ? ? defined on the space X ? ? ? ? k ,such that if a bundle ?x 1 ,x 2 ,...x L ? ? ?y 1 ,y 2 ,...y L ? for all ?y 1 ,y 2 ,...y L ? such that p i y i ? w then the bundle x 1 ,x 2 ,...x k?1 , ? i?k L p i x i ? ? ?y 1 ,y 2 ,...y k?1 ,z? for all bundles: ? i?1 k?1 p i y i ? ?z ? w. This preference ordering ? ? is rational and continuous if the underlying preference ordering ? is rational and continuous, so there must exist a utility function ??x 1 ,x 2 ,...x k?1 ,z? that represents these preferences. Moreover, the function ??x 1 ,x 2 ,...x k?1 ,z? will have all the standard characteristics of a utility function, the walrasian demand functions x i ?p 1 ,p 2 ,...p k?1 ,?,w? will have all the properties of usual demand functions and the compensated demand function h i ?p 1 ,p 2 ,...p k?1 ,?,u? will have all the usual properties of a compensated demand functions. Labor Supply (1) Certainly among the most important decisions people make, (2) filled with policy relevance and correlations with policy variables across time and space (3) interesting in that income effects actually may be big (not usually so) (4) a little twist on what we have been doing because "wealth" is endogenous Start with two budget sets– cash and time: WH ? Y ? ? i p i x i where W is wage, H is hours working, Y is unearned income, p i is prices and x i is commodities purchased. This is just our usual budget constraint, but then we also have: T ? ? j t j ? H where T is total time budget and t j represents a variety of leisure type activity. Using Walras’ Law (so the budget sets hold with equality), they can be combined so that: WT ? Y ? W? j t j ? ? i p i x i To start, consider the utility function u ? U?x 1 ,x 2 ...x L ,t 1 ,t 2, ...t L ? Let’s use the composite commodity theorem to aggregate non-work time, and commodities so u ? U?c,t? The budget set is WT ? Y ? Wt ? pc or c ? W?T?t??Y p Since I’m interested in labor supply, let’s use the notation S ? T ? t so we get t ? T ? S and c ? WS?Y p so we can write: U? WS?Y p ,T ? S? which has first order condition U c ? WS?Y p ,T ? S? W p ? U t ? WS?Y p ,T ? S? ? 0 Using the implicit function theorem we can write: U c WS ? ?Z??Y p ,T ? S ? ?Z? W p ? U t WS ? ?Z??Y p ,T ? where Z represents any exogenous variable Y, p, W or T. Differentiation then yields: ?S ? ?Y ? U cc W p 2 ?U ct ?U cc W 2 p 2 ?U tt ?2U tc W p and ?S ? ?W ? U c 1 p ?U cc W p 2 ?U ct ?U cc W 2 p 2 ?U tt ?2U tc W p How would you model a straight income tax (proportional)? Among the most interesting things about labor supply is that the budget set may not be linear. For example, there might be a fixed cost of going to work. What impact would that have on outcomes? There might be a minimum number of hours required for other reasons. Perhaps most commonly, there might be a tax rate that varies with income. What sort of taxes should have the least impact on hours worked– what sort of taxes should have the most impact. One way to formalize utility is to say: u ? U?Z 1 ?x 1 ,x 2 ...x L ,t 1 ,t 2, ...t L ?,...Z K ?x 1 ,x 2 ... where utility come’s directly form the Z’s which are produced using commodities and time. This yields the first order conditions: ? i?1 K ?U ?Z i ?Z i ?x i ? ?p i ? i?1 K ?U ?Z i ?Z i ?t i ? ?W Let’s assume no joint production right now, so that: Z i ? Z i ?x i ,t i ? Then we have ?U ?Z i ?Z i ?x i ? ?p i ?U ?Z i ?Z i ?t i ? ?W which means that ?Z i ?x i ?Z i ?t i ? p i W The use of time and goods in producing a particular commodity is determined only by the technological tradeoffs between the two. Furthermore, we can even think about each commodity separately, and minimizing costs for producing each commodity. In that case the job is to minimize Wt ? px s.t. Z?t,x? ? Z dropping the i subscripts for convenience. This yields the same first order condition: Z x Z t ? p W which we can then differentiate to get: WZ xx ?x ?w ? WZ xt ?t ?w ? Z xx ? pZ xt ?x ?w ? pZ tt ?t ?w and using the equality Z?t,x? ? Z we know ? Z x Z t ?x ?w ? ?t ?w or ? p W ?x ?w ? ?t ?w Plugging this in, we get: WZ xx ?x ?w ? WZ xt p W ?x ?w ? Z xx ? pZ xt ?x ?w ? pZ tt p W or ?x ?w ? WZ xx ? W 2 Z xx ?p 2 Z tt ?2pWZ xt and ?x ?w ? ?pZ xx ? W 2 Z xx ?p 2 Z tt ?2pWZ xt The denominator is positive– so we have some results.