Economics 2010a Fall 2003 Edward L. Glaeser Lecture 5 5. Aggregating Consumers a. Consumer Heterogeneity and a Discrete Good b. The Properties of Aggregate Demand c. The Existence of a Representative Consumer d. Externalities e. The Social Multiplier f. Equalizing Differentials with Heterogeneity g. Welfare Losses with Heterogeneous Consumers h. Application: Price Controls and Aggregation A general lesson– aggregate outcomes do not always resemble individual outcomes. Let’s start with a discrete commodity, of which only one unit can be consumed. Normalize this to be commodity 1. Proposition: Suppose that the preference relation ? on X ? ?x 1 ? ?0,1?,?x 2 ,...x L ? ? ? ? L?1 ? ,is rational, continuous and locally non-satiated on the commodities other than commodity 1, and if x ? ? x ? B 0 : x ? y for all y ? B 0 , where B 0 ? x ? X : x 1 ? 0 and ? i?2 L p i x i ? w , and x ?? ?p 1 ? ? x ? B 0 ?p 1 ? : x ? y for all y ? B 0 where B 0 ?p 1 ? ? x ? X : x 1 ? 1 and ? i?2 L p i x i ? w ? p 1 . If x ? ? ?1,0,0...0? then there exists a ? p 1 ? ?0,w? for all p 1 ? ? p 1 , x ? ? x ?? ?p 1 ? and x ? ? x ?? ?p 1 ? for all p 1 ? ? p 1 . If preferences are continuous then where x ? ? x ?? ? ? p 1 ?. Proof: It is obviously true that there exist some values of p 1 ? ?0,w? in which x ? ? x ?? ?p 1 ? and some values for x ? ? x ?? ?p 1 ?, at the least x ? ? x ?? ?0? and x ? ? x ?? ?w?. Furthermore, non-satiation tells us that x ? ? x ?? ? ? p 1 ? implies that x ? ? x ?? ?p 1 ? for all p 1 ? ? p 1 , and that x ? ? x ?? ? ? p 1 ? implies that x ? ? x ?? ?p 1 ? for all p 1 ? ? p 1 . Hence, since the set of p 1 for which x ? ? x ?? ?p 1 ? is bounded above there exists an upper bound ? p 1,A . And since the set of p 1 for which x ? ? x ?? ?p 1 ? is bounded below there exists a greatest lower bound ? p 1,B . As it would be impossible to have x ?? ?p 1 ? ? x ? ? x ?? ?p 1 ?, it must be that ? p 1,A ? ? p 1,B .If ? p 1,A ? ? p 1,B then there exists a continuum of values of p 1 at which x ? ? x ?? ?p 1 ? and x ? ? x ?? ?p 1 ?, i.e. for which x ? ? x ?? ?p 1 ? but local non-satiation rules that out, so it must be that ? p 1,A ? ? p 1,B ? ? p 1 and by construction for all p 1 ? ? p 1 , and x ? ? x ?? ?p 1 ? and x ? ? x ?? ?p 1 ? for all p 1 ? ? p 1 . Continuity gives you the existence of a price where consumers are indifferent. Consider, the sequence x ?? ?p 1 ? 1 n ? and the sequence x ? , where x ?? ?p 1 ? 1 n ? ? x ? for all n, continuity implies that x ?? ?p 1 ? ? x ? . Likewise consider the sequences x ?? ?p 1 ? 1 n ? and x ? , where x ?? ?p 1 ? 1 n ? ? x ? for all n, so x ?? ?p 1 ? ? x ? which together imply that x ?? ?p 1 ? ? x ? . Thus each person has a price at which he is indifferent between consuming commodity one and not doing so. In this case, individual demand is always downward sloping– no income effects or cross-partials to worry about. Why? What would happen if we dropped continuity or local non-satiation? Assume that there are J consumers, each with preferences ? j and wealth w i ,facing prices for all goods i?1, p ? 0,which generates a cutoff price ? p 1 i for each consumer. Let D?p 1 ,p,w? ? ? the number of consumers s.t. p 1 ? ? p 1 i This function is certainly weakly downward sloping– as p 1 falls the set of consmers for which p 1 ? ? p 1 i must rise. If we want something like continuity, we must assume that a distribution of ? p 1 i in the population that has positive density for all values ? p 1 i between zero and some value p which denotes the highest price at which anyone buys anything. If this distribution is denoted f? ? p 1 ?, then aggregate demand is: D?p 1 ,p,w? ? ? ? p 1 ?p 1 p f? ? p 1 ?d ? p 1 ? 1 ? F? ? p 1 ? and ?D ?p 1 ? ?f? ? p 1 ?. So aggregate demand is downward sloping, continuous, and will be concave or convex depending on the shape of the density. For example if ? p 1 is uniformly distributed on the interval ?p,p?then demand equals p?p 1 p?p – the familiar linear demand curve. A demand function with constant price elasticity occurs if reservation prices are distribution so that: f? ? p 1 ? ? ? p 1 ???1 and F? ? p 1 ? ? 1 ? ?1 ? ? p 1 ?? ?. One way to think about this is that demand curves slope down for at least two reasons: (1) diminishing returns at the individual level (for continuous goods) and (2) heterogeneity of demand across people (for discrete goods). In many cases, the latter is easier to work with and closer to reality. Back to MWG: The Properties of Aggregate Demand Let x j ?p 1 ,p 2 ,...,p L ,w 1 ,w 2 ,...,w J ? denote the demand for the jth consumer, out of a total of J consumers. Aggregate demand x?p 1 ,p 2 ,...,p L ,w 1 ,w 2 ,...,w J ? ? ? ? j?1 J x j ?p 1 ,p 2 ,...,p L ,w j ? We are interested in two specific issues: (1) to what extent does aggregate demand display similar properties (e.g. law of demand) to individual demand, and (2) to what extent can we use a representative consumer framework to capture aggregate outcomes. Definition 4.C.1: The aggregate demand function x?p,w? satisfies the weak axiom if p ? x?p ? ,w ? ? ? w and x?p ? ,w ? ? ? x?p,w? implies p ? ? x?p,w? ? w ? . This may not hold– Individual demand functions may satisfy the weak axiom, but the aggregate may not. Note to Students, etc., the next examples may be helpful, and I included them because MWG does little on this, but I will go over them quickly and I don’t see this material as crucial. An example of the weak axiom failing: Assume that there are two consumers each of which has wealth w ? 1 and each of which faces the same prices for two goods (1 and 2). A little example: Initially the price for both goods is one. We consider the change so that the price for good one is 1 ? ? and the price for good two is 1 ? ?, ? ? 0. First agent has a utility function x 1 ? max?x 1 ,?? ? ?x 2 , where ? is quite small. Second agent has a utility function min??x 1 ,?1 ? ??x 2 ?. At price vector (1, 1)– agent 1 consumes ? units of the first good and 1 ? ? units on the second good. Agent 2 consumes 1 ? ? units of the first good and ? units of the second good. At price vector ?1 ? ?,1? ?? the first agent consumes ? units of the first good and 1???1??? 1?? units of the second good. Agent 2 consumes exactly 1?? 1???2?? units of the first good and ? 1???2?? units of the second good. Question # 1: Is the old aggregate bundle affordable at the new prices? Certainly– the old aggregate consumption was 1 and 1 which is still doable. Question # 2: Is the new bundle affordable at the old price: This also is a possibility– aggegate consumption at old prices comes to 1?2?? 1?? ? 1 1???2?? , which can be rewritten: 2 ? 4?? ? 2?? 2 ? 4? 2 ? 2 ?1 ? ? ? 2????1 ? ?? For this to be affordable at the old prices it must be that 1 ? 1?2????? 2 ?2? 2 ? 2 ?1???2????1??? or 1 ? ? 2 ? 2?? ? 2?? 2 ? 1 ? 2?? ? ?? 2 ? 2? 2 ? 2 or 0 ? ?1 ? 2???1 ? ?? This calculation suggests that as long as ? is greater than .5, the weak axiom fails. So let’s assume that ? ?.75and let ? ? 1/3. In this case, the switch in prices pushed consumer one from consuming (.75, .25) to (.75, 0). The switch in prices pushed consumed two from consuming (.25, .75) to consuming (.3, .9) Aggregate consumption has changed from (1, 1) to (1.05,.9). Which was, after all, affordable at the old price. How did I rig this example? The key is that the individual who receives a negative income shock by the change in prices, needs to respond very sharply to this income shock (which means cutting back spending on the cheaper good). The individual who receives a positive income needs to respond slowly, which means increasing spending on the more expensive good. Endowing the big consumer of the good that is getting cheaper with leontief preferences, ensures that there is almost no substitution into the cheap good. Endowing the big consumer of the good that is getting more expensive with very elastic preferences for the other good ensured that the income effect (really a budget set effect) would dominate in that case. The key to WA failure lies in the income effects of price changes. This is really beyond and above the call of duty– read if for enlightenment, but it is not core material. Let’s do it with algebra with two goods and with two (types?) consumers. Consider the situation where: w ? p 1 0 ? x 1 0 ? p 2 0 ? x 2 0 w ? p 1 1 ? x 1 1 ? p 2 1 ? x 2 1 , where ? x i j refers to the aggregate consumption of good i in situation j. Furthermore we assume prices change in such a way that the old bundle remains affordable, i.e.: w ? p 1 1 ? x 1 0 ? p 2 1 ? x 2 0 , We are interested in showing that there are some cases where w ? p 1 0 ? x 1 1 ? p 2 0 ? x 2 1 , The new bundle was affordable at the old prices. This is equivalent to saying: ?p 1 1 ? p 1 0 ? ? x 1 1 ? ? x 1 0 ? ?p 2 1 ? p 2 0 ? ? x 2 1 ? ? x 2 0 ? 0 Change in prices times change in quantities is positive. But as ?p 1 1 ? p 1 0 ? ? x 1 0 ? ?p 2 1 ? p 2 0 ? ? x 2 0 ? 0 or ? ? x 1 0 ? x 2 0 ?p 1 1 ? p 1 0 ? ? ?p 2 1 ? p 2 0 ? We can certainly focus on the case where ?p 1 1 ? p 1 0 ? ? 0 and rewrite this as: ? x 1 1 ? ? x 1 0 ? x 1 0 ? ? x 2 1 ? ? x 2 0 ? x 2 0 In other words, the percentage growth (decline) in the good with rising prices must be greater than the percentage growth (decline) in the good with falling prices. It seems pretty clear that income effects are going to have to drive this, given that by price effects alone, good two should increase and good one should decrease. Now I’m going to try to understand this by using first order taylor series approximations for ? x 1 1 ? ? x 1 0 and ? x 2 1 ? ? x 2 0 For any individual i– a slutsky decomposition holds: x 1,i 1 ? x 1,i 0 ? ?h 1 i ?p 1 ? x 1 i ?x 1 i ?w ?p 1 1 ? p 1 0 ? ? ? ?h 1 i ?p 2 ? x 2 i ?x 1 i ?w ?p 2 1 ? p 2 0 ? or using ? ? x 1 0 ? x 2 0 ?p 1 1 ? p 1 0 ? ? ?p 2 1 ? p 2 0 ?.Ifweuse A and B to refer to the two people, we get: ?h 1 A ?p 1 ? x 1 A ?x 1 A ?w ?p 1 1 ? p 1 0 ? ? ? x 1 0 ? x 2 0 ?p 1 1 ? p 1 0 ? ?h 1 A ?p 2 ? x 2 A ?x 1 A ?w ? ?h 1 B ?p 1 ? x 1 B ?x 1 B ?w ?p 1 1 ? p 1 0 ? ? ? x 1 0 ? x 2 0 ?p 1 1 ? p 1 0 ? ?h 1 B ?p 2 ? x 2 B ?x 1 B ?w ? ? x 1 0 ? x 2 0 ?h 2 A ?p 1 ? x 1 A ?x 2 A ?w ?p 1 1 ? p 1 0 ? ? ? x 1 0 ? x 2 0 2 ?p 1 1 ? p 1 0 ? ?h 2 A ?p 2 ? x 2 A ?x 2 A ?w ? ? x 1 0 ? x 2 0 ?h 2 B ?p 1 ? x 1 B ?x 2 B ?w ?p 1 1 ? p 1 0 ? ? ? x 1 0 ? x 2 0 2 ?p 1 1 ? p 1 0 ? ?h 2 B ?p 2 ? x 2 B ?x 2 B ?w or using the equality that p 1 ?h j i ?p 1 ? p 2 ?h j i ?p 2 ? 0 and p 1 ?x 1 i ?w ? p 2 ?x 2 i ?w ? 1 ?h 1 A ?p 1 p 1 ? x 1 0 ?p 2 ? x 2 0 p 2 ? x 2 0 2 ? ? x 1 0 x 2 A ? ? x 2 0 x 1 A ? x 2 0 p 1 ? x 1 0 ?p 2 ? x 2 0 p 2 ? x 2 0 ?x 1 A ?w ? ?h 1 B ?p 1 p 1 ? x 1 0 ?p 2 ? x 2 0 p 2 ? x 2 0 2 ? ? x 1 0 x 2 B ? ? x 2 0 x 1 B ? x 2 0 p 1 ? x 1 0 ?p 2 ? x 2 0 p 2 ? x 2 0 ?x 1 B ?w ? ? x 1 0 p 2 ? x 2 0 ? x 1 0 x 2 A ? ? x 2 0 x 1 A ? x 2 0 ? ? x 1 0 p 2 ? x 2 0 ? x 1 0 x 2 B ? ? x 2 0 x 1 B ? x 2 0 ? 0 Normalize so that A consumes relatively more of good one and B consumes relatively more of good two– then we have: x 2 A ? x 2 0 ? x 1 A ? x 1 0 ?x 1 A ?w ? x 2 B ? x 2 0 ? x 1 B ? x 1 0 ?x 1 B ?w ? ? p 1 ? x 1 0 ?p 2 ? x 2 0 p 1 p 2 ? x 1 0 ? x 2 0 p 1 ?h 1 A ?p 1 ? p 1 ?h 1 B ?p 1 Person B get richer (good 2 got cheaper) so this person has to have a big income elasticity of good 1 with respect to income. Person A got poorer (good 1 got more expensive) so this person has to have a low or negative income elasticity of good one. MWG Definition 4.C.2: The individual demand function x i ?p,w i ?, satisfies the uncompensated law of demand property if ?p ? ? p? ? ?x i ?p ? ,w i ? ? x i ?p ? ,w i ?? ? 0 for any p, p’ and w i with strict inequality if x i ?p ? ,w i ? ? x i ?p ? ,w i ? The aggregate demand function x?p,w? satisfies the uncompensated law of demand property if ?p ? ? p? ? ?x?p ? ,w? ? x i ?p,w?? ? 0 for any p, p’ and w with strict inequality if x i ?p ? ,w? ? x i ?p,w? MWG Proposition 4.C.1: If every consumer’s Walrasian demand function satisfies the uncompensated law of demand property, so does the aggregate demand x?p 1 ,.,p L ,w? ? ? j?1 J x j ?p 1 ,.,p L ,? j w?, where ? j?1 J ? j ? 1,? j ? 0 . As a result, aggregate demand satisfies the weak axiom. Proof– this is just adding. We know that ?p ? ? p? ? ?x i ?p ? ,w i ? ? x i ?p ? ,w i ?? ? 0 (with strict equality if the consumption vectors aren’t equal) so just sum this across consumers. For the connection between the uncompensated law of demand and the weak axiom, take two vectors ?p,w? and ?p ? ,w ? ? with x?p ? ,w ? ? ? x?p,w?, and p ? x?p ? ,w ? ? ? w. Define p ?? ? w w ? p ? so x?p ? ,w ? ? ? x?p ?? ,w? (WHY?). Consider only the case where x?p ? ,w ? ? ? x?p,w? (the equality case works easily). We know: ?p ?? ? p? ? ?x?p ?? ,w? ? x?p,w?? ? 0, or p ?? ? x?p ?? ,w? ? p ? x?p,w? ? p ? x?p ?? ,w? ? p ?? ? x?p,w? Since p ?? ? x?p ?? ,w? ? w and p ? x?p,w? ? w ,we have p ? x?p ? ,w ? ? ? p ? x?p ?? ,w? ? w Hence p ?? ? x?p,w? ? w or p ? ? x?p,w? ? w ? Ways to get the uncompensated law of demand: Proposition 3.C.4: If ? i is homothetic, then x i ?p,w i ? satisfies the uncompensated law of demand property. Claim: Homothetic preferences imply that ?x i ?p,w i ? ? x i ?p,?w i ? Proof: We consider only the case of single valued demand functions. If ?x i ?p,w i ? ? x i ?p,?w i ?,as?x i ?p,w i ? is affordable at wealth ?w i , we know x i ?p,?w i ? ? ?x i ?p,w i ?. Furthermore, if x i ?p,?w i ? ? ?x i ?p,w i ?, there must exist some scalar ? ? ?0,1? for which ?x i ?p,?w i ? ? ?x i ?p,w i ?, which implies (because of homotheticity) that ? ? x i ?p,?w i ? ? x i ?p,w i ?,whichinturn implies that p ? ? ? x i ?p,?w i ? ? w i ,whichis only reconcilable with p ? x i ?p,?w i ? ? ?w i ,if ? ? 1 which is a contradiction. Hence ?x i ?p,w i ? ? x i ?p,?w i ?. In the case of homothetic preferences: ?x i,j ?p,w i ? ?w i w i x i,j ?p,w i ? ? 1 (follows from differentiation). To show the law of uncompensated demand, in the continuously differentiable case, we want to show: ? j?1 L dp j ? k?1 L ?x i,j ?p,w i ? ?p k ?p k ? 0 which is equivalent to saying that the matrix ? p x i,j ?p,w i ?, which has elements ?x i,j ?p,w i ? ?p k is negative definite. The slutsky decomposition gives us that ?x i,j ?p,w i ? ?p k ? ?h i,j ?p,u i ? ?p k ? x i,k ?x i,j ?p,w i ? ?w or ? p x i,j ?p,w i ? ? ? p h i,j ?p,u i ? ? 1 w i x i ?p,w i ?x i ?p,w i ? T Both other these matrices are negative semi-definite, except when dp is proportional to p in which case, dp ?? p h i,j ?p,u i ? ? dp ? 0 (why)orif dp ? x i,j ?p,w i ? in which case, 1 w i x i ?p,w i ?x i ?p,w i ? T ? 0 Alternatively, I can rewrite the inequality as ? j?1 L dp j ? k?1 L ?h i,j ?p,u i ? ?p k ? x i,k ?x i,j ?p,w i ? ?w ?p k ? ? j?1 L dp j ? k?1 L ?h i,j ?p,u i ? ?p k ?p k ? ? j?1 L dp j ? k?1 L x i,j ?p,w i ?x i,k ?p,w i ? w i ?p k The first term is negative (this is just the slutsky being negative semi-definite– and definite for non-proportional price changes). The second term is 1 w i times ? k?1 L x i,k ?p,w i ??p k squared, which of course, must be positive. Other ways to get uncompensated law of demand are discussed in Proposition 4.C.3 and 4.C.4 (only income heterogeneity and identical preferences and uniform wealth distribution). Representative Consumer Theorem MWG Definition 4.D.1: A positive representive consumer exists if there is a rational preference relation ? on ? ? L such that the aggregated demand function x?p,w? is precisely the Walrasian demand function generated by this preference relation. That is x?p,w? ? x whenver x ? x?p,w? and p ? x ? w Definition 4.D.2: A Bergson-Samuelson social welfare function is a function W : ? I ? ? that assigns a utility value to each possible vector ?u 1 ,u 2 ,..u I ? ? ? I of utility levels for the I consumers in the economy. The benevolent social planner’s problem: Max W?v 1 ?p,w 1 ?,v 2 ?p,w 2 ?,...v I ?p,w I ?? such that ? i?1 I w i ? w Proposition 4.D.1 Suppose that for each level of prices p and aggregate wealth w , the wealth distribution ?w 1 ?p,w?,w 2 ?p,w?,...w I ?p,w?? solves the problem of maximizing 4.D.1, then the value function v?p,w? is an indirect utility function of a positive representative consumer for the aggregate demand function x?p,w? ? ? i?1 I x i ?p,w i ?p,w?? Definition 4.D.3: The positive representive consumer ? for the aggregate demand x?p,w? ? ? i?1 I x i ?p,w i ?p,w?? is a normative representative consumer relative to the social welfare function W(.) if for every ?p,w? the distribution of wealth ?w 1 ?p,w?,w 2 ?p,w?,...w I ?p,w?? solves the problem of maximizing 4.D.1, and therefore the value function of problem 4.D.1 is an indirect utility function for ?. Externalities Definition 11.B.1: An externality is present whenever the well-being of a consumer or the production possibilities of a firm are directly affected by the actions of another agent in the economy. Note– directly means no pecuniary externalities (i.e. externalities that work through the price system). We will not consider the social welfare implications of externalities today, but rather their impact on aggregation issues, which can be considerable. To simplify the issue assume that utility has the following form: u ? x 1 ? v?x 2 , ? x 2 ? ? w?x 3 ,...x L ? Where ? x 2 is the average level of consumption in society– assume that the individual is sufficiently small so that he ignores his impact on this quantity. We assume that the price of the first good is one. In this case, we know that the first order conditions tell us that ?v?x 2 , ? x 2 ? ?x 2 ? p 2 The standard treatment of externalities might assume that v?x 2 , ? x 2 ? ? v A ?x 2 ? ? v B ? ? x 2 ? In which case, ?v A ?x 2 ? ? ?x 2 ? p 2 The standard point is that if there is say, measure one, of identical consumers then imposing a tax on consumption and refunding it improves welfare– Consumers now maximize: u ? y ? ? j?2 L p 2 x j ? tx 2 ? t ? x 2 ? ?v A ?x 2 ? ? v B ? ? x 2 ? ? w?x 3 ,...x L ? which implies ?v A ?x 2 ? ? ?x 2 ? p 2 ? t, or ?x 2 ? ?t ? 1 v A ?? ?x 2 ? ? Since homogeneity implies that x 2 ? ? x 2 , net tax payments are zero, and overall utility will be u ? y ? ? j?2 L p j x j ? ?v A ?x 2 ? ? ? v B ?x 2 ? ? ? w?x 3 ,...x L ? And we can differentiate to get: ?u ?t ? ?x 2 ? ?t ??p 2 ? v A ? ?x 2 ? ? ? v B ? ?x 2 ? ?? ? ? ?x 2 ? ?t ?t ? v B ? ?x 2 ? ?? Which is positive until the point where t ? v B ? ?x 2 ? ?, which is the optimal tax. More interesting is the case where ? 2 v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? 0 The former case is the classic Pigovian tax situation, but from the perspective of positive economics, it may not be all that interesting. The case where ? 2 v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? 0 implies interesting things about aggregate price effects. The Social Multiplier Continue with the previous utility function (quasi-linear to get rid of income effects– separable to get rid of cross-price elasticities). We know that in this case: ?v?x 2 , ? x 2 ? ?x 2 ? p 2 And this implies that if one individual (holding average consumption constant) experiences a change in price, the derivative of quantity with respect to price will be: ?x 2 ?p 2 ? ? 2 v?x 2 , ? x 2 ? ?x 2 2 ?1 – this is negative by assumption. Now what happens if we allow the entire market to react to the change in price: ? 2 v?x 2 , ? x 2 ? ?x 2 2 ?x 2 ?p 2 ? ?v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? ? x 2 ?p 2 ? 1 But if consumers are identical, ? ? x 2 ?p 2 ? ?x 2 ?p 2 So ?x 2 ?p 2 ? ? 2 v?x 2 , ? x 2 ? ?x 2 2 ? ?v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ?1 If ?v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? 0, what does this mean? What does this do to the demand curve? If ?v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? 0, how would you interpret this? What does this do to the demand curve? In principle, if ?v?x 2 , ? x 2 ? ?x 2 ? ? x 2 ? ? ? 2 v?x 2 , ? x 2 ? ?x 2 2 ,insome region, the demand curves may bend backwards– how would you interpret this? What does all this suggest about using aggregate data to learn about individual attributes? Equalizing Differentials with Heterogeneity In the labor market context– consider the utility function u?w?z?,z,??, where z is a job characteristic, and ? is a personal characteristic that differs among people. The distribution of jobs is characterized by a density g?z?, the distribution of individual characteristics is characterized by a density f???. Both of these are treated as being exogenous. Each individual selects a job so that: u w ?w?z ? ?,z ? ,??w ? ?z ? ? ? u z ?w?z ? ?,z ? ,?? ? 0, thus as before wages must compensate for changes in utility. If we assume that u z? ?w?z?,z,?? ? u w? ?w?z?,z,??u z ?w?z ? ?,z ? ,?? u w ?w?z ? ?,z ? ,?? then ?z ? ?? ? 0. How do I know that this is the condition? What is the interpretation of these terms? Let’s make our lives a little easier and assume u?w?z?,z,?? ? v?w?z?? ? ??z,??–inwhich case the condition for ?z ? ?? ? 0,isjust ? z? ?z,?? ? 0. Fine– assume this condition so ?z ? ?? ? 0, and individuals with higher levels of ? will choose jobs with higher levels of z. Now let’s describe an equilibrium. For all values of z, there must exist a value of ? denoted ? ??z? such that: (1) ? ?? ? ??z? f?a?d? ? ? ?? z g?z?d?,or ? ? ? ?z? ? g?z? f? ? ??z?? and (2) v ? ?w?z??w ? ?z? ? ? z ?z,?? ? 0. The first condition ensures that supply equals demand– the second condition ensures that each person is choosing his job optimally. We know that w ? ?z? ? ? ? z ?z,?? v ? ?w?z?? But what do we know about the second derivative, i.e. the convexity or concavity of the wage profile. Differentiating the first order condition v ? ?w?z??w ? ?z? ? ? z ?z, ? ??z?? ? 0 with respect to z yields: v ? ?w?z??w ?? ?z? ? v ?? ?w?z??w ? ?z? 2 ? ?? zz ?z, ? ??z?? ? ? z? ?z, ? ??z?? g?z? f? ? ??z?? ? 0 or w ?? ?z? ? 1 v ? ?w?z?? ??v ?? ?w?z?? ? z ?z, ? ??z?? v ? ?w?z?? 2 ? ?? zz ?z, ? ??z?? ? ? z? ?z, ? ??z?? g?z? f? ? ??z?? ? There are three terms: (1) coming from diminishing marginal utility of income– this pushes towards convexity– why? – (2) a term related to ? zz ?z, ? ??z?? the degree to which the marginal utility from z increases or decreases with more z– when the marginal utilty from z declines with more z, then w(z) must be more convex– why? (3) a term related to ? z? ?z, ? ??z?? g?z? f? ? ??z?? ,or sorting– this must push towards concavity– why? What happens as there is more heterogeneity of types of people relative to jobs? How does this relate to the estimation of compensating differentials from wages? Welfare Losses with Heterogeneous Consumers You are not getting much welfare economics from me, but I thought I would just throw this out. Definition 16.B.2 (slightly modified): A feasible allocation, x, is Pareto optimal if there is no other feasible allocation x ? , where x i ? ? i x i for all i and x i ? ? i x i for some i. At this point define feasible so that p ? x ? ? i w i Definition: given a family u i ?.? of continuous utility functions representing the preferences ? i of the I consumers, the utility possibility set is defined as: U ? ?u 1 ,u 2 ,...u I ? ? ? I : there is a feasible allocation, x, such that u i ? i u i ?x i ? for all i?1, 2, ..I This is the set of all outcomes that are either on the Pareto frontier or worse than the pareto frontier. Proposition 16.E.2: If the utility possibility set U is convex, then for any Pareto optimal allocation, ? ? u 1 , ? u 2 ,... ? u I ? there is a vector of welfare weights ?? 1 ,? 2 ,...? I ? ? 0 , ? ? 0, such that ? ? ? u ? ? ? u for all u ? U, that is such ? u is a solution to the social welfare maximization problem. In other words, any pareto optimal can be justified as the solution to maximizing the pareto problem ? ? u?x? such that p ? x ? ? i w i This is the general way that we approach aggregation and social welfare– write down a bunch of welfare weights and then start maximizing. Application: Price Controls and Aggregation Things like price controls have very different impacts when demand comes from decreasing marginal utility of consumption within consumers, or from heterogeneity across consumers. The key is that prices serve as valuable allocation tools, and this allocation mechanism is particularly easy to understand in the discrete demand case. Assume the following utility function for a product. u?x 1 ,x 2 ? ? x 1 ? x 2 ? 1 2 x 2 2 The price of the first good is one. This yields an individual demand curve x 2 ? 1 ? p 2 Everyone is assumed to be identical– and there is measure one of consumers. Let’s assume (I know I haven’t covered this yet, but let’s pretend)– that there is a fixed number (again measure one) of suppliers each with costs 1 2 s 2 2 – and they maximize p 2 s 2 ? 1 2 s 2 2 and this leads to supply of s 2 ? p 2 Without price controls– p 2 ? 1 2 , x 2 ? 1 2 , firm profits are 1 8 , Individual utility equals w ? 1 8 If we impose a price control of p, the new quantity supplied will equal p, and firm profits are .5p 2 . Thelossinprofitscanbe written: 1 2 ? 1 2 ? p? 2 ? p? 1 2 ? p? ? 1 8 ?.5p 2 In the case of measure one homogeneous consumers, every consumer restricts consumption back to p, and consumer utility equals w ? p 2 ? p ? 1 2 p 2 ? w ? p ? 3 2 p 2 The change in consumer welfare can be written: ?.5? p?p ? 1 2 ? 1 2 ? p? 2 ? p ? 3 2 p 2 ? 1 8 This has the familiar interpretation. In the case of a discrete good, demanded by a continum of consumers, we assume that each consumer has utility x 1 ? v i I 2 where I 2 is an indicator function that takes on a value of one if the person consumes the good and zero otherwise. To make things comparable, assume that v i is uniformly distributed on the unit interval. Thus the share or number of people that want the good at a given price equals x 2 ? 1 ? p 2 In the absence of the price control, the price again equals 1 2 , and again firm profits equal 1 8 . Consumer welfare is found by integrating: ? v i ?p 1 ?w ? p ? v i ?dF?v i ? ? ? v i ?0 p wdF?v i ? Note the uniform distribution on the unit interval means dF?v i ? ? dv i So average utility over the population without price control equals w ? 1/8 Exactly the same as before. But now introduce a price control and rationing randomly. At a price p, exactly p units will be produced and firm profits are again .5p 2 . Thelossinprofitscanbe again written 1 2 ? 1 2 ? p? 2 ? p? 1 2 ? p? ? 1 8 ?.5p 2 But what happens to demand? Now everyone for whom v i ? p wants the good, for a total demand of 1 ? p. But total supply is p so only a share p 1?p gets the good. Assuming that your probability of getting the good is independent of your desire for the good (perhaps an extreme assumption). Expected consumer welfare is now: ? v i ?p 1 w ? p 1?p ?v i ? p? dv i ? ? v i ?0 p wdF?v i ? This can be rewritten: w ? p 2 ? p 1?p ? v i ?p 1 v i dv i or w ? p?1?p? 2 . The change in consumer welfare equals p?1?p? 2 ? 1 8 which can be rewritten ?.5? p?p ? 1 2 ? 1 2 ? p? 2 ? p?1?2p? 2 ? p?1?p? 2 ? 1 8 What is this funny new term? Under perfect rationing, the average consumer’s utility for the good equals an average of 1 and 1 ? p, or 1 ? p 2 . Under random rationing, the average consumer’s utility for the good equals an average of 1 and p or 1?p 2 . The difference between 1 ? p 2 and 1?p 2 equals 1?2p 2 and this gets multiplied by p because the loss per unit consumed is multiplied by the number of units consumed.