Lecture 5: Applications of Consumer Theory

Transcription

1 Lecture 5: Applcatons of Consumer Theory Alexander Woltzky MIT

2 Applcatons of Consumer Theory Consumer theory s very elegant, but also very abstract. Ths lecture: three classc topcs that brng consumer theory closer to economc applcatons: 1. Welfare effects of prce changes. 2. Constructng prce ndces. 3. Aggregatng consumer demand. 2

3 Welfare Evaluaton of Prce Changes What s effect on consumer s welfare of prce change from p to p? Examples: taxes and subsdes, ntroducton of new good Obvous answer: change n utlty v p, w v (p, w ) Problem: whch ndrect utlty functon? Utlty just way of representng preferences. 3 Dfferent ndrect utlty functons gve dfferent value of v (p, w ) v (p, w ).

4 Money Metrc Indrect Utlty Class of ndrect utlty functons that let us measure effect of prce change n dollar unts: money metrc ndrect utlty functons. Construct from expendture functon: Start from any ndrect utlty functon v, any prce vectorp» 0, consder e (p, v (p, w )) e (p, u) s strctly ncreasng n u = e (p, v (p, w )) s strctly ncreasng transformaton of v (p, w ) = e (p, v (p, w )) s tself an ndrect utlty functon! 4 Measure welfare effect of prce change n dollar terms by ( ( e p, v p )), w e ( p, v (p, w ))

5 Money Metrc Indrect Utlty e (p, v (p, w )) e (p, v (p, w )) s ndependent of choce of v: for every v, e ( p, v (p, w )) = mn p x x :u(x ) max y B (p,w ) u(y ) = mn p x. x :x ty for all y B (p,w ) = all money metrc ndrect utlty functons wth same p are equvalent Lettng u = v (p, w ), u = v (p, w ), the dfference ( ) e p, u e (p, u) s ndependent of the utlty representaton. 5 How much more money does consumer need to get new utlty rather than old utlty, when prces gven by p?

6 Equvalent Varaton and Compensatng Varaton Only remanng ssue choce of p. Two natural choces: ntal prce vector p, new prce vector p. Lead to two best-known ways of measurng welfare effect of prce change: equvalent varaton (EV) and compensatng varaton (CV) 6

7 Equvalent Varaton EV measures requred expendture change at orgnal prces: ( ) EV = e p, u e (p, u) = e p, u w EV = amount of money consumer would need to be gven before prce change to make her as well off as would be after prce change. Consumer ndfferent between ether gettng EV or facng prce change. EV = amount of money that s equvalent to prce change. 7

8 Compensatng Varaton CV measures requred expendture change at new prces: ( ) ( ) ( ) CV = e p, u e p, u = w e p, u CV = amount of money consumer would need to lose after prce change to make her as well off as was before prce change. Consumer ndfferent between 1. gettng both (mnus) CV and facng prce change, and 2. gettng nether. CV = amount of money needed to compensate for prce change. 8

9 EV vs. CV EV and CV are dfferent. Can be ranked f 1. prce change affects only one good, and 2. good s ether normal or nferor over relevant range of prces. Follows from connecton between EV/CV and Hcksan demand: 9 ( ) EV = e p, u w ( ) ( ) = e p, u e p, u p ( ) = h p, u dp p ( ) CV = w e p, u ( = e (p, u) e p ), u p = h (p, u) dp p

10 EV vs. CV Suppose p > p (so u > u). p ( ) EV = h p, u dp p p CV = h (p, u) dp p If good normal, then h (p, u ) > h (p, u), so EV > CV. If good nferor, then h (p, u ) < h (p, u), so EV < CV. If no wealth effect for good, then EV = CV. 10 Graphcally, EV s area to left of h (, u ), CV s area to left of h (, u).

11 Marshallan Consumer Surplus Marshallan consumer surplus (CS) = area to left of Marshallan demand curve x (, w ): CS = p p x (p, w ) dp For changes n prce of one good, mn {EV, CV } CS max {EV, CV } Proof. x (p, w ) = h (p, e (p, u)), x (p, w ) = h (p, e (p, u )), so Marshallan demand curve cuts across regon between h (, u) and h (, u ). 11 If wealth effects small, then EV, CV, and CS are smlar.

12 Estmatng Welfare from New Goods CV for new good: h (p, u) dp p How to estmate demand at very hgh prce? If prce drops to 0 atp, can estmate p p h (p, u) dp so just have to estmatep/demand around p. See Hausman and Newey (1995, 2011) for recent approaches to estmatng welfare from new goods. 12

13 Prce Indces An mportant applcaton of measures of welfare changes s constructon of prce ndces: measures of changes n prce level (or nfaton). Important for estmatng real GDP growth, determnng socal securty payments, negotatng long-term labor contracts, etc.. 13

14 Laspeyres and Paasche Indces Problem s to construct ndex of prce change from perod 0 to perod 1, where: n perod 0, see prces p and consumpton x n perod 1, see prces p and consumpton x Laspeyeres ndex: rato of prce of orgnal basket of goods n perod 1 to prce n perod 0: p x p x Paasche ndex: rato of prce of new basket of goods n perod 1 to prce n perod 0: p x p x 14 Problem: no one cares about p x or p x

15 Ideal Indces Ideal ndces are constructed from money metrc ndrect utlty functons. Measure how much more expensve t gets to attan utlty u: e (p, u) Ideal (u) = e (p, u) Ex. u could equal v (p, w ) or v (p, w ) 15

16 Bases n Prce Indces Ideal ndces let us formalze popular vew that Lespeyres overstates nfaton, and Paasche understates nfaton : p x p x e (p, u) Laspeyres = = = Ideal (u) p x e (p, u) e (p, u) p x e (p, u ) e (p, u ) ( Paasche = = = Ideal u ) p x p x e (p, u ) Problem s called substtuton bas: Laspeyres and Paasche don t take nto account that, when prces change, consumers substtute to cheaper goods. 16

17 Substtuton Bas n Practce 1996 Boskn commsson report: CPI overstated by about 1.1 percentage ponts per year. Sources of bas: Substtuton bas (0.4%): CPI used Laspeyres ndex, updated basket of goods very nfrequently. Outlet bas (0.1%): CPI treated dfferent goods at dfferent stores as dfferent. Mssed swtch to cheaper stores. New goods bas (0.6%): CPI only tracked changes n prce from when new goods were added to basket, not orgnal drop from prce. 17 An area of research s gettng better estmates of new goods bas. Some economsts thnk 0.6% s way too low.

18 Demand Aggregaton Consumer theory concerns behavor of a sngle consumer. Often care about aggregate behavor of consumers. Ex. to construct deal prce ndex for US economy, would need aggregate expendture functon for US populaton Does consumer theory also apply to aggregate demand and welfare? 18

19 Demand Aggregaton Three questons: 1. Does aggregate demand depend only on p and aggregate wealth w = w, or does dstrbuton of wealth also matter? 2. Does postve theory of ndvdual demand also apply to aggregate demand? (Is there a postve representatve consumer?) 3. Do welfare measures derved from aggregate demand mean anythng? (Is there a normatve representatve consumer?) 19

20 Aggregate Demand Suppose there are I consumers. Consumer has Marshallan demand x ( p, w ). Aggregate demand X = sum of ndvdual demands: ( ) I ( ) 1 I X p, w,..., w = x p, w =1 20

21 Aggregate Demand and Aggregate Wealth When does aggregate demand depend only on p and w? When does there exst X : R n + R R n such that ( ) ( ) X p, w 1,..., w I I = X p, w for all p, w =1 Clearly, ths holds ff every possble redstrbuton of wealth among consumers leaves aggregate demand unchanged. Turns out that ths holds ff preferences are quashomothetc, a class that generalzes both homothetc and quaslnear preferences. 21

22 Homothetc Preferences Defnton Preferences are homothetc f, for every x, y R n and α > 0, x t y αx t αy Graphcally, preferences are homothetc f slope of ndfference curves s constant along rays begnnng at the orgn. Homothetc preferences are represented by utlty functons that are homogeneous of degree 1: u (αx) = αu (x) for all x Demand s homogeneous of degree 1 n ncome: x (p, αw ) = αx (p, w ) 22 Have ndrect utlty functon of form: v (p, w ) = b (p) w

23 Quaslnear Preferences Recall that preferences are quaslnear (n good 1) f admt utlty representaton of the form u (x) = x 1 + f (x 2,..., x n ) Assumng some ncome s spend on the numerare good, have ndrect utlty functon of form: v (p, w ) = a (p) + w 23

24 Quashomothetc Preferences Defnton Preferences are quashomothetc (or Gorman form) f they admt an ndrect utlty functon of the form: v (p, w ) = a (p) + b (p) w 24 Theorem Aggregate demand depends only on aggregate wealth ff preferences admt Gorman form ndrect utlty functons wth the same functon b for every consumer: that s, there exst functons a : R n + R and b : R n + R such that, for all, ( v p, w ) = a (p) + b (p) w.

25 Quashomothetc Preferences Theorem Aggregate demand depends only on aggregate wealth ff preferences admt Gorman form ndrect utlty functons wth the same functon b for every consumer: that s, there exst functons a : R n + R and b : R n + R such that, for all, ( v p, w ) = a (p) + b (p) w. 25 Intuton: Aggregate demand depends on aggregate wealth ff not affected by any redstrbuton of wealth. Ths holds f wealth effects are the same across ndvduals and across wealth levels. Wealth effects same across wealth levels = quashomothetc preferences. Wealth effects same across ndvduals = same functon b for everyone.

26 Postve Representatve Consumer? Exsts f aggregate demand depends only on aggregate wealth: then aggregate demand s same as f all wealth held by consumer 1, so consumer 1 s representatve consumer. Hard to wrte down condtons much weaker than ths that mply exstence of representatve consumer. So, representatve consumer exsts only f not much heterogenety, especally heterogenety n wealth effects. Illustrate wth example wth well-behaved preferences but no representatve consumer. 26

27 Example Two consumers are buyng apples and bananas. Each consumer has wealth w = 4. Consumer 1 lkes apples more. Consumer 2 lkes bananas more. Nether has much taste for more than two unts of same frut. 1 2 agg p = (1, 2) : x = (2, 1), x = (0, 2), x = (2, 3). 1 2 agg pˆ = (2, 1) : ˆx = (2, 0), xˆ = (1, 2), x = (3, 2). Apples are cheap = more bananas get bought. Bananas are cheap = more apples get bought. 27 Cannot result from optmzaton by a ratonal representatve consumer (volates WARP).

28 Normatve Representatve Consumer? 28 A Bergson-Samuelson socal welfare functon s a functon W : R I R that maps vectors of ndvdual utltes nto a socal utlty. When does choosng consumpton for each consumer to maxmze W (subject to aggregate budget constrant p x w ) lead to same aggregate consumpton that results when each consumer maxmzes her utlty separately? Not very often. One problem: consumpton vector that maxmzes W depends on choce of utlty representaton. If consumer 1 lkes apples, consumer 2 lkes bananas, and scale up consumer 1 s utlty by 100, then decentralzed aggregate demand s constant, whle maxmzng W nvolves buyng more apples.

29 Representatve Consumer wth Gorman Form Preferences Wth Gorman form preferences (wth same b), each consumer consumers goods other than numerare untl ther margnal utlty falls to b (p), puts rest of wealth n numerare. = decentralzed aggregate demand maxmzes I =1 a (p) + b (p) w The same aggregate demand functon maxmzes utltaran socal welfare ( ) I W u 1,..., u I = u =1 29 Wth Gorman form preferences, decentralzed consumer optmzaton leads to the allocaton that maxmzes utltaran socal welfare. Not true more generally.

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