Copyright (C) 2008 David K. Levine This document is an open textbook; you can redistribute it and/or modify it under the terms of the Creative

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1 Copyrght (C) 008 Davd K. Levne Ths document s an open textbook; you can redstrbute t and/or modfy t under the terms of the Creatve Commons Attrbuton Lcense.

2 Compettve Equlbrum wth Pure Exchange n traders k goods Economc Fundamentals x consumpton by trader of good x denote the vector, bundle or basket of goods consumed by trader trader s preferences for consumng dfferent goods gven by her utlty functon u ( x ) trader endowed wth x of good there s no producton n ths economy, t s a pure exchange economy traders smply exchange goods wth each other the economy lasts only one perod

3 Market Insttutons we assume "the law of one prce" traders scope out opportuntes to such an extent that each good s sold (and purchased) at only one prce p the prce of good p lst of all prces of all goods, or the prce vector we assume compettve behavor traders do not perceve that they have any nfluence over market prces theory of the result of tradng n ths economy: compettve equlbrum (compettve equlbrum s not the Nash equlbrum of the compettve game ) compettve equlbrum prces wrtten as ˆp are (by defnton) prces at whch every trader can smultaneously satsfy her desre to trade at those prces

4 Demand x ( p, m) demand by trader for good when prces are p and money ncome s m the soluton to the problem max x u ( x ) subect to k px mor = p x m 3

5 Excess Demand n pure exchange economy money ncome generated by sellng endowment [doesn t cost anythng extra to sell your endowment then buy t back, snce the prces at whch you buy and sell are the same] demand to buy or net, or excess demand m ( ) ( ) x x x = z p, x p, p. ths can be negatve, as bg as x 4

6 Cobb Douglas Example = α ux (, x) Ax x β demand from Lagrangean α β λ + Ax x ( p x p x ) frst order condtons α β λ Aαx x p = 0 Aβx x = 0 α β λp rearrange and dvde α β λ α β λp Aαx x p Aβx x = cancellng terms αx p = βx p 5

7 αx βx p = cross multply αpx = βpx p plug nto the budget constrant px + px = m to get px + ( β/ α) px = m = α m, α m x = + α + β p or x α β p excess demand z α px + px = x α + β p for smplcty take α + β = then p z = x x, z α ( α) p, x β px + px = x α + β p = α αx p ( ) px 6

8 Aggregate or Market Excess Demand n z( p) z( p) = n market for each good demand to buy cannot exceed zero there s no producton or outsder to provde supply to the market one traders excess demand must be anothers excess supply ˆp compettve equlbrum prces are determned by z ( pˆ ) 0 for every good =,,, k 7

9 Cobb-Douglas Economy Two consumers both have dentcal preferences wth α = / Endowments are z = z z z p = so z p = p p = so z x = (, 0), x = (0,) p = + p p = p we aren t gong to solve ths yet 8

10 Propertes of Demand Indvdual demand: two key propertes Homogeneous of degree zero x ( λp, λ m) = x ( p, m) (relatonshp to nflaton, dollars versus quarters) α m x = α + β p Satsfes the budget constrant p k = α px( p, m) = m m β m + p = α + β p α + β p?? 9

11 Indvdual excess demand: two key propertes Homogeneous of degree zero z ( λ p) = z ( p) p z x x = α ( α) p Walras s Law k = proof: pz ( p) = 0 k (, ) = k px(, ) p p x px px = k k pz( ) p px = = k pz( ) p m = m = p x p m = + = + = + 0

12 Aggregate excess demand: two key propertes Homogeneous of degree zero z ( λ p) = z ( p) z p = + p Walras s Law k = pz ( p) = 0 p p p + + = p p p??

13 Solvng for Equlbrum there are k dfferent excess demand condtons z( p ) = 0 and there are k dfferent prces p, p, k but one excess demand condton s redundant suppose z( p ) = 0 for =,, k, then from Walras s law zk( p ) = 0 on the other hand, f z( p ) = 0 for all =,, k then so does z ( λ p) = 0 so many compettve equlbra can solve only for relatve prces usng k excess demand equatons

14 The Numerare may arbtrarly set the prce of one good to called the numerare good, all prces are measured relatve to that good (for example money s numerare) 3

15 Cobb-Douglas Example Pck one equaton z p = + = p so p/ p = 0 pck the other equaton z p = p get the same answer of course f we choose good as numerare then we have p =, p = how do we fnd ndvdual demands? 4

16 The Frst Welfare Theorem Suppose we have a compettve equlbrum wth prces p and ndvdual demands x s ths pareto effcent? That s: can we fnd x socally feasble that makes nobody worse off and at least one person better off? n n x = = That s: can we fnd x so that u ( x ) u ( x ) for everybody (all ) and for somebody (some ) u ( x ) > u ( x )? Observaton: f u ( x ) > u ( x ) then p x > p x Why?? Further observaton: u ( x ) u ( x ) then spend your extra ncome to buy more) p x p x (otherwse 5

17 Our concluson: f u ( x ) u ( x ) for everybody (all ) and for somebody (some ) u ( x ) > u ( x ), then p x p x for all and p x > p x for some add these together: n n p x = = > p x on the other hand n n x = = x, so addng over dfferent goods k n k n p x p x = = = = whch says that n n p x = = p x 6

18 relatonshp to the core mplcatons for nternatonal trade the edgeworth box the second welfare theorem the compettve mechansm 7

19 Fnance Trade n perod 0 clams to consumpton n perod k dfferent states of nature n perod, probablty of state s π consumpton n state s c tme 0 prce of consumpton n state s p budget constrant k = pc martngale prces p = p / π m budget constrant n martngale prces k Epc = π p c m = 8

20 Securtes Securty a pays r n state Examples: Arrow securty on state pays n state 0 n all other states Prce of an arrow securty p or p Bond pays n all states Prce of a bond k = p or Ep arbtrage prcng = law of one prce 9

21 Spannng Two states =,, k = Stock pays (,) prce q, Stock pays (,) prce q What s the prce of a bond? Buy both stocks: get (3, 3) bond pays (,) so /3 rd of both stocks q = ( q + q )/3 b what does spannng mean? k dfferent assets that are ndependent can determne all asset prces n terms of a spannng set of assets 0

22 Short Sales Stock pays (,) prce q, Stock pays (3,) prce q What s the prce of a bond? a(,) + b(3,) = (,) 3 a = b a 3 = b 3 = ( ) = check (,) (3,) = (4,) (3,) = (,)

23 so buyng unts of stock and unts of stock s the same as buyng a bond: cost q q what does t mean to buy unt of stock? observaton q q > 0 so we can conclude that q > q /

24 The Holdup Problem entrepreneur (nventor, merchant) creates value of ρ ρ s drawn from a unform dstrbuton over [0, ] and s prvate nformaton to the entrepreneur case : the nnovator receves a fracton of the socal total φρ case : the nnovator receves the entre socal total ρ but must pay N exstng rghts holders for the rght to create value examples: the slk road patents and copyrghts polluton 3

25 effcency = the good s always produced n case the good s always produced n case rghts holder set prce p for hs rght and gets an expected revenue of p ( ( ) ) = ( / N + ) N p p p entrepreneur pays N N + N < ρ N + what happens as N to clear the needed rghts, so creaton f as technologes grow more and more complex requrng more and more specalzed nputs,monopoly power nduced by patents and copyrght becomes more and more socally damagng 4