Lecture Notes, January 11, 2010

Transcription

1 Economcs 200B UCSD Wnter 2010 Lecture otes, January 11, 2010 Partal equlbrum comparatve statcs Partal equlbrum: Market for one good only wth supply and demand as a functon of prce. Prce s defned as the soluton to the equaton. z(p) D(p) S(p) 0 The mplct assumpton s ceters parbus, other thngs beng equal (all other prces held fxed). Suppose there s a shft parameter, α, that descrbes changes n the demand and supply functons. Then the defnton of equlbrum now looks lke: z(p, α ) D(p, α) S(p, α ) 0. Consder changes n α. What happens to p? Totally dfferentate z wth respect to α. We have dz dp + 0 dα dα α Assumng 0 we have dp 1 z Dα S α dα α Dp S p α p z (the denomnator Dp Spof ths expresson s the Jacoban of the system). January 11,

2 Economcs 200B UCSD Wnter 2010 Then suppose that α represents an upward shft n demand and that the usual slopes apply to D and S. Dp < 0,Sp > 0. We have ( ) ( 0) ( ) ( ) dp dα + Just what you d expect. An upward shft n demand ncreases prce. Example: The tax ncdence problem Who really pays a tax leved on sellers? Let α excse tax, p o - α prce receved by seller, p o prce pad by buyer D (p, α) D (p, 0), S(p, α) S(p - α, 0) D α (p, α) 0, S α (p, α) S p. ( S) Sp dα D S D S D S o dp D S α α p p p p p p p Consder the case S p >> 0, D p 0 ; elastc supply, nelastc demand. o dp Then 1. Interpretaton: Prce to seller s unaffected by mposton of the dα tax α. The tax s shfted to buyers. Comparatve Statcs, Implct Functon Theorem Characterze market equlbrum, subject to a parameter α, by market clearng n z (p, α). January 11,

3 Economcs 200B UCSD Wnter 2010 z 1(p, α) 0 z (p, α) 0 z (p, α) 0 Prces p are endogenously determned by the market clearng condton. Then as α shfts, market-clearng values of p wll change as well. Assumng everythng n sght s dfferentable and well defned, we have, 1 1 dp1 z1 dz 1 1 d α α dα 0 z dp z dz 0 z dp z dα d 1 α α + 0 j dα α The expresson j 1 s the Jacoban of the market clearng January 11,

4 Economcs 200B UCSD Wnter 2010 equaton system. Solvng for dp1 dα dp dα dp dα we have 1 dp d p1 p α α dp d p α j α dp z dα 1 α Ths expresson s well defned when the Jacoban s non-sngular. Ths s an applcaton of the Implct Functon Theorem. See also Regular Economes. HELP: There s a shortage of good, ntellgent, relatvely smple transparent, comparatve statcs problems sutable for a problem set or exam. Please suggest your favorte to Ross. Reward: 3 browne ponts plus your queston may show up where t wll do you the most good. January 11,

5 Economcs 200B UCSD Wnter 2010 Mas-Colell notaton, ch. 10. Consumer Surplus and Compensaton Tests What we say n Econ 1: Compettve Equlbrum optmzes trangle area of total surplus. Du Put: Ecole des Ponts et Chaussées Valung a brdge across the Sene Embarrassng varety of consumer surplus measures equvalent varaton; compensatng varaton resultng from ncome effects. MasColell & Alfred Marshall: Assume neglgble ncome effects and that margnal utlty of ncome s constant. Ths mples valdty of partal equlbrum (ceters parbus --- other thngs beng equal) treatment. Results to be demonstrated: Proposton: 1. Welfare optmzaton (Pareto effcency subject to ncome redstrbuton) s equvalent to maxmzng Marshallan Surplus Consumer Surplus + Producer Surplus Consumer Surplus + Profts 2. (1FTWE) Compettve Equlbrum allocaton s Pareto effcent (Marshallan Surplus maxmzng). 3. (2FTWE) Any Pareto effcent allocaton can be supported as a compettve equlbrum, subject to a redstrbuton of ncome. Model: H, j F good m Hcksan composte of all goods but one wth prces held constant (partal equlbrum, ceters parbus other thngs beng equal) m s numerare, prce set equal to unty, 1. good, prce of good s p, market determned Producton c j (q) frm j's cost functon January 11,

6 Economcs 200B UCSD Wnter 2010 q j j's nput requrement (n m) to produce q of output of frm j Households m 's consumpton of m x 's consumpton of ω 's endowment of m, u 's utlty functon m + ϕ (x ) quas-lnearty, partal equlbrum, constant margnal utlty of ncome (ths s equvalent to assumng other thngs beng equal, all prces except s held constant; mplyng constant margnal rates of substtuton across all goods other than, hence vald aggregaton). Frms Proft of frm j at prce p s defned as π j p q j - c j (q j ) 1 θ j 0, θ j s 's ownershp share of frm j, θ j 1 Compettve equlbrum p o, x o, q jo so that p o c j '(q jo ), all j, p o ϕ '(x o ), all, (ncome condtons) p o x o + m ω j j + θπ, all, and x j q (market clearng). Determnaton of the (effcent/equlbrum) quantty of good n MasColell, Whnston & Green s quas-lnear model The only thng that determnes the gross quantty of n ths model s the frst order condton ϕ '(x )c j '(q j ) for all n H, all j n F, (assumng nteror soluton for x, q j ). There s no effect from the total endowment of m, ω. The January 11,

7 Economcs 200B UCSD Wnter 2010 reason for ths s that we purposely omt any nonnegatvty condton on m. Thus total m used as nputs for producng may be more than total endowment. If that happens some households end up wth large negatve holdngs of m. The ntal endowment ω s very mportant n determnng the compettve equlbrum dstrbuton of welfare --- snce t represents ntal wealth, but t has no effect on the equlbrum quanttes of held ndvdually, x. Ths s a massvely oversmplfed model. The purpose s to emphasze the noton of the relaton of compettve equlbrum and effcency to consumer and producer surplus. It does that effectvely at the cost of great oversmplfcaton. Welfare Economcs The quas-lnear form of u makes welfare economcs very smple. ote that j any attanable plan wll have the property that x q. The lnear form of u says that the utlty possblty fronter s a straght lne. Choose q j effcently and then dstrbute resultng to max sum of ϕ (x ), then redstrbute ω for desred mx of utltes. Any attanable Pareto effcent allocaton of resources and consumpton (gnorng boundary condtons) s characterzed as choosng x, q j, so that, j x q, to maxmze u [m + ϕ (x )] [ϕ (x ) + ω -px + ( θ j π j )] [ϕ (x ) + ω -px +{ θ j (p q j - c j (q j )}] ϕ (x ) - px + ω January 11,

8 Economcs 200B UCSD Wnter 2010 But + ( θ j pq j ) - ( θ j c j (q j )) ϕ (x ) - px + ω + pq j - c j (q j ) Consumer surplus + endowment + proft ϕ (x ) + ω - c j (q j ) ω s a constant, so maxmzng u mples maxmzng ϕ (x ) - c j (q j ) consumer surplus + producer surplus Marshallan surplus. Welfare Maxmzaton n quas-lnear model: Maxmze S(x 1, x 2,..., x #H ; q 1,..., q #F ) ϕ (x ) - c j (q j ) subject to L x ϕ (x ) - q j c j (q j ) - λ( x - q j ) L x ϕ ' - λ 0 L q -c j ' + λ 0 Therefore the Frst Order Condton for Pareto Effcency s ϕ ' c j ' Frst Fundamental Theorem of Welfare Economcs n quas-lnear model: January 11,

9 Economcs 200B UCSD Wnter 2010 ϕ ' c j ' p o s the characterzaton of compettve equlbrum so Compettve Equlbrum s Pareto Effcent. Second Fundamental Theorem of Welfare Economcs: Any attanable Pareto effcent allocaton can be sustaned as a compettve equlbrum, ϕ ' c j ' p o, subject to a redstrbuton of ω. Compensaton tests for publc works: Pareto preferablty Increase n Marshallan surplus (possbly wthout compensaton). ote theory of the second best n the presence of dstortonary taxaton, Auerbach. January 11,