Welfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium?

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1 APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Welfare Propertes of General Equlbrum What can be sad about optmalty propertes of resource allocaton mpled by general equlbrum? Any crteron used to compare resource allocatons must be based on value judgements - typcally use the Pareto crteron (see Fgure) Why s ths crteron adopted? (a) Suppose levels of ndvdual well-beng h=,...,h are gven as U h, and suppose some set of explct value judgements gves rse to a socal orderng over U h, gven by the swf: W=W(U,...,U H ) () Suppose that an ncrease n the well-beng of one ndvdual s a good thng and a decrease s bad, the well-beng of no-one else changng, mplyng: W U h > 0 for all h ()

2 The Pareto Crteron U a 3 a 4 a a U - weak Pareto crteron states a s superor to a f and only f both ndvduals and are better off at a - strct Pareto crteron states a 3 s preferred to a f everyone s at least as well off, and at least one person s better off - provdes only a partal orderng, e.g., a 4 cannot be compared to a by ths crteron

3 Ths can be used to show that a resource allocaton that maxmzes W must be a Pareto optmum Take an allocaton that s not a Pareto optmum, we know t s possble to change ths and ncrease some U h, wth no U h decreasng, hence, W must ncrease by () Proves that an allocaton that s not Pareto optmal wll not maxmze W, hence, a welfare maxmum must le n a set that s Pareto optmal (b) Under certan condtons, equlbrum of a market economy s Pareto optmal,.e., the Frst Welfare theorem Characterstcs of a Pareto Optmum Focus on a consumer, frm, good, nput econo my, and assume a central planner that wants to establsh the condtons that wll be satsfed by a Pareto optmal allocaton

4 The model of the economy s: (a) Utlty functons: U h = U h (x h,x h,z h ) =, (3) x h are quanttes of goods consumed, gvng postve utlty, and z h s nput h suppled by household h, generatng negatve utlty; ntal endowments of each nput are ž h, h=, (b) Producton functons: x f (z,z ), (4) where x s output of each good, and each frm produces only one good (c) Market Clearng Condtons: h x h x 0, (5) z h z h 0 h, (6)

5 Need to fnd necessary condtons for resource allocaton satsfyng constrants, and whch maxmzes one household s utlty functon subject to a gven level of utlty for the other household Defne the problem: max U ( x,x,z ) (7) s.t. U ( x,x,z ) (8) Lagrange multplers µ,, h and can be assocated wth (4),(5), (6), and (8) respectvely, the necessary condtons for Pareto optmalty are: U 0 U 3 0 U 0 U 3 0, µ 0, µ f h h 0 h, total output condton nput allocaton condton

6 Condtons for Pareto optmalty gven by takng approprate ratos: U U U U (9) U h 3 U h h f h h,, (0) f f f f () U U U U µ h /f h µ h /f h f h f h () - (9) states households margnal rates of substtuton between goods have to be equal - (0) states household s margnal rate of substtuton between good and nput z h has to equal nput s margnal product - () states frms margnal rates of techncal substtuton between nputs have to be equal

7 - () states that equalzed margnal rates of substtuton have to be equal to the margnal rate of transformaton / and / are shadow prce ratos, (9) - () are presented graphcally n Fgure (a) At, margnal rates of substtuton are equal, and equal to the margnal rate of transformaton; total consumpton also equals total producton (b) At, margnal rates of techncal substtuton are equal, frms are on soquants correspondng to x * and x *, and nput allocatons sum to what s made avalable by households z * and z * (c) At, margnal rate of substtuton between x and z s equal to the margnal product of z n producng x (could be expressed n terms of good )(m shows valuaton of ncrement of z n terms of x consumer would have to be pad to supply z ) (d) At, margnal rate of substtuton between x and z s equal to the margnal product of z n producng x (could be expressed n terms of good )(m has smlar nterpretaton to m )

8 Frst Welfare Theorem: A compettve equlbrum s a Pareto Optmum Ths theorem ndcates that equlbrum prce sgnals are suffcent to coordnate decentralzed economc actvtes n a way that s Pareto optmal By ndvdual maxmzaton behavor, each economc agent responds to prces by equatng margnal rate of substtuton (margnal rate of techncal substtuton) to these prces As all agents face the same prces, all margnal rates are equated to each other n equlbrum. Combned wth market equlbra, these equaltes characterze Pareto optma n a convex envronment,.e., non-ncreasng returns and convex preferences * see the appendx for a proof of the theorem

9 Second Welfare Theorem: Gven a Pareto optmum, there exsts a set of prces that wll lead to a compettve equlbrum Second theorem has mportant polcy mplcaton: By makng economy compettve, selectng equlbrum to be decentralzed, and by usng lump-sum transfers to ensure each household has enough ncome to afford ther allocaton, polcy maker can acheve a Pareto optmum (Fgure ) Normatve queston of whch allocaton to acheve s separated from how to acheve that allocaton Problems wth Welfare theorems (a) Frst Welfare theorem: - n presence of market falures, theorem no longer applcable to any economy where they are present (theory of second-best) - hghly nequtable allocatons may be optmal A socety or an economy can be Pareto optmal and stll be perfectly dsgustng (A. Sen, 970)

10 (b) There are three key problems wth the Second Welfare theorem: - wth market falures, there wll not be a Pareto optmum - requres convexty assu mptons to hold - reles on the use of lump-sum transfers, but these may be costly to collect, characterstcs on whch they are based are unobse rvable, gvng households an ncentve to make false re velatons It s therefore best to treat the Second Theorem as beng of consderable theoretcal nterest but of very lmted practcal rele vance (G. M yles, 995)

11 Appendx (Welfare Propertes of General Equlbrum) Proof of the Frst Welfare Theorem Defntons: () Feasblty: An array of consumpton vectors {x,...,x h,...,x H } s feasble f x h X h, all h, and there exsts an array of producton vectors {y,...,y j,...,y m }, each y j Y j, such that: x y + where: H x h H x h, h m h, y j y j..e., an allocaton of consumpton bundles to consumers s feasble f t can be produced wth ntal endowment and producton technology () Pareto Optmalty: A feasble consumpton array {x h *} s Pareto optmal f there does not exst another array {x h } such that:

12 U h ( x h ) U h (x h ), h,...,h, U h (x h ) > U h (x h ), for at least one h. {x h *} s Pareto optmal f there s no alternatve feasble array that gves each household as much utlty as {x h *}, and gves strctly more utlty to at least one household () Compettve Equlbrum (CE): An array [p*,{x h *}, {y j *}] s a compettve equlbrum f: m x h X h, p x h p h h j p y j, h,...,h. (3) j y j Y j, j,...,m (4) and: () U h (x h ) U h (x h ) for all x h X h such that p x h p h m j h j p y j,

13 () p y j p y j,all y j Y j, () x y (3) and (4) requre household demands to be both affordable and n ther consumpton sets, and frms choces are n ther producton sets () mples households maxmze utlty, () mples that frms maxmze profts, and () that the equlbrum s feasble To develop frst theorem, suppose household h has locally non-satated preferences, and let x h * be ther consumpton plan. Lemma: Let x h * be a locally non-satatng choce for household h at prces p*. Then: (a) U h ( x h ) > U h (x h ) p x h > p x h, (b) U h (x h ) U h ( x h ) p x h p x h.

14 Proof: If (a) were false, then x h would have satsfed the household s budget constrant and would have been chosen nstead of x h *. To prove (b), suppose p*x h < p*x h *. As x h * s not a pont of local sataton, then nether s x h. There then exsts x h at a dstance from x h wth U h ( x h ) > U h ( x h ) = U h ( x h *). Suppose s small such that p*x h < p*x h *. Ths then contradcts the assumpton that x h * was the optmal choce. Frst Welfare Theorem: Let [p*,{x h *},{y j *}] be a compettve equlbrum wth no household locally satated at {x h *}. Then [{x h *},{y j *}] s a Pareto optmum Proof: Suppose [{x h *},{y j *}] s not a Pareto optmum, then there exsts [{x h },{y j }] wth x h X h, y j Y j, and:

15 () x y () U h ( x h ) U h (x h ) all h, () U h ( x h ) > U h (x h ) some h. Gven () and (), (a) and (b) mply: H h H p x h > p x h. h Under local non-sataton, () of CE gves p*x*= p*y*+p*, so t follows that p*x > p*y*+p*. Proft maxmzaton, () of CE, mples that p*y j *p*y j all y j Y j, and, specfcally p*y j *p*y j. Summng over j, p*y*p*y. Hence, p*x> p*y+p*. From ths nequalty, t follows that [{x h },{y j }] s not feasble, provng the theorem by contradcton