2. Equlibrium and Efficiency

Transcription

1 . Equlbrum and Effcency

2 . Introducton competton and effcency Smt s nvsble and model of compettve economy combne ndependent decson-makng of consumers and frms nto a complete model of te economy exstence of compettve equlbrum compettve equlbrum=effcent coordnatng role played by prces

3 . Economc Models Wy employ models to make predctons? experments --> dffcult economc system s too large and too complex formal model = logcal consstency mportant common features <-- polces objectves of te ndvdual agents n te economy constrants tey face aggregate ndvdual decsons to markets equlbrum --> partal or general trade-off between specfed and general

4 .3 Compettve Economes compettve ndvdual agents do not consder ter actons to ave any effect on prces prces = fxed or parametrc neglgble n sze relatve to te market prces values --> relatve values of dfferent goods sgnals --> coordnatng te decsons of agents economc effcency

5 secondary feature symmetrc nformaton uncertanty = all agents are equally unnformed any agent as no nformatonal advantage two forms of compettve model excange economy producton economy

6 .3. Excange Economy trade of commodtes between two partes for mutual advantage two-consumer, two-good excange economy all results extend to large numbers endowment = ntal stock of te goods (constant) excange (market prces = excange rates ) compettve = prce-takng beavor desrable consumpton plans

7 consumer endowment ω = ( ω, ω ), ω 0 : s ntal stock of good prces p, p x = ( x, x ), x budget constrant px + px = pω + pω preference = utlty functon U = U ( x, x ) : s consumpton plan margnal utlty s postve for bot goods ndfferent curve as convex sape 0

8 Edgewort box feasble plan x rectangle lengt + x = ω + ω, =, soutwest corner: consumer s orgn norteast corner: consumer s orgn ω + ω, ω + ω

9 x ω + ω x x x ω ω + ω x

10 Equlbrum budget constrants consumer : relatve to te norteast corner consumer : relatve to te soutwest corner same endowment pont gradent p p equlbrum prces utlty maxmzaton for bot consumers supply = demand

11 x x ω

12 good : excess demand p goes up good : excess supply p goes down x x ω

13 x x ω

14 consumers demand functons max of utlty subject to budget constrant demand demand = supply (market clearng) x ( p equlbrum, p x = x ( p, p) ) + x ( p, p ) = ω + ω, te prces lead to a budget lne on wc te ndfference curves as a pont of common tangency number of prces, number of ndependent equatons =,

15 only relatve prces matter budget constrant always passes troug te endowment pont gradent s determned by te prce rato demand and supply depend on te relatve prce but not on te absolute values x p, p) = x ( λp, λp ), λ > ( omogeneous of degree 0 any scalng up or down wll also be equlbrum prces prce normalzaton numerare = unt of account 0

16 Walras s law nterdependence between te two equlbrum equatons excess demand total value of excess demands, for any prces,, = + = x x Z ω ω 0 ] [ ] [ = + = + = + = = p p x p x p x x p Z p Z p ω ω ω ω

17 Walras s law (contnued) f Z =0, ten Z =0 f Z =0, ten Z =0 snce p Z +p Z =0 equlbrum n one market mples equlbrum n te oter coose good as te numerare (p =) and plot te excess demand for good as a functon of p f te ED of s zero, ten te ED of s zero ---> equlbrum extenson of many consumers and goods number of equlbrum condtons

18 Z (, p) p

19 .3. Producton and Excange addton of producton to excange economy ntal endowments (e.g. labor) consumpton goods produced from n. end. ntermedates: produced from n. end. and nput for anoter Arrow-Debreu economy model of compettve economy producer (frm): producton tecnology consumer: preference, ntal endowments sareoldngs of te frms

20 frm (producer) producton set avalable producton tecnology = knowledge nputs ---> outputs Y j : frm j s producton set nput: negatve = subtracton from te stock of tat good avalable for consumpton output: postve producton plan: prces: proft: j j y =, y = p =, p = j j j π = p y + p y = 3

21 Good Y j 3 - Good

22 Process of proft maxmzaton soproft curve <--- are gven all producton plans tat gve a specfed level of proft π π = p + y p y π =0 <---> nner product of p and y = 0 π >0 <---> nner product > 0 π <0 <---> nner product < 0 proft maxmzaton p, p y* : tangency pont between te gest attanable soproft curve and producton set

23 π = 0 π < 0 π > 0 * y Good p p Good π p

24 economy wt n goods m frms producton plan cosen to maxmze profts s.t. producton set supply functon level of proft aggregate supply j y = negatve or posstve j j ( y,..., y ) n j j y = y ( p), p = ( p,..., pn) j π = n = p y j Y ( p) = j ( p) = π ( p) m j= y j ( p)

25 prvate ownersp economy H consumers: ntal endowments ω sareoldngs n frms budget constrant n θ j 0 p = + x pω = = j= cange n prces ---> endowment n m θ j j π H = dvdend ncome θ j =

26 demand functon aggregate demand equlbrum <---> demand = supply (not fxed) excess demand=0 demand (supply) s determned by relatve prces ---> Z s omogeneous of degree zero normalzaton, numerare ( p) x x = = = H p x p X ) ( ) ( n p p Y p X p Z H,..., 0, ) ( ) ( ) ( ) ( = = = = ω

27 .4 Effcency of Competton effcency f more cannot be aceved, ten te outcome s effcent <--- ndvdual decson-maker dfferent decson-makers aggregaton of preferences frst-best only te producton tecnology and te lmted endowments restrct te coce of DM second-best oter constrants could be lmts

28 .4. Sngle Consumer ndvdual preferences = socal preferences best outcome = frst-best no constrant on polcy coce sngle frm & sngle consumer dagram = Robnson Crusoe economy Robnson acts as a frm carryng out producton and as a consumer of te product of te frm ) Robnson = socal planner for coordnaton ) proft maxmzaton & utlty maxmzaton

29 Good good =numerare ~ x * π p Good

30 budget constrant p[ x ω ] + p[ x ω] = π or p ~ x + p ~ x = π were s net consumpton x~ utlty maxmzaton ---> ~ x * * * equlbrum for te economy ---> ~ x = y net consumpton = supply proft maxmzaton = utlty maxmzaton effcent

31 Good ~ * * x = y π p Good

32 caracterzaton of frst-best allocaton tangency pont between te two curves margnal rate of substtuton MRS =U /U margnal rate of transformaton MRT (=MC /MC ) MRS =MRT margnal value = margnal cost at te frst-best te market aceves effcency utlty maxmzaton ---> MRS =p /p proft maxmzaton ---> MRT =p /p compettve equlbrum MRS =p /p = MRT

33 constant return to scale effcent producton fronter =stragt lne troug te orgn only equlbrum : zero proft postve ---> nfntely large proft soproft curve wt π =0 equal to effcent PF two furter mplcatons equlbrum prce rato s determned by te zeroproft condton and s ndependent of demand snce te proft ncome of te consumer s zero, te budget constrant passes troug te orgn

34 Good ~ * * x = y p Good

35 .4. Pareto-Effcency more tan one consumer Pareto-effcency no unexploted economc gans consder weter t s possble to undertake a reallocaton of resources tat can beneft at least one consumer wtout armng any oter f possble, tere would exst unexploted gans f not, te ntal poston s Pareto-effcent effcent allocaton of resources

36 feasblty: consumpton, producton, n. endowment x = y + π Defnton : A feasble consp. allocaton xˆ s Pareto-effcent f tere does not exst an alternatve feasble allocaton x suc tat. Allocaton x gves all consumers at least as muc utlty as xˆ.. Allocaton x gves at least one consumer more utlty tan xˆ. xˆ s Pareto-effcent f tere s no x tat can make someone better off wtout makng anyone worse off

37 .4.3 Effcency n an Excange Economy Two Teorems of Welfare Economcs Frst Teorem a compettve equlbrum s Pareto effcent Second Teorem any Pareto-effcent allocaton can be decentralzed as a compettve equlbrum encouragement of competton Edgewort box dagram a s not a Pareto-effcent allocaton b s Pareto-preferred to a

38 d c b a

39 Pareto-effcent allocaton c begnnng at c, any cange n te allocaton must lower te utlty of at least one of te consumers tangency between te two ndfference curves MRS =MRS Pareto-effcent allocaton s not unque ---> contract curve

40 e ω contract curve

41 compettve equlbrum a prce lne troug te ntal endowment pont ω tat s tangental to bot ndfference curves at te same pont compettve equlbrum = Pareto-effcent utlty maxmzaton ---> MRS =p /p compettve equlbrum MRS =p /p = MRS Teorem (Frst Teorem of Welfare Economcs) Te allocaton of commodtes at a compettve equlbrum s Pareto-effcent.

42 e' ω ω'

43 Second Teorem weter any cosen Pareto-effcent allocaton can be made nto CE by coosng sutable locaton for te ntal endowment possble f ndfference curves are convex a budget lne troug te Pareto-ef.allo. exsts consumers potmal coce decentralzaton Teorem (Second Teorem of Welfare Economcs) Wt convex preferences, any Pareto-effcent allocaton can be made a compettve equlbrum.

44 .4.4 Extenson to Producton producton ---> supply s varable consumer s ncome dffers Robnson Crusoe economy: same more tan one consumer utlty maxmzaton ---> MRS j =p /p j proft maxmzaton ---> MRT jm =p /p j compettve equlbrum effcency n consumpton MRS j =p /p j = MRS j effcency n producton MRT jm =p /p j = MRT jm MRS j = MRT j m --->Frst Teorem

45 Second Teorem W: feasble output plans w=y+ω Z: quanttes for Pareto-mprovement f bot are convex, a common tangent exsts total consumpton ˆ ˆ x = x + xˆ

46 W ˆx ˆx xˆ Z Prce lne Feasble set

47 .5 Lump-Sum Taxaton decentralzaton mecansm Second Teorem lump-sum transfers lump sum no cange n a consumer s beavor can affect te sze of te transfer dfferentated from ncome or commodty taxes redstrbutve nstrument

48 e ω' ω ˆx

49 ntal endowment ω Pareto effcent e equlbrum prces pˆ ncome at te ntal pont ˆ pω budget constrant M pˆ ω = px ˆ ˆ lump-sum taxes consumer pays a tax p ˆx ~ T = consumer pays a negatve tax ( T, T ) par of taxes = transfer of endowment effcency n acevng dstrbutonal objectves T pˆ ω = px ˆ ~ = T