Economics 201B–Second HalfLecture 10, 4/15/10, Revised 5/5/10Debreu’s Theorem on Determinacy of EquilibriumDefinition 1 Let F : RL−1+× RLI+→ RL−1be defined byF (ˆp, ω)=ˆz(ˆp) when the endowment is ωThe Equilibrium Price Correspondence E : RLI+→ RL−1++isdefined byE(ω)=ˆp ∈ RL−1++: F (ˆp, ω)=0 Proposition 2 The Equilibrium Price Cor respondence has closedgraph.Proof: A version of this is on Problem Set 5.Remark 3 If ωn→ ω, it follows that the aggregate endowment¯ωn→ ¯ω.If¯ω ∈ RL++, then an elaboration of the proof of theboundary condition on excess demand shows thatn∈NE(ωn)iscontained in a compact subset of RL−1++,soinfactE is upperhemi-continuous at every ω such that ¯ω ∈ RL++.Corollary 4 (Debreu) Fix 1,...,Iso thatDi(p, ω) is a C1function of p, ωiand aggregate excess demand satisfies the hypotheses of theDebreu-Gale-Kuhn-Nikaido Le mma. Then there is a closedset Ω⊂ RLI+of Lebesgue measure zero such that wheneverω0∈ RLI+\ Ω,• the economy with preferences 1,...,Iand endowmentω is regular, so E(ω) is finite and odd;1• if E(ω0)={ˆp∗1,...,ˆp∗N}, then there is an open set W con-taining ω0and C1functions h1,...,hNsuch that, for allω ∈ W ,E(ω)={h1(ω),...,hN(ω)}so E is upper hemicontinuous and lower hemicontinuousat ω.Proof:• Claim: For all ω  0 and price ˆp ∈ RL−1++,andforeachi,rank DωiF (ˆp, ω) ≥ L − 1Why? Letp =(ˆp, 1)Form an orthonormal basis V = {v1,...,vL} of RLsuch thatv1= p/|p|;thus,{v2,...,vL} will be an orthonormal basis ofthe hyperplaneH = {x ∈ RL: p · x =0}of all vectors perpendicular to p.LetEidenote the excessdemand of agent i.∗∗Ei(p, ω)=L =1(Ei(p, ω) · v ) v =L =2(Ei(p, ω) · v ) v since Ei(p, ω) · p = 0 by Walras’ Law. Changing ωito ωi+v ( =2,...,L) leaves the budget set unchanged, and henceleaves Di(p, ωi) unchanged, hence changes Ei(p, ω)by−v .2Then using the basis V for the domain and range,DωiEi(p, ω)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝00 00··· 00? −100··· 00?0−10··· 00.....................?0 00··· 0 −1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠The terms in first column other than the first entry come fromthe income effects in the Slutsky decomposition; we don’t needto determine them. Obviously,rank DωiEi(p, ω)=L − 1• The rank of the Jacobian matrix is independent of the basis,so when computed with respect to the standard basis,rank DωiEi(p, ω)=L − 1But in the standard basis, DωiF (ˆp, ω) consists of the first L−1rows of DωiEi(p, ω). By Walras’ Law, the last row of thematrix is a linear combination of the first L − 1 rows, sorank DωiF (ˆp, ω)=L − 1• Since the range of F is RL−1,L − 1 ≥ rank DF (ˆp, ω)≥ rank DωiF (ˆp, ω)≥ L − 1sorank DF(ˆp, ω)=L − 13• LetΩ=⎧⎨⎩ω ∈ RLI++: ∃ˆp∈RL−1++F (ˆp, ω)=0, det Dˆpˆz(ˆp, ω)=0⎫⎬⎭denote the set of endowments for which the resulting econ-omy is not regular. By the Transversality Theorem, ΩhasLebesgue measure zero. Suppose we are given a sequenceωn∈ Ωwith ωn→ ω ∈ RLI++.Chooseˆpn∈ E(ωn)suchthat det Dˆpnˆz(ˆpn,ωn) = 0. By Remark 3, there is a compactsubsetˆK of RL−1++such that ∪n∈NE(ωn) ⊂ˆK.Thus,wecanfind a subsequence ˆpnkconverging to ˆp ∈ RLI++.det Dˆpˆz(ˆp, ω) = limk→∞det Dˆpnkˆz(ˆpnk,ωnk)=0so Ωis relatively closed in RL−1++.• LetΩ=Ω∪RLI+\ RLI++RLI+\ RLI++is a set of Lebesgue measure zero, so Ωis set ofLebesgue measure zero. Clearly Ωis closed.• If ω0∈ Ω, the economy is regular, so E(ω0) is finite and odd.– LetE(ω0)={ˆp∗1,...,ˆp∗N}By the Implicit Function Theorem, there are open setsVn,Wnwith ˆp∗n∈ Vnand ω0∈ Wnand C1functionshn: Wn→ RL−1++such that for ω ∈ Wn,E(ω) ∩ Vn= {hn(ω)}– E is lower hemicontinuous at ω by the Transversality The-orem as we stated it. This also follows directly from theimplicit functions in the previous bullet.4– LetW0= W1∩···∩WN,V= V1∪···∪VNW0is open and ω0∈ W0.Forω ∈ W0,E(ω) ∩ V = {h1(ω),...,hN(ω)}– By Remark 3, E is upper hemicontinuous at ω0.– **Since E is upper hemicontinuous at ω0and E(ω0) ⊆ V ,thereisanopensetW1with ω0∈ W1such that for ω ∈ W1,E(ω) ⊆ V .Thus,forω ∈ W0∩ W1,E(ω)=E(ω) ∩ V = {h1(ω),...,hN(ω)}Limitations:• The assumption that demand is C1is strong, but fixable (Cheng,Mas-Colell).• Since the boundary of RLI+has Lebesgue measure zero, theformulation effectively assumesω ∈ RLI++– Terrible assumption, most agents are endowed with fewgoods.– Natural Conjecture: You can set certain endowments =0and, as long as you have enough degrees of freedom in thenonzero endowments, Debreu’s Theorem still holds. False:example due to Minehart.– Solution: Perturb preferences as well as endowments. Needgenericity notion on infinite-dimensional spaces. Debreu’s5Theorem holds generically in a topological notion of gener-icity (Mas-Colell) and a measure-theoretic notion of gener-icity (Anderson & Zame).• For Finance, commodity differentiation, choice under uncer-tainty, need version of theorem for infinite-dimensional com-modity spaces. Shannon and Zame showed that close ana-logue to Debreu’s Theorem holds. The consumption set oftenhas empty interior in these infinite-dimensional settings, so dif-ferentiability is problematic; Shannon and Zame find that thefunctions defining the movement of the equilibrium prices areLipschitz.Quick Romp Through 17.E,F,H• 17.ETheorem 5 (Sonnenschein-Mantel-Debreu) Let K bea compact subset of Δ0. Given f : K → RLsatisfying– continuity– Walras’ Law with Equality (p · f(p)=0)there is an exchange economy with L consumers whose ex-cess demand function, restricted to K,equalsf.Proof: Elementary, but far from transparent. Individual pref-erences may be made arbitrarily nice.Corollary 6 There are no comparative statics re sults forWalrasian Equilibrium in the Arrow-Debreu model; moreassumptions are needed.6• 17.F, Uniqueness:There are no results known under believeable assumptions onindividual preferences.• 17.H, Tatonnement Stability:dˆpdt=ˆz(ˆp)onRL−1++dpdt= E(p)onΔ02=p ∈ RL++: p2=1 We would like to know that the solutions converge to the equi-librium price. Scarf gave an example of a non-pathologicalexchange economy in which the solutions all circle around