A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 1/45 GR5211 Microeconomic Theory - General Equilibrium A General Exchange Economy Wouter Vergote Columbia University 2020–2021 Djm a A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 2/45 (Varian 17 and MWG 16) Roadmap 1 A Pure Exchange Economy 2 Excess Demand 3 Existence 4 Welfare A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 3/45 A General Exchange Economy (Varian 17 and MWG 16) Roadmap 1 A Pure Exchange Economy 2 Excess Demand 3 Existence 4 Welfare A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 4/45 A Pure Exchange Economy Consider a model of a decentralized economy E with the following ingredients: Commodities: l = 1, ..., L Consumers: i = 1, ..., I have preferences (
utility function ui which maps consumption bundles xi = (x1i, ...., xLi) 2 Xi into R hold an initial endowment vector li 2 RL+ Let (l1, ....,lI) and lˆ = Õi li 2 RL+ ⑤ of the L goods . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 5/45 A pure Exchange Economy Walrasian Equilibrium Definition AWalrasian Equilibrium for the economy E is an allocation (x⇤1 , ..., x⇤I ) and a price vector p⇤ 2 RL++ i: 1 (Utility Maximization): x⇤i = argmaxui (xi); s.t. p⇤xi = p⇤li, 8i = 1, ..., I 2 (Market Clearing):’ i x⇤il 6 ’ i lil, 8l = 1, ..., L D im Iynmnassmun.ms 'out Tony;.de . Toma.YET ⇒ Yj; - A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 6/45 A Pure Exchange Economy Generalization Remember we assumed that each consumer i holds an initial endowment vector li 2 RL+: li = ©≠≠≠≠≠´ li1 . . . liL ™ÆÆÆÆƨ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 7/45 A Pure Exchange Economy Generalization Then l = (l1, ....,lI) is the distribution of endowments The total endowment of each commodity l is, for all l 2 {1, ..., L}: lˆl = ’ i lil 2 R+ and lˆ = Õ i li 2 RL+ = ©≠≠≠≠≠´ Õ i li1 . . .Õ i liL ™ÆÆÆÆƨ = ©≠≠≠≠≠´ lˆ1 . . . lˆL ™ÆÆÆÆƨ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 8/45 A Pure Exchange Economy Generalization For any price vector p = (p1, ..., pL) consumer i has a Marshallian demand: xi (p, p ·li) 2 RL+ xi (p, p ·li) = ©≠≠≠≠≠´ xi1(p, p ·li) . . . xiL(p, p ·li) ™ÆÆÆÆƨ Damon .mn.mu . - optimizing bundles . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 9/45 A General Exchange Economy (Varian 17 and MWG 16) Roadmap 1 A Pure Exchange Economy 2 Excess Demand 3 Existence 4 Welfare A FIFTY, FT math and convex , then the point will be Pareto optimal 2) Walrus Law- tf at Mekka dear , then the remaining markets clears A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 10/45 A Pure Exchange Economy Excess demand This allows us to write down the excess demand vector, zi (p) = xi (p, p ·li) li 2 RL+, for each consumer i: zi (p) = ©≠≠≠≠≠´ xi1(p, p ·li) li1 . . . xiL(p, p ·li) liL ™ÆÆÆÆƨ A ° ÷:÷÷÷÷:÷÷÷÷:* A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 11/45 A Pure Exchange Economy Excess Demand This aggregate excess demand vector, z (p) is obtained by summing zi (p) over consumer i: zi (p) = xi (p, p ·li) li 2 RL+ z (p) = ’ i zi (p) = ©≠≠≠≠≠´ Õ i xi1(p, p ·li) lˆ1 . . .Õ i xiL(p, p ·li) lˆL ™ÆÆÆÆƨ A - " in:: tinnitus . then antheridium's = dismal FI: :It is amitabh in the economy A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 12/45 A Pure Exchange Economy Walrasian Equilibrium In this economy we can define a Walrasian equilibrium by means of the vector of aggregate excess demands in the following manner: Definition The Walrasian Equilibrium of a pure exchange economy is defined by a vector of prices p⇤ and an induced feasible allocation x⇤ = (x⇤1 , ..., x⇤I ) such that all markets clear; if and only if: z (p⇤) 6 0 B O agg excess demand ↳ opium demand A impunity defined ahem my , is dymdWe got rid of utility maximany here because of the A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 13/45 A Pure Exchange Economy Generalization Assumptions: ui ( · ) is continuous and increasing li 0 for all i Consequences: z ( · ) is continuous (sum of continuous functions) zi ( · ) is homogeneous of degree zero in prices xi (p, p ·li) is homogeneous of degree zero in prices endowments are constant z (p) ⌘ Õi zi (p) is homogeneous of degree zero. ⑤ All agents have smithy positive endowments of ency good A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 14/45 A Pure Exchange Economy Walras’ Law Lemma Walras’ Law: For any price vectore p it must be that p · z (p) = 0 Proof: this result follows from binding budget constraints: p · z (p) = p ’ i xi (p, p ·li) ’ i li ! = ’ i (p · xi (p, p ·li) p ·li) = 0 since xi (p, p ·li) satisfies the budget constraint p · xi (p, p ·li) = p ·li for each agent i. A Valueofexussdlmandiszno " ° µ=pwibwyetm" " * A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 15/45 A Pure Exchange Economy Equilibrium Prices are non-negative Lemma If p⇤ is a walrasian equilibrium price vector then p⇤l > 0 for all goods l : 1, ..., L. Proof: if not, there is no solution to the utility maximization problem. A utility inmates in consumption hymns negatives consumer can choose an ignite amount A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 16/45 A Pure Exchange Economy Free Goods Lemma Free goods: if p⇤ is a walrasian equilibrium price vector and zl (p⇤) < 0 then good l must be a free good: p⇤l = 0. Proof: p⇤ is a walrasian equilibrium price vector, hence z (p⇤) 6 0. Since prices are nonnegative: p⇤ · z (p⇤) 6 0. If zl (p⇤) < 0 and p⇤l > 0 it follows p⇤l · zl (p⇤) < 0 , a contradiction. Why? A shroud =o by Walras Law ÷÷¥÷÷¥÷¥ ÷÷÷÷÷¥¥÷÷÷÷÷) A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 17/45 A Pure Exchange Economy Free Goods Lemma Free goods: if p⇤ is a walrasian equilibrium price vector and zl (p⇤) < 0 then good l must be a free good: p⇤l = 0. Proof: p⇤ is a walrasian equilibrium price vector, hence z (p⇤) 6 0. Since prices are nonnegative: p⇤ · z (p⇤) 6 0. If zl (p⇤) < 0 and p⇤l > 0 it follows p⇤l · zl (p⇤) < 0 , a contradiction. Why? Walras’ Law A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 18/45 A Pure Exchange Economy Desirable Goods Definition A good l is desirable when the aggregate excess demand is strictly positive when its price equals zero. Assumption: Suppose that all goods are desirable pl = 0) zl (p) > 0 for all l : 1, ..., L. A Hpe -- o ⇒ Zelp) 20 A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 19/45 A Pure Exchange Economy Walrasian Equilibrium Lemma If p⇤ is a walrasian equilibrium price vector and all goods are desirable, then z (p⇤) = 0. Proof: Suppose that zl (p⇤) < 0 then l is a free good and p⇤l = 0. But since l is desirable, then zl (p⇤) > 0. A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 20/45 A Pure Exchange Economy Walrasian Equilibrium Lemma If p⇤ is a walrasian equilibrium price vector and p⇤l > 0, then zl (p⇤) = 0. Proof: This follows from Walras’ law and weakly positive prices p⇤: p⇤ · z (p⇤) 6 0. Since equilibrium prices are nonnegative for all commodities and p⇤ · z (p⇤) 6 0, it follows that If zl (p⇤) = 0 when p⇤l > 0. Finding W.E. means solving z (p) = 0 A MY" """ " A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 21/45 A General Exchange Economy (Varian 17 and MWG 16) Roadmap 1 A Pure Exchange Economy 2 Excess Demand 3 Existence 4 Welfare A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 22/45 A Pure Exchange Economy Existence Existence: is there a p⇤ that clears all markets? Assumptions: 1 z (p) is single valued 2 z (p) is continuous 3 z (p) is bounded 4 z (p) is homogenous of degree 0. 5 Walras’ Law: pz (p) = 0 A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 23/45 A Pure Exchange Economy Existence Theorem Suppose z (p) satisfies assumptions 1-5, then there exists a vector of prices p⇤ and an induced allocation x⇤ = (x⇤1 , ..., x⇤I ) such that all markets clear: z (p⇤) 6 0 A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 24/45 A Pure Exchange Economy Existence Proof: Normalize prices: denote by 4 the unit simplex in RL+: 4 = ( p 2 RL+ : L’ l=1 pl = 1 ) Note that 4 is a closed, bounded and convex set. Define on 4 the function g(p) : 4 ! 4 by, for all l = 1, ..., L: gl (p) = pl + max (0, zl (p)) 1 + ÕLl=1 max (0, zl (p)) for all l = 1, ..., L. A ✓ divide efhtpwnj.nu, 'm the ✓ mapping puns fixed Points . good e → kind Zecp) ¥:*: : : ::::: ⇒ yelps = ¥k E Pe . Making good less expensive . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 25/45 A Pure Exchange Economy Existence g(p) is continuous (why?) and belongs to 4 (why?). economic interpretation: this mapping tend to increase the price of commodities in excess demand A ( assumptive ' Igel p) -- I A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 26/45 A Pure Exchange Economy Existence Definition (Fixed Point) Consider a mapping F : S ! S, any x such that x = F (x) is a fixed point of the mapping F . Theorem (Brouwer Fixed Point Theorem) Let S be a compact and convex set, and F : S ! S a continuous mapping from S into itself. Then F has at least one fixed point in S. A ✓ f '-9-79 here . Tp s- tglpp A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 27/45 A Pure Exchange Economy Existence Proof: Proof Continued Hence g(p) has a fixed point p⇤ such that gl (p⇤) = p⇤l + max (0, zl (p⇤)) 1 + ÕLl=1 max (0, zl (p⇤)) = p⇤l for all l = 1, ..., L. We show that p⇤ is a W.E. A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 28/45 A Pure Exchange Economy Existence Proof: Proof Continued Suppose not, then there exists good l such that zl (p⇤) = Õi xil (p⇤, p⇤ ·l) Õi lil > 0 and p⇤l > 0 since if p ⇤ l = 0 then gl (p⇤) > p⇤l = 0, a contradictionÕL l=1 max (0, zl (p⇤)) > zl (p⇤) > 0. Suppose there are K such goods for which zk (p⇤) > 0 and p⇤k > 0. We then have for good l that: p⇤l + zl (p⇤) 1 + ÕKk=1 zk (p⇤) = p⇤l zl (p⇤) = p⇤l K’ k=1 zk (p⇤) AD if we *neo - gecp*s=ot¥Y > Og to ⇐ gap =p . . Frothy positive excess dead 1) Multiply Pe ' on both sites c) Sum over K . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 29/45 A Pure Exchange Economy Existence Proof: Proof Continued It must be the case that p⇤j = 0 and zj (p⇤)p⇤j = 0 for all j such that zj (p⇤) 6 0. Why? Now consider zl (p⇤)p⇤l over all l : 1, .., L we have: K’ k=1 zk (p⇤)p⇤k + L\K’ j=1 zj (p⇤)p⇤j = K’ k=1 zk (p⇤) ( K’ k=1 p⇤k 2) L’ l=1 zl (p⇤)p⇤l = K’ k=1 zk (p⇤) ( K’ k=1 p⇤k 2) > 0 contradicting Walras’ law: ÕL l=1 zl (p⇤)p⇤l = 0. A warm 0 multiply by pet ⑤ 70 & ¥Cp* ) E O A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 30/45 A Pure Exchange Economy Existence Some Remarks This result is fairly general: hinges on continuity of aggregate excess demand One important caveat: how are we guaranteed that z (p) is defined for all prices in the simplex? this assumption can be relaxed and then the proof relies on a slightly more technical argument (Kakutani’s fixed point theorem - MWG Proposition 17.C.1) A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 31/45 A General Exchange Economy (Varian 17 and MWG 16) Roadmap 1 A Pure Exchange Economy 2 Excess Demand 3 Existence 4 Welfare A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 32/45 A Pure Exchange Economy Pareto Eiciency Definition A feasible allocation (x1, ..., xI) isWeakly Pareto Optimal if there does not exist another feasible allocation (x 01, ..., x 0 I ) preferred by all agents i : 1, ..., I. Definition A feasible allocation (x1, ..., xI) is Strongly Pareto Optimal if there does not exist another feasible allocation (x 01, ..., x 0 I ) such that all agents weakly prefer x 0 over x , and some agent stricly prefers x 0 over x . A story ⇒Weak A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 33/45 A Pure Exchange Economy Pareto Eiciency Lemma When preferences are continuous and strongly monotonic, then weak and strong Pareto optimality are equivalent Proof. We just need to show that weakly pareto optimal allocations are strongly pareto optimal. Suppose an allocation x is not strongly Pareto optimal. Then consider feasible allocation x 0 which strongly pareto dominates x . ⇤ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 34/45 A Pure Exchange Economy Pareto Eiciency Proof. Since preferences are continous we can scale down the consumption bundle of agent i slightly to (1 Y)x 0i for small Y > 0 such that she prefers (1 Y)x 0i over xi . Then add Yx 0 i I 1 to the allocations of the other over x . Hence x is not weakly Pareto optimal. ⇤ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 35/45 A Pure Exchange Economy Welfare For the welfare theorems we will assume that agents have strongly monotonic and continous preferences. Under these assumptions and ignoring free goods we can restate the definition of a Walrasian Equilibrium in the following way: Definition The Walrasian Equilibrium of a pure exchange economy is defined by a vector of prices p⇤ and an induced feasible allocation x⇤ = (x⇤1 , ..., x⇤I ) such that all markets clear; i: 1 Õ i x⇤i (p⇤, p⇤ ·li) 6 Õ i li 2 If xi is preferred by agent i to x⇤i then’ l p⇤l xil > ’ l p⇤l x ⇤ il (p⇤, p⇤ ·li) = ’ l p⇤llil A 0 TohY 't > everything -armlet h f god l A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 36/45 A Pure Exchange Economy 1 WT Theorem Suppose that (x⇤, p⇤) is a Walrasian Equilibrium of a pure exchange economy, then x⇤ = (x⇤1 , ..., x⇤I ) is Pareto Eicient Proof. Suppose not, then there exists a feasible allocation x = (x1, ..., xI) such that: p⇤xi > p⇤x⇤i (p⇤, p⇤ ·li) = p⇤l⇤i for some agent i (xi preferred to x⇤i ), and p⇤xj > p⇤x⇤j (p⇤, p⇤ ·lj) = p⇤l⇤j for all j < i. ⇤ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 37/45 A Pure Exchange Economy 1st WT Proof. Summing over all agents and using feasibility of allocation x we obtain: p⇤ Õ i li = p⇤ Õ i xi > p⇤ Õ i li a contradiction ⇤ A A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 38/45 A Pure Exchange Economy 2nd WT Second Welfare Theorem Theorem Suppose that x⇤ is a Pareto Eicient allocation in which each agents holds a positive amount of each commodity (l 0) and that the preferences of all agents are continous, convex and monotonic. Then x⇤ is a Walrasian Equilibrium for the initial endowments l⇤ = x⇤ = (x⇤1 , ..., x⇤I ). A *** t÷÷÷ """ A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 39/45 A Pure Exchange Economy 2nd WT - Proof Proof. Let Bi = xi 2 RL : xi i x⇤i be the beer than set of allocation x⇤ for agent i. Then define the set B of allocations that make all agents beer o compared to x⇤: B = ’ i Bi = ( v : v = I’ i=1 xi with xi 2 Bi ) Note that B is convex. Define l = ÕI i=1 x ⇤ i . It must be that l 8 B. (why?). ⇤ A → " iii: swftetmeffmageIIITeu.us Xi. # Bi fits ⇒ WEB . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 40/45 A Pure Exchange Economy 2nd WT - Proof Proof. By the separating hyperplane theorem there exists a p < 0 such that pv > pl Rearranging we have: p(v ’ i x⇤i ) > 0 ⇤ A W - - ? Xi * A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 41/45 A Pure Exchange Economy 2nd WT - Proof Proof. OK, but is p an equilibrium price vector? Needs to be non-negative, and If yj 2 Bj then pyj > px⇤j . ⇤ ÷ A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 42/45 A Pure Exchange Economy 2nd WT - Proof Proof. Showing non-negativity of p is the easy part due to monotonicity: Let ]l 2 RL be a unit vector in RL such that the l th component is equal to 1. Then consider for all l the allocation l + ]l . Then l + ]l 2 B (why?) hence we write: p(l + ]l l) > 0 for all l : 1, ..., L. We then have that pl > 0 for all l : 1, ..., L. ⇤ µ By monotourney pinnies: in i. w ; ( h . . - - . R) (! )→eth component A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 43/45 A Pure Exchange Economy 2nd WT - Proof Proof. If yj 2 Bj then pyj > px⇤j . Follows from strong monotonicity (see in class) ⇤ A better then x . ✓ then agutjcanuetaffd . Step 1) Suppose yjE$j and pyjcpxj * . → show pyjzpxj ' 2) Show Pyj > Pij ① Suppose Pyjckj . Shaw off and nedurwhte Construct a new allocation z ( stoutly preferred . Zj - CI -f) Yj . where 0 > o , CI - o )yj > x'j . by ① Continuity ② Monotounity . ¥¥÷. - la . . - * . . + %. "s . . p¥Ezi7p÷Exi" ( by separating hyperplane thrown) - plci-osyj.li#jxi*+CI-yFfT, ] Z p[xj* t Ej Xi '' ] . plyj t #jxi " ] Z pfxj 't i¥ixi* ) . pyj z pxj . • ② Suppose pyj = page . Since yj > Xi ' , ta s - t ayjsxjt.ae co , 17 . From step I , pay; Z pxj " and we know Kj 's> 0 , pxj ' ZO 'ht - am :p ;÷÷¥i: autumn . ⇒ play; - Xj . ) co payjspxj " . ⇒ So pyj > pxj " must be true . A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 44/45 A Pure Exchange Economy Second Welfare Theorem Some Remarks Crucial assumptions: convex and continous preferences The assumption of strong monotonicity can be relaxed (local non-satiation) The assumption strictly positive endowments can also be relaxed. Of course, any relaxiation comes at the expense of more technical arguments (e.g. concept of quasi-equilibrium in MWG) A y when preferences are " " " B " °" " A General Exchange Economy A Pure Exchange Economy Excess Demand Existence Welfare Ch.4 – 45/45 A Pure Exchange Economy Generalization Conclusion Pure-exchange model Focusing on economies where goods cannot be produced and agents consume and exchange their endowments Many results (existence of equilibrium, optimality, etc.) can be well understood only by focusing on exchange. Other important issues Uniqueness: How many Walrasian equilibria are there? Stability: How does an economy (as distinct from an economist) “find” equilibria? Empirical validation: Can we take the Walrasian model to the data? What are empirically-testable implication of GET? The road ahead Equilibrium with production: some simple examples Addressing uniqueness, stability, and empirical validation (using the pure-exchange model again) A 欢迎咨询51作业君
