1. General Equilibrium

Advanced Microeconomics II Ning Sun Shanghai University of Finance & Economics School of Economics March, 2010 N Sun (SHUFE) General Equilibrium March, 2010 1 / 38 Lecture Note 1. General Equilibrium Ning Sun Shanghai University of Finance & Economics School of Economics March 4, 2010 N Sun (SHUFE) General Equilibrium March 4, 2010 2 / 38 5.1 Equilibrium in Exchange Pure exchange economies with two consumers and two goods Each consumer i(= 1, 2) has a utility function ui and an endowment ei = (ei1, e i 2) ∈ Rn++. Edgeworth box Allocation, Pareto efficient allocation, contract curve, core allocation, competitive equilibrium N Sun (SHUFE) General Equilibrium March 4, 2010 3 / 38 Pure Exchange Economies A pure exchange economy is an economy with no production. A general pure exchange economy with I consumers and n goods. Let I = {1, · · · , I} denote the set of consumers. Each consumer i ∈ I has a preference relation �i on his consumption set Xi = Rn+, and an endowment ei = (ei1, · · · , ein) ∈ Rn+. A pure exchange economy can be represented by E = (�i, ei)i∈I . N Sun (SHUFE) General Equilibrium March 4, 2010 4 / 38 Let e := (e1, · · · , en) denote the economy’s endowment vector. Let e¯ := ∑ i∈I e i denotes the social endowment of the economy. An allocation in this economy is x := (x1, · · · , xn), where xi ∈ Rn+ denotes consumer i′s consumption bundle according to the allocation. The set of feasible allocation in this economy is given by F (e) := { x | ∑i∈I xi = e¯}. Definition 5.1 (Weakly) Pareto-Efficient Allocation (i) A feasible allocation x ∈ F (e), is Pareto efficient if there is no other feasible allocation y ∈ F (e), such that yi�i xi for all consumer i ∈ I, with at least one preference strict. (ii) A feasible allocation x ∈ F (e), is weakly Pareto efficient if there is no other feasible allocation y ∈ F (e), such that yi�i xi for all consumer i ∈ I. N Sun (SHUFE) General Equilibrium March 4, 2010 5 / 38 Definition 5.2 Blocking Coalition Let S ⊂ I denote a coalition of consumers. We say that S blocks x ∈ F (e) if there is an allocation y such that (i) ∑ i∈S y i = ∑ i∈S e i. (ii) yi�i xi for all i ∈ S, with at least one preference strict. Definition 5.3 The Core of an Exchange Economy The core of an exchange economy with endowment e, denoted by C(e), is the set of all unblocked feasible allocations. Question: Does there always exist a feasible allocation in the core? N Sun (SHUFE) General Equilibrium March 4, 2010 6 / 38 5.2 Equilibrium in Competitive Market Systems We consider a perfectly competitive market system. The properties of a perfectly competitive market: (i) Decentralized nature (prevailing prices, price taker) (ii) Interdependence N Sun (SHUFE) General Equilibrium March 4, 2010 7 / 38 5.2.1 Existence of Equilibrium in an Exchange Economy Assumption 5.1 Consumer Utility Utility ui is continuous, strongly increasing, and strictly quasi-concave on Rn+. On competitive markets, each consumer takes prices as given. If p� 0 is the vector of market prices, then each consumer solves maxxi∈Rn+ u i(xi) (5.2) s.t. p · xi ≤ p · ei. The solution xi(p, p · ei) to (5.2) is consumer i′s demanded bundle, which depends on market prices and the consumer’s endowment income. N Sun (SHUFE) General Equilibrium March 4, 2010 8 / 38 The Berge Theorem (Theorem of Maximum) Consider the canonical optimization problem maxx f(x, y) (CSy) s.t. x ∈ Γ(y), in which x ∈ X ⊆ Rl and y ∈ Y ⊆ Rk. Assume that (A1) f is a continuous function on X × Y ; (A2) Γ is a (compact-valued) continuous correspondence from Y to X (which means Γ is both u.h.c. and l.h.c.) The Berge Theorem: Assume (A1) and (A2) hold for (CSy). Then The value function V (y) is continuous in y; The optimal solution correspondence G(y) is u.h.c. in y. N Sun (SHUFE) General Equilibrium March 4, 2010 9 / 38 Theorem 5.1 Basic Properties of Demand If ui satisfies Assumption 5.1, then for each p� 0, the consumer’s problem (5.2) has a unique solution xi(p, p · ei). In addition, xi(p, p · ei) is continuous in p on Rn++. Definition 5.4 Excess demand The aggregate excess demand function for good k is the real-valued function, zk(p) := ∑ i∈I xik(p, p · ei)− ∑ i∈I eik. The aggregate excess demand function is the vector-valued function, z(p) := (z1(p), · · · , zn(p)). When zk(p) > (or,<)0, there is excess demand (or, supply) for good k. N Sun (SHUFE) General Equilibrium March 4, 2010 10 / 38 Theorem 5.2 Properties of Aggregate Excess Demand Function If for each consumer i, ui satisfies Assumption 5.1, then for all p� 0, (1) Continuity: z(·) is continuous at p; (2) Homogeneity: z(λp) = z(p) for all λ > 0; (3) Walras’ law: p · z(p) = 0. Definition 5.5 Walrasian Equilibrium A vector p∗ ∈ Rn++ is called a Walrasian equilibrium (price vector), whenever z(p∗) = 0. References on general equilibrium theory: McKenzie (1954), Arrow and Debreu (1954), Negishi (1961), Debreu and Scarf (1963), Gale and Mas-Colell (1975). N Sun (SHUFE) General Equilibrium March 4, 2010 11 / 38 Theorem 5.3 Aggregate Excess Demand Function and Equilibrium Suppose z(p) satisfies the following three conditions. (1) z(·) is continuous on Rn++ (2) p · z(p) = 0 for all p� 0. (3) If {pm} is a sequence of price vectors in Rn++ converging to p¯ 6= 0, and p¯k = 0 for some good k, then for some good k′ with p¯k′ = 0, the associated sequence of excess demands in the market for good k′, zk′(pm), is unbounded above. Then there is a price vector p∗ � 0 such that z(p∗) = 0. N Sun (SHUFE) General Equilibrium March 4, 2010 12 / 38 Proof of Theorem 5.3: For each good k, let z¯k(p) = min(zk(p), 1) for all p� 0, and let z¯(p) = (z¯1(p), · · · , z¯n(p)). Thus, for all p� 0, it holds p · z¯(p) ≤ p · z(p) = 0. (P.1) Fix any � ∈ (0, 1), and let S� = { p | ∑n k=1 pk = 1 and pk ≥ �1 + 2n for all k } . For every fixed � ∈ (0, 1), S� is compact, convex, and nonempty. For each good k and every p ∈ S�, define fk(p) as follows: fk(p) = �+ pk +max(z¯k(p), 0) n�+ 1 + ∑n m=1max(z¯m(p), 0) , and let f(p) = (f1(p), · · · , fn(p)). It is easy to check that f is a vector-valued function form S� to S�. N Sun (SHUFE) General Equilibrium March 4, 2010 13 / 38 Note that each fk is continuous on S� because by condition (1). Therefore, f is a continuous function mapping S� into itself. Thus, by Brouwer’s fixed point theorem, we see that for every � ∈ (0, 1) there exists p� ∈ S� such that f(p�) = p�. But this means that for each k it holds p�k [ n�+ ∑n m=1 max(z¯m(p�), 0) ] = �+max(z¯k(p�), 0). (P.2) WLOG, suppose p� → p∗ as �→ 0. Of course, p∗ ≥ 0 and p∗ 6= 0. Moreover, by condition (3), we can show that p∗ � 0. Thus, p� → p∗ � 0 as �→ 0. Because z¯(·) inherits continuity on Rn++ from z(·), we may take the limit as �→ 0 in (P.2) to obtain p∗k ∑n m=1 max(z¯m(p∗), 0) = max(z¯k(p∗), 0) (P.3) for all k = 1, · · · , n. N Sun (SHUFE) General Equilibrium March 4, 2010 14 / 38 Multiplying both sides by z¯k(p∗) and summing over k yields p∗ · z¯(p∗) (∑n m=1 max(z¯m(p∗), 0) ) = ∑n k=1 z¯k(p∗)max(z¯k(p∗), 0) By (P.1), we have p∗ · z¯(p∗) ≤ 0. Therefore, the left-hand side of the preceding equation is non-positive, and so the right-hand side must be non-positive as well. This implies that z¯k(p∗) ≤ 0 for all k. Note zk(p∗) = min(zk(p∗), 1) = z¯k(p∗) when z¯k(p∗) ≤ 0 . Consequently, zk(p∗) ≤ 0 for all k = 1, · · · , n. Therefore, we have z(p∗) ≤ 0, and p∗ � 0. But condition (2) (Walras’s law) states that p∗ · z(p∗) = 0. Consequently, every component of z(p∗) must be zero. That is, z(p∗) = 0. Q.E.D. Brouwer’s Fixed Point Theorem: Suppose that X is a compact and convex set in some Rl and f is a continuous function from X to X. Then there is some x∗ ∈ X with f(x∗) = x∗. N Sun (SHUFE) General Equilibrium March 4, 2010 15 / 38 Theorem 5.4 Utility and Aggregate Excess Demand If each consumer’s utility function satisfies Assumption 5.1, and if the aggregate endowment of each good is strictly positive (i.e., e¯ = ∑ i∈I e i � 0), then the aggregate excess demand function satisfies conditions 1 through 3 of Theorem 5.3. Proof: Conditions (1) and (2) follow from Theorem 5.2. Thus, it remains only to verify condition (3). Consider a sequence of strictly positive price vectors {pm}, converging to p¯ 6= 0 with p¯k = 0 for some good k. Because e¯ = ∑ i∈I e i � 0, there must be at least one consumer i for whom p¯ · ei > 0. Consider this consumer i′s demand xi(pm, pm · ei), along the sequence of prices. Now, assume by way of contradiction that this sequence {xi(pm, pm · ei)} is bounded. Then, there must be a convergent subsequence. So we may WLOG assume that xi(pm, pm · ei)→ x∗. N Sun (SHUFE) General Equilibrium March 4, 2010 16 / 38 To simplify the notation, let xm = xi(pm, pm · ei) for every m. Now, because xm maximize ui subject to i′s budget constraint given by prices pm, and because ui is strongly increasing, it must hold that pm · xm = pm · ei for every m. Taking the limit as m→∞ yields p¯ · x∗ = p¯ · ei > 0. (P.1) Let xˆ = x∗ + ek. Then because ui is strongly increasing on Rn+, ui(xˆ) > ui(x∗). Note that p¯ · xˆ = p¯ · ei > 0 because p¯k = 0. By the continuity of ui, there is a t ∈ (0, 1) such that ui(txˆ) > ui(x∗) but p¯ · (txˆ) < p¯ · ei. But since pm → p¯, xm → x∗ and ui is continuous, this implies that for m large enough, ui(txˆ) > ui(xm) but pm · (txˆ) < pm · ei, contradicting the fact that xm is the optimal solution of the consumer’s problem at prices pm. Consequently, consumer i′s sequence of demand vectors {xm} must be unbounded. N Sun (SHUFE) General Equilibrium March 4, 2010 17 / 38 It follows from the above result that there must be some good k′ such that {xmk′} is unbounded. But because i′s income converges to p¯ · ei, the sequence of i′s income {p · ei} is bounded. Therefore, we must have pmk′ → p¯k′ = 0, because this is the only way that the demand for good k′ can be unbounded. Finally, because the aggregate supply of good k′ is fixed and equal to e¯k′ , and because all consumers demand a nonnegative amount of good k′, the aggregate excess demand {zk′(pm)} is unbounded, as desired. Q.E.D. Theorem 5.5 Existence of walrasian Equilibrium If each consumer’s utility function satisfies Assumption 5.1, and ∑ i∈I e i � 0, then there exists at least one price vector p∗ � 0, such that z(p∗) = 0. N Sun (SHUFE) General Equilibrium March 4, 2010 18 / 38 5.2.2 Efficiency Definition 5.6 Walrasian Equilibrium Allocations (WEAs) Let p∗ be a Walrasian equilibrium (price vector) for some economy with initial endowment e, and let x(p∗) := ( x1(p∗, p∗ · e1), · · · , xI(p∗, p∗ · eI)), where component i gives the n-vector of goods demanded and received by consumer i at prices p∗. Then x(p∗) is called a Walrasian equilibrium allocation, or WEA. Lemma 5.1 Let p∗ be a Walrasian equilibrium for some economy with initial endowment e. Let x(p∗) be the associated WEA. Then x(p∗) ∈ F (e). N Sun (SHUFE) General Equilibrium March 4, 2010 19 / 38 Lemma 5.2 Suppose that ui is continuous and strictly increasing on Rn+, that consumer i′s demand is well-defined at p ≥ 0 and equal to xˆi, and that xi ∈ Rn+. (i) If ui(xi) > ui(xˆi), then p · xi > p · xˆi. (ii) If ui(xi) ≥ ui(xˆi), then p · xi ≥ p · xˆi. Definition 5.7 The Set of WEAs For any economy with endowment e, let W (e) denote the set of Walrasian equilibrium allocations. N Sun (SHUFE) General Equilibrium March 4, 2010 20 / 38 Theorem 5.6 Core and Equilibria in Competitive Economies Consider an exchange economy E = (�i, ei)i∈I . If each consumer’s utility function ui is continuous and strictly increasing on Rn+, then every Walrasian equilibrium allocation is in the core. That is, W (e) ⊂ C(e). Proof: The theorem claims that if x(p∗) is a WEA for equilibrium prices p∗, then x(p∗) ∈ C(e). To prove it, suppose x(p∗) is a WEA, but x(p∗) /∈ C(e). At first, by the assumption x(p∗) /∈ C(e), there is some coalition S and another allocation y such that∑ i∈S y i = ∑ i∈S e i, and (P.1) ui(yi) ≥ ui(xi) for all i ∈ S, (P.2) with at least one inequality strict. (P.1) implies p∗ · ∑ i∈S yi = p∗ · ∑ i∈S ei. (P.3) N Sun (SHUFE) General Equilibrium March 4, 2010 21 / 38 But form (P.2) and Lemma 5.2, we know that for each i ∈ S, it must hold p∗ · yi ≥ p∗ · xi(p∗, p∗ · ei) = p∗ · ei, with at least one inequality strict. Summing over all consumers in S, we obtain p∗ · ∑ i∈S yi > p∗ · ∑ i∈S ei, contradicting (P.3). Thus, x(p∗) ∈ C(e), and the theorem is proved. Q.E.D. Theorem 5.7 First Welfare Theorem Under the hypotheses of Theorem 5.6, every Walrasian equilibrium allocation is Pareto efficient. Proof: The proof follows immediately from Theorem 5.6 and the observation that all core allocation are Pareto efficient. Q.E.D. N Sun (SHUFE) General Equilibrium March 4, 2010 22 / 38 Theorem 5.8 Second Welfare Theorem Consider an exchange economy E = (�i, ei)i∈I with aggregate endowment∑ i∈I e i � 0, and with each utility function ui satisfies Assumption 5.1. Suppose that x¯ is a Pareto-efficient allocation for E , and that endowments are redistributed so that the new endowment vector is x¯. Then x¯ is a competitive equilibrium allocation of the resulting exchange economy (�i, x¯i)i∈I . Proof: Corollary 5.1 Another Look at the Second Welfare Theorem Under the assumptions of the preceding theorem, if x¯ is Pareto-efficient, x¯ is a WEA for some Walrasian equilibrium prices p¯ after redistribution of initial endowments to any allocation e∗ ∈ F (e), such that p¯ · e∗i = p¯ · x¯i. N Sun (SHUFE) General Equilibrium March 4, 2010 23 / 38 5.3 Equilibrium in Production 5.3.1 Producers There is a fixed number J of firms indexed by the set J = {1, · · · , J}. Each firm j ∈ J possesses a production set Y j . Assumption 5.2 The Individual Firm (1) 0 ∈ Y j ⊂ Rn; (2) Y j ∩Rn+ = {0}, (and −Rn+ ⊂ Y j , free disposal); (3) Y j is closed and (bounded, but need not); (4) Y j is strongly convex. That is, for all distinct y1, y2 ∈ Y j and all t ∈ (0, 1), there exists y¯ ∈ Y j such that y¯ ≥ ty1+(1− t)y2 and equality does not hold. N Sun (SHUFE) General Equilibrium March 4, 2010 24 / 38 Profit Maximization On competitive markets, each producer also takes prices as given. Facing market prices p ≥ 0, each producer chooses a production plan in his production set Y j to maximize his profit. Thus, given a price vector p, each firm solves the problem max p · yj (5.3) s.t. yj ∈ Y j . Let, pij(p) := maxyj∈Y j p · yj denote firm j′s profit function. By Berge’s theorem, pij(p) is continuous on Rn+. Theorem 5.9 Basic Properties of Supply and Profits If Y j satisfies Assumption 5.2, then for every given price p� 0, the solution of the firm’s problem (5.3) is unique and denoted by yj(p). Moreover, yj(p)is continuous on Rn++. In addition, pij(p) is well-defined and continuous on Rn+. N Sun (SHUFE) General Equilibrium March 4, 2010 25 / 38 Aggregate production possible set Y := ∑ j∈J Y j = { y = ∑ j∈J y j , where yj ∈ Y j } . Theorem 5.10 Properties of Y If each Y j satisfies Assumption 5.2, then the aggregate production possible set Y satisfies conditions 1, 3 and 4 of Assumption 5.2. Note that: It is possible that Y j ∩Rn+ 6= {0}. Therefore, we sometimes need further assume that: Y ∩Rn+ = {0} (impossibility of free production); Y ∩ (−Y ) = {0} (irreversibility, i.e., the production process cannot be reversed.). N Sun (SHUFE) General Equilibrium March 4, 2010 26 / 38 Theorem 5.11 Aggregate Profit Maximization For any prices p ≥ 0, we have that y¯ ∈ Y such that p · y¯ ≥ p · y for all y ∈ Y, if and only if for some y¯j ∈ Y j , j ∈ J , we may write y¯ =∑j∈J y¯j , and p · y¯j ≥ p · yj for all yj ∈j , j ∈ J . Note that: This theorem says that y¯ ∈ Y maximizes aggregate profit, if and only if it can be decomposed into individual firm profit-maximizing production plans. N Sun (SHUFE) General Equilibrium March 4, 2010 27 / 38 Proof: Suppose y¯ = ∑ j∈J y¯ j ∈ Y maximizes aggregate profits at price p, where y¯j ∈ Y j for each j ∈ J . If y¯k does not maximizes profits for firm k, then there is some y˜k ∈ Y k such that p · y˜k > p · y¯k. Thus, let y˜ =∑j 6=ky¯j + y˜k ∈ Y . Then we have p · y˜ > p · y¯, contradicting the assumption that y¯ maximizes aggregate profit at price p. Next, suppose feasible production plans y¯1, · · · , y¯J maximize profits at price p for the individual firms j ∈ J , and y¯ =∑j∈J y¯j ∈ Y . Then, for any y = ∑ j∈J y j ∈ Y , where yj ∈ Y j for each j ∈ J , we have p · y¯ = p · (∑j∈J y¯j) =∑j∈J p · y¯j ≥ ∑j∈J p · yj = p · (∑j∈J yj) = p · y, so, y¯ maximizes aggregate profits at price p, completing the proof. Q.E.D. N Sun (SHUFE) General Equilibrium March 4, 2010 28 / 38 5.3.2 Consumers There is a fixed number I of consumers indexed by the set I = {1, · · · , I}; Each consumer i ∈ I has a preference relation �i (or, a utility function ui) on his consumption set Rn+, and an endowment ei ∈ Rn+; Each consumer i ∈ I owns the shares θij of the profit of the jth firm for all j ∈ J . Of course, it holds that 0 ≤ θij ≤ 1 for all i ∈ I, j ∈ J , and∑ i∈Iθ ij = 1 for all j ∈ J . A private ownership economy can be represented by: E = ( (�i, ei)i∈I , (Y j)j∈J , (θij)i∈I,j∈J ) . N Sun (SHUFE) General Equilibrium March 4, 2010 29 / 38 In a private ownership economy, a consumer’s income can arise from two sources—from selling his endowment, and from his shares in the profits of firms. Therefore, facing the market prices p ≥ 0, consumer i′s income mi(p) = p · ei + ∑ j∈J θ ijpij(p), and so, consumer i′s utility maximizing problem is maxxi∈Rn+ u i(xi) (5.5) s.t. p · xi ≤ mi(p). Since under Assumption 5.2 each firm will earn nonnegative profits, we see mi(p) ≥ 0. By Theorem 5.9, we further see that mi(p) is continuous on Rn+. N Sun (SHUFE) General Equilibrium March 4, 2010 30 / 38 Theorem 5.12 Basic Property of Demand with Profit Shares If each Y j satisfies Assumption 5.2 and if ui satisfies Assumption 5.1, then a solution to the consumer’s problem (5.5) exists and is unique for all p� 0. Denoting it by xi(p,mi(p)), we have further that xi(p,mi(p)) is continuous on Rn++. In addition, mi(p) is continuous on Rn+. N Sun (SHUFE) General Equilibrium March 4, 2010 31 / 38 5.3.3 Equilibrium For the market prices p� 0, the aggregate excess demand (vector) is z(p) := ∑ i∈Ix i(p,mi(p))− ∑ j∈J y j(p)− ∑ i∈Ie i. A Walrasian equilibrium price is a market price vector p∗ � 0 that clears all markets, i.e., z(p∗) = 0. Generally, Definition: Walrasian (or, competitve) Equilibrium A Walrasian equilibrium of the private ownership economy E is an (I + J + 1)-tuple ((x∗i), (y∗j), p∗) of vectors of Rn such that (1) For each consumer i ∈ I, if xi�ix∗i, then p∗ · xi > p · ei +∑j∈J θijp∗ · y∗j ; (2) For each firm j ∈ J , y∗j ∈ Y j and p∗ · y∗j ≥ p∗ · yj for all yj ∈ Y j ; (3) ∑ i∈Ix ∗i = ∑ i∈Ie i + ∑ j∈J y ∗j . N Sun (SHUFE) General Equilibrium March 4, 2010 32 / 38 Theorem 5.13 Existence of Walrasian Equilibrium with Production Consider the production economy E = ((ui, ei)i∈I , (Y j)j∈J , (θij)i∈I,j∈J ). If each ui satisfies Assumption 5.1, each Y j satisfies Assumption 5.2, and y + ∑ i∈Ie i � 0 for some aggregate production vector y ∈ Y , then there exists at least one Walrasian equilibrium price p∗ � 0. Proof: We will show the aggregate excess demand function z(p) satisfies the conditions of Theorem 5.3. First, by Theorem 5.9 and Theorem 5.12, we see that the aggregate excess demand function z(p) is continuous on Rn++. That is, the first condition of Theorem (5.3) is satisfied. Next, since each ui is strongly increasing, p · xi(p,mi(p)) = mi(p). And so, the second condition p · z(p) = 0 of Theorem (5.3) is satisfied. Thus, we only need to show the third condition is satisfied. In fact, we can mimic the proof of Theorem 5.4 to show this. N Sun (SHUFE) General Equilibrium Marc