Advanced Microeconomics II by Jinwoo Kim October 6, 2010 Contents I General Equilibrium and Social Welfare 1 General Equilibrium Theory 3 3 1.1 Basic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Pareto Efficient Allocations . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Walrasian Equilibrium and Core . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Example: 2 × 2 Pure Exchange Economy . . . . . . . . . . . . . . . . . . . 9 2 Equilibrium Analysis 13 2.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Welfare Properties of WE . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Uniqueness and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Equivalence between Core and WE . . . . . . . . . . . . . . . . . . . . . . 22 3 Equilibrium in Production Economy 26 3.1 Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Utility maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 3.4 First and Second Welfare Theorems . . . . . . . . . . . . . . . . . . . . . 4 General Equilibrium under Uncertainty 29 32 4.1 Arrow-Debreu Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 Asset Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5 Public Goods and Externality 36 5.1 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Externality and Lindahl Equilibrium . . . . . . . . . . . . . . . . . . . . . 39 6 Social Choice and Welfare 42 6.1 Arrow’s Impossibility Theorem . . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 Some Possibility Results: Pairwise Majority Voting in Restricted Domain 44 2 Part I General Equilibrium and Social Welfare 1 General Equilibrium Theory 1.1 Basic model • Consider an economy in which there are n goods traded. • The initial resources, or endowment, of the economy is given as a vector ē = (ē1 , · · · , ēn ) ∈ Rn+ . Consumers • Assume that there are I consumers: – I = {1, · · · , I} : Set of consumers. – xik : Consumer i’s consumption of good k, xi = (xi1 , · · · , xin ) ∈ X i , where X i is i’s consumption set and assumed to be Rn+ . – eik : i’s endowment for good. Let ei = (ei1 , · · · , ein ) and e = (e1 , · · · , eI ). Note ∑ that ēk = i∈I eik . – ≽i : i’s preference defined on X i , which we assume can be represented by a function ui : X i → R. • Assumption C: For any i ∈ I, (i) ui is continuous. (ii) ui is strictly quasiconcave: For all x, y ∈ Rn+ , ui (λx + (1 − λ)y) > min{ui (x), ui (y)}, ∀λ ∈ (0, 1). (iii) ui is strictly monotone: If x > y (that is xk ≥ yk , ∀k and xk > yk for some k), then ui (x) > ui (y). 3 Producers • Assume that there are J producers: – J ={1, · · · , J}: Set of producers. – ykj : Firm j’s output (or input) for good k, y j = (y1j , · · · , ynj ) ∈ Rn : j’s production plan. ∗ If ykj > (<)0, then good k is produced (used) as output (input). – Y j ⊂ Rn : Set of feasible production plan for j, called production possibility set. ∗ Call any production plan y j ∈ Y j feasible. ∗ Assume that there exists a function F j : Rn → R such that Y j = {y ∈ Rn | F j (y) ≤ 0}. ∗ {y ∈ Rn | F j (y) = 0} : Production possibility frontier for firm j. ∑ – Y ≡ {y y = j∈J y j , where y j ∈ Y j , ∀j ∈ J }: Aggregate production possibility set. • Assumption P: For any j ∈ J , (i) 0 ∈ Y j . (ii) Y j is closed and bounded. (iii) Y j is strictly convex. Example 1.1. Suppose that there are one input and one output. .y2 . Production . Possibility Set .y1 .O 4 Feasible Allocations • An allocation in this economy is denoted by (x, y) = (x1 , · · · , xI , y 1 , · · · , y J ) ∈ RnI + × RnJ . • Given e = (e1 , · · · , eI ), an allocation (x, y) is feasible if ∑ xik ≤ ēk + i∈I ∑ ykj , ∀k = 1, · · · , n. (1) j∈J • Let F (e, Y ) denote the set of all feasible allocations i.e. set of all (x, y)’s satisfying (1). 1.2 Pareto Efficient Allocations Definition 1.2. An allocation (x, y) ∈ F (e, Y ) is Pareto efficient (PE) if there does not exist any other allocation (x̃, ỹ) ∈ F (e, Y ) such that ui (x̃i ) ≥ ui (xi ), ∀i ∈ I with at least one strict inequality. • With fixed utility levels, ū2 , · · · , ūI , let us solve the following maximization problem: max nJ (x,y)∈RnI + ×R u1 (x11 , · · · , x1n ) s.t. ui (xi1 , · · · , xin ) ≥ ūi , ∑ ∑ j xik ≤ ēk + yk , i∈I F j (y1j , · · · (2) i = 2, · · · , I (3) k = 1, · · · , n (4) j = 1, · · · , J (5) j∈J , ynj ) ≤ 0, – One can show the following: An allocation (x, y) is Pareto efficient if and only if it solve (2) for some utility levels, ū2 , · · · , ūI . 5 First Order Conditions for Pareto Efficiency • Letting δ i ,µk , and γ j denote a nonnegative multiplier for (3), (4), (5), respectively, the Lagrangian function for problem (2) can be set up as L =u1 (x11 , · · · , x1n ) + I ∑ δ i [ui (xi1 , · · · , xin ) − ūi ] i=2 + n ∑ ∑ µk [ēk + j∈J k=1 + J ∑ ykj − ∑ xik ] i∈I γ j [−F j (y1j , · · · , ynj )]. j=1 – Define δ 1 ≡ 1. Assuming the interior solution (i.e. xi > 0, ∀i), the first order conditions for maximizing the Lagrangian are ∂ui δ = µk , ∀i, k or δ i ∇ui = µ, ∀i i ∂xk ∂F j γ j j = µk , ∀j, k or γ j ∇F j = µ, ∀j, ∂yk i i i j j 1 n 1 n (6) (7) ∂u ∂u j , · · · , ∂F ), and µ = (µ1 , · · · , µn ). where ∇ui = ( ∂x = ( ∂F i , · · · , ∂xi ), ∇F ∂y j ∂y j – The conditions (6) and (7) imply the followings: ′ ′ µk ∂ui /∂xik ∂ui /∂xik ′ ′ = = ′ for all i, i , k, k ′ i i i i µk′ ∂u /∂xk′ ∂u /∂xk′ that is, MRS for any pair of goods must be equalized across consumers. ′ ′ µk ∂F j /∂ykj ∂F j /∂ykj ′ ′ = = ′ for all j, j , k, k µk′ ∂F j /∂ykj ′ ∂F j ′ /∂ykj ′ that is, MRT for any pair of goods must be equalized across producers. ∂F j /∂ykj ∂ui /∂xik = for all i, j, k, k ′ ∂ui /∂xik′ ∂F j /∂ykj ′ that is, MRS and MRT for any pair of goods must be equalized across consumers and producers. 6 1.3 Walrasian Equilibrium and Core From here on, we focus on the exchange economy in which there is no production i.e. Y j = {0}, ∀j ∈ J . We denote an exchange economy by E = (ui , ei )i∈I. Walrasian (or Competitive) Equilibrium • Let us introduce a perfectly competitive market system: – Consumers see themselves as not being able to affect prevailing prices in the markets. – Consumers only need to look at the market prices and not worry about what other consumers might demand or how demands are satisfied. • Suppose that price of good k is given as pk > 0 with p = (p1 , · · · , pn ) ∈ Rn++ being a price vector. – p · ei = ∑n k=1 pk eik : Market value of i’s endowment i.e. i’s wealth. – Consumer i’s budget set is B i (p) = {xi ∈ Rn+ | p · xi ≤ p · ei }. • Given market price vector p ∈ Rn++ , each consumer i has to solve ui (xi ) s.t. xi ∈ B i (p). max n i x ∈R+ (8) – Let xi (p, p·ei ) = (xi1 (p, p·ei ), · · · , xin (p, p·ei )) denote the optimal bundle/bundles (or demand function/correspondence) that solves (8). – Note that xi (p, p · ei ) is not continuous in p in all of Rn+ since demand will be infinite if some price is zero. • The aggregate excess demand function for good k is defined as zk (p) ≡ ∑ i∈I 7 xik (p, p · ei ) − ēk . – If zk (p) > (<)0, we say that good k is in excess demand (supply). – Define z(p) ≡ (z1 (p), , · · · , zn (p)). Definition 1.3. Walrasian equilibrium (WE) is a price vector p∗ ∈ Rn++ such that zk (p∗ ) = 0, ∀k. Core The core is another equilibrium concept that has its foundation in the cooperative game theory and assumes more centralized market than the Walrasian equilibrium does. • Let S ⊂ I denote a coalition of consumers. We say S blocks x if there is an allocation x̃ such that ∑ ∑ (i) i∈S x̃i = i∈S ei (ii) x̃i ≽ xi for all i ∈ S with at least one preference strict. – Note that a feasible allocation x is Pareto efficient if and only if it is not blocked by S= I. Definition 1.4. A feasible allocation x is in the core if and only if x is not blocked by any coalition of consumers, i.e. the core is the set of allocations that are not blocked by any coalitions. The following result shows the relationship between allocations in WE and core. Theorem 1.5. If each ui is strictly monotone, then any WE allocation belongs to the core. Proof. Letting p∗ and x denote WE price and allocation vectors, suppose to the contrary that x does not belong to the core. Then, we can find a coalition S ⊂ I and another allocation x̃ such that ∑ ∑ x̃i = ei and ui (x̃i ) ≥ ui (xi ), ∀i ∈ S with at least one inequality strict. i∈S i∈S By the strict monotonicity, this implies p∗ · x̃i ≥ p∗ · ei , ∀i ∈ S with at least one inequality strict, 8 which can be added up across consumers to yield ∑ ∑ p∗ · x̃i > p∗ · ei . This, however, contradicts with 1.4 i∈S ∑ i∈S x̃i = i∈S ∑ i∈S ei . Example: 2 × 2 Pure Exchange Economy Suppose that n = 2 and I = 2 and call a feasible allocation nonwasteful if it satisfies (1) with equality. • The set of nonwasteful allocations can be depicted by an Edgeworth box. .e21 .O2 .p1 /p2 . e. 22 endowment . point e. 12 .p1 /p2 .O1 e. 11 • How to find the set of PE allocations – Fix an indifference curve for consumer 2 and identify allocation(s) on that curve that maximizes consumer 1’s utility. – Do this for all possible indifference curves of consumer 2, which will give us the set of all PE allocations, often called contract curve. 9 .O2 .Contract curve .O. 1 • How to find the WE allocations – Trace out optimal bundles for each consumer i by varying p = (p1 , p2 ) and obtain an offer curve denoted OCi . .O2 .OC 1 .e .O1 . – Any intersection of OC1 and OC2 , which is different from the endowment point, corresponds to an equilibrium allocation. 10 .O2 .OC 1 .OC 2 .E .e .O1 . – Note that WE allocation is always PE (First Welfare Theorem). • The core is the segment of contract curve that lies within the lens formed by two indifference curves. .O2 . ontract C .curve .e .Core .O. 1 Example 1.6 (Calculating WE). Consider an exchange economy with two consumers and two goods: e1 = (1, 0) and e2 = (0, 1) 11 u1 (x1 ) = (x11 )a (x12 )1−a and u2 (x2 ) = (x21 )b (x22 )1−b , a, b ∈ (0.1). Setting p1 = 1 for normalization, consumer 1 solves max u1 (x1 ) subject to x11 + p2 x12 = 1, 1 x which yields x11 (1, p2 ) = a and x12 (1, p2 ) = 1−a . p2 Likewise, for consumer 2, we obtain x21 (1, p2 ) = bp2 and x22 (1, p2 ) = 1 − b The market clearing condition is z1 (1, p2 ) = x11 (1, p2 ) + x21 (1, p2 ) − 1 = a + bp2 − 1 = 0, which yields p2 = 1−a . b Note that given p2 = 1−a , b the market for good 2 is cleared, i.e. z2 (1, p2 ) = 0, as can be expected by the Walras’ Law. Example 1.7 (Calculating PE allocation). Consider the same setup as in Example 1.6. We can set up the Lagrangian for calculating the PE allocations as L =(x11 )a (x12 )1−a + δ 2 [(x21 )b (x22 )1−b − ū2 ] + µ1 [1 − x11 − x21 ] + µ2 [1 − x12 − x22 ], whose first-order conditions, assuming the interior solution, are a(x11 )a−1 (x12 )1−a = µ1 = δ 2 b(x21 )b−1 (x22 )1−b (9) (1 − a)(x11 )a (x12 )−a = µ2 = δ 2 (1 − b)(x11 )a (x22 )−b (10) x11 + x21 = 1 = x12 + x22 . (11) Dividing the RHS and LHS of (9) and (10) side by side, we obtain a x12 b x22 b 1 − x12 = = , 1 − a x11 1 − b x21 1 − b 1 − x11 where the second equality follows from applying (11). Rearranging this yields a(1 − b)x12 . (12) b(1 − a) + (a − b)x12 So the set of PE allocations or contract curve contains all the points in the Edgeworth x11 = box that satisfy (12). Try to see for yourself what the contract curves look like for varying values of a and b. 12 2 Equilibrium Analysis 2.1 Existence The proof of WE existence relies much on the properties possessed by the excess demand functions, which thus we first explore. Properties of Excess Demand Functions • Under Assumption C, the following basic properties hold: For any i ∈ I, (P0) Uniqueness: The problem (8) has a unique solution, i.e. xi (p, p · ei ) is a function of p. – Follows from the strict quasiconcavity of ui . (P1) Continuity: The demand function xi (p, p · ei ), and thus excess demand function z i (p), is continuous in p on Rn++ . – Follows from the continuity of ui and Berge’s maximum theorem. (P2) Homogeneity: z i (p) is homogeneous of degree 0 i.e. z i (λp) = z i (p), ∀p, ∀λ > 0. – Follows from the fact that B i (λp) = B i (p). • Under Assumption C, we can prove some further properties: (P3) Walras’ Law: p · z(p) = 0, ∀p ∈ Rn++ . Proof. Write down the budget constraint for each consumer i ∈ I n ∑ pk [xik (p, p · ei ) − eik ] = 0. k=1 Summing across consumers yields 0= n ∑∑ pk [xik (p, p · ei ) − eik ] i∈I k=1 13 = = n ∑ k=1 n ∑ pk [ ∑ ] [xik (p, p · ei ) − eik ] i∈I pk zk (p) = p · z(p). k=1 – This result implies that if n−1 markets clear, then the remaining 1 market must also clear. (P4) Bounded below: zk (p) ≥ ēk , ∀k (Obvious). (P5) Unbounded above: Suppose that ē ≫ 0. Then, if {pm }∞ m=1 is a sequence of price vectors in Rn++ converging to p̄ ̸= 0 and p̄k = 0 for some k, then we must have zk′ (pm ) → ∞ for some k ′ with p̄k′ = 0. ∑ Proof. Since p̄ · [ i∈I ei ] = p̄ · ē > 0, there is at least one i ∈ I such that p̄ · ei > 0. Now let xm ≡ xi (pm , pm · ei ). Suppose for a contradiction that ∗ m ∗ {xm }∞ m=1 is bounded. Then, there must exist some x ≪ ∞ such that x → x . Note that as m → ∞, pm · xm = pm · ei → p̄ · x∗ = p̄ · ei > 0. Let x̂ = (x∗1 , · · · , x∗k + 1, · · · , x∗n ) and then ui (x̂) > ui (x∗ ) (by strict monotonicity) and p̄ · x̂ = p̄ · ei > 0. Since ui is continuous, there exists t ∈ (0, 1) such that ui (tx̂) > ui (x∗ ) and pm · (tx̂) < pm · ei . Thus, for large enough m, ui (tx̂) > ui (xm ) and pm · (tx̂) < pm · ei , which is a contradiction since xm maximizes i’s utility given price vector pm . So, there must be some k ′ such that {xm k′ } is unbounded. Also, pm · xm = pm · ei → p̄ · ei < ∞ implies p̄k′ = lim pm k′ = 0. m→∞ 14 Proof of Existence • The existence in two goods case can be proved in the following steps: 1. By (P2), we can normalize one of two prices to 1 say p2 = 1. 2. By (P3), we can focus on one market, say good 1 market. 3. For some p′1 w 0, we must have z1 (p′1 , 1) > 0 by (P5). 4. For some p′′1 w ∞, we must have z1 (p′′1 , 1) < 0 by (P3) and (P5). 5. By (P2) and mean value theorem, there is some p∗1 ∈ (p′1 , p′′1 ) such that z1 (p∗1 , 1) = 0. .z1 (·) . .O p. ′1 p. ∗∗ 1 p. ′′1 p. ∗1 p. ∗∗∗ 1 .p1 The above argument, however, cannot be used when there are more than two goods. So we rely on the fixed point theorem, the version of Brower. Theorem 2.1 (Borwer’s Fixed Point Theorem). Let A be a nonempty, compact, and convex subset of Rn and f : A → A be continuous. Then, f has a fixed point, that is there exists x∗ ∈ A such that f (x∗ ) = x∗ . Using the fixed point theorem and the properties above, we are able to prove that WE always exists. Theorem 2.2 (Existence of WE). If Assumption C holds and ē ≫ 0, then there exists at least one price vector, p∗ ≫ 0, such that z(p∗ ) = 0. 15 Proof. For each k, let z̄k (p) ≡ min(zk (p), 1), ∀p ≫ 0 and let z̄(p) ≡ (z̄1 (p), · · · , z̄n (p)). So z̄k (p) ≤ 1, ∀p, ∀k. Fix ϵ ∈ (0, 1) and let Sϵ ≡ {p | n ∑ pk = 1 and pk ≥ k=1 ϵ , ∀k}. 1 + 2n Note that Sϵ is compact, convex, and nonempty. For each k and every p ∈ Sϵ , define fk (p) as fk (p) ≡ ϵ + pk + max{0, z̄k (p)} ∑ nϵ + 1 + nm=1 max{0, z̄m (p)} and let f (p) = (f1 (p), · · · , fn (p)). Hence, for all p ∈ Sϵ , n ∑ fk (p) = 1 and fk (p) ≥ k=1 ϵ ϵ ≥ , nϵ + 1 + n 1 + 2n which implies that f is a mapping from Sϵ to itself. Since each z̄k is continuous in p on Sϵ , fk and f are also continuous in p on Sϵ . Appeal to the fixed point theorem to find pϵ ∈ S ϵ such that f (pϵ ) = pϵ or fk (pϵ ) = pϵk , ∀k, which means [ ] n ∑ pϵk nϵ + max{0, z̄m (pϵ )} = ϵ + max{0, z̄k (pϵ )}. (13) m=1 Now, let ϵ → 0 and find a sequence of ϵ such that pϵ → p∗ for some p∗ (since the sequence belongs to a compact set). We argue that p∗ ≫ 0. Suppose not for a contradiction. Then, p∗k = 0 for some k and, by (P5), zk′ (pϵ ) → ∞ for some k ′ with p∗k′ = 0, which contradicts (13). Thus, pϵ → p∗ ≫ 0 as ϵ → 0, which implies by the continuity of z̄ that for all k p∗k n ∑ max{0, z̄m (p∗ )} = max{0, z̄k (p∗ )}. m=1 Multiplying zk (p∗ ) to both sides and summing up for all k’s, we obtain ( n ) n ∑ ∑ ∗ ∗ ϵ 0 = p · z(p ) max{0, z̄m (p )} = zk (p∗ ) max{0, z̄k (p∗ )}. m=1 (14) k=1 Since zk and z̄k have the same sign and p∗ · z(p∗ ) = 0, we must have zk (p∗ ) = 0, ∀k due to (14). 16 • WE may not exist in cases – Preferences are not strictly monotone: Suppose that u1 (x1, x2 ) = x1 and u2 (x1, x2 ) = x1 x2 and e1 = (1, 0) and e2 = (0, 1). It is straightforward to verify that given p2 = 1, there exists no price p1 that clears the markets. – Preferences are nonconvex .O2 z. 12 (·) .p2 = 1 .slope = p′′′ 1 . .O .slope = p′1 . ′1 p . ′′1 p . ′′′ p 1 . .z11 (·) .e .slope = p′′1 . .O 1 . Corollary 2.3. If Assumption C holds and ē ≫ 0, then the core is nonempty. Proof. This is immediate from combining Theorem 1.5 and Theorem 2.2. 2.2 Welfare Properties of WE First Welfare Theorem The first welfare theorem is a positive answer to the question of whether the competitive market mechanism always yields an efficient allocation. Theorem 2.4 (First Welfare Theorem). If each ui is strictly increasing, then every WE allocation is PE. Proof. Immediate from Theorem 1.5 and the fact that all core allocations are PE. 17 Second Welfare Theorem The second welfare theorem is the converse of the first welfare theorem, i.e. any PE allocation can be achieved as an equilibrium allocation in a competitive market (with appropriate redistribution). Theorem 2.5 (Second Welfare Theorem). Consider an exchange economy E = (ui , ei )i∈I satisfying Assumption C and ē ≫ 0. Then, for every PE allocation x̂, there are a price vector p and a redistribution of wealth (w1 , · · · , wI ) ∈ RI such that ∑ (i) i∈I wi = 0 (budget balance) and (ii) For every i, x̂i ∈ arg max ui (xi ) subject to p · xi ≤ p · ei + wi (utility maximization). Proof. Since x̂ is feasible, we have ∑ i∈I x̂i = ē ≫ 0. Thus, by Theorem 2.2, we can find a WE price vector p and WE allocation x̄ for the economy Ê = (ui , x̂i )i∈I . Now set wi = p · (x̂i − ei ), which satisfies (i) since ∑ i∈I wi = p · ∑ (x̂i − ei ) = 0. i∈I To see that (ii) is satisfied, note first that ui (x̄i ) ≥ ui (x̂i ), ∀i ∈ I since in the economy ˆ consumers must be maximizing their utilities with endowments (x̂i )i∈I . Since x̂ is PE, E, however, we must have ui (x̄i ) = ui (x̂i ), which implies (by the strict quasiconcavity of ui ) that x̂i = x̄i . Since p and x̄ are WE prices and allocation for the economy Ê, we have for all i x̂i = x̄i ∈ arg max ui (xi ) subject to p · xi ≤ p · x̂i = p · ei + wi , as desired. Example 2.6. The following graph illustrates how the second welfare theorem can fail when the preference is nonconvex: 18 .O2 . .O1 2.3 Uniqueness and Stability Uniqueness When there are more than one equilibria, it is harder for us as economists to provide an unambiguous prediction about what will happen in the economy. Also, the second welfare theorem loses its power since, if the economy resulting from the redistribution admits multiple equilibria, then some unwanted allocation may arise in equilibrium. Here we look for conditions under which WE is unique. Definition 2.7. The excess demand function z(·) satisfies the weak axiom of revealed preference (WARP) if for any pair of price vectors p and p′ , we have z(p) ̸= z(p′ ) and p · z(p′ ) ≤ 0 implies p′ · z(p) > 0. • Letting x ≡ ∑ i∈I xi (p, p · ei ) and x′ ≡ ∑ i∈I xi (p′ , p′ · ei ), WA can be rewritten as p · x ≥ p · x′ and x ̸= x′ implies p′ · x > p′ · x′ . – So, WARP requires that the aggregate demand behaves as if it is generated by a single consumer. 19 – Can show that WARP is satisfied if the indirect utility function for each consumer i with income mi , denoted v i (p, mi ), takes a Gorman form v i (p, mi ) = ai (p) + b(p)mi . Proposition 2.8. If z(·) satisfies the WARP, then the set of WE is convex (and so, if the set of WE is finite, there can be at most one WE). Note: In a generic economy, the number of WE is finite and thus WE, if exists, is unique. Proof. Suppose for a contradiction that for some p, p′ , and α ∈ (0, 1), we have z(p) = z(p′ ) = 0 ̸= z(αp + (1 − α)p′ ). Let p′′ = αp + (1 − α)p′ . Since z(p) = 0 ̸= z(p′′ ) and p′′ · z(p) = 0, WA implies p · z(p′′ ) > 0. (15) Also, since z(p′ ) = 0 ̸= z(p′′ ) and p′′ ·z(p′ )˙ = 0, WA implies p′ · z(p′′ ) > 0. (16) Combining (15) and (16) yields p′′ · z(p′′ ) = αp · z(p′′ ) + (1 − α)p′ · z(p′′ ) > 0, which contradicts the Walras’ Law. Here I introduce another condition that guarantees a uniqueness of WE. Definition 2.9. The function z(·) has the gross substitute (GS) property if whenever p′ and p are such that, for some ℓ, p′ℓ > pℓ and p′k = pk for k ̸= ℓ, we have zk (p′ ) > zk (p) for k ̸= ℓ. Proposition 2.10. An aggregate excess demand function z(·) that satisfies the GS property has at most one WE. Proof. Suppose for a contradiction that for some p∗ and p ̸= αp∗ for any α > 0, we have z(p∗ ) = z(p) = 0. Since z(·) is homogeneous of degree 0, we can find some α > 0 and p′ = αp∗ such that p′ ≥ p and p′ℓ = pℓ for some ℓ. Now consider changing p′ to obtain p in n − 1 steps, lowering (or keeping unchanged) the price of every good k ̸= ℓ one at a time. By GS, the excess demand of good ℓ cannot decrease in any step, and, because p ̸= p′ , it will actually increase in at least one step. Hence, zℓ (p) > zℓ (p′ ), a contradiction. 20 Tatonnement Stability Let us suppose that an economy is in disequilibrium state. Can we be sure that the economy will eventually evolve into some equilibrium state? • Let us assume that the dynamic path of price takes the following form of differential equation: For every k, dpk = ck zk (p), (17) dt where dpk /dt is the rate of change of the price for good k and ck > 0 is a constant affecting the speed of adjustment. – This adjustment process can be better considered as a tentative trial-and-error process taken by an auctioneer who tries to find the equilibrium prices. – According to (17), the excess demand (supply) for good k causes its price to move upward (downward). – Suppose that there is a unique WE p∗ . We say that p∗ is globally stable if every trajectory satisfying (17) converges to p∗ (or αp∗ for some α > 0 ). Proposition 2.11. Suppose that z(p∗ ) = 0 and p∗ · z(p) > 0 for every p not proportional to p∗ . Then, p∗ is globally stable. Proof. We borrow a result from Lyapunov’s stability theory. To do so, consider a dynamical system ẋ = f (x) where f : Rn → Rn , and a unique equilibrium point x∗ satisfying f (x∗ ) = 0. Lyapunov Function: A differentiable function V : Rn → R is called Lyapunov function if it satisfies (i) V (x∗ ) = 0; (ii) V (x) > 0 for all x ̸= x∗ ; and (iii) dV (x(t)) dt < 0 for all x(t) ̸= x∗ , i.e. ∇V (x(t)) · f (x(t)) < 0 for x(t) ̸= x∗ . We can prove Lyapunov’s Theorem: If one can find a Lyapunov function for a dynamical system, then the unique equilibrium x∗ is globally stable. Remark. The condition in Proposition 2.11 that p∗ · z(p) > 0 for every p not proportional to p∗ is satisfied if GS property holds or if WARP holds for a generic the economy. 21 Proof. To apply this result, let us set a candidate Lyavnov function as n ∑ 1 (pk − p∗k )2 . V (p) = c k k=1 Obviously, this function satifies (i) and (ii) above. As for (iii), we have n ∑ dV (p(t)) 1 dpk (t) =2 (pk (t) − p∗k ) dt ck dt k=1 = n ∑ 1 (pk (t) − p∗k )ck zk (p(t)) c k k=1 = −p∗ · z(p(t)) < 0 if p(t) ̸= p∗ , as desired. 2.4 Equivalence between Core and WE In general, the set of core is larger than that of WE. Edgeworth conjectured that the difference between the two would disappear if the economy gets ‘large’, which is where the assumption of price-taking behavior makes most sense. Here, we follow Debreu and Scarf to formalize the idea of ‘large’ economy and establish the equivalence between core and WE. Definition 2.12 (r-Replica Economy). Given an exchange economy E = (ui , ei )i∈I , the r-fold replica economy, denoted Er , is the economy with r consumers of each type for a total of rI consumers. We assume that the preference of each type satisfies Assumption C. • An allocation in Er is denoted as x = (x11 , · · · , x1r , · · · , xI1 , · · · , xIr ), where xiq denotes the bundle for the qth consumer of type i. – The allocation is feasible if I ∑ r ∑ i=1 q=1 22 xiq = rē. Theorem 2.13 (Equal Treatment in the Core). If x belongs to the core of Er , then for ′ every i, xiq = xiq , ∀q, q ′ . ′ Proof. Suppose for a contradiction (and without loss of generality) that x1q ̸= x1q for some q and q ′ . We show that x can be blocked by a coalition of I distinct types of consumers, each of whom is treated worst within its type. Suppose wlog that S = {11, · · · , I1} is that coalition i.e. for each i = 1, · · · , I, ui (xi1 ) ≤ ui (xiq ), ∀q = 1, · · · , r. Define x̂i = 1 r ∑r q=1 i xiq and give x̂i to each consumer i1 in S. Note that by the strict quasi-concavity of u , we have 1 ∑ iq u (x̂ ) = u ( x ) ≥ min ui (xiq ) = ui (xi1 ) q r q=1 r i i i ′ and also u1 (x̂1 ) > u1 (x11 ) since x1q ̸= x1q for some q and q ′ . So every consumer in S is weakly better off while type 1 consumer in S is strictly better off. We will be done if (x̂i )i∈I turns out to be feasible within S, which is true since I ∑ i=1 1 ∑ ∑ iq 1 x̂ = x = rē = ē, r i=1 q=1 r I r i where the second equality is due to the feasibility of x in Er . Thanks to Theorem 2.13, we may (and will do so) let (xi )i∈I denote an allocation in the core of any replica economy. We want to show that as r gets large, the core of Er shrinks to the set of WE in the original economy E. Before that, we first look at the example of Edgeworth box economy. Example 2.14. Suppose that there are two types of consumers. Some allocation in the core of E is not in the core of E2 . Consider, for instance, allocation x as below, where x̂ is a middle point between x and e: 23 .O2 .x .x̂ .e .Contract curve ..O1 Consider a candidate blocking coalition S = {11, 12, 21}, and give a bundle x̂1 = 12 (e1 + x1 ) to each of 11 and 12, a bundle x̂2 = x2 to 21. So, type 1 consumers in S are better off while type 2 consumer is indifferent. Also, this allocation is feasible within S since 2x̂1 + x̂2 = (e1 + x1 ) + x2 = e1 + (x1 + x2 ) = e1 + (e1 + e2 ) = 2e1 + e2 . Theorem 2.15. Suppose that for each i, ui is differentiable and ∇ui (x) ≫ 0, ∀x ∈ Rn++ . If (xi )i∈I ≫ 0 belongs to the core of Er for every r, then it is a WE of E. Proof. We first prove the following claim. Claim. For each i ∈ I, ui ((1 − t)xi + tei ) ≤ ui (xi ), ∀t ∈ [0, 1]. Proof. Suppose not for a contradiction. Then, we must have some type i and t̄ ∈ (0, 1) such that ui ((1 − t̄)xi + t̄ei ) > ui (xi ). The strict quasi-concavity of ui then implies ui ((1 − t)xi + tei ) > ui (xi ), ∀t ∈ (0, t̄]. 24 Thus, we can find a positive integer r large enough that 1 r ∈ (0, t̄] and thus 1 1 ui ((1 − )xi + ei ) > ui (xi ). r r (18) Consider Er and a blocking coalition S that consists of all type i consumers and (r − 1) of type j consumer for all j ̸= i. Give a bundle x̂i = (1 − 1r )xi + 1r ei to each of type i and a bundle x̂j = xj to each of type j ̸= i, which makes type 1 better (due to (18)) while making other types indifferent. Also, this allocation is feasible within S since ∑ ∑ ∑ rx̂i + (r − 1) x̂j = (r − 1)xi + ei + (r − 1) xj = (r − 1)ē + ei = rei + (r − 1) ej , j̸=i j̸=i j̸=i which establishes a contradiction. From the above claim, for each i ∈ I, ui ((1 − t)xi + tei ) − ui (xi ) = ∇ui (xi ) · (ei − xi ). t→0+ t 0 ≥ lim (19) Since xi is in the core of E and thus Pareto efficient, the first order condition (6) implies that the gradient vectors are proportional at (xi )i∈I across consumers, i.e. for all i ̸= 1, ∇ui (xi ) = λi ∇u1 (x1 ) for some λi > 0. Letting p∗ ≡ ∇u1 (x1 ) and λ1 ≡ 1, we have ∇ui (xi ) = λi p∗ , ∀i ∈ I. (20) p∗ · xi ≥ p∗ · ei , ∀i ∈ I. (21) Then, (19) and (20) imply The inequality here must be satisfied as equality, for otherwise adding up the inequalities across consumers yields p∗ · which is impossible since ∑ xi > p∗ · ē, i∈I ∑ i∈I xi = ē. Now, (20) and (21) as equality correspond to the first-order (necessary) condition for the utility maximization, which is also sufficient given the assumption that each consumer’s preference is strictly quasi-concave. Thus, p∗ and (xi )i∈I constitute WE prices and allocation. 25 3 Equilibrium in Production Economy 3.1 Profit Maximization Given a price vector p ≫ 0, each firm j ∈ J solves max p · y j . y j ∈Y j Let y j (p) ≡ arg maxyj ∈Y j p · y j and π j (p) = p · y j (p). Example 3.1. Continuing on the Example 1.1, the profit maximization problem can be illustrated by the following graph: .y2 .p1 y1 + p2 y2 = π(p) . .y (p) π(p) .p 2 Production . Possibility Set π(p) .p 1 .y1 .O Theorem 3.2. If Y j satisfies Assumption P, then for any p ≫ 0, y j (p) is unique and continuous on Rn+ . Also, π j (p) is continuous on Rn+ . Theorem 3.3 (Aggregate Profit Maximization ). Fix prices at p. Then, p· ȳ ≥ p·y, ∀y ∈ Y ∑ if and only if there is some ȳ j ∈ Y j for each j ∈ J such that ȳ = j∈J ȳ j and p · ȳ j ≥ p · y j , ∀y j ∈ Y j . 3.2 Utility maximization • The firms are owned by consumers: 26 – θij ∈ [0, 1]: Consumer i’s share in firm j → ∑ i∈I θij = 1, ∀j ∈ J . – A production economy is described by a vector (ui , ei , θij , Y j )i∈I,j∈J . • Given prices p, each consumer i has two sources of income: – Endowment income =p · ei ∑ – Share income= j∈J θij π j (p). ∑ – Let mi (p) ≡ p · ei + j∈J θij π j (p) denote the total income of consumer i. • Consumer i solves max ui (xi ) subject to p · xi ≤ mi (p), xi ∈Rn + whose solution is given as xi (p, mi (p)). 3.3 Equilibrium • The aggregate excess demands are defined as zk (p) = ∑ xik (p, mi (p)) − ∑ i∈I eik − i∈I ∑ ykj (p) and j∈J z(p) = (z1 (p), · · · , zn (p)). Theorem 3.4 (Existence of competitive equilibrium ). Consider the economy (ui , ei , θij , Y j )i∈I,j∈J . ∑ Under Assumption C, Assumption P, and ē + y ≫ 0 for some y = j∈J y j ∈ Y, there exists at lest one price vector p∗ such that z(p∗ ) = 0. Proof. It suffices to establish that z(p) satisfies the properties (P0) to (P5). The only nontrivial part is to prove that z(p) is “unbounded above” that is, for any sequence of price vectors pm → p̄ ̸= 0 with p̄k = 0 for some k, we must have zk′ (pm ) → ∞ for some k ′ . The proof can be done following the same steps as in the exchange economy once it is shown that ∃i ∈ I such that mi (p̄) > 0 (or some consumer has a positive income given p̄). To this end, note that ∑ i∈I mi (p̄) = ∑ ( p̄ · ei + i∈I ∑ j∈J 27 ) θij π j (p̄) = ∑ p̄ · ei + i∈I ≥ ∑ ∑ π j (p̄) j∈J p̄ · ei + i∈I ∑ p̄ · y j j∈J = p̄ · (ē + y) > 0, which implies that ∃i such that mi (p̄) > 0. Example 3.5 (Robinson Crusoe Economy ). Robinson, who lives in a desert island, woks in the daytime (Robinson producer or RP) and consumes “consumption” and “leisure” in the remaining time (Robinson consumer or RC). • RC’s utility: U (h, y) = h1−β y β , β ∈ (0, 1), where h denotes the amount of leisure time and y the amount of consumption. – RC’s initial endowment of time and consumption is T and 0, respectively. – RC owns the firm for which he is the only provider of labor. • RP’s production function: y = ℓα , α ∈ (0, 1), where ℓ ≥ 0 denotes the supply of labor. Letting p and w denote the price of consumption and wage, resp., RP solves max pℓα − wℓ (22) max h1−β y β subject to py + wh = wT + π(w, p). (23) ℓ≥0 while RC solves h,y Solving (22) yields ℓ(w, p) = 1 ( αp ) 1−α w Solving (23) yields h(w, p) = ( and π(w, p) = w (1−β)(wT +π(w,p)) . w 1−α α )( Thus, setting p = 1 for normalization, the market clearing requires ℓ(w, 1) + h(w, 1) = T or 1 ( α ) 1−α w + (1 − β)(wT + π(w, 1)) = T, w 28 1 αp ) 1−α . w from which ( w=α αβ + (1 − β) αβ )1−α T 1−α . Graphically, y. C y. P Iso-Profit Line . = Budget Line . .E Production . Possibility Set .h(w, 1) .−ℓ .π (w, 1) .ℓ(w, 1) .w .OC .OP .h .T 3.4 First and Second Welfare Theorems Theorem 3.6 (First Welfare Theorem with Production ). If each ui is strictly increasing on Rn+ , then any competitive equilibrium allocation must be Pareto efficient. Proof. Suppose that (x, y) is a competitive equilibrium but not Pareto efficient. Then, ∃(x̂, ŷ) ∈ F (e, Y ) such that ui (x̂i ) ≥ ui (xi ), ∀i ∈ I with at least one strict inequality, which implies that letting p∗ denote the competitive equilibrium price vector, p∗ · x̂i ≥ p∗ · xi , ∀i ∈ I with at least one strict inequality ∑ ∑ ⇒ p∗ · x̂i > p∗ · xi ⇒ p∗ · ( ∑ i∈I i∈I ei + ∑ i∈I ) ŷ j > p∗ · j∈J ⇒ p∗ · ∑ j∈J ( ∑ ei + i∈I ŷ j > p∗ · ∑ yj , j∈J 29 ∑ j∈J ) yj which implies p∗ · ŷ j > p∗ · y j for some j ∈ J , contradicting the profit maximization. Theorem 3.7 (Second Welfare Theorem with Production ). Assume (i) Assumption C ∑ and Assumption P (ii) y + i∈I ei ≫ 0 for some y ∈ Y , and (iii) (x̂, ŷ) is Pareto efficient. ∑ Then, ∃T1 , · · · , TI with i∈I Ti = 0 and p̄ such that 1. x̂i ∈ arg maxxi ∈Rn+ ui (xi ) such that p̄ · xi ≤ mi (p̄) + Ti , ∀i ∈ I 2. ŷ j ∈ arg maxyj ∈Y j p̄ · y j , ∀j ∈ J . Proof. Let Ȳ j ≡ Y j − ŷ j and then each Ȳ j satisfies A4, as can be easily verified. Consider a (hypothetical) economy (ui , x̂i , θij , Ȳ j )i∈I,j∈J . Then, ∃ a competitive equilibrium p̄ ≫ 0 and (x̄, ȳ) for this economy. We first prove the following claim. Claim. (x̄i , ȳ j + ŷ j )i∈I,j∈J ∈ F (e, Y ). Proof. Note first that ȳ j + ŷ j ∈ Y j , as can be easily seen. From the fact that (x̄, ȳ) ∈ F (x̂, Ȳ ), we also have ∑ x̄i = i∈I ∑ x̂i + i∈I = ∑ = i∈I ȳ j j∈J x̂i − i∈I ∑ ∑ ∑ j∈J ei + ∑ ŷ j + ∑ (ȳ j + ŷ j ) j∈J (ȳ j + ŷ j ), j∈J as desired. The above claim, together with the Pareto efficiency of (x̂, ŷ) in the original economy, implies that ui (x̄i ) = ui (x̂i ), ∀i ∈ I so x̄i = x̂i , ∀i ∈ I. Thus, x̂i = x̄i ∈ arg max ui (xi ) subject to p̄ · xi ≤ p̄ · x̂i + n i x ∈R+ ∑ θij p̄ · ȳ j , ∀i ∈ I, (24) j∈J ∑ which implies that j∈J θij p̄ · ȳ j = 0. Since this is true for all consumer i ∈ I, we must ∑ have j∈J p̄ · ȳ j = 0 so p̄ · ȳ j = 0, ∀j ∈ J . Then, 0 = p̄ · ȳ j ≥ p̄ · (y j − ŷ j ), ∀y j ∈ Y j , ∀j ∈ J ⇒ p̄ · ŷ j ≥ p̄ · y j , ∀y j ∈ Y j , ∀j ∈ J . 30 (25) Now, letting Ti = p̄ · x̂i − mi (p̄), (24) and (25) imply x̂i = arg max ui (xi ) subject to p̄ · xi ≤ mi (p̄) + Ti = p̄ · x̂i , ∀i ∈ I n i x ∈R+ ŷ j = arg max p̄ · y j , ∀j ∈ J . j j y ∈Y It remains to verify that ∑ Ti ∑ Ti = 0, which holds since ( ) ∑ ∑ ∑∑ = p̄ · x̂i − p̄ · ei + θij π j (p̄) i∈I i∈I i∈I = ∑ i∈I p̄ · x̂i − i∈I = ∑ ( ∑ p̄ · ei + ∑∑ i∈I p̄ · x̂ − i ( ∑ i∈I = p̄ · i∈I j∈J ( ∑ i∈I x̂i − i∈I 31 θij p̄ · ŷ j i∈I j∈J p̄ · e + i i∈I ∑ ) ∑ p̄ · ŷ ) j j∈J ei − ∑ j∈J ) ŷ j = 0. 4 General Equilibrium under Uncertainty We now apply the general equilibrium framework to economic situations involving uncertainty, which may be about technologies, endowments, or preferences. • Let s ∈ S = {1, · · · , S} denote the state of the world, which will realize at time t = 1. – At t = 0, each consumer i is uncertain about s and believes it will occur with probability πsi .. – x = (x1 , · · · , xS ) ∈ RnS + : Contingent consumption vector (or contingent commodity), where xs = (x1s , · · · , xns ) ∈ Rn+ is the commodity vector to be consumed if state s occurs at t = 1. • Each consumer i’s endowment and preference may depend on the state of the world. – ei = (ei1 , · · · , eiS ): Consumer i’s endowment vector. – ≽i : Consumer i’s (rational) preference relation, which is represented by a utility function U i defined on RnS + . ∗ For instance, if the consumer i has Bernoulli (state-dependent) utility function uis (xs ) in state , then his preference will be given as follow: x ≽i x′ if and only if U i (x) = ∑ πsi uis (xs ) ≥ s∈S 4.1 ∑ πsi uis (x′s ) = U i (x′ ). s∈S Arrow-Debreu Equilibrium Consider an exchange economy with no production for simplicity. Definition 4.1. An allocation x∗ = (x1∗ , · · · , xI∗ ) ∈ RnSI and prices p = (p11 , · · · , pnS ) ∈ RnS constitute an Arrow-Debreu equilibrium (ADE) if: (i) ∀i ∈ I, xi∗ ≽i xi for any xi ∈ {xi | p · xi ≤ p · ei }. ∑ ∑ i (ii) i∈I xi∗ s = i∈I es , ∀s ∈ S. Example 4.2. Suppose that I = 2, n = 1, and S = 2. Also, each consumer has a stateindependent Bernoulli utility function ui while e1 = (1, 0) and e2 = (0, 1) so that there 32 is no aggregate risk. At the (interior) optimum xi for consumer i = 1, 2, the first-order conditions yield / / 1 1 1 1 2 2 p1 du2 (x22 ) 1 du (x1 ) 1 du (x2 ) 1 2 2 du (x1 ) π1 π2 = M RS12 = = M RS12 = π1 π22 . dx dx p2 dx dx (26) Assuming that πs1 = πs2 = πs , ∀s = 1, 2, we have xi1 = xi2 , ∀i = 1, 2 and p1 /p2 = π1 /π2 . As shown here, the contingent commodity serves the purpose of transferring wealth across the states of the world. Example 4.3. Consider the same economy as in Example 4.2 except that e1 + e2 = (2, 1) so there is an aggregate risk. Assuming πs1 = πs2 = πs , ∀s = 1, 2, (26) implies that xi1 > xi2 , ∀i = 1, 2, which in turn implies p1 /p2 < π1 /π2 . 4.2 Asset Market In reality, not all goods perform the wealth-transferring role. There are some commodities called assets, or securities, that are designed specifically for transferring wealth across states. Definition 4.4. A unit of asset or security is the title to receive an amount rs units of good 1 at t = 1 if state s ∈ S occurs. Thus, an asset is denoted by r = (r1 , · · · , rS ) ∈ RS . Example 4.5. Here are the examples of securities: 1. r = (1, · · · , 1) : Noncontingent delivery of one unit of good 1 →Commodity futures. 2. r = (0, · · · , 0, 1, 0, · · · , 0) : Contingent delivery of one unit of good 1 at a certain state (Arrow security). 3. r(c) = (max{0, r1 − c}, · · · , max{0, rS − c}) for a primary asset r ∈ RS : Call option with the primary asset r and strike price c→ Option to exercise r at price c ∈ R (assuming that the spot-market price of good 1 is normalized to 1 in every state). • At t = 0, only the asset market opens in which K assets are traded. – q = (q1 , · · · , qK ) : Asset price vector. 33 – z = (z1 , · · · , zK ) ∈ RK : Asset trade vector or portfolio. • At t = 1, the spot market opens at each state s ∈ S in which all goods are traded. – ps = (p1s , · · · , pns ) : Spot prices at state s. – xs = (x1s , · · · , xns ) : Spot trades at state s. Definition 4.6. A collection {q, (ps )s∈S , (z i∗ )i∈I , (xi∗ )i∈I } constitutes a Radner equilibrium (RE) if: (i) ∀i ∈ I, (z i∗ , xi∗ ) solves i max U (x) subject to (a) x∈RnS + z∈RK K ∑ qk z k ≤ 0 k=1 (b) ps · xs ≤ ps · eis + k ∑ p1s zk rsk , ∀s ∈ S. (27) k=1 (ii) ∑ i∈I zki∗ ≤ 0, ∀k and ∑ i∈I xi∗ s ≤ ∑ i∈I eis , ∀s ∈ S. Normalize p1s = 1 for each s ∈ S and then (27) can be written p1 · (x1 − ei1 ) r11 · · · r1K z1 . . .. .. ≤ .. ... = R · z, .. . . i pS · (xS − eS ) rS1 · · · rSK zK where R is a return matrix. Theorem 4.7. Suppose that rank R = S or there are at least S assets with linearly independent returns. Then, the set of allocations in ADE is the same as that of allocations in RE. Proof. Refer to the MWG, p. 705. Example 4.8. Suppose that n = 2, I = 2, S = 2, and π 1 = π 2 . Suppose also that u1 = u2 = u and e12 = e21 = (0, 0) while ē = e11 = e22 ≫ 0. In this economy, two assets are available: r1 = (1, 0) and r2 = (0, 1). 34 .α .O2 .E = 12 ē .O1 .α . 1 i∗ In ADE, we must have xi∗ 1 = x2 = 2 ē, i = 1, 2. The corresponding RE is as follows: q1 = q2 = 1 and z21∗ = z12∗ = α for some α > 0 while z11∗ = z22∗ = −α. 35 5 Public Goods and Externality 5.1 Public Goods • There are one private good and one pubic good (PG): Letting y i and xi denote the amount of private and public goods for consumer i ∈ I, – Consumer i’s utility: ui (y i , X), where X = ∑ i∈I xi – Consumer i’s endowment: ei = (wi , 0). – G(y, X) ≥ 0 : Production technology; ∂G ∂y > 0 and ∂G ∂X < 0. – The set of feasible allocations is given as } { ∑ ∑ wi − y i , X) ≥ 0 . F (e) = (y i , X)i∈I G( i∈I i∈I Pareto Efficiency • A Pareto efficient allocation solves max (y 1 ,··· ,y I ,X) u1 (y 1 , X) subject to ui (y i , X) ≥ ūi , ∀i = 2, · · · , I (µi ) ∑ ∑ G( wi − y i , X) ≥ 0. (λ) i∈I i∈I – Assuming an interior solution and letting µ1 = 1, the first-order condition is given as yi : X: ⇒ ∂ui ∂G −λ =0 ∂y ∂y ∑ ∂ui ∂G µi +λ =0 ∂X ∂X i∈I ( ) / ∑ ∂ui / ∂ui ∂G ∂G ⇒λ + =0 ∂X ∂y ∂X ∂y i∈I / ∑ ∂ui / ∂ui ∂G ∂G =− , ∂X ∂y ∂X ∂y i∈I µi which is so called “Samuelson Condition”. 36 (28) Competitive Equilibrium and Inefficiency Normalize the price of private good to 1 and let p denote the price of PG. • An allocation (y i∗ , xi∗ )i∈I is a competitive equilibrium allocation if ∑ (i) xi∗ ∈ arg maxxi ∈[0,wi /p] ui (wi − pxi , xi + j̸=i xj∗ ) ∑ ∑ ∑ (ii) ( i∈I wi − i∈I y i∗ , i∈I xi∗ ) ∈ arg max(y,X) pX − y subject to G(y, X) ≥ 0. – From (i), the first-order condition is given as / ∂G ∂G − =p ∂X ∂y (29) – From (ii), the first-order condition (or Kuhn-Tucker condition) is given as ∂ui ∂ui (−p) + ≤0 ∂y ∂X (30) i∗ = 0 if x > 0. – Combining (30) and (29) yields / / ∂ui ∂ui ∂G ∂G ≤ p=− ∂X ∂y ∂X ∂y / ∂G ∂G = p=− if xi∗ > 0, ∂X ∂y which is different from (28). So. a competitive equilibrium is not PE in general. Lindahl Equilibrium and Efficiency Lindahl suggested an idea that the inefficiency caused by the PG can be fixed by creating a market for each individual consumer. • To explain, take an example with two consumers and the production technology given as G(y, X) = y − cX : – Let each consumer i pay an individualized price q i ∈ [0, 1] for the PG. – So, the producer of PG collects q 1 + q 2 for selling 1 unit of PG, which implies q 1 + q 2 = c at the equilibrium.(why?) 37 – Stipulate that the transaction of PG only occurs if two consumers agree on the amount of PG. – Then, each consumer i solves max ui (y i , X) subject to y i + q i X = wi , (y i ,X) which yields the first-order condition / ∂ui ∂ui ≤ qi ∂X ∂y = q i if X > 0. – In the equilibrium, it must be that ∑ ∂ui / ∂ui = q1 + q2 = c ∂X ∂y i=1,2 so the efficiency is restored. In general case where q = (q 1 , · · · , q I ) be a vector of individualized prices for PG, Definition 5.1. A collection {(q iL )i∈I , (y iL )i∈I , X L }is a Lindahl equilibrium if letting q L = ∑ iL i∈I q ∑ ∑ (i) ( i∈I wi − i∈I y iL , X L ) ∈ arg max(y,X) q L X − y subject to G(y, X) ≥ 0 (ii) For all i ∈ I, (y iL , X L ) ∈ arg max(yi ,X) ui (y i , X) subject to y i +q iL X ≤ wi +θi π(q L ), where π is the (maximized) profit function. Theorem 5.2. A Lindahl equilibrium is Pareto efficient. Proof. Suppose not and then we must have some allocation (ŷ 1 , · · · , ŷ I , X̂) such that ui (ŷ i , X̂) ≥ ui (y iL , xL ), ∀i ∈ I with at least one strict inequality ⇒ ŷ i + q iL X̂ ≥ wi + θi π(q L ), ∀i ∈ I with at least one strict inequality ∑ ∑ wi + π(q L ) ⇒ ŷ i + q L X̂ > i∈I i∈I ∑ ∑ ∑ or ( q iL )X̂ − ( wi − ŷ i ) > π(q L ), i∈I i∈I i∈I a contradiction. 38 5.2 Externality and Lindahl Equilibrium • Suppose that there are two consumers who both consume “nuts” and one of whom, say consumer 1, consumes “smoking”: – B i : Amount of nuts consumed by consumer i; S : Amount of smoking by consumer 1 – wi : Endowment of nuts for consumer i – ui (B i , S) : Consumer i’s utility; ∂ui ∂B > 0, ∀i = 1, 2 and ∂u2 ∂S < 0. – Graphically, .S .S .B .A .O1 . .ω 1 .O2 .ω 2 • In an autarky, – if consumer 1 (2 resp.) has the property right, then B (A resp.) results; – neither A nor B is PE. • The competitive equilibrium allocation is also inefficient if two consumers face the common price for smoking.(why?) • Introduce the Lindahl market where 2 has the property right: 39 – q i : Price of smoking for consumer i – Since the production of smoking takes no cost, we must have q 1 + q 2 = 0 in the equilibrium. – In the Lindahl equilibrium, (B iL , S L ) ∈ arg max ui (B i , S) subject to B i + q iL S ≤ wi . Graphically, .S .S .E .O1 .O2 .A .−1/q 1L = 1/q 2L . – When 1 has the property right, 40 .S .S .B .E ′ .O2 .O1 . 41 6 Social Choice and Welfare • Let X denote the set of social alternatives. – R : Set of all rational preference relations on X – P : Set of all strict preference relations. Then, P ⊂ R – A : Admissible domain of preference profiles. Assume that either A = RI or A = P I that is the largest domains. Definition 6.1. A social welfare function F : A → R is a rule to assign a social preference relation F (%1 , · · · , %I ) ∈ R to any profile of individual rational preference relations (%1 , · · · , %I ) ∈ A. – If xF (%1 , · · · , %I )y, then we say x is socially at least as good as y – If xF (%1 , · · · , %I )y but not yF (%1 , · · · , %I )x, then we say x is socially preferred y, in which case we write xFp (%1 , · · · , %I )y. 6.1 Arrow’s Impossibility Theorem • Consider the following properties for a social welfare function F : – (Weak) Paretian: xFp (%1 , · · · , %I )y whenever x ≻i y, ∀i ∈ I – Independence of irrelevant alternatives (IIA): For any pair of alternatives x, y ∈ X and for any pair of preference profiles (%1 , · · · , %I ) , (%′1 , · · · , %′I ) ∈ A, if for all i ∈ I, x %i y ⇐⇒ x %′i y, then xF (%1 , · · · , %I )y ⇐⇒ xF (%′1 , · · · , %′I )y. – Dictatorial: There is an agent h such that for any x, y ∈ X and any profile (%1 , · · · , %I ) ∈ A, xFp (%1 , · · · , %I )y whenever x ≻h y. Theorem 6.2 (Arrow’s Impossibility Theorem). Suppose that |X| ≥ 3 and that either A = RI or A = P I . Then, every social welfare function that is Paretian and satisfies IIA is necessarily dictatorial. 42 Proof. Refer to pages 796-799 of MGW. Example 6.3 (Pairwise Majority Voting and Condorcet Paradox). The pairwise majority voting (PMV) is defined as follows: xF (%1 , · · · , %I )y if #{i ∈ I : x ≻i y} ≥ #{i ∈ I : y ≻i x}. (31) Though widely used, this social preference is not transitive. Consider the following example: I = {1, 2, 3} and X = {x, y, z}; Preferences are given as x ≻1 y ≻1 z y ≻2 z ≻2 x and z ≻3 x ≻3 y It is easy to check that x is socially preferred to y,y to z, and z to x, violating the transitivity. Example 6.4 (Borda Rule). Given a preference relation %i ∈ R, assign a number ci (x) to each alternative x ∈ X: If x ∈ X is kth ranked in the ordering of %i , then assign ci (x) = k. (If there are ties among alternatives, then assign the average rank of those alternatives.) The Borda rule is defined as follows: xF (%1 , · · · , %I )y if ∑ ci (x) ≤ i∈I ∑ ci (y). i∈I It is easy to check that for any profile (%1 , · · · , %I ), the social preference F (%1 , · · · , %I ) is complete and transitive. Also, F is Paretian. However, F violates IIA, as shown in the following example: I = {1, 2} and X = {x, y, z}; One preference profile is given as x ≻1 z ≻1 y and y ≻2 x ≻2 z while the other as x ≻′1 y ≻′1 z and y ≻′2 z ≻′2 x. Note that the ranking between x and y is preserved across two profiles. However, under the former profile, c(x) = 3 < 4 = c(y) while under the latter, c(x) = 4 > 3 = c(y), violating IIA. 43 6.2 Some Possibility Results: Pairwise Majority Voting in Restricted Domain • Suppose that X is linearly ordered, that is elements of X are ordered as real numbers are ordered. • Given a profile (%1 , · · · , %I ), let F̂ (%1 , · · · , %I ) denote the social preference from the pairwise majority voting, which is defined in Example 6.3. Definition 6.5. The preference relation % is single-peaked if there exists a peak x ∈ X such that if x ≥ z > y, then z ≻ y and if y > z ≥ x, then z ≻ y. Let Rs = {%∈ R : % is single-peaked} and xi denote i’s peak. Definition 6.6. Agent h ∈ I is a median agent (or voter) for the profile (%1 , · · · , %I ) if #{i ∈ I : xi ≥ xh } ≥ I I and #{i ∈ I : xh ≥ xi } ≥ . 2 2 Note that it is always possible to find a median agent. Proposition 6.7. Suppose that each agent has a single-peaked preference. Let h ∈ I be a median agent. Then, xh F̂ (%1 , · · · , %I )y for every y ∈ X. Note: An alternative that is not defeated by any other alternative in the PMV is called a Condorcet winner. So the result says that Condorcet winner always exists and it is the median agent’s peak. 44 Proof. Take any y < xh and we show that y does not defeat xh . (The same argument applies to y > xh .) Consider any agent i ∈ I whose peak xi is larger than or equal to xh . That is, xi ≥ xh > y so the single-peakedness implies xh ≻i y. Since h is a median agent, we have #{i ∈ I : xi ≥ xh } ≥ I 2 so #{i ∈ I : xh ≻i y} ≥ I2 , which means that (31) holds between xh and y. Certainly, the PMV is Paretian and satisfies IIA. (Verify this for yourself.) Also, the social preference relation generated by the PMV is complete. Proposition 6.8. Suppose that I is odd. Then, given any profile (%1 , · · · , %I ) of singlepeaked and strict preferences, F̂ (%1 , · · · , %I ) is transitive. Proof. Suppose that xF̂ (%1 , · · · , %I )y and y F̂ (%1 , · · · , %I )z. Consider a set X ′ = {x, y, z}. Restrict each agent’s preference to X ′ and it is still single-peaked. So there exists a Condorcet winner, which can neither y nor z, and thus has to be x. This let us conclude xF̂ (%1 , · · · , %I )z. 45