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Advanced Microeconomics II Advanced Microeconomics II by Jinwoo Kim October 6, 2010 Contents I General Equilibrium and Social Welfare 3 1 General Equilibrium Theory 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 ..................... 29 4 General Equilibrium under Uncertainty 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 e¯ = (¯e1; ··· ; e¯n) 2 Rn +: Consumers • Assume that there are I consumers: – I = f1; ··· ;Ig : Set of consumers. i i i ··· i 2 i i – xk : Consumer i’s consumption of good k, x = (x1; ; xn) X ; where X is Rn i’s consumption set and assumed to be +. – ei : i’s endowment for good. Let ei = (ei ; ··· ; ei ) and e = (e1; ··· ; eI ). Note k P 1 n i that e¯k = i2I ek: – ≽i: i’s preference defined on Xi, which we assume can be represented by a function ui : Xi ! R. • Assumption C: For any i 2 I; (i) ui is continuous. i 2 Rn (ii) u is strictly quasiconcave: For all x; y +; ui(λx + (1 − λ)y) > minfui(x); ui(y)g; 8λ 2 (0; 1): i (iii) u is strictly monotone: If x > y (that is xk ≥ yk; 8k and xk > yk for some k), then ui(x) > ui(y): 3 Producers • Assume that there are J producers: – J =f1; ··· ;Jg: Set of producers. j j j ··· j 2 Rn – yk : Firm j’s output (or input) for good k, y = (y1; ; yn) : j’s production plan. ∗ j If yk > (<)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 yj 2 Y j feasible. ∗ Assume that there exists a function F j : Rn ! R such that Y j = fy 2 Rn j F j(y) ≤ 0g: ∗ fy 2 Rn j F j(y) = 0g : Production possibility frontier for firm j. P ≡ f j j 2 j 8 2 J g – Y y y = j2J y ; where y Y ; j : Aggregate production possibil- ity set. • Assumption P: For any j 2 J ; (i) 0 2 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. y. 2 . Production . Possibility Set y. 1 O. 4 Feasible Allocations • 1 ··· I 1 ··· J 2 RnI × An allocation in this economy is denoted by (x; y) = (x ; ; x ; y ; ; y ) + RnJ : • Given e = (e1; ··· ; eI ); an allocation (x; y) is feasible if X X i ≤ j 8 ··· xk e¯k + yk; k = 1; ; n: (1) i2I j2J • 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) 2 F (e; Y ) is Pareto efficient (PE) if there does not exist any other allocation (~x; y~) 2 F (e; Y ) such that ui(~xi) ≥ ui(xi); 8i 2 I with at least one strict inequality. • With fixed utility levels, u¯2; ··· ; u¯I , let us solve the following maximization problem: 1 1 ··· 1 max u (x1; ; xn) (2) 2RnI ×RnJ (x;y) + s.t. ui(xi ; ··· ; xi ) ≥ u¯i; i = 2; ··· ;I (3) X 1 n X i ≤ j ··· xk e¯k + yk; k = 1; ; n (4) i2I j2J j j ··· j ≤ ··· F (y1; ; yn) 0; j = 1; ;J (5) – One can show the following: An allocation (x; y) is Pareto efficient if and only if it solve (2) for some utility levels, u¯2; ··· ; u¯I . 5 First Order Conditions for Pareto Efficiency i j • Letting δ ,µk; and γ denote a nonnegative multiplier for (3), (4), (5), respectively, the Lagrangian function for problem (2) can be set up as XI L 1 1 ··· 1 i i i ··· i − i =u (x1; ; xn) + δ [u (x1; ; xn) u¯ ] i=2 Xn X X j − i + µk[¯ek + yk xk] k=1 j2J i2I XJ j − j j ··· j + γ [ F (y1; ; yn)]: j=1 – Define δ1 ≡ 1. Assuming the interior solution (i.e. xi > 0; 8i), the first order conditions for maximizing the Lagrangian are @ui i 8 ir i 8 δ i = µk; i; k or δ u = µ, i (6) @xk @F j j 8 jr j 8 γ j = µk; j; k or γ F = µ, j; (7) @yk r i @ui ··· @ui r j @F j ··· @F j ··· where u = ( @xi ; ; @xi ), F = ( j ; ; j ); and µ = (µ1; ; µn): 1 n @y1 @yn – The conditions (6) and (7) imply the followings: i i i0 i0 µk @u =@xk @u =@xk 0 0 = = 0 0 for all i; i ; k; k 0 i i i i µk @u =@xk0 @u =@xk0 that is, MRS for any pair of goods must be equalized across consumers. 0 j j j0 j µk @F =@yk @F =@yk 0 0 = j = 0 0 for all j; j ; k; k µ 0 j j j k @F =@yk0 @F =@yk0 that is, MRT for any pair of goods must be equalized across producers. @ui=@xi @F j=@yj k k for all 0 i i = j i; j; k; k @u =@x 0 j k @F =@yk0 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. j i i Y = f0g; 8j 2 J . We denote an exchange economy by E = (u ; e )i2I: 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. • ··· 2 Rn Suppose that price of good k is given as pk > 0 with p = (p1; ; pn) ++ being a price vector. P · i n i – p e = k=1 pkek: Market value of i’s endowment i.e. i’s wealth. – Consumer i’s budget set is i f i 2 Rn j · i ≤ · ig B (p) = x + p x p e : • 2 Rn Given market price vector p ++; each consumer i has to solve max ui(xi) s.t. xi 2 Bi(p): (8) i2Rn x + i · i i · i ··· i · i – Let x (p; p e ) = (x1(p; p e ); ; xn(p; p e )) denote the optimal bundle/bundles (or demand function/correspondence) that solves (8). i · i Rn – Note that x (p; p e ) is not continuous in p in all of + since demand will be infinite if some price is zero. • The aggregate excess demand function for good k is defined as X ≡ i · i − zk(p) xk(p; p e ) e¯k: i2I 7 – If zk(p) > (<)0, we say that good k is in excess demand (supply). – Define z(p) ≡ (z1(p);; ··· ; zn(p)). ∗ 2 Rn ∗ Definition 1.3. Walrasian equilibrium (WE) is a price vector p ++ such that zk(p ) = 0; 8k: 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 x~ such that P P i i (i) i2S x~ = i2S e (ii) x~i ≽ xi for all i 2 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 x~ such that X X x~i = ei and ui(~xi) ≥ ui(xi); 8i 2 S with at least one inequality strict. i2S i2S By the strict monotonicity, this implies p∗ · x~i ≥ p∗ · ei; 8i 2 S with at least one inequality strict; 8 which can be added up across consumers to yield X X p∗ · x~i > p∗ · ei: i2S i2S P P i i This, however, contradicts with i2S x~ = i2S e .
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