General Equilibrium. General Equilibrium, Berardino. Cesi, MSc Tor Vergata

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1 General Equilibrium

2 Equilibrium in Consumption

3 GE begins (1/3) 2-Individual/ 2-good Exchange economy (No production, no transaction costs, full information..) Endowment (Nature): e Private property/ NO Social Planner Redistribution: Voluntary exchange (blocking some trades) SMS SMS

4 GE begins (2/3) Contract curve and Pareto efficiency (exchange equilibrium of the process of voluntary exchange) MRS i, j Not all Pareto efficient allocations are equilibria (i.e. allocation d)

5 GE begins (3/3) What does exchange mean? (no Blocking) Each individual face the following maximization problem:

6 General equilibrium (1/3) : Exchange economy Consumers: i T; wwww T = 1,, I with preferences i over allocations of n goods with endowment e i = e i i 1,, e n and e e 1,, e I Allocations x x i x i i 1,, x n x 1,, x I with consumer i s bundle given by Conditions for allocations to be a barter equilibrium: a) Feasibility: x F e with F e x i T x i = i T e i : assignment of goods not exceed the available amount

7 General equilibrium (2/3) : b) Pareto efficiency: An allocation is Pareto efficient if it is not possible to make someone strictly better off without making someone else worse off A feasible allocation x F e is Pareto efficient if there are no other feasible allocations, y F e, such that y i i x i for all consumers, i, with at least one preference strict The equilibrium must be Pareto efficient, but not all Pareto efficient allocation are equilibria Some Pareto efficient allocations will be blocked

8 General equilibrium (3/3) : c) Unblocked Allocation must be unblocked by any coalition (S) of consumers With I>2, consumers may form coalitions to block proposed allocations S blocks x F e if there is an allocation y such that: i S y i = i S e i y i i x i for all i S, with at least one preference strict The set of all equilibrium allocations (satisfying a)-c)) is defined as the Core of an Exchange Economy (C e ) with endowment e.

9 Perfectly Competitive Market Consumers: Don t know each other (preferences), only prices matter Sufficiently insignificant on every market: price takers (no market power) Price is known Decentralized markets: each agent (seller/buyer) acts in his own selfinterest while ignoring the actions of others Self-interest oriented Individual utility maximizer at that price No production No uncertainty Equilibrium: There exists a price (vector) at which markets for goods k (simultaneously) clear (demand-supply matching) WE show the Existence of this price vector

10 Existence (1/17) Basic assumptions (consumers) i T; wwww T = 1,, I with preferences i on the n consumption set given by the utility function u i n (on ) u i is: 1. Continuous 2. Strongly increasing 3. Strictly quasiconcave At the (given) price vector p p 1,, p n 0 Consumer solves: max u x i n R + i ( x i ) R + s.t. p x i p e i R + with solution x i p, p e i, by 1-3, unique for each p 0 and n continuous in p on n-good/ n-market R ++

11 Existence (2/17) Excess demand functions Aggregate Excess demand function for good k (real) z k i ( ) ( i p x ) k p, p e i I i I z k p > 0: Aggregate demand for good k exceeds the aggregate endowment for good k z k p < 0: excess supply of good k Aggregate excess demand function (vector) e i k z ( p) ( z ( p),..., z ( p) ) 1 n

12 Existence (3/17) Given u i satisfying (1-3), then for all p 0, z p satisfies: Continuity at p Homogeneity of degree zero in prices: z λp = z p λ > 0 Walras law: p z p = 0 Walras law says that at any set of positive prices, the aggregate excess demand is zero (follows from u i strongly increasing and binding budget constraint) Excess demand in the system of markets must be matched by excess supply of equal value, at given prices, somewhere else in the system 2-good case: p 1 z 1 p = p 2 z 2 p z 1 p > 0 z 2 p < 0 z 1 p = 0 z 2 p = 0 By the Walras law if, at same prices, n-1 markets are in equilibrium then the nth market must also be in equilibrium

13 Existence (4/17) We have a general equilibrium when z k p = 0, partial equilibrium in the single market k (demand equal to supply in the market k); Or z p = 0 (demand equal to supply in every market) Walrasian prices p : prices equalizing demand and supply in every market * n p R + + A vector is called Walrasian equilibrium if z p = 0

14 Existence (5/17) THEOREM 1 (Existence) Suppose z satisfying the following conditions: 1) z(.) is continuous on n R ++ 2) p z p = 0 for all p 0 3) if p m n is a sequence of price vectors in R ++ converging to p 0, and p k = 0 for some good k, then for some good k, the associate sequence of excess demands in the market k, z kk p m, is unbounded above Then there is a price vector p 0 such that z p = 0 Condition 3 says that if the prices of some but not all goods are arbitrarily close to zero the excess demand for at least one of those goods is arbitrarily high (to be shown) Proof: We need the fixed-point theorem (Brouwer). Brouwer s fixed-point Theorem Let SϲR n be a non-empty compact and convex set. Let f: S S be a continuous function. Then there exists at least one fixed point of f in S. That is, there exists at least one x S such that x = f(x ).

15 Existence (6/17) f is a continuous mapping from a, b into a, b (itself). Thus, f crosses the 45 line at least once within the square a, b x a, b nothing about uniqueness

16 Existence (7/17) Proof First part: We start by showing the existence of p in a relatively small set of prices, then we will extend this result by extending this set. (hint: keep in mind the fixed point theorem, f: S S) Define: z k p = mmm z k p, 1 for all p 0 (bounded above by 1) and z p = z 1 p,.., z n p n S ε = p p k = 1 aaa p k ε k=1 k with εε 0,1 1+2n Compact (closed and bounded) Convex Non-empty: i.e. p= 2+1 n, n 1 n, 1+2n because ε < 1 εs ε

17 Existence (8/17) For each p S ε and for each k, let s define f k p as: f k p = ε+p k+mmm 0,z k p nε+1+ n m=1 mmm 0,z m p and f p = f 1 p,, f n p n Then k=1 f k p = 1 and f k p f k p ε 1+2n given ε < 1( ε nn+1+n 1 < f is continuous function mapping the set S ε into itself Is f k p continuous on S ε? Check: z k p is continuous on S ε (the same holds for z k p ) ε nn+1+n 1 because z m p 1 m ε 1+2n ). This implies f: S ε S ε denomiator and numerator of f k p are continuous on S ε denominator bounded away from zero (at least equal to 1)

18 Existence (9/17) Assume n=2, then f k p ε 1+2n becomes 1 > f k p ε 5 f maps S ε into S ε (it recalls the fixed point theorem f: S ε S ε ) f p 1 S ε f p ε 5 ε 5 p 1 p S ε

19 Existence (10/17) We apply FPT: p ε S ε f p ε = p ε or f k p ε =p k ε Then by applying the above result to the definition of f p (and rearranging), we have: i) p ε k nε n + m= 1 max ( ( ε 0, z p ) = ε + max( 0, z ( p ε ) m But the proof is not over yet.what do we know so far? Simply: a price vector p ε satisfying (i) for every εε 0,1 k

20 Existence (11/17) Second part: let ε approach to zero, then it gives a sequence of price vector p ε satisfying (i). Check out: sequence p ε is bounded because p ε S ε (every price is between 0 and 1). Thus p ε must converge. Assume to pick up the sequence p ε converging to a vector defined (by chance) as p What we do know about this vector: since all p k sum up to 1, then we must have p 0 with p 0. We need to show that p 0 For this proof we use condition (3).

21 Existence (12/17) We show that p 0 by contradiction: assume p 0 does not hold Then for some k we must have p * k = 0 But by condition (3) there must exist some k * with p such that z kk p ε k ' = 0 is unbounded above (remind: we have ε 0) Since p ε p * ε, p 0 implies 0, therefore for some k=k i) p ε k nε n + m= 1 k ' = max p k ' ( ( ε 0, z p ) = ε + max( 0, z ( p ε ) m zero bounded above 0 k By unboundedness of z kk p ε (from condition (3)). CONTRADICTION: 0 (positive value)=ε + 1 Cannot exist a NON strictly positive price vector p for market k (then neither for k)

22 Existence (13/17) We have shown that p ε p 0 as ε 0. Last step: showing the second part of the statement, z p We use the continuity of z. on n R ++ obtained from z(.) = 0 lim ε 0 multiplying by z k (p ) and summing over k p * ε ε ( 0, z ( p ) = ε + max( 0 z ( p ) n ε p k nε + max m, m= 1 * k n m= 1 ( ( * ) ( ( * 0, z p max 0, z p ) p max = m k k for all k. ( *) ( ( * ) ( *) ( ( * p max 0, z p = z p max 0, z p ) z n m= 1 m n k = 1 k k By the Walras law the LHS is zero, then together with p 0 implies z k p = 0.

23 Existence (14/17) Let s check when aggregate excess demand z p satisfies (3) Theorem When u i is Continuous, Strongly increasing and Strictly quasiconcave and I i=1 e i 0 then z p satisfies (1)-(3). Proof 1) Let s focus on (3). Consider the sequence of strictly positive price vectors p m p 0 such that p k =0 for some good k. From I i=1 e i 0 we have p I i=1 e i > 0, then p I i=1 e i = I i=1 p e i > 0. Thus there must be at least one consumer i with p e i >0. 2) Part. By contradiction we assume the demand of consumer i along the sequence p m, x i p m, p m e i, is bounded : Then it must converge and assume x i p m, p m e i x m x Since individual is utility (strongly increasing) maximizer (under constraint) we have: p m x m = p m e i for every m, with: Budget constraint: lll m (p m x m = p m e i ) p x = p e i >0

24 Existence (15/17) Let s introduce x = x + 0,, 0, 1 k, 0,, 0, since u i is n strongly increasing on we have: 1) u i x > u i x Recalling that = 0, that implies: 2) p x = p e i >0 p k R + By continuity of u i we have that (1) e (2) imply, t 0,1 u i tx > u i x p tx < p e i Because p m p, x m x and continuity of u i, for large m u i tx > u i x m p m tx < p m e i.contradicting that x m solve the maximization problem at p m Thus sequence of demand x m p m, p m e i is unbounded There exist some good k such that m { } x k ' is unbounded above.

25 Existence (16/17) Because i s income converge to p e i then the sequence of i s income p m e i is bounded. The condition for the demand of k to be unbounded above and m affordable is Then: Thus: p k ' 0 m p k = lim p ' m k ' = Aggregate supply for k is fixed (endowment) 0 All consumers demand a non-negative amount of k then because consumer i s demand for k is unbounded above...aggregate excess demand for k is unbounded above along the sequence p m.

26 Existence (17/17) Theorem for the Existence of GE (revised conclusion) When u i is i) continuous, ii) strong increasing and iii) strictly n quasiconcave on and I R i=1 e i + 0 then there exists at least one price vector, p 0, such that z p = 0 Intuition: We know that i)-iii) make z(.) continuous, homogeneous of degree zero and respecting Walras s law I i=1 e i 0 makes z(.) satisfy (3) in theorem 1.

27 WE and Pareto efficiency (1/6) WEA: Pareto efficient allocation

28 WE and Pareto efficiency (2/6) Is a WEA Pareto efficient? Is the feasible bundle received in the WEA the most preferred in the consumer s budget set at the Walrasian prices? + Is the set of WEA on the Contract Curve? = Pareto Efficiency

29 WE and Pareto efficiency (3/6) Definition of WEA Let p be a Walrasian equilibrium for some economy with initial endowment e, and let: x p x 1 p, p e 1,, x I p, p e I, Then x p is called a Walrasian equilibrium allocation (WEA). a) A WEA must be (clearly) feasible Formally: Let p be a Walrasian equilibrium with initial endowment e. Let x p be the associate WEA. Then x p F e b) A WEA must be the most preferred (any other WEA feasible and preferred must be too expensive) Formally-Lemma 1: Assume u i n is strictly increasing on R, that consumer s demand at p 0 is x i, and that x i n + R + i. If u i x i > u i x i, then p x i > p x i ii. If u i x i u i x i, then p x i p x i

30 WE and Pareto efficiency (4/6) Proof: Part i, assume (ii) holds. Proof by contradiction on (i). Assume (i) does not hold: Assume u i x i > u i x i, then p x i < p x i It is possible a small increase on x i toward x i still lower than x i But this would imply u i x i > u i x i > u i x i and p x i < p x i Replacing x i with x i contradicts (2) Part 2, see the book.

31 WE and Pareto efficiency (5/6) W e : Set of WEA No uniqueness

32 WE and Pareto efficiency (6/6) and about the Pareto efficiency (remind the CC and the Core)? Consider an exchange economy u i, e i increasing on, then W e C e i T. If ui is strictly Proof. By contradiction. Assume W e C e. Since x p is WEA then x p F e (is feasible). However: x p n R + C e a (blocking) coalition S and allocation y such that: 1) y i i S = e i i S and 2) u i y i u i x i p, p e i for all i S, with at least one inequality strict. Also we have: (i) p i S y i = p i S e i By 2) and Lemma 1, for each i S, we must have p y i p x i p, p e i =p e i (with at least one strict) Summing over all i in S: p i S y i > p i S e i (that contradicts (i)) (at least one must get more than the endowment to be a member of the coalition)

33 Welfare (1/4) From the previous results: Core is a non-empty set; all Core allocations are Pareto efficient. Thus 1 Welfare Theorem: every WEA is Pareto Efficient Trade off: efficiency (Pareto) vs. Equity

34 Welfare (2/4) 2 Welfare Theorem: Consider an economy u i, e i with i T aggregate endowment I i=1 e i 0, and u i continuous, strong increasing and strict quasiconcave in n R. Suppose that x is Pareto efficient for u i, e i + and the endowment is i T redistributed so that the new endowment is x. Then x is a WEA of the resulting exchange economy u i, x i i T Graphical proof (intuition): assume by contradiction that x was not a WE, but it would contradict Pareto efficiency

35 Welfare (3/4) Formal Proof What we have: if x is Pareto efficient, then it must be feasible in the economy u i, x i, that is I i T i=1 x i = i=1 e i 0, then this economy has a WEA, let s say x. What is remained to show: x = x Since x is WEA, then: x i is the consumer i demand function with endowment x i, then this implies: a) u i x i u i x i for all i T x must be feasible for the economy u i, x i i T I i=1 x i = I i=1 e i = i=1 I x i Then x is feasible for the original economy u i, e i i T But then every inequality in (a) (for every i) must be binding, otherwise x cannot be Pareto efficient. u i x i = u i x i must implies x i = x i because otherwise by strict quasiconcavity of u i, consumer would choose x α = αx i + 1 α x i with u i x α > u i x i = u i x i, contradicting that x i is utility-maximizing in the WE.

36 Welfare (4/4) Corollary of 2 WT: if x is Pareto efficient then x is a WEA for some Walrasian equilibrium p after redistribution of initial endowment to any allocation e F e, such that, p e i = p x i for every i I Redistribution + free market=pareto efficiency Less equity concerns

37 Equilibrium in Production

38 Producers (1/3) Fixed number of firm, I 1,.., J Production plan of each firm: y j j Input y k < 0 output > 0 j y k n R + Production possibility set Y j, j I 0 Y j R n Y j is closed and bounded Y j is strongly convex No constant and increasing return to scale (profitmaximizing production plant is unique)

39 Producers (2/3) Firm j s maximization problem: Since objective function is continuous and the constraint set is bounded and closed, a maximum for the profit function will exist, given by: The profit maximizing production plan, y j Unique by strong convexity, whenever p 0 n Continuous on Π j ( p) Π is continuous on j R ++ maxp y y j Y j j ( p) = maxp y y n R + j Y j j p, is: Firm s supply function (Remind Production Theory)

40 Producers (3/3) Aggregate production possibilities Y y y = y j j I, wwwww y j Y j Y respects all the assumptions on Y j. Aggregate profit p y has a unique maximum over Y when p 0 Define y p : aggregate profit-maximizing plan (continuous in p) Theorem: Aggregate profit maximization (y mmmmmmmmm AA ) For any price p 0, we have: p y p y for all y Y If and only if for some y j Y j, j I, we may write i) y = j I y j, and ii) p y j p y j for all y j Y j, j I Intuition: y maximizes aggregate profit iff it can be decomposed into individual firm profit-maximizing production plans

41 Consumers Consumers may supply goods and services by labour-commodity Consumer i s share in firm j (profit) 0 θ ii 1, with i T = 1 for all j I Consumer i s maximization problem: θ ii max u x i n R + ( i x ) s. t.: p x i p e i ( p) = m ( p) Solution x i p, m i p (unique) exists whenever p 0: Continuous in p (because m i p is continuous in p) on i + j J ij θ Π j n R ++

42 Equilibrium Let s define the refined economy u i, e i, θ ii, Y j i T,j I Aggregate excess demand for commodity k: z k i i ( p ) x p, m ( p) i I k ( ) j y ( ) k p j J i I e i k And the aggregate excess demand z ( p) ( z ( p),..., z ( p) ) 1 Theorem. If u i is continuous, strongly increasing and quasiconcave, 0 Y j R n, Y j is closed, bounded and strong convex, y + i T e i 0 for some y j I Y j, then there exist at least a price p 0 such that z p = 0. n

43 Example (1/5) 1-consumer, 1-firm economy Y = h, y 0 h b, aaa 0 y h α Where b>0 and αα 0,1 Y is closed, bounded, strongly convex and includes 0 Consumption, 2-good case 2 R + u h, y = h 1 β y β ; βε 0,1 Endowment (hours): e T, 0, with b>t (h<b)

44 Example (2/5)

45 Example (3/5) Solution (find p 0 or (w, p )) Firm s supply function Consumer s demand function Market clearing condition Firm: Where y = h α and h > 0. Foc: αph α 1 w = 0 h f = αp w max y py 1 1 α ; y f = αα w max h ph α 1 α ; π w, p α wh = 1 α α w αα w 1 1 α

46 Example (4/5) Consumer max u h, y ( h, y) s.t. pp + ww = ww + π w, p (net of profit) In fact from the endowment: w, p T, 0 Solutions: h c = 1 β ww+π w,p w ; y c = β ww+π w,p p = ww Since WE prices are positive and Excess demand is homogeneous of degree zero we assume p = 1 By Walras law we only need to find w ssss tttt h c + h f = T: w = α 1 β 1 α ααt 1 α > 0

47 Example (5/5) w p B: Technologically possible but infeasible (more than T) C: Feasible but not utility maximizer for the consumer C Pareto Efficient WEA

48 Welfare with production (1/7) WEA Pareto efficient in the economy u i, e i, θ ii, Y j i T,j I WEA is feasible: x, y = x 1,, x I, = y 1,, y j is feasible if x i for all i, y j Y j for all j, and i T x i = e i + y i 1 Welfare Theorem with Production: If u i is strictly increasing n on then every WEA is Pareto Efficient. R + i T j I n R +

49 Welfare with production (2/7) Proof. By contradiction. Assume x, y is a WEA at p but not Pareto efficient. The WEA is also feasible: i T x i = i T e i + j I y j Pareto inefficiency implies (x, y ), feasible, such that: u i x i u i x i with at least one with strict inequality, and p x i p x i Summing over i: p i T x i > p i T x i (at least one gets more) By feasibility: p i T e i + j I y j > p i T e i + j I y j After rearranging, p j I y j > p j I y j, for some j this would imply: p y j > p y j, but this contradicts that in the WE y j maximizes firm j s profit at p

50 Welfare with production (3/7) Is a Pareto efficient allocation also WEA? 2 Welfare Theorem with production: Suppose an economy with: a) u i continuous, strong increasing and strict quasiconcave b) 0 Y j R n, Y j closed, bounded and strongly convex c) y + i T e i 0 for some aggregate production vector y d) A Pareto efficient allocation (x, y ) Then there exist income transfers T 1,, T I T i = 0 i T and p such that: 1. x i maximizes u i x i s.t. p x i m i p + T i, i T 2. y i maximizes p y i s.t. y j Y j, j I

51 Welfare with production (4/7) Proof Let s construct: 1. a new production set for each firm j still satisfying (b): Y j Y j y i 2. A new economy E = u i, x i, θ ii, Y j i T,j I Lemma: since the original economy E = u i, e i, θ ii, Y j, has a WE, then also E satisfies all the necessary conditions for the existence of the WE with production (see revised theorem for the existence). Thus economy E has a WEA x, y at p 0. Next step: check whether x, y can also be a WE of the original economy,e = u i, e i, θ ii, Y j, first we need to show x i =x i. Let s use feasibility. Remind: x, y is feasible for the original economy by definition Is x,? also feasible for the original economy? Let s check. Each consumer can afford his endowment because 0 Y j implies that equilibrium profits of every firm j are non-negative. Thus: Consumption side: u i x i u i x i

52 Welfare with production (5/7) Production side We now show that a x, y is feasible also in the original economy Note that for now we have: a) y j Y j is such that y j =y j -y i for some y j Y j (see the definition of Y j ) b) x, y is feasible in E because it is a WEA in E a) + feasibility (a+b) imply: i T x i = i T x i + j I y j = i T x i + j I y j y j = i T x i j I y j + j I y j = i T e i + j I y j y is feasible in the original economy E. (still not enough ). Then x, Since x, y is Pareto efficient, then we must have: u i x i = u i x i By quasiconcavity it implies: x i =x i

53 Welfare with production (6/7) Thus we have: x i maximizes u i x i s.t. p x i p x i + j I θ ii p y j u i x i is strongly increasing then budget constraint equalizes at x i =x i. This implies: j I θ ii p y j =0 (zero profit condition for each firm j). y j =0 maximizes firm j s profits at p under Y j in E Since Y j Y j y j then y j maximizes p y j s.t. y j Y j, j I production side OK: the pareto efficient vector y j characterize a WE also in the original economy. Transfer side (T): Back to the consumer side. We have shown that: (*) x i maximizes u i x i s.t. p x i p x i Note that the transfer at the original economy to get (*) is such that T i +m i p p x i or T i p x i -m i p, with consumer s income at the original economy: m i p = p e i + θ ii j I p y j

54 Welfare with production (7/7) Last part to show: transfers must sum up to zero. i T T i = i T p x i i T p e i i T j I θ ij p y j From i T = 1 i T T i = i T p x i i T p e i j I p y j θ ii See that feasibility of x, y implies: i T p x i = i T p e i + j I p y j i T T i =0

Notes on General Equilibrium

Notes on General Equilibrium Alejandro Saporiti Alejandro Saporiti (Copyright) General Equilibrium 1 / 42 General equilibrium Reference: Jehle and Reny, Advanced Microeconomic Theory, 3rd ed., Pearson