Market Equilibrium and the Core

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1 Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, / 19

2 Market Exchange: Basics Let us introduce price in our pure exchange economy. Let, Assume There be N individuals and M goods e i = (e1 i,..., ei M ) denote endowment for individual i p i denote the price of ith good; p i > 0 for all i = 1,.., M. So the price vector is p = (p 1,..., p M ) >> 0. each good has a market and each individual is price-taker. For each individual, Total value of the initial endowment depends on the price vector An economic agent can buy any bundle of goods However, the total value of the bundle bought cannot exceed the total value of her endowment. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

3 Market Exchange: 2 2 economy I For person 1, the set of feasible allocations/consumptions is the set of y 1 = (y1 1, y 2 1 ) such that: p 1 y p 2 y 1 2 p 1 e p 2 e 1 2. Assuming monotonic preferences, Person 1 maximizes utility by choosing bundle x 1 = (x 1 1, x 1 2 ) s.t. p 1 x1 1 + p 2 x2 1 = p 1 e1 1 + p 2 e2 1 Person 2 maximizes utility s.t. p 1 x1 2 + p 2 x2 2 = p 1 e1 2 + p 2 e2. 2 Recall, within the Edgeworth box, for each allocation (x 1, x 2 ), we have x1 1 + x1 2 = e1 1 + e1, 2 and x2 1 + x2 2 = e2 1 + e2. 2 Ram Singh (DSE) Market Equilibrium September 22/25, / 19

4 Market Exchange: 2 2 economy II Note: The budget line for person 2 is: p 1 x1 2 + p 2x2 2 = p 1e1 2 + p 2e2 2. However, p 1 e1 2 + p 2 e2 2 = p 1 x1 2 + p 2 x2 2, i.e., p 1 e1 2 + p 2 e2 2 = p 1 (e1 1 + e1 2 x1 1 ) + p 2 (e2 1 + e2 2 x2 1 ), i.e., 0 = p 1 (e1 1 x1 1 ) + p 2 (e2 1 x2 1 ), i.e., p 1 x1 1 + p 2 x2 1 = p 1 e1 1 + p 2 e2, 1 which is the budget line for the 1 person. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

5 Preferences and Utilities: Assumptions We assume: Preference relations to be continuous, strictly monotonic, and strictly convex The utility functions to be continuous, strictly monotonic and strictly quasi-concave However, several of the results will hold under weaker conditions. Question What is the role of assumption that the utility functions are strictly quasi-concave? Let u 1 (.) denote the utility function for person 1 u 2 (.) denote the utility function for person 2 Ram Singh (DSE) Market Equilibrium September 22/25, / 19

6 Competitive Equilibrium: 2 2 economy I An allocation is ˆx = (ˆx 1, ˆx 2 ) along with a price vector p = (p 1, p 2 ) is competitive equilibrium, if 1 ˆx 1 = (ˆx 1 1, ˆx 1 2 ) maximizes u1 (.) subject to p 1 x p 2x 1 2 = p 1e p 2e ˆx 2 = (ˆx 2 1, ˆx 2 2 ) maximizes u2 (.) subject to p 1 x p 2x 2 2 = p 1e p 2e ˆx ˆx 2 1 = e1 1 + e2 1 4 ˆx ˆx 2 2 = e1 2 + e2 2 For well-behaved utilities: 1. Implies : In equi. IC of person 1 will be tangent to her budget line. 2. Implies : In equi. IC of person 2 will be tangent to his budget line We know that: both of the demanded bundles, i.e., ˆx 1 and ˆx 2 lie on the same line. Why? 3 and 4 imply that the demanded bundles, i.e., ˆx 1 and ˆx 2 coincide. Why? Ram Singh (DSE) Market Equilibrium September 22/25, / 19

7 Competitive Equilibrium: 2 2 economy II Therefore, at the equilibrium allocation, ˆx = (ˆx 1, ˆx 2 ), the ICs are tangent to each other Therefore, the equilibrium allocation ˆx = (ˆx 1, ˆx 2 ) is Pareto Optimum. Question Suppose, ˆx = (ˆx 1, ˆx 2 ) is a Competitive (market) equilibrium allocation Are unilateral deviations from ˆx = (ˆx 1, ˆx 2 ) profitable? Does the eq. allocation ˆx = (ˆx 1, ˆx 2 ) belong to the core? Ram Singh (DSE) Market Equilibrium September 22/25, / 19

8 Competitive Equilibrium: N M economy I Consider a N M economy denoted by (u i (.), e), where e = (e 1, e 2,..., e N ). An allocation ˆx = (ˆx 1,..., ˆx N ) along with a price vector p = (p 1,..., p M ) is a competitive equilibrium, if the following conditions are satisfied: First: For each i = 1,..., N, ˆx i maximizes u i (.), subject to p.x i = p.e i. That is, ˆx i solves max x i {u i (x i )} (1) subject to p 1 x i p Mx i M = p 1e i p Me i M. Second: For all j = 1,..., M N i=1 ˆx i j = N ej i (2) i=1 Ram Singh (DSE) Market Equilibrium September 22/25, / 19

9 Competitive Equilibrium: N M economy II Definition (ˆx; p), i.e., (ˆx 1,..., ˆx N ; p) is called a Competitive or Walrasian equilibrium, if (ˆx i, p) together satisfy (1) and (2) simultaneously, for all i = 1,..., N. Definition The set of Walrasian/Competitive Equilibria, W (u i (.), e i ) N M, is given by W (u i (.), e i ) N M = {x = (x 1,..., x N ) p such that (x i, p) satisfy (1) and (2), } simultaneously, for all i = 1,..., N. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

10 Competitive Equilibrium: N M economy III Remark We will show that: Walrasian/Competitive equilibrium may not exist. However, If utilities fns are continuous, strictly increasing and strictly quasi-concave, there does exist at least one equilibrium. In general there can be more than one Competitive equilibrium. Walrasian/Competitive equilibrium depends on the vector of initial endowments, i.e., e. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

11 Some Observations I Let ˆx = (ˆx 1,..., ˆx N ) be a Competitive equilibrium allocation. Proposition Suppose, (ˆx, p) is a competitive equilibrium. Then, ˆx = (ˆx 1,..., ˆx N ) is a feasible allocation. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

12 Some Observations II Proposition Suppose, (ˆx, p) is a competitive equilibrium. Take a bundle y i. If u i (y i ) > u i (ˆx i ), then p.y i > p.e i. Formally, u i (y i ) > u i (ˆx i ) p.y i > p.e i, i.e., M M u i (y i ) > u i (ˆx i ) p j yj i > p j ej i j=1 j=1 Definition Consider any x = (x 1,..., x M ) and y = (y 1,..., y M ) such that y j > x j for all j = 1,..., M. Function u i is monotonic if u i (y) > u i (x). Ram Singh (DSE) Market Equilibrium September 22/25, / 19

13 Some Observations III Proposition Suppose, (ˆx, p) is a competitive equilibrium, and u i is monotonic. Take a bundle y i. If u i (y i ) u i (ˆx i ), then p.y i p.e i. Formally, u i (y i ) u i (ˆx i ) p.y i p.e i i.e., p.y i < p.e i u i (y i ) < u i (ˆx i ) Ram Singh (DSE) Market Equilibrium September 22/25, / 19

14 Competitive Equilibrium and Core I Let W (u i (.), e i ) N M denote the set of Walrasian/competitive allocations. C(u i (.), e i ) N M denote the set of Core allocations. For a 2 2 economy, suppose an allocation ˆx = (ˆx 1, ˆx 2 ) along with a price vector p = (p 1, p 2 ) is competitive equilibrium. Then, Individual i prefers x i at least as much as e i Indifference curves of the individuals are tangent to each other Allocation ˆx = (ˆx 1, ˆx 2 ) is Pareto Optimum In view of the above, allocation ˆx = (ˆx 1, ˆx 2 ) is in the Core. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

15 Competitive Equilibrium and Core II So, for a 2 2 economy, x W (u i (.), e i ) x C(u i (.), e i ). Theorem Consider an exchange economy (u i (.), e i ) N M, where individual preferences are monotonic, i.e., u i is increasing. If x is a WEA, then x C(u i (.), e i ) N M. Formally, W (u i (.), e i ) N M C(u i (.), e i ) N M. Proof: Take any x WEA. Let, x along with the price vector p be a WE. Suppose x C(e). Therefore, there exists a blocking coalition against x. That is, Ram Singh (DSE) Market Equilibrium September 22/25, / 19

16 Competitive Equilibrium and Core III there exists a set S N and an allocation say y, s.t. y i = e i (3) i S i S Moreover, u i (y i ) u i (x i ) for all i S (4) and for some i S u i (y i ) > u i (x i ). (5) (3) implies p. i S y i = p. i S e i (6) Ram Singh (DSE) Market Equilibrium September 22/25, / 19

17 Competitive Equilibrium and Core IV (4) implies p.y i p.x i = p.e i, for all i S (7) (5) implies: for some i S p.y i > p.x i = p.e i. (8) (7) and (8) together give us: p. y i > p. e i (9) i S i S But, (6) and (9) are mutually contradictory. Therefore, x C(e). Ram Singh (DSE) Market Equilibrium September 22/25, / 19

18 Competitive Equilibrium and Pareto Optimality So, we have proved the First Fundamental Theorem of Welfare Economics: Theorem Consider an exchange economy (u i, e i ) i {1,..,N}, where u i is strictly increasing, for all i = 1,.., N. Every WEA is Pareto optimum. Ram Singh (DSE) Market Equilibrium September 22/25, / 19

19 Competitive Equilibrium: Merits and Demerits Question Is the price/market economy better than the barter economy, in terms of its functioning? Is the price/market economy better than the barter economy, in terms of the outcome achieved? Question What are the limitations of a market economy? Can these limitations be overcome? Ram Singh (DSE) Market Equilibrium September 22/25, / 19

Fundamental Theorems of Welfare Economics

Fundamental Theorems of Welfare Economics Ram Singh Lecture 6 September 29, 2015 Ram Singh: (DSE) General Equilibrium Analysis September 29, 2015 1 / 14 First Fundamental Theorem The First Fundamental