In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics 1 44715 (1396-97 1 st term) - Group 1 Dr. S. Farshad Fatemi Chapter 10: Competitive Markets
So far we studied how individual consumers and firms make their decisions. This chapter considers an economy in which the consumers and firms interact with each other while they compete for goods and services. The main assumption here is that the economic agents are price-takers (remember this was one of our main assumptions in Microeconomics 1). Microeconomics 1 Dr. F. Fatemi Page 2
Consider an economy: Consumers: Goods: Consumer i s consumption bundle: i = 1,, I l = 1,, L x i = (x 1i,, x Li ) R L Consumer i s utility function: u i (. ) Initial endowment of good l: ω l 0 Firms: Firm s production vector: Consumer i s share from firm s profit: = 1,, J y = y 1,, y L R L θ i i θ i = 1 Price vector: p = (p 1,, p L ) R L Microeconomics 1 Dr. F. Fatemi Page 3
Pareto Optimality and Competitive Equilibria Pareto, an Italian economist, in early 20 th Century developed one of the most fundamental notions of efficiency in economics. Definition (MWG 10.B.1): An economic allocation x 1,, x I, y 1,, y J is feasible if x li i ω l + y l for l Microeconomics 1 Dr. F. Fatemi Page 4
Definition (MWG 10.B.1): A feasible allocation x 1,, x I, y 1,, y J is Pareto optimal if there is no other feasible allocation x 1,, x I, y 1,, y J such that i u i (x i ) u i (x i ) and i u i (x i ) > u i (x i ) Note: Pareto optimality is ust a positive measure of efficiency not a normative one. It says nothing about the desirability of allocation from distribution point of view. Checking for Pareto optimality ensures us that there is no waste in allocation and the economy is at an efficient point. Microeconomics 1 Dr. F. Fatemi Page 5
Definition (MWG 10.B.1): The allocation x 1,, x I, y 1,, y J and the price vector p R L constitute a competitive (or Walrasian) equilibrium if the following conditions are satisfied: - Profit maximization: y argmax y Y p. y - Utility maximization: argmax u i (x i ) ; i x x i X i i s. t. p. x i p. ω i + θ i p. y - Market clearing: l x li i = ω l + y l Microeconomics 1 Dr. F. Fatemi Page 6
As it is clear what is important about a market clearing price vector is price ratios. In other words, if the allocation x 1,, x I, y 1,, y J and the price vector p is a competitive equilibrium, the same is true for x 1,, x I, y 1,, y J and αp for α > 0. Lemma (MWG 10.B.1): If the allocation x 1,, x I, y 1,, y J and price vector p 0 satisfy the market clearing condition for L 1 goods and all consumers budget constraints with equality; then the market clears for the last good as well. Microeconomics 1 Dr. F. Fatemi Page 7
Partial Equilibrium Competitive Analysis Partial equilibrium studies the market for one (or several goods). The market for that good is considered to constitute a small fraction of economy. This assumption is useful on two fronts: 1) When the consumers expenditure on this market is a small fraction of the total expenditure, we can expect the wealth effect to be small. 2) Again because of the size of this market, the prices of other goods remain approximately unaffected by changes in this market. Microeconomics 1 Dr. F. Fatemi Page 8
We study a two good market: good l consumer i s consumption from it is denoted by x i the numeraire (all other goods); with consumption m i for consumer i Consumer i s utility function takes the quasilinear form: u i (. ): R R + R & u i (x i, m i ) = m i + φ i (x i ) φ i (. ) is bounded above, φ i (0) = 0, and x i φ i (x i ) > 0, φ i (x i ) < 0 The price of numeraire is normalized to 1 and the price of good l is p. Microeconomics 1 Dr. F. Fatemi Page 9
J firms can produce good l from good m, each has a production set like: Y = z, q : q 0 & z c q c (. ) is the firm s cost function, and q 0 c q > 0, c q 0 Consumers have no initial endowment from good. Consumer i s initial endowment from good m is ω mi > 0 (ω m = i ω mi ). Microeconomics 1 Dr. F. Fatemi Page 10
Profit maximization problem: The FOC: max q 0 p. q c q p c q, with equality if q > 0 In equilibrium firm s usage of good m is: Utility maximization problem: z = c q max m i R x i R + m i + φ i (x i ) s. t. m i + p. x i ω im + θ i p. q c q Microeconomics 1 Dr. F. Fatemi Page 11
In any solution the budget constraint is holding with equality, then: max x i R + φ i (x i ) p. x i + ω mi + θ i p. q c q The FOC: φ i (x i ) p, with equality if x i > 0 Consumer i s equilibrium consumption of good m is: m i = ω mi + θ i p. q c q p. x i Microeconomics 1 Dr. F. Fatemi Page 12
Then the allocation x 1,, x I, q 1,, q J and the price p is a competitive equilibrium if and only if: p c q, with equality if q > 0 i φ i (x i ) p, with equality if x i > 0 x i i = q Microeconomics 1 Dr. F. Fatemi Page 13
In any interior equilibrium: Marginal cost of each firm equals eq. price Marginal utility of each consumer equals eq. price Due to quasilinear form of utility functions, the eq. allocation and price are independent of distribution of endowment and ownership shares. At the equilibrium level of aggregate output the marginal social cost of good l is exactly equal to its marginal social benefit. Microeconomics 1 Dr. F. Fatemi Page 14
Comparative Statistics Analysis In many cases the researcher would like to investigate the effect of an exogenous factor on the competitive equilibrium. In general; the functional forms of consumers utility and firms cost, we had so far would change to: φ i (x i, α) where α is a vector of exogenous factors affecting utility c q, β where β is a vector of exogenous factors affecting production cost Microeconomics 1 Dr. F. Fatemi Page 15
Furthermore; consumers and firms face different prices than the market price as a result of government s taxes/subsidies, then: p i(p, t) where t is the taxes (or subsidies) effective for consumer i p (p, t) where t is the taxes (or subsidies) effective for firm For example: a tax per unit of t i on good l for consumer i: p i(p, t) = p + t i a percentage tax of t i on good l for consumer i: p i(p, t) = p(1 + t i ) Microeconomics 1 Dr. F. Fatemi Page 16
The Fundamental Welfare Theorems in a Partial Equilibrium Context In this section, the Pareto optimality conditions will be applied on the competitive equilibria. When consumer preferences are quasilinear the boundary of utility possibility set is linear. Each of the points on the boundary is associated with a different allocation of the numeraire good. Microeconomics 1 Dr. F. Fatemi Page 17
Suppose the consumption and production level of good l is fixed at: x 1,, x I, q 1,, q J Therefore, the total amount of the numeraire (which can be distributed between the consumers) is: ω m c q The set of possible utilities is (by different distributions of the numeraire): (u 1,, u I ): u i i φ i (x i) i + ω m c q Microeconomics 1 Dr. F. Fatemi Page 18
Any change in the consumption and production level of good l only results in parallel shift of the boundary of the set. Therefore, every Pareto optimal allocation should be located on the boundary, which is as far out as possible. The optimal solution can be obtained from max φ i (x i ) + ω m c q (x 1,,x I ) 0 i q 1,,q J 0 Subect to x i i = q Solving for the FOCs: Microeconomics 1 Dr. F. Fatemi Page 19
μ c q, with equality if q > 0 i φ i (x i ) μ, with equality if x i > 0 x i i = q And by setting μ = p, it is exactly the same conditions which was calculated for the competitive equilibrium. Proposition (MWG 10.D.1) The 1 st Fundamental Theorem of Welfare Economics: If the allocation x 1,, x I, q 1,, q J and the price p is a competitive equilibrium, then this is Pareto optimal. Microeconomics 1 Dr. F. Fatemi Page 20
We need to remind ourselves about the condition under which this theorem holds: Markets are complete (a market for each good) All participants are price-takers If these conditions are not satisfied, then markets fail to deliver a Pareto optimal outcome. This is what we call market failure and will return to it. Proposition (MWG 10.D.2) The 2 nd Fundamental Theorem of Welfare Economics: For any Pareto optimal level of utility (u 1,, u I ), there are transfers of the numeraire commodity (T 1,, T I ), satisfying Microeconomics 1 Dr. F. Fatemi Page 21
T i, such that a competitive equilibrium reached from the endowments (ω m1 + T 1,, ω mi + T I ) yields precisely the utilities (u 1,, u I ). A critical requirement for this theorem is the convexity of preferences and production sets. This is not necessary for the 1 st theorem (MWG Ch.16). Microeconomics 1 Dr. F. Fatemi Page 22
Welfare Analysis in the Partial Equilibrium Model The set of utility vectors achievable through reallocation of numeraire: (u 1,, u I ): u i i ω m + φ i (x i ) i c q The Marshallian aggregate surplus: S x 1,, x I, q 1,, q J = φ i (x i ) i c q Microeconomics 1 Dr. F. Fatemi Page 23
Consider an initial consumption and production vector x 1,, x I, q 1,, q J ; where x = i x i and q = q. Now, Assume a differential change in consumption and production vector dx 1,, dx I, dq 1,, dq J satisfying i dx i change in Marshallian aggregate surplus is: = dq = dx ; the ds = dφ i(x i ) dx dx i i i dc q dq dq Microeconomics 1 Dr. F. Fatemi Page 24
In a competitive environment dφ i (x i ) dx i = φ i (x i ) = P(x), and dc q dq = c q = C (q) ds = P(x) dx i i C (q) dq Since = q : ds = [P(x) C (x)]dx Microeconomics 1 Dr. F. Fatemi Page 25
Then the total value of aggregate surplus can be interpreted as a function of inverse demand function and industry s marginal cost: x S(x) = S 0 + [P(s) C (s)]ds 0 S 0 is the value of aggregate surplus when there is no production (and consumption). If c (0) = 0 then S 0 = 0. Aggregate surplus can be divided to two various components: S = CS + Π Microeconomics 1 Dr. F. Fatemi Page 26
Aggregate consumer surplus (if the p is the price paid by all consumers): CS(p ) = φ i x i (p ) i p x i (p ) x(p ) CS(p ) = [P(s) p ]ds 0 Aggregate Profit: Π(p ) = p q(p ) c q (p ) q(p ) Π(p ) = Π 0 + [p C (s)]ds 0 Microeconomics 1 Dr. F. Fatemi Page 27
Free-Entry and Long-Run Competitive Equilibria In the short-run, usually firms cannot exit and enter the market. The free-entry condition is a reasonable assumption when we think of the markets in long-run. Some of the costs for the firm might be fixed in the short-run, but in the long-run all costs can be considered as variable. Firms are considered identical entities with similar technologies and there are potentially an infinite number of them. Microeconomics 1 Dr. F. Fatemi Page 28
The concept behind the long-run competitive equilibrium is the entry of the potential entrants in case of existence of positive profit. The firms continue to enter until the profit vanishes. Definition (MWG 10.F.1): Given an aggregate demand function x(p) and a cost function c(q) for each potentially active firm having c(0) = 0, a triple (p, q, J ) is a long-run competitive equilibrium if: i) q = argmax q 0 p q c(q) ii) x (p ) = J q iii) p q c (q ) = 0 Microeconomics 1 Dr. F. Fatemi Page 29
If q(. ) is the supply correspondence of an individual firm with cost function c(. ) and profit function π(. ), the long-run aggregate supply correspondence will be Q(p) = if π(p) > 0 Jq 0 ; for some J N and q q(p) if π(p) = 0 If c(q) = cq (CRS) the long-run eq price is p c If c(. ) is increasing and strictly convex no long-run eq will exist If c(. ) has a strictly positive efficient scale q > 0 the long-run eq p = c = c(q ) ; and q J = x(c ) q Microeconomics 1 Dr. F. Fatemi Page 30