Elements of Economic Analysis II Lecture VII: Equilibrium in a Competitive Market

Transcription

1 Elements of Economic Analysis II Lecture VII: Equilibrium in a Competitive Market Kai Hao Yang 10/31/ Partial Equilibrium in a Competitive Market In the previous lecture, e derived the aggregate supply function for an industry under various conditions Hoever, this is not sufficient to describe the economy In particular, as the supply function only specifies ho much ill the industry produce given a market price of the output, e do not have sufficient information to give a prediction about the actual aggregate output and market price ill be To this end, e ill first study a partial equilibrium model that enables us to analyze the behavior of the good market that the industry can produce hen other things are held equal Specifically, suppose that there is a demand function for the good y, denoted by D(p) Assume that D(p) is strictly decreasing and differentiable Recall that if the industry has entry barrier, the aggregate supply, Y (p) both in the long run and in the short run, is strictly increasing As such, the market clearing condition requires D(p) = Y (p) Since Y is strictly increasing and D is strictly decreasing, there is a unique solution p p is then called the equilibrium price and D(p ) = Y (p ) is the equilibrium quantity On the other hand, if an industry has free entry, the supply function is flat and therefore the equilibrium price p must be the loest point of average cost, hich means that the Department of Economics, University of Chicago; khyang@uchicagoedu 1

2 2 market demand complete determines the equilibrium quantity and the firms must earn zero profit in equilibrium It is noteorthy that after introducing such partial equilibrium analysis, even hen the firms have constant return to scale technology, e can still make some predictions Formally, suppose that an industry has constant return to scale technology and all the firms have the same production function F (L, K), ith associated cost function c(y) = cy Let D(p) be the demand for the output Firms profit maximization and market clearing imply = pf L (L, K) r = pf K (L, K) (1) D(p) = F (L, K) Therefore, for given and r, (1) is a system ith three equations and three unknons and it admits a solution (L D (, r), K D (, r), y (, r)) even if F has constant return to scale The system in (1) is useful for deriving the ell-knon Marshall s la, hich states that ε L = s L ε D }{{} Scale Effect s K σ }{{} Substitution Effect here ε L is the labor demand elasticity, ε D is the demand elasticity of the output, s L = LD (, r) y (, r), s K = rkd (, r) y (, r) are labor and capital share in equilibrium, respectively, and d ( ) ( K L ) ( L ) ( K ) F d L F K F K FL (L,K)=(L D (,r),k D (,r)) is the elasticity of substitution at (L D (, r), K D (, r)) 2 General Equilibrium ith Production: Robinson Crusoe So far, e have been studying the equilibrium in the market in hich an industry/a firm is producing output Specifically, all the other market sectors including the labor market and the capital market, as ell as markets for other goods that this industry/firm is not

3 3 producing but is related due to sustainability or complementarity are held fixed This is often referred as a partial equilibrium model in the sense that it only considers one market at a time Hoever, it is obvious that in the real orld economy, markets are intertined ith each others and simply analyzing one market hile holding the other markets fixed may sometimes be insufficient As such, e ill revisit the general equilibrium model that e learned before, but introduce production at the same time To begin ith, let us first consider the simplest possible situation Consider an economy here there is only one agent (Mr Robinson) ith one commodity (say, coconut) Suppose that Robinson can produce F (L) units of coconuts for every L hours he orks, and that he likes consuming coconut and leisure (ie hours that he is not orking) Specifically, Robinson s preference is represented by a utility function u(h, C) and the production technology is described by a production function F (L) Finally, suppose that Robinson is endoed ith T = 24 hours each day that he can choose beteen orking and enjoying leisure Clearly, Robinson s problem can be summarized by max u(h, C) st C = F (L), H = T L (2) H,C,L Hoever, instead of analyzing problem (2) directly, e can image that Robinson is trading ith himself in this economy Specifically, e can regard him as a consumer, ho solves a utility maximization problem and derives the demand of coconut and the supply for labor, taking the market price of coconut and age as given At the same time, e can also regards Robinson as a producer, ho maximizes profit by choosing the optimal amount of labor to hire, also taking the market price of coconut and the age as given and the profit becomes the income for the consumer Formally, as a producer, the Robinson-firm s problem is given by max pf (L) L (3) L [0,T ]

4 4 As before, e let L D (, p) denote the solution of problem (3), hich is the labor demand in this economy and let C S (, p) = F (L D (, p)) denote the optimal amount of output, hich is the commodity supply of this economy Also, let π(, p) be pc S (, p) L D (, p) be the firm s maximized profit On the other hand, as a consumer, the Robinson-consumer s problem is given by: max u(h, C) st H + pc T + π(, p) (4) C 0,H [0,T ] and e ill let C D (, p) and H D (, p) be the solutions to (4), hich stand for the demand for consumption and the demand for leisure, respectively Before detailed analyses, let us first notice that the problem in (3) is exactly a firm s profit maximization problem e studied before, in hich the amount of capital is held fixed at 1 and the problem in (4) is very similar the consumer problem e learned in ECON 200, except that in addition the value of the endoments (time), the consumer also has income from the profit of the firm Using the same idea that a competitive equilibrium must be the prices such that supply equals to demand in all the markets, e ant to find a price and a age level such that the commodity market clears hich means that the demand for coconut equals to the supply of coconut and that the factor market also clears hich means that the total demand for time equals to the total endoment; or equivalently, that labor supply (ie T leisure demand) equals to labor demand That is, e ish to find and p such that C D (, p) = C S (, p), L D (, p) + H D (, p) = T (5) Equation (5) is a system of to equations and to unknons and therefore e can solve for the system an obtain the equilibrium price p and age generically For example, suppose that u(h, C) = H 1 β C β and that F (L) = L α for some α, β (0, 1), using the Lagrangian method, e can solve the consumer s problem and derive the (Marshallian) demands: C D (, p) = β(t + π(, p)), H D (, p) = p (1 β)(t + π(, p)) On the other hand, the first-order condition for the firm s problem (3) gives L D (, p) =

5 5 and therefore and C S (, p) = L D (, p) α = ) α π(, p) = pc S (, p) L D (, p) = p Together, (5) can be reritten as [ β T + 1 α p α + 1 β [ ) α ) ] 1 = T + 1 α α ) α = 1 α α ] = T (6) As before, e can notice that this system is homogeneous ith degree zero That is, for any (, p ) that solves (6), (λ, λp ) must also solve (6) for any λ > 0 Therefore, e may normalize p = 1 and use one of the market clearing condition the solve for, hich gives us ( ) 1 β(1 α) = α αβt As such, for any p > 0, ( ( ) 1 β(1 α) pα, p) αβt is a competitive equilibrium (or general equilibrium, Walarsian equilibrium) of the Robinson economy Finally, notice that if e plug in any competitive equilibrium (, p ) into and the labor demand L D (, p ), such L D (, p ) ill exactly be the solution for the simple problem given by (2), hich implies that any competitive equilibrium is efficient 3 General Equilibrium ith Production: To-Agents Economy No consider a slightly more complicated economy Suppose that Mr Robinson found a company, Wilson, on the island and that they jointly on a firm ho can produce coconut by using labor and capital (say, ooden sticks) Again, suppose that both Robinson and Wilson has T = 24 hours a day and they care about their consumption of coconuts, C, and their leisure time, H

6 6 More formally, let 1 be Robinson and let 2 be Wilson, their preferences are represented by utility functions u 1 (H 1, C 1 ) and u 2 (H 2, C 2 ) Both of them are endoed ith T units of time and K 1, K 2 units of capitals, respectively On the other hand, suppose that the firm s technology can be described by a production function F (L, K) and that Robinson has θ 1 (0, 1) share of the firm hile Wilson has θ 2 = 1 θ 1 share of the firm This is a complete description of this simple economy by Given market prices, r,p, recall that the firm s profit maximization problem is given max pf (L, K) L rk (7) L 0,K 0 and that L D (, r, p) and K D (, r, p) denotes the firm s labor and capital demand, hile C S (, r, p) = F (L D (, r, p), K D (, r, p)) denotes the firm s supply of the coconut and π(, r, p) = pc S (, r, p) L D (, r, p) rk D (, r, p) denotes the firm s profit On the other hand, since Robinson and Wilson on θ 1 and θ 2 share of the firm, respectively, given any market prices, r, p, θ 1 π(, r, p) ill be added to Robinson s income hereas θ 2 π(, r, p) ill be added to Wilson s income Together ith their value of endoments, as consumers, their utility maximization problem is given by max H 1 0C 1 0 u1 (H 1, C 1 ) st H 1 + pc 1 T + rk 1 + θ 1 π(, r, p) max H 2 0C 2 0 u2 (H 2, C 2 ) st H 2 + pc 2 T + rk 2 + θ 2 π(, r, p), (8) here H D 1 (, r, p), C D 1 (, r, p), H D 2 (, r, p), C 2 (, r, p) denote 1 and 2 s (Marshallian) demand functions, respectively

7 7 Again, a competitive equilibrium are the prices such that total demand equals to total supply in all the markets That is, e ish to find prices, r and p such that C D 1 (, r, p) + C D 2 (, r, p) = C S (, r, p) H D 1 (, r, p) + H D 2 (, r, p) + L D (, r, p) = 2T (9) K D (, r, p) = K 1 + K 2 Solving the system (9) can then give us the competitive equilibria in this economy For example, if u 1 (H, C) = u 2 (H, C) = H 1 γ C γ and F (L, K) = L α K β, for some α, β, γ (0, 1) Using the Lagrangian method, e have C 1 (, r, p) = γ(t + rk 1 + θ 1 π(, r, p)) p H 1 (, r, p) = (1 γ)(t + rk 1 + θ 1 π(, r, p)) C 2 (, r, p) = γ(t + rk 2 + θ 2 π(, r, p)) p H 2 (, r, p) = (1 γ)(t + rk 1 + θ 1 π(, r, p)) On the other hand, in Problem Set 3, e derived the labor and capital demands L D (, r, p), K D (, r, p) for such technology and the expression of profit π(, r, p), using these expressions and substitute them into (9), e can then solve for the competitive equilibria (, r, p ) in this economy 4 General Equilibrium: Definition and Properties* With the to specific examples above, e can no introduce a more general formulation Consider an economy ith I consumers and J firms Suppose that there are n commodities in the economy (including both the inputs and the outputs) Suppose that for each consumer i, i s preference is represented by a utility function u i : R n + R and that for each firm j, j has a production function F j that corresponds to a production possibility set Y j R n Finally, suppose that each consumer i is endoed ith e i = (e i 1,, e i n) units of the commodities and has θj i share of firm j

8 8 Given any market prices p = (p 1,, p n ), the firm s profit maximization problem is then given by max p y, (10) y Y j ith the maximized profit being denoted by π j (p) On the other hand, consumer i s problem is max u i (x) st p x p e i + x R n + J θjπ i j (p) (11) We say that p is a competitive equilibrium/general equilibrium/walarsian equilibrum in this economy if there exists (x i ) I i=1 R I+n and (y ) J j=1 R J+n such that: 1 (Optimality) x i solves (11) for all i and y j solves (10) for all j j=1 2 (Market Clearing) N i=1 x i }{{} Total Demand = I e i + i=1 J j=1 y j } {{ } Total Supply In terms of our example of a to-agent economy I = 2, J = 1 There are three commodities, n = 3, including coconut, time (labor/leisure), capital Each consumer is endoed ith T units of time, K i units of capital A price vector is given by (, r, p) Consumers only care about their consumption of coconuts and time The firm s production possibility set is given by: Y := {(C, L, K) R 3 C F (L, K)} and therefore the firm s profit maximization problem max pf (L, K) L rk L,K is equivalent to a problem that chooses a combination in the production possibility set to maximize net revenue: max pc L rk (C,L,K) Y There are several noticeable properties for the competitive equilibrium defined above We ill summarize but ithout proving these properties belo:

9 9 (Homogeneity of Degree Zero) If p is a competitive equilibrium, then λp is also a competitive equilibrium, for all λ > 0 (Warlas La) Suppose that u i is increasing Then for any competitive equilibrium p and the associated allocations (x i ) I i=1 and (yj) J j=1, ( N x i i=1 I J e i + yj) p = 0 i=1 j=1 (First Welfare Theorem) Suppose that u i is increasing for all i Then any competitive equilibrium is Pareto efficient (Second Welfare Theorem) Under certain regularity conditions, for any Pareto efficient allocation, there exists money transfers such that a competitive equilibrium under the economy after transfers gives exactly the same allocation