Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 13 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. In this question you will practice writing down the expressions for profit objectives in terms of output for the four types of firms: competitive, market power, monopsony and dual monopoly. Suppose a firm uses labor and fixed capital to produce output with a CobbDouglas technology: Q = L1/2 k1/2 Note how capital is fixed since K = k. The firm currently has k = 10. For your convenience, here are the profit equations for the four types of firms: notice that the profit equation can be expressed in terms of primitives (wages, capital lease rates, labor and capital) and from that in terms of cost function: Firm Type Competitive Market Power Monopsony Dual Monopoly Equation in Primitives = Pq PK k PL L(q) = P(q) q PK k PL L(q) = P q PK k PL(L) L(q) = P(q) q PK k PL(L) L(q) Equation in Cost Function = PQ TFC TVC(Q) = P(Q) Q TFC TVC(Q) = P Q TFC TVC(Q) = P(Q) Q TFC TVC(Q) Answer each parts (b) (e) independently of other parts. You should really think about what the firm is taking as "given" or exogenous. Think about what sets each type of firm apart from 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain others and how you'd go about writing and solving the profit equation. (a) Given a target output q and fixed capital of k = 10, do the Cost Minimization Problem (CMP) to solve for the optimal labor. Answer: From the CMP we know that optimal labor is chosen to minimize cost of producing target output. That is: Choose L to: min L = C [L k q] min L = TVC + TFC [L k q] min L = PL L + PK k [L k q] FOCs: L/L = 0 PL L1 k = 0 PL = L1 k L/ = 0 L k = q L = q / k L = [q / k ]1/ Now in this question = = and so: L = [q / k1/2 ]1/(1/2) L = [q / (10)1/2]2 L = q2/10 Observe how neither the output price nor the labor market demand equation came into play. In this simple problem, given fixed capital and target output, optimal labor is simply "read off" the isoquant. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (b) Suppose the firm is competitive, P = $20, PL = $10 and PK = $10. Write down the profit equation in terms of output. Answer: The primitive profit equation is: = Pq PK k PL L(q) Observe how price of output does not depend on output and wages do not depend on number of workers. That is, the firm is a price taker in the inputs and output market. = 20 q (10) (10) 10 L(q) But we know what L is in terms of q: from part (a) above L = q2/10 and thus: = 20 q (10) (10) 10 (q2/10) = 20q 100 q2 The profit equation in terms of cost functions is: = PQ TFC TVC(Q) Notice that in: = 20q 100 q2: TFC = 100 and TVC(Q) = q2 (c) Suppose the firm has market power where P = 100 20Q, PL = $10 and PK = $10. Write down the profit equation in terms of output. Answer: The primitive profit equation is: = P(q) q PK k PL L(q) Observe how price of output depends on output and wages do not depend on number of workers. That is, the firm is a price taker in the inputs but not the output market. = (100 20q)q (10) (10) 10 L(q) We know what L is in terms of q: from part (a) above L = q2/10 and thus: = (100 20q)q (10)(10) 10(q2/10) 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain = 100q 20q2 100 q2 = 100q 21q2 100 (d) Suppose the firm is a monopsony where P = 20, PK = $10 and wages in the labor market are given by: PL = 100 20 L. Write down the profit equation in terms of output. Answer: The primitive profit equation is: = P q PK k PL(L)L(q) Observe how price of output does not depend on output and wages depend on number of workers. That is, the firm is a price taker in the output and capital market but not the labor market. = 20 q (10) (10) (100 20 L) L(q) We know what L is in terms of q: from part (a) above L = q2/10 and thus: = 20 q (10) (10) (100 20 L) q2/10 = 20 q 100 (100 20 q2/10) q2/10 = 20 q 100 (100 2 q2) q2/10 = 20 q 100 (10q2 q4/5) = 20 q 100 10q2 + q4/5 (e) Suppose the firm is a dual monopoly where P = 100 20Q, PK = $10 and wages in the labor market are given by: PL = 100 20 L. Write down the profit equation in terms of output. Answer: The primitive profit equation is: = P(q) q PK k PL(L)L(q) Observe how price of output depend on output and wages depends on number of workers. That is, the firm is a price taker in the capital markets only. = (100 20q) q (10) (10) (100 20 L) L(q) We know what L is in terms of q: from part (a) above L = q2/10 and thus: 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain = 100q 20q2 (10) (10) (100 20 L) q2/10 = 100q 20q2 100 (100 20 q2/10) q2/10 = 100q 20q2 100 (100 2 q2) q2/10 = 100q 20q2 100 (10q2 q4/5) = 100q 20q2 100 10q2 + q4/5 = 100q 30q2 100 + q4/5 Notice how, given technology and, as the case may be, the market demand and labor demand equations, a firm can always express it's profits in terms of output q. This allows the firm to "solve" for the profit maximizing output by setting M = d/dq = 0. 5 ...
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 Fall '08
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 Economics, Microeconomics, profit equation, S. Ajaz Hussain

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