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Student Number: QUIZ 4 Section D Question 1. [5 marks] Consider the production function Q = 10 K 2 L2 a) [3 marks] Letting w be the wage rate and r be the rental rate of capital, derive the cost-minimizing quantities of labor and capital as a function of w , r , and Q in the long run. Answer: Set up the tangency condition to determine the relationship between capital and labor. MPL w = MPK r 10 K 2 L w = 10 KL2 r K w = L r w K= L r Plug this result into the production function to determine the optimal quantities of capital and labor. 2 w Q = 10 L L2 r w2 4 Q = 10 L r L4 = Qr 10 w 1 2 Q 4 r L= 10 w The optimal level of capital is 1 2 1 w Q 4 r K = r 10 w 1 Q 4 w K = 10 r 1 1 2 2 b) [2 marks] Derive the long-run total cost function as a function of w , r , and Q . Answer: To find the long-run total cost function, plug the optimal quantities of capital and labor from part a) into the total cost function. TC = wL + rK 1 Q 4 r TC = w 10 w 1 2 1 Q 4 w + r 10 r 1 2 1 Q 4 TC = rw 5 Question 2. [5 marks] In a certain market in the long-run, each firm and potential entrant has a long-run total cost curve, TC = 8Q 3 − 4Q 2 + 20Q where Q is thousands of units per year. Market demand is given by D( P) = 20,000 − 1000( P ) . a) [3 marks] Find the long-run equilibrium price and quantity per firm. Answer: In the long-run equilibrium, each firm will produce where P = AC = MC . Thus, 24Q 2 − 8Q + 20 = 8Q 2 − 4Q + 20 ⇒ Q = 0.25 Since each firm produces where P = MC , price will be P = MC ⇒ P = 24(0.25)^ 2 − 8(0.25) + 20 = 19.5 b) [2 marks] Find the number of firms operating in the long-run in this market? Answer: First find the total market demand. D ( P) = 20,000 − 1000(19.5) = 500 Since total market demand is 250 and each firm is produce 0.25 units, the total number of firms in the market in equilibrium will be N= 500 = 2000 .25