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 15 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. Question 1 In this question you will practice shocks to a competitive industry. Suppose all existing firms in a competitive industry have CobbDouglas technologies: Q = L1/2 k1/2 Note how capital is fixed since K = k. Suppose PL = $10, PK = $10, k = 10 and P = $20. (a) Calculate numerically and algebraically each firm's profit maximizing output and labor subject to the constraint that output produced equals the target output. Answer: The problem is: choose q and L to maximize subject to constraint: L1/2 k1/2 = q Being a constrained optimization problem, set up the Lagrangian. I will do the numerical and algebraic problems side by side: Numerical Answer Algebraic Answer 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Choose q and L to: max L = R C [L1/2 k1/2 q] max L = R TVC TFC [L1/2 k1/2 q] max L = Pq PL L PK k [L1/2 k1/2 q] max L = 20 q 10 L 10 k [L1/2 k1/2 q] FOCs: L/q = 0 20 + = 0 20 = L/L = 0 10 (/2) L1/2 k1/2 = 0 10 = (/2) L1/2 k1/2 To see what this says, note that = 20 (=P) and that MPL = (1/2) L1/2 k1/2. Thus, this FOC simply says: PL = P MPL which as you know is the optimal L rule from ECO 100. L/ = 0 k1/2 L1/2 = q This simply states that actual output = target output. From the second FOC we have: 10 = (/2) L1/2 k1/2 Choose q and L to: max L = R C [L k q] max L = R TVC TFC [L k q] max L = Pq PL L PK k [L k q] max L = Pq PL L PK k [L k q] FOCs: L/q = 0 P + = 0 P = L/L = 0 PL L1 k = 0 PL = L1 k To see what this says, note that = P and that MPL = L1 k . Thus, this FOC simply says: PL = P MPL which as you know is the optimal L rule from ECO 100. L/ = 0 L k = q This simply states that actual output = target output. From the third FOC we have: 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain 10 = (20/2) L1/2 (10)1/2 L k = q 1/2 1/2 10 = L L = q / k L = 10 L = [q / k ]1/ To calculate q, exploit the last FOC: q = L1/2 k1/2 q = (10) (10)1/2 q = 10 (b) From your answer in part (a), calculate the actual dollar cost and derive the optimal short run cost function. Hint: a cost function gives the cost of producing target output using optimal labor and fixed capital. As such, it should be expressed in terms of q only. Use the algebraic answer from above. Answer: The actual dollar cost is simply: The cost function is: C = TFC + TVC C = TFC + TVC C = PK k + PL L C = PK k + PL L C = (10)(10) + 10 (10) Instead of substituting actual value of labor, use labor expressed in terms of output. From C = $200 above this is: L = [q / k ]1/ L =[q / 101/2 ]1/(1/2) L =q2/10 L = q2/10 Thus: C = PK k + PL L C = 10 (10) + 10 q2/10 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C = 100 + q2 Check if this answer is right. We know that q = 10. Thus: C = 100 + 100 = $200 which is identical to the numerical answer. (c) If the industry is in the long run, then confirm that the output price of $20 is the long run equilibrium price. Answer: If the industry is in the long run, all existing firms should be earning zero profits: Now: = R C = PQ AC Q = (P AC)Q Thus for profits to be 0, we need: P = AC Since AC is Ushaped, this really means: P = min AC 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now all existing firms have cost function: C = 100 + q2 This implies: AC = C/q AC = 100/q + q AC is minimized when dAC/dq = 0. This is: dAC/dq = 100/q2 + 1 = 0 100/q2 = 1 100 = q2 q = 10 AC is minimized at output of 10. At this output, the AC is: AC = 100/q + q AC = 100/10 + 10 AC = $20 This has to be the long run equilibrium price. Hence, $20 is the long run equilibrium price: 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (d) Suppose the market demand curve is P = 100 0.04Q. How many firms are in this industry? Answer: The long run equilibrium price is $20. Thus, market demand is: P = 100 0.04Q 20 = 100 0.04Q 0.04 Q = 80 Q = 80/0.04 Q = 2,000 With each firm producing 10 units, this implies there must be 2,000/10 = 200 firms in the market: 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (e) Derive the long run supply equation. Answer: In perfect competition, and only perfect competition, the firm's supply curve is the MC curve. Now: C = 100 + q2 MC = dC/dq MC = 2q Notice, how TFC does not enter the MC expression thus, TFC is not a component of the supply curve. Now, any firm produces where: MR = MC In perfect competition, P = MR and thus: P = MC P = 2q q = P/2 Now, total supply is output supplies by all firms. With 200 firms in the market: Market Supply = 200*Each firm's supply 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Q = 200(P/2) Q = 100P This is the market supply curve. (f) Confirm your answer in part (e) by solving for market equilibrium price and output. Answer: In equilibrium: Q demanded = Q supplied Rearranging the demand curve, we have: P = 100 0.04Q 0.04Q = 100 P Q = 100/0.04 P/0.04 Q = 2,500 25P Thus, setting Q demanded equal to Q supplied: 2,500 25 P = 100 P 125 P = 2,500 P = $20 And Q = 100P implies: Q = 100(20) Q = 2,000 This confirms our answer in part (e): 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (g) Suppose market demand experiences a positive shock where the new demand equation is: P = 200 0.04Q Suppose this is an increasing cost industry assume the expansion of this industry raises the price of capital to $40. Further assume that all potential entrants have the same technology as the incumbent firms. What is the equilibrium price, the optimal number of workers and the number of new (if any) firms? Assume existing firms continue to hold capital at k =10. You may want to think about why this question tells you that the new firms have the same technology as incumbent firms. Answer: Before you answer this question, why is it important that new firms have the same technology as existing firms? If new firms had a superior technology so that A > 1 in Q = A L1/2 k1/2, then the new firms will have lower AC curves. Thus, even if the industry is an increasing cost industry in the sense that capital lease rates increase as the industry expands the new equilibrium could be lower than $20 due to new firms being more efficient than existing firms. Given that new firms are identical to existing firms, we know that when all is said and done, the new price of capital will be $40. To calculate the new long run equilibrium price, we need to calculate how much AC has increased and the new minimum AC. Repeating the calculations from above: firms will choose q and L to maximize subject to constraint: L1/2 k1/2 = q Being a constrained optimization problem, set up the Lagrangian. 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Algebraic Answer Choose q and L to: max L = R C [L k q] max L = R TVC TFC [L k q] max L = Pq PL L PK k [L k q] max L = Pq PL L PK k [L k q] FOCs: L/q = 0 P + = 0 P = L/L = 0 PL L1 k = 0 PL = L1 k To see what this says, note that = P and that MPL = L1 k . Thus, this FOC simply says: PL = P MPL which as you know is the optimal L rule from ECO 100. L/ = 0 L k = q This simply states that actual output = target output. From the third FOC we have: 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain L k = q L = q / k L = [q / k ]1/ From this: The cost function is: C = TFC + TVC C = PK k + PL L Instead of substituting actual value of labor, use labor expressed in terms of output. From above this is: L = [q / k ]1/ L =[q / 101/2 ]1/(1/2) L =q2/10 L = q2/10 Thus: C = PK k + PL L C = 40 (10) + 10 q2/10 C = 400 + q2 Check if this answer is right. We know that q = 20. Thus: C = 400 + 400 = $800 which is identical to the numerical answer. Now the new cost function due to this being an increasing cost industry is: 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain C = 400 + q2 The price has to = min AC. Now, AC = C/q or: AC = 400/q + q AC is minimized when dAC/dq = 0: dAC/dq = 400/q2 + 1 = 0 400/ q2 = 1 q = 20 Which implies that: AC = 400/20 + 20 AC = $40 Thus, the equilibrium price will be $40. You can now solve for the new optimal number of workers. From the second FOC above: k1/2 L1/2 = q (10)1/2 L1/2 = 20 L = 40 Observe how the higher price of capital has led to an increased use of labor. You should really look at this result carefully. When we did optimal inputs, then even if PK increased, the company would use the same labor. This was because there the target output was the same. Thus, with fixed capital and target output, the company would use the same labor even if the isocost line has become flatter (assuming Labor is in the X axis and capital on the Y). So why is labor use rising? After all, the firm is in the short run and has k = 10. How can the company use more labor when PK is rising? How is it that the company is able to "substitute" more labor as capital become more expensive? The answer is that the target output has changed: the firm produces 20 units instead of 10 units. Thus, even though capital is fixed at 10 units, the greater output necessitates more labor. 12 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain To calculate q, exploit the last FOC: q = L1/2 k1/2 q = (10) (40)1/2 q = 20 To calculate the number of new entrants, substitute this price into the new demand curve: P = 200 0.04Q 40 = 200 0.04Q 160 = 0.04Q Q = 160/0.04 Q = 4,000 Since all firms existing and new will produce at the lowest point of the AC curve (q = 20), this implies there will be a total of 4,000/20 = 200 firms. The industry initially had 200 firms, which implies that the demand shock has led to no new firms entering the industry. Here's what happened: the initial demand shock resulted in greater market output which exerted upward pressure on the price of capital. The higher costs stemming from higher price of capital and greater use of labor resulted in all existing firms making zero profits. Because every incumbent firm is making zero profits, it gives potential entrants no incentive to enter. 13 ...
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 Fall '08
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 Economics, Microeconomics, Supply And Demand, Department of Economics, S. Ajaz Hussain, Ajaz Hussain

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