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 14 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 examine a company with a Ushaped MC curve. Given that a perfectly competitive firm (a price taker) has a flat demand curve and therefore a MR curve that is also flat (and equal to the demand curve), there will be two points where MR = MC. You will practice and confirm that only one of these points maximizes profits and is an "equilibrium". This way you will see why ECO 100 only worked with the "right" side of a Ushaped MC curve. Suppose a perfectly competitive firm has the cost function: C = 25 log q + 0.5q2 Here the log is a "natural log" (to the base e). (a) Derive the MC and interpret it. Is it Ushaped? Answer: MC, marginal cost, is dC/dq or the cost of producing an additional unit. Now this is: C = 25 log q + 0.5q2 MC = dC/dq = 25/q + q Here we have used the fact that d logq/dq = 1/q. 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain MC is the cost of producing the next unit and by definition only encompasses variable costs. For example, if q = 1, the cost of producing the second unit is MC = 25/1 + 1 = $26. Similarly, if q = 5, the cost of producing the sixth unit is MC = 25/5 + 5 = $10. Observe that MC is Ushaped. To see this, you have to check that when q = 0 MC is very large and when q is large, MC is very small. For example, if q = 0 then: as q 0 then MC = + 0 = (i.e. a "very large number"). Similarly, if q then MC = 0 + = (i.e. a "very large number"). Therefore, MC is Ushaped. (b) Suppose the market price is $15. Calculate the output or outputs where MR = MC. If you need to use it, here is the quadratic formula: for ax2 + bx + c = 0, x = [b +/ square root of (b2 4ac)]/2a. Answer: You should suspect that you will get two solutions for outputs where MR = MC, since MR is a horizontal line and MC is Ushaped: In perfect competition, P = MR and thus: MR = 15. Setting MR = MC: 15 = 25/q + q 15q = 25 + q2 q2 15q + 25 = 0 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now apply the quadratic rule: if ax2 + bx + c = 0 then: x = [b +/ square root of (b2 4ac)]/2a. Here x is quantity ("q"), a = 1, b = 15 and c = 25. Thus: q = [(15) +/ square root of (152 4(1)(25))]/2(1) q = [ 15 +/ square root of (225 100))]/2 The solution is either q = 1.91 or q = 13.09: (c) We have seen that the smaller quantity where MR = MC cannot be an equilibrium and in fact does not maximize profits, whereas the larger quantity where MR = MC is an equilibrium and in fact maximizes profits. Prove this. Answer: Let's prove that the output of 1.91, even though one where MR = MC, is not an equilibrium. Suppose the company produces a little less than 1.91. At that point, MC > MR, which means that if you produce more the cost is greater than the revenue. Thus you should not produce more and you will not return to the output of 1.91. In fact, you should produce less until you have zero output. Should you produce more than 1.91 units? Yes. Since MR > MC, if you produce more the cost is smaller than the revenue. Thus you should produce more and more and you will not return to the output of 1.91. 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now let's prove that the output of 13.09, also a MR = MC output, is an equilibrium. Suppose the company produces a little less than 13.09. At that point, MR > MC, which means that if you produce more the cost is less than the revenue. Thus you should produce more and you will return to the output of 13.09. Should you produce more than 13.09 units? No. Since MC > MR, if you produce more the cost is smaller than the revenue. Thus you should produce less and you will return to the output of 13.09: The company's profits are: = R C = Pq 25 log q 0.5q2 = 15q 25 log q 0.5q2 The profits from q = 13.09 are: = 15(13.09) 25 log (13.09) 0.5(13.09)2 $46.4 The profits from q = 1.91 are: = 15(1.91) 25 log (1.91) 0.5(1.91)2 $10.6 Note how the output of 13.09 the equilibrium MR = MC point yields greater profits than the output of 1.91 the disequilibrium MR = MC point. Nice. 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Question 2 The table below shows the MC of major producers of Copper. From this, one can graph the global supply curve of copper: (a) Suppose Russia is a "rational" Copper producer. Will it supply copper if the price of copper is 0.8? What if the price was 0.4? 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Being a rational producer, Russia will produce Copper so long as MR MC. In perfect competition, P = MR, so that Russia will be supplier so long as P 0.65. Thus, if the price is 0.8, it will supply copper, but if the price is 0.4 it will cease production. (b) Suppose Russia is an "irrational" Copper producer. How much copper will it supply if the price of copper is 0.8? What if the price was 0.4? Answer: Being an irrational producer, Russia will produce Copper at any price. In practice, this happens when irrational producer supply products not for economic reasons but for other reasons such as employment, national security and national "interests". This makes analysis of commodities harder as one has to know how much irrational production there will be. This is something we will do in the Aluminum case. 6 ...
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 Fall '08
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 Economics, Microeconomics, S. Ajaz Hussain

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