Introductory Microeconomics (ES10001) Exercise 6: Suggested Solutions 1. Match each lettered concept with the appropriate numbered phrase: (a) Input; (b) short run average variable cost (SAVC); (c) U-Shaped average cost curve; (d) capital-intensive; (e) production function; (f) short-run average fixed costs (SAFC); (g) fixed costs; (h) constant returns to scale; (i) long run average cost (LAC) (j) law of diminishing returns; (k) short-run marginal costs (SMC); (l) marginal product of capital (MPK); (m) economies of (increasing returns to) scale; (n) long-run; (o) short-run average total cost (SATC); (p) variable costs; (q) long-run marginal costs (LMC); (r) short-run; (s) labour intensive; (t) long-run total cost (LTC); (u) minimum efficient scale; (v) fixed factor of production; (w) marginal product of labour; (x) diseconomies of (decreasing returns to) scale. (1) the specification of the maximum output that can be produced from any given amount of inputs; (2) the total cost of producing a given output level when the firm is able to adjust all inputs proportionately; (3) the period long enough for the firm to adjust all its inputs to a change in conditions; (4) a factor of production any good or service used to produce output; (5) the output level at which further economies of scale become unimportant for the individual firm and the average cost curve becomes horizontal; (6) the increase in output obtained by adding one unit of labour, holding constant the input of all other factors; (7) the situation in which long-run average costs increase as output rises; (8) a production technique using a lot of labour but relatively little capital; (9) the increase in short run total costs (and in short run variable costs) as output is increased by one unit; (10) the cost per unit of producing a given output level when the firm is able to adjust all inputs optimally; (11) costs that change as output changes; (12) the situation where, beyond some level of the variable input, further increases in the variable input lead to a steadily decreasing marginal product of that input; (13) short-run variable cost per unit of output; (14) short-run fixed cost per unit of output; (15) the increase in long-run total costs if output is permanently increased by one unit; (16) a production technique using a lot of capital but relatively little labour; (17) costs that do not vary with output levels; (18) the situation when long-run average costs are constant as output rises; (19) short-run total cost per unit of output; (20) the situation when long-run average costs decrease as output rises; (21) a long-run average cost curve faced by a firm confronted with first increasing and then decreasing returns to scale; (22) a factor whose input level cannot be varied in the short-run; (23) the increase in output obtained by adding one unit of capital, holding constant the input of all other factors; (24) the period in which the firm can make only partial adjustments of its inputs to a change in conditions. (1) e (5) u (9) k (13) b (17) g (21) c (2) t (6) w (10) i (14) f (18) h (22) v (3) n (7) x (11) p (15) q (19) o (23) l (4) a (8) s (12) j (16) d (20) m (24) r 2. A firm uses ten units of labour and twenty units of capital to produce ten units of output. The marginal product of labour is equal to 0.5. If there are constant returns to scale then the marginal product of capital must be at least equal to 0.5: TRUE / FALSE Solution: Constant returns to scale implies that a 10 per cent increase in capital input together with a ten per cent increase in labour input yields a ten per cent increase in output. Thus, an additional two units of capital and one unit of labour produce one more unit of output. By definition, however, if the marginal product of labour is 0.5 then an additional unit of labour 1
applied on it s own raises output by 0.5 units. The contribution of the two units of capital is therefore to add another 0.5 units of output, implying that the marginal product of capital must be 0.25. Note: This is clearly an approximate procedure since we have averages marginal product over two units of capital and evaluated it at a slightly higher output than 10. However, it so happens that in this instance the answer obtained is exactly correct. To see this, Euler s theorem may be used. This states that for constant returns production function: MP L L + MP K K = Q( K, L) MP K = Q K, L MP K = ( ) MP L L K 10 0.5( 10) = 0.25 20 3. A firm has selected the output level at which it wishes to produce. Having checked the marginal condition, the firm is now considering the average condition as it applies in the short run and in the long run. Cost conditions are such that LAC = 12, SATC = 17, SAFC = 6. Tick the appropriate short- and long-run decisions for the firm at each stated price: Short-Run Decision Long Run Decision Price a Profit a Loss Close Down a Profit a Loss Close Down 18.00 Yes Yes 5.00 Yes Yes 7.00 Yes Yes 13.00 Yes Yes 11.50 Yes Yes Table 1 4. Explain why it might make sense for a firm to sell goods that it can only sell at a loss. Can it keep on doing this forever? Solution: Define short run profit as π SR ( q) = TR q ( ) TVC ( q) + TFC ( ) TVC q ( ) = TFC. with π 0 SR Providing production adds more to TR than TVC [i.e. TR q ( )] then it is worth the firm producing a positive supply of output, even if total profits remain negative. Such a situation is not sustainable in the long run when entry and exit apply. 5. Shifty s brother, Thrifty, owns a junkyard used for disposing off cars. Thrifty can use one of two methods to deal with the cars that come to his junkyard. One way is to use a hydraulic press that can automatically compress discarded cars. The press costs 600 per year to own and it costs a further 3 per car to operate. The alternative to the hydraulic press is to simply use a shovel and get an associate to bury a car in a landfill site. A shovel costs 30 and its usable life is one year only. The person who uses the shovel for burying cars must be paid 15 per car. (Total 50 marks) (a) Assuming that the yearly output is Q, what is the total cost function for each method? (b) Calculate the average cost function and the marginal cost function for each method. 2
(c) If Thrifty s output is 40 cars per year, which method should he employ? What is his output is 50 cars per year? Let Q denote the number of cars processed, H the hydraulic press and S the shovel. (a) Price of the hydraulic press per year = 600. Cost per car while using the press = 3 per car Cost in functional form C(H) = VC(H)+FC =3Q+600 Price of a shovel = 30. Cost per car to use a shovel = 15 per car Cost in functional form C(S) = VC(S) + FC =15Q + 30 (b) Thus: AC(H) = C(H)/Q = 3+(600/Q) MC(H) = d[c(h)]/dq = 3 AC(S) = C(S)/Q = 15+(30/Q) MC(S) = d[c(s)]/dq = 15 (c) The method chosen depends upon the total cost that Thrifty runs up. He will prefer the method that results in a lower cost. If Q = 40 C(H) = 3Q+600 = 20+600 = 720 C(S) = 15Q+30 = 600+30 = 630 Thus, he will use a shovel if Q=40. If Q = 50 C(H) = 3Q+600 = 150+600 = 750 C(S) = 15Q+30 = 750+30 = 780 He will use a hydraulic press if Q = 50. 6. You win a free ticket to see an Eric Clapton concert tonight. You can t resell it. Bob Dylan is performing on the same night and his concert is the only other activity you are considering. A Dylan ticket costs $40 and on any given day you would be willing to pay as much as $50 to see him perform. (In other words, if Dylan tickets sold for more than $50, you would pass on the opportunity to see him even if you had nothing else to do.) There is no other cost of seeing either performer. What is your opportunity cost of attending the Clapton concert? A. $0 B. $10 C. $40 D. $50 E. None of the above 3
Solution: The only thing of value you must sacrifice to attend the Clapton concert is seeing the Dylan concert. By not attending the Dylan concert, you miss out on a performance that would have been worth $50 to you, but you also avoid having to spend $40 for the Dylan ticket. So the value of what you give up by not seeing him is $50 $40 = $10. If seeing Clapton is worth at least $10 to Dylan. Opportunity cost is, by consensus, one of the two or three most important ideas in introductory economics. Yet there is persuasive evidence that most students do not master this concept in any fundamental way. The economists Ferraro and Taylor (2005) posed the Clapton/Dylan question to groups of students to see whether they could answer it. As noted, the correct answer is $10, the value of what you sacrifice by not attending the Dylan concert. Yet when Ferraro and Taylor posed this question to 270 undergraduates who had previously taken a course in economics, only 7.4 per cent of them answered it correctly. Since there were only four choices, students who picked at random would have had a correct response rate of 25 per cent. A little bit of knowledge seems to be a dangerous thing here! When Ferraro and Taylor posed the same question to eighty eight students who had never taken an economics course, 17.2percent answered it correctly more than twice the correct response rate as for former economics students, but still less than chance. Why didn t the economics students perform better? One reason posited is that opportunity cost is only one of several hundred concepts that professors throw at students during the typical introductory course, it simply goes by in a blur. If students don t spend enough time on it and use it repeatedly indifferent examples, it never really sinks in. But Ferraro and Taylor suggest another possibility: the instructors who teach economics may not have mastered the basic opportunity cost concept themselves. When the researchers posed the same question to a sample of 199 professional economists at the annual American Economic Association meetings in 2005,only 21.6 per cent chose the correct answer; 25.1 per cent thought the opportunity cost of attending the Clapton concert was $0, 25.6 per cent thought it was $40, and 27.6 per cent thought it was $50. 7. Dodgy Enterprises is deciding how many chocolate teapots, q, to produce for Christmas 2017. Its fixed costs for this production run are FC, and it experiences first increasing returns to scale, then decreasing returns to scale. The firm calculates: Total Variable Cost, TVC q ( ) ; ( ) TVC ( q) q ; ( ) TVC ( q) + FC ; ( ) dtc ( q) dq ; ( ) TC ( q) q. Average Variable Cost, AVC q Total Cost,TC q Marginal Cost, MC q Average Cost, AC q It prepares graphs showing these five costs (vertical axis) against q (horizontal axis). Which of the following statements is false? A. MC q ( ) is the gradient of TC ( q) ( ) = AVC q B. If q 1 q 1 C. If q 2 D. If q 3 decreasing returns to scale at q 3 E. AC q ( ) and AVC ( q) converge as q ( ), then the gradient of AVC q ( ) is zero at ( ) = AC( q), then the gradient of AC( q) is zero at q 2 ( ) = AVC q ( ), then the firm is experiencing 4
Solution: Decreasing returns to scale implies upward sloping LAC. Reference Ferraro, P. J. and L. O Taylor (2005). Do Economists Recognize an Opportunity Cost When They See One? A Dismal Performance from the Dismal Science. The B.E. Journals in Economic Analysis & Policy, 4(1) 7. 5