First Quiz Solutions Managerial Economics: Eco 685. a. (a) The production function is Cobb-Douglas. Therefore we can use the output elasticity

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1 First Quiz Solutions Managerial Economics: Eco 685 Question 1 (10 points) a. (a) The production function is Cobb-Douglas. Therefore we can use the output elasticity formula: output elasticity = a+b = = 2.10 > 1. (1) The output elasticity greater than one, which is increasing returns to scale. A bigger firm can produce more output at the same cost. (b) The output elasticity is = 0.94 which is less than one so we have decreasing returns to scale. (c) The output elasticity is = 1.35 which is greater than one so we have increasing returns to scale. (d) The output elasticity is = 1 which is equal to one, so we have constant returns. (e) We do not have a Cobb-Douglas production function, so we have to solve the problem the long way. If two firms with E = G = 1 merge, we have E = G = 2 in one merged firm. Thus merged production is: Q merged = = 66. (2) Conversely if we have two separate firms each with E = G = 1, total production is: Q apart = 2Q = 2 ( ) 12 = = 65. (3) So doubling the inputs more than doubles outputs and we have increasing returns to scale. Notice the results are different than the engineers and technicians problem because we have a new term 2 G E. This term means that when we increase programmers and engineers together, we gain additional output (they complement each other). This effect more than offsets the diminishing marginal product, when E and G are relatively small. b. No. We have decreasing returns and so T-Mobile should get smaller. Interestingly, this data came from economists studying a proposed merger between T-Mobile and ATT that was trying to get approval for from the federal government. T-Mobile argued that the merger would allow the firm to take advantage of scale economies, and pass some of the costs savings to consumers in terms of lower prices. So the research did not support T-Mobile s assertion and ended up being denied by the Justice Department. Question 2 (20 points) 1

2 a. IusedMRTS = MP K /MP L (doingthereverseisfine). Writtenthisway, thedefinition is the number of units of L which replace 1 stitching machine (K) while keeping production constant. From the above data, 12 workers replace one stitching machine, so MRTS = 12. b. Using the same variable in the numerator, I get P K /P L = $725/$145 = 5. Machines are 5 times as expensive, but are also more productive. c. We have MRTS = MP K /MP L > P K /P L and so K is the relatively productive and relatively cheap input. We want to add K and reduce L. We can add one in the numeratorandreducesmrts oftheinputinthedenominator. Thus, addonemachine and eliminate 12 workers. d. The extra machine costs $745 and we save 12 $145 in labor. The total savings are: 12 $145 $725 = $1015. (4) e. Yes, automation will continue until the marginal product of capital starts to diminish because we start to have many machines with few workers to operate them. Question 3 (10 points) a. False. Profits could be positive on units 1-19, leading to positive total profits even though profits are negative for the 20th unit (P < MC(20)). b. True. We can sell the 20th unit for P and the cost of the 20th unit is MC(20), so if P < MC(20) profits on the 20th unit are negative. c. True. Since MRP (20) > ME, profits for the 20th unit are positive. Therefore, we should increase use of the input until MRP = ME. d. False. Since MP x /MP y = 1/3, input x is only 1/3 as productive as input y (MP x = (1/3)MP y ). However, x is still attractive since it is 4 times cheaper. e. False. Input x is 5 times as productive. Therefore, 1 unit of x can be substituted for 5 units of y without changing production (not the reverse). Question 4 (15 points) a. The missing marginal revenue products can be computed from the marginal products. For example, in 2016 we are given a marginal product of 0.08 and the price of Q is $30, MRP = MP MR = 0.08 $30 = $2.40 (5) 2

3 We can also use the same equation to recover the marginal product for the planned windmill: MRP = MP MR, (6) 1.8 = MP 30, MP = (7) For the 2018 windmill size, we can use the marginal product formula: MP = Q D, (8) 9 6 x 150 = 0.12, (9) 3 = 0.12x , (10) 21 = 0.12x, x = 175. (11) We can also use the marginal product formula to recover the missing outputs for 2007 and the planned windmill. For the 2007 windmill: MP = Q D, (12) 0.08 = 6 x , (13) 3 = 6 x, x = 3. (14) Here is the filled in table: 3

4 Year Windmill Diameter (meters) Energy Generated (megawatts) marginal product marginal revenue product NA NA $ $3.60 planned $1.80 Table 1: Energy production from windmills. b. No, the marginal product is increasing from 0.08 to c. Yes, the marginal product is decreasing from 0.12 to Windmill sizes have been increasing for some time and are now truly huge (keep in mind that 225m is larger than many skyscrapers. Indeed, the arm of the windmill is about the size of 2.5 football fields. However, the gains from increasing the diameter appear to be starting to diminish. d. Increasing returns. Doubling the 112.5m diameter windmill results in a 225m windmill, which has more than double the output (3 to 12 is 4 times the output). However, doubling from 225m to 450m is unlikely to produce the same gains. Question 5 (8 points) a. False, the accounting profits may be positive, just less than the value of the next best option. b. True, since GM is pursuing the strategy, not the merger. c. True, opportunity costs include the value of time that could be spent on the next best option. d. False, opportunity costs include both accounting costs and the value of the next best option. Question 6 (5 points) Firms in many industries are smaller in Peru and Uruguay than Chile. In fact 69% of firms in Peru have less than 20 employees, versus on 31% in Chile. This indicates that decreasing returns to scale are occurring in Peru/Uruguay but not Chile. We have 3 reasons for decreasing returns: extra layers of management, poor information flow, individual workers have little impact on profits, and regulation. The only one of these that would be different across countries in identical industries is regulation. Indeed, the regulation associated with 4

5 hiring new employees is very complex, leading many firms in Peru to have only a single employee. Question 7 (12 points) a. We have: TC = TFC +TVC = Q+ 1 2 Q2. (15) For marginal costs, we have: TC = Q+ 1 2 Q2. (16) MC = dtc dq = 25+Q. (17) Plug in for 30 kids: MC = = 55. (18) b. The price is $45 so we have P < MC. The 30th kid brings in $45 in revenue, but costs an extra $55 to service. We lose $10 on this kid. c. Since we calculated marginal cost in (a), we just set P = MC: $45 = 25+Q, Q = 20. (19) Question 8 (12 points) A wealth management firm hires senior financial advisors (S) and junior advisors (J). Let Q be the number of clients the firm services. Production is: Q = 10J 1 10 J2 +45S 1 2 S2. (20) Suppose senior advisers are paid $100K per year and junior advisors are paid $20K per year. a. How many senior and junior advisers should be hired if the budget is $340K? b. Suppose the firm currently has 4 senior advisers. How many junior advisors are optimal? 5

6 a. We are maximizing production while keeping costs at $340K. Since we are optimizing, we set MRTS equal to the price ratio. MRTS = MP S MP J. (21) Since we have the production function: MP S = dq ds = 45 S. (22) MP J = dq dj = 10 1 J. (23) 5 Putting everything together: MRTS = MP S = 45 S MP J 10 1 (24) J. 5 Next the price ratio is: Thus, P S = $100,000 = 5. (25) P J $20,000 MRTS = P S P J, (26) 45 S J = 5, (27) 45 S = 5 (10 15 ) J, (28) 45 S = 50 J, (29) J = S +5. (30) 6

7 We want to hire 5 more junior advisers than senior advisors. We have only 1 equation and 2 unknowns. So we must use the budget to make sure we do not exceed costs: $340K = $100K S +$20K J, (31) 17 = 5S +J, (32) Now plug in the MRTS equation: 17 = 5S +S +5, S = 2, J = 7. (33) We want to hire 2 senior advisors and 7 junior advisors. b. The MRTS equation still holds here. We know we want 5 more junior advisors than senior advisors. So if we are given that S = 4, we want 9 junior advisors. 7