Johns Hopkins University School of advanced international studies Microeconomics - SA XXXXXXXXXXInstructor: Paul Piveteau Fall 2017 Problem set 2 Question XXXXXXXXXXpoints For each of the following production functions: 1. F(K, L) = K + 2L 2. F(K, L) = min{K, 2L} 3. F(K, L) = K 1 2L (a) Find the marginal product of labor (MPL) and the marginal product of capital (MPK). Does the production function exhibit diminishing marginal productivity of labor and capital? Explain. (b) In a diagram, draw the isoquants when output is equal to 2, 4, and 8. (c) Does the production function exhibit increasing, constant or decreasing returns to scale? Justify your answer. Question XXXXXXXXXXpoints A company produces computers using machines (K) and workers (L). Their production is the following F(K, L) = K 1 3L1 6 such that L workers and K machines allow the company to produce F(K, L) computer units. The usage of these inputs is compensated by paying an annual wage w=20 for each worker and an annual rent of r=80 for each machine. The firm just signed a lease to rent 100 machines and is therefore stuck with these 100 machines for the year. 1. The firm received a total computer order of 10 units for the year. What is the number of workers the firm needs to fill up this order? Why is this number independent from the wage of the workers? 2. By only comparing the current MRTS and the price ratio, do you think they should have acquired more or less machines? 1 A big client just announced it would be willing to buy a number q of computers each year in the next five years. Therefore, the company decides to only produce for this client in the next years. 3. What will be the numbers of machines and workers that will minimize the total cost of the firm? In other words, find the optimal number of machines and workers to produce a quantity q. 4. Show that the resulting cost of production is C(q)=2 2 3 60q2 ? 95q2. Plot this cost function for quantities ranging from 10 to XXXXXXXXXXWould the firm make profit if the unit price of computers was 2000 and the order was 20 units? What if the unit price was 2000 but the order was 40 units? How do you explain this answer? The CEO of the company is worried about the current political situation. Because he is a fan of the Canadian prime minister, he is considering moving the production of his computers to Canada. Price of labor would be cheaper (w=10) but the rental cost of machines is more expensive (r= XXXXXXXXXXWhat would be the cost function of the firm if producing in Canada? Given this function, should the company move to Canada? Question XXXXXXXXXXpoints We are in 1979, Travis is a young New Yorker considering a career as taxi driver. By driving some friends around, he realized that his ability to deliver rides follow the production function: q = F(L, G)=2L1 4G1 4 where q is the quantity of rides, L his working time and G the amount of gas in gallons. Therefore, using his time and some gas, he can produce a quantity q of rides. In addition to these inputs, Travis needs to rent a taxi license which costs a fixed amount F = 80 that needs to be paid at the beginning of the year. We assume that the price of an unit of his time is w = 4 and the price of a gallon of gas is pg = 1. Part 1 1. What would be the cost function of Travis if he becomes a taxi? (Remember that the cost function depends on the quantity q produced and not the units L and G of each input.) 2. From the cost function of Travis, what are the fixed and variable costs? Compute the average cost function and the marginal cost function. 3. The current market price of a ride is p=20. What would be the optimal number of rides Travis provide? What is the profit he would obtain from his taxi business and therefore, shall he start driving a taxi? 4. Plot on a graph the marginal cost function, average cost function and the profit of Travis. 2 Part 2 Travis made the same calculations and therefore started his business. He has been making a good amount of money but, in the spring of 1979, the energy crisis hits and the price of gas increases from pg = 1 to pg = 4. 1. What is the new cost function of Travis? 2. With a price p=20, what is the new optimal number of rides provided by Travis? 3. Shall he keep driving his taxi? Would it be worth it to renew his taxi license at the end of the year if the cost of gas stays the same? 4. In the long run, for these input prices, by how much the price of a ride will need to increase? In other words, assuming that all the taxis have the same production function, what will be the long-run price of a ride? 5. Plot on a graph the marginal cost function, average cost function and the long-run price on the market. Part 3 We are in 2016, Travis is 60 and is still driving his taxi. Hubert is 20 and decides to also start driving. However, Hubert uses his smart-phone to pick up passengers. His production function is similar to Travis but he does not have to pay the taxi license. Instead, he only needs to register on his application which costs him a fixed cost F=10 every year. The costs of inputs are still w = 4 and pg = 4. 1. With a price p=20, what is the optimal number of rides provided by Hubert? What is his profit? 2. As more and more young drivers like Hubert use a smart-phone to drive, what will be the long run price on the taxi market? Can Travis have a profitable business at this long-run price? 3. Plot on a graph the two marginal cost functions, average cost functions and the loss that would make Travis if he decides to keep driving with a taxi license at the new long-run price. 3