Linear regression

  Question 1 - 50 marks A firm produces y units of output using the production function f(K, L) = L 1 2 K 1 4 , which it can sell for a price p. The firm pays wages w for the labour and a rate r for the capital used. Suppose the firm faces a fixed level of K, denoted by K. a) What is the optimal level of labour used by a profit maximising firm? What is the optimal output level? [6 marks] b) What are the total costs of the firm? How do they change as wages w increase? How are they affected by an increase in r? Provide intuition for your result. [8 marks] Suppose from now on that the firm plans for the long run, where it can choose both capital and labour. c) What is the marginal rate of technical substitution? What is the optimal input combination to produce a fixed level of output y? Calculate the derived demands for capital and labour. [10 marks] d) What are the total costs given the derived demands? Find the optimal amount of y. [10 marks] e) Would you generally expect the short run or the long run costs to be higher? Support your answer with an appropriate graph. [8 marks] f) Does there always exist an optimal output level? Define the concepts of marginal product and returns to scale and relate your answer to these concepts. [8 marks] Question 2 - 25 marks Consider a competitive market setting in a constant cost industry. Firms are identical, that is they face the same prices, and there are no barriers to entry or exit. The market demand is given by Y = 10 − 2p, where Y is the output produced by all firms in the market. The total cost of each firm is c(y) = 1 2 y 2 + 2. a) Define a constant cost industry and explain how it differs from an increasing cost and decreasing cost industry. What is the average cost and the marginal cost of a single firm? [7 marks] b) Assume that each firm makes zero profit in the long run. What is the output level y of a single firm? How does the number of firms in the industry, k, depend on the demand curve? How many firms are there? [8 marks] Question 2 continues on next page ECN 211 (2019) Page 3 c) Now there is an increase in the market demand. Show graphically how prices and output adjust in the short run. How do prices and output adjust in the long run? What are the implications for the long run industry supply curve? [10 marks] Question 3 - 25 marks Suppose that three roommates, Ann (A), Bob (B) and Charlie (C) need to decide whether to hire a cleaner for a certain amount of time G, which increases everyone’s utility. An individual i = {A, B, C} also benefits from the remaining income xi , which is defined as xi = mi − gi , where gi denotes i 0 s contribution to the cleaner’s salary and mi is the overall income. The utility function for each individual i is then given by ui(xi , G) = xi + αi ln G a) Interpret the utility function. How much of their income would each individual be willing to give up for the cleaner if the cleaner costs p? Calculate the marginal rate of substitution between the public and the private good and interpret. [6 marks] b) What is the optimal amount of the public good that should be provided? What amount of time for the cleaner does each individual choose? Compare and explain why these differences arise. [6 marks] c) Explain the concept of single-peaked preferences and show that all of the agent’s preferences have this property.