Question 1 (45 marks) A firm's production function is q=ak¹/3L¹/3 where q is the quantity of final production, K is the

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Question 1 45 Marks A Firm S Production Function Is Q Ak 3l 3 Where Q Is The Quantity Of Final Production K Is The 1 (128.42 KiB) Viewed 11 times

Question 1 (45 marks) A firm's production function is q=ak¹/3L¹/3 where q is the quantity of final production, K is the quantity of capital, and I is the quantity of labour, a > 0. Suppose that each unit of capital costs r, and each unit of labor costs w. (a) (5 marks) Does this production function exhibit increasing, decreasing, or constant returns to scale? Justify your answer. (b) (5 marks) Find the optimal quantities of capital and labour for this firm as a function of r, w, and q (where q is the quantity to be produced). (c) (5 marks) Based on your answer in part (b), derive the firm's demand function for capital (i.e., the quantity of capital that the firm would use as a function of r, w and q). Given constant values of w and q, find the slope of the (inverse) demand curve for capital (K as a function of r). Is the (inverse) demand curve for capital downward or upward sloping? (d) (5 marks) Is the demand for capital elastic or inelastic with respect to its own-price? Show your work. (e) (5 marks) Suppose w = r = ³/². Derive the (long-run) total, average and marginal cost functions. (f) (5 marks) How do average costs change when the output increases? Explain why this is the case by comparing marginal costs and average costs, and provide an intuitive explanation. (g) (5 marks) Write the equation of the (inverse) supply curve of this firm (with price P as a function of the quantity q). Draw this supply curve (with P in the vertical axis, and q in the horizontal axis). (h) (5 marks) Suppose that there are 81 identical companies in total in the market, all of them with the same production function of this question. There are no prospects of additional firms that could enter the market. Write the equation of the market (inverse) supply curve (with price P as a function of the quantity Q). Draw this supply curve (with P in the vertical axis and Q in the horizontal axis), where is the total quantity produced in the market. (i) (5 marks) Suppose that the price in the market is P = 27. Obtain the production of each firm, qi, AND the production in the market, Q.