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ps6_sol_fall11 - Department of Economics Columbia...

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Department of Economics W3211 Columbia University Fall 2011 SO LUTIONS T O Probl e m S e t 6 Int e rm e diat e Mi c ro ec onomi c s Prof . S e yhan E Arkona c 1 . Sam and Erica are starting a new restaurant in Portland, Oregon. While Sam plans to do the cooking himself, he will need to employ workers and machinery to produce food. He estimates his production function as: q = 15L .25 K Sam is able to accumulate $10,000 to finance the business. Workers cost $10 and capital costs $50. (a) If Sam wishes to produce the most output with the finances available, how much labor and capital should Sam employ. Use a Lagrangian to solve this problem. (b) Does this bundle of capital and labor also minimize the costs? Explain using a graph. Answer: (a) The Lagrangian is L = 15L .25 K + λ[10000 - 10L - 50K] The first-order conditions are: L L = .25(15)L -.75 K - 10λ = 0 L K = 15L .25 - 50λ = 0 L λ = 10000 - 10L - 50K = 0 Using the first two of these conditions we derive the optimal ratio of capital to labor: K/4L = 10/50 Rearranging to get K = 4L/5. Substitute into the third condition above to get 10,000 = 10L + 40L = 50L or L = 200, K = 160. The amount of output is q = 9025.4 (b) Yes, the first two conditions yield the MRTS = w/r condition, which is the same as for the cost-minimization problem. If a firm is producing the maximum output for a given amount of costs, then it must also be producing that amount of output at the lowest cost (duality).
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2 . What is the last dollar rule for cost-minimization? Provide a brief explanation (in words) as well as the corresponding mathematical equality. If the firm is producing at a point where the isocost line is st ee p e r than the isoquant, what does the last dollar rule imply (i.e., where is the last dollar most productive, L or K) and how should the firm alter its capital and labor in the long run? Answer: The last dollar a firm spends on capital should have the same impact on output as the last dollar a firm spends on labor in order to be minimizing costs: MP L /w = MP K /r If the isocost is steeper than the isoquant, then MRTS < w/r This implies MP L /w < MP K /r, in which case the last dollar is more productive when employing capital. The firm should increase the amount of capital and decrease the amount of labor in order to minimize its costs. 3 . Suppose Ralph hires workers at his supermarket at a wage of $12/hour. Ralph currently has 10 checkstands (i.e., capital) with a rental rate of $10/hour. Production of customers served (i.e., output) is determined by the hourly production function f ( L , K ) = 0. 5 L 3 / 4 K 2 For the questions that follow, the number of checkstands is fixed. Show your work clearly. (a) If Ralph wants to serve 400 customers per hour, how many workers must he employ? How much will it cost to serve 400 customers per hour? (b) Derive Ralph’s short -run cost function with the 10 checkstands.
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