COMM

5 points b suppose that you decide to take the new

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(5 points) (b) Suppose that you decide to take the new job. What is the maximum amount that you would pay to eliminate the uncertainty in the income from the new job? Solution: Expected income from new job = .6*50 + .4*30 = 42 U(42 – RP) = 20.345 [10*(42 – RP] .5 = 20.345 RP = \$0.61 (5 points) (c) Illustrate your results in a diagram. 40 20 30 50 42 20.345 U w .61 11

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QUESTION 4 A Vancouver company produces robotic controls using the production function Q = 8K – 0.5K 2 + 2L – 0.1L 2 Q = units of output K = units of capital L = units of labour Because of a city by-law, it is not possible for L to exceed 10 units and for K to exceed 8 units. The wage rate for labour is \$6 per unit of L and the rental cost of capital is \$12 per unit of K. (3 marks) a) In the short run suppose K is fixed at 2 units. Calculate the marginal product of labour (MP L ) and the average product of labour (AP L ). Show that MP L diminishes as L increases. Without doing any further calculations, is it possible to know whether MP L is strictly greater than or less than AP L for any value of L between 0 and 10? Briefly explain. With K = 2, Q = 14 + 2L – 0.1L 2 . So, MP L = dQ/dL = 2 – 0.2L and AP L = Q/L = 14/L + 2 – 0.1L. It is directly evident that MP L = 2 – 0.2L decreases with higher L. Because the production function has diminishing MP L for all L between 0 and 10, MP L must be strictly less than AP L for any value of L between 0 and 10. (7 marks) b) In the short run with K fixed at 2 units, what is the change in output and what is the change in total cost if L is increased from 4 to 5 units? Repeat your answer if L is changed from 5 to 6 units. Using these derived values, what can you conclude about the marginal cost of producing more product when L is allowed to vary between 4 and 6 units? With K = 2, Q = 14 + 2L – 0.1L 2 . Thus, L = 4 implies Q = 14 + 2(4) – 0.1(4) 2 = 20.4 and Cost = 4(6) + 2(12) = 48 because 6 is the wage rate and 12 is the rental cost of capital. L = 5 implies Q = 14 + 2(5) – 0.1(5) 2 = 21.5 and Cost = 5(6) + 2(12) = 54. Finally, L = 6 implies Q = 14 + 2(6) – 0.1(6) 2 = 22.4 and Cost = 6(6) + 2(12) = 60. Therefore, Q changes by 21.5 – 20.4 = 1.1 when L increases from 4 to 5, and Q changes by 22.4 – 21.5 = .9 when L increases from 5 to 6. In both cases, cost changes by 6. Marginal cost can be written as: MC = Cost/ Q. Therefore, MC = 6/1.1 = 5.454 starting from L = 4 and MC = 6/.9 = 6.666 starting from L =5. Consequently, over this range of L, MC is higher for higher levels of output. (4 marks) c) In the long run where K can be freely chosen, derive an expression for the marginal rate of technical substitution (MRTS) of labour for capital. If the firm is observed to employ L = 5 in the long run, , what must the level of K be if the firm is minimizing cost? 12
MRTS = MP L /MP K . MP L = 2 – 0.2L and MP K = 8 – K. Therefore, MRTS = (2 – 0.2L)/ (8 – K). Cost minimization requires setting MRTS equal to ratio of the wage rate and the rental cost of capital. Thus, for any target level of output, cost minimization implies (2 – 0.2L)/(8 – K) = 6/12. With L = 5, this equation can be solved to yield K = 6 as the cost minimizing level of capital.

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• Spring '09
• Brian

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