Assignment #2 Production economics Exercise 1: Production function In the Deep Creek Mining Company example described in class suppose that capital is fixed at 500 brake horse power, while labor is a variable input. a) Complete the following table Labor input L (n. of workers) 1 2 3 4 5 6 7 8 9 10 b) Plot the (i) total product, (ii) marginal product and (iii) average product functions c) Determine the boundaries of the three stages of production Exercise 2. Optimal input level in short‐run The amount of fish caught per week on a trawler is a function of the crew size assigned to operate the boat. Based on past data, the following production schedule was developed: Crew size (number of workers) 2 3 4 5 6 7 8 9 10 11 12 a) Over what ranges of workers are there (i) increasing, (ii) constant, (iii) decreasing and (iv) negative returns? b) How large a crew should be used if the trawler owner is interested in maximizing the total amount of fish caught?
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Total product Q

Marginal product MPL

Average product APL

Amount of fish caught per week (hundreds of lbs.) 3 6 11 19 24 28 31 33 34 34 33

c) How large a crew should be used if the trawler owner is maximizing the average amount of fish caught per worker? d) Suppose that the trawler owner can sell all the fish caught for $75 per 100 pounds and can hire as many crew members as desired by paying them $150 per week. Determine the optimal crew size if the trawler owner wants to maximize profits. Exercise 3. Optimal input levels in long‐run A firm uses two variable inputs, labor (L) and raw materials (M), in producing its output. At its current level of output: CL = $10/unit MPL = 25 CM = $2/unit MPM = 4 a) Determine whether the firm is operating efficiently, given that its objective is to minimize the cost of producing the given level of output. b) Determine what changes (if any) in the relative proportions of labor and raw material need to be made to operate efficiently. Exercise 4. Optimal use of labor Consider the following short‐run production function for a firm, where L is labor, and Q is output. Q = 20∙L ‐ .25∙L2 Suppose that the good is sold for $5 per unit. Also assume that the firm can obtain as much of labor (L), as it needs at $81.25 per unit. a) Determine the marginal revenue product of labor. (This is a function. Start from 1 up to 10 workers) b) Determine the marginal factor cost function. c) Determine the optimal value of L, given the objective is to maximize profits. Exercise 5. Returns‐to‐scale Consider the following production function for FFS railways: Q = L∙Eß∙W where L = labor input in man‐hours; E = electricity in KWh; W = capital input in number of wagons; and Q = output measured in millions of kilometers. Suppose that the parameters (, ß, and ) of this model were estimated using a log linear regression model with annual data for the past 40 years. The following results were obtained:  = .42 ß = .37  = .31 a) What type of returns to scale appears to characterize this railways system? b) Suppose that all inputs increase by 5% next year. Determine the approximate percentage change in output. [Hint: multiply by 1.05 …..]
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