A firm estimates its cubic production function of the following form: Q=AL +BL a

Question

A firm estimates its cubic production function of the following form: Q=AL +BL and obtains the following results: 32 ??? ? ? DEPENDENT VARIABLE: Q OBSERVATIONS: 32 VARIABLE L3 L2 R?SQUARE F?RATIO 0.7547 92.31 PARAMETER STANDARD ESTIMATE ERROR ?0.0016 0.0005 0.4000 0.0950 P?VALUE ON F 0.0001 T?RATIO P?VALUE ?3.20 0.0032 4.21 0.0002 ????? a. The equations for total product, average product, and marginal product are: TP = ___________________________ AP = ___________________________ MP = ___________________________ b. The estimated values of A and B are statistically significant at the (exact) levels, ________ and _________, respectively. c. At _______ units of labor usage, marginal product of labor begins to diminish. When the wage rate is $300, answer the following questions. (Remember that AP = Q/L; AVC = w/AP; and SMC = w/MP.) d. Average product of labor reaches its maximum value at ________ units of labor. e. At the output for part d, average variable cost is $______________ and marginal cost is $____________. f. When the rate of labor usage is 100 units of labor, output is _______ units. Average variable cost is $_________ and marginal cost is $__________. ?Chapter 10: Production and Cost Estimation 217 2. Suppose Heritage Corporation believes that its total variable costs follow a cubic specification and so it estimates its average variable costs using the following specification: AVC = a + bQ + cQ2 The regression analysis produces the following computer output: ?????????DEPENDENT VARIABLE: AVC OBSERVATIONS: 45 VARIABLE INTERCEPT Q Q2 R?SQUARE F?RATIO 0.6145 33.47 PARAMETER STANDARD ESTIMATE ERROR 175.0 25.00 ?3.20 0.80 0.08 0.01 P?VALUE ON F 0.0001 T?RATIO P?VALUE 7.00 0.0001 ?4.00 0.0003 8.00 0.0001 ??????????????a. Do the estimated coefficients have the required signs to yield a -shaped AVC curve? Discuss the significance using the p-values. b. Heritage Corporation

Explanation / Answer

a. Q=-0.05*L^3+0.6*L^2 AP=-0.05*L^2+0.6*L MP = 3(–0.05)L^2+2*(0.6L)=-0.15*L^2+1.2L b. The signs of both parameters are correct: A is negative, B is positive. The p-values indicate significance at better than the 2 percent level for both parameter estimates. c. La = –B/2A = –0.6/–0.1 = 6 AP reaches its maximum value when 6 units of labor are employed. d. Q = –0.05(6)^3 + 0.6(6)^2 = 10.8 At 10.8 units of output, AP reaches its maximum value e. APmax = –0.05(6)^2 + 0.6(6) = 1.8 (or APmax = Q/L = 10.8/6 = 1.8) So, AVC = w/AP = 36/1.8 = $20 Since AP = MP when AP is at its maximum value, AVC = SMC = $20 at L = 6 and Q = 10.8. f. When L = 7, AP = 1.75 and MP = 1.05. Thus, AVC = 36/1.75 = $20.57 and SMC = 36/1.05 = $34.29.

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