Q2: Consider the following Cobb- Douglas production function for the bus transpo
ID: 1198295 • Letter: Q
Question
Q2: Consider the following Cobb- Douglas production function for the bus transportation system in a particular city: Q= aL^1 F^b2 K^b3 Where L is the labor input in worker hours; F is the fuel input in gallons; K is the capital input in number of buses; Q is the output measured in millions of bus miles Suppose that the parameters (a,b1, b2, b3) of this model were estimated using annual data for the past 25 years. The following results were obtained: a= 0.0012 b1= 0.45 b2: 0.20 b3: 0.30 A. Determine the (i) labor, (ii) fuel, and (iii) capital input production elasticities. B. Suppose that labor input (worker hours) is increased by 2 percent next year (other inputs held constant). Determine the approximate percentage change in output. C. Suppose the capital input (number of buses) is decreased by 3 percent next year (when certain older buses are taken out of service). Assuming that the other inputs are held constant, determine the approximate percentage change in output. D. What type of return to scale appears to characterize this bus transportation system? (Ignore the issue of statistical significance.) E. Discuss some of the methodological and measurement problems one might encounter in using time-series data to estimate the parameters of this model.
Explanation / Answer
Q= 0.0012*L^0.45 F^0.2 K^0.3
A. Elasticity = Marginl productivity of input / Average productivity of input = dQ/d(input) / Q/input
input production elasticity (labor) = 0.0012*0.45*L^0.45F^0.2K^0.2/L / 0.0012*L^0.45 F^0.2 K^0.3/L = 0.45
input production elasticity (fuel) = 0.0012*L^0.45*0.2*F^0.2K^0.2/F / 0.0012*L^0.45 F^0.2 K^0.3/F = 0.2
input production elasticity (capital) = 0.0012L^0.45F^0.2*0.3*K^0.2/K / 0.0012*L^0.45 F^0.2 K^0.3/K = 0.3
B. If labor input is increased by 2% this means L is now 1.02L
New Q = 0.0012*1.02^0.45*L^0.45F^0.2K^0.2 = 1.01( 0.0012*L^0.45 F^0.2 K^0.3) = 1.01Q
C. If capital input is decreased by 3% this means K is now 0.97K
New Q = 0.0012*L^0.45F^0.2*0.97^0.3K^0.2 = 0.99( 0.0012*L^0.45 F^0.2 K^0.3) = 0.99Q
D. As the effect of change in Q is less than change in input factor so here we have decreasing returns to scale
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