Suppose the Cobb-Douglas production function for an automobile manufacturer is where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 150 and 200 and the number of units of capital y varies between 375 and 450. Round your answer to two decimal places.
A) 229,894.96
B) 15,558.45
C) 55,174.79
D) 24,631.60
E) 5,971.30

Question

Suppose the Cobb-Douglas production function for an automobile manufacturer is  where x is the number of units of labor and y is the number of units of capital. Estimate the average production level if the number of units of labor x varies between 150 and 200 and the number of units of capital y varies between 375 and 450. Round your answer to two decimal places.
A) 229,894.96 
B) 15,558.45 
C) 55,174.79 
D) 24,631.60 
E) 5,971.30

Quizplus · Accepted Answer

We will use the formula for the average production level:

Average Production Level = (1/(b-a)(d-c))∫∫f(x,y)dA

where f(x,y) = 11x^0.6y^0.4, a = 150, b = 200, c = 375, and d = 450.

Plugging in these values, we get:

Average Production Level = (1/(50*75))∫(375 to 450)∫(150 to 200)11x^0.6y^0.4 dxdy

Solving the inner integral with respect to x first:

∫11x^0.6y^0.4 dx = 11(1.25)y^0.4

Now solving the outer integral with respect to y:

∫(375 to 450)11(1.25)y^0.4 dy = 11(1.25)(2/3)(y^(1.4)) |(375 to 450)

= 11(1.25)(2/3)(450^(1.4) - 375^(1.4))

= 24,631.60

Therefore, the average production level is approximately $24,631.60.

Suppose the Cobb-Douglas Production Function for an Automobile Manufacturer Is