Question

The Cobb-Douglas production function for an automobile manufacturer is \(f(x, y)=100 x^{0.6} y^{0.4}\) 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 200 and 250 and the number of units of capital \(y\) varies between 300 and 325 .

Short answer

To estimate the average production level, substitute the minimum and maximum values of \(x\) and \(y\) into the Cobb-Douglas function to compute the corresponding production levels, and then take the average of these two results.

Step-by-step solution

Determine the Minimum and Maximum Values of Labor and Capital

From the problem statement, we know that the units of labor, \(x\), varies between 200 and 250, and the units of capital, \(y\), varies between 300 and 325. So, the minimum value of \(x\) is 200, the maximum value of \(x\) is 250, the minimum value of \(y\) is 300, and the maximum value of \(y\) is 325.

Substitute the Values and Compute Cobb-Douglas Function Output

Next, we need to substitute these values into the Cobb-Douglas function and compute the output. We need to do this for both the minimum and maximum values of \(x\) and \(y\). So, \(f(200,300) = 100 \times 200^{0.6} \times 300^{0.4}\) and \(f(250,325) = 100 \times 250^{0.6} \times 325^{0.4}\). Calculate these values.

Compute the Average Production

Finally, find the average production by calculating the arithmetic mean of the two results obtained in the previous step. This can be done by adding the two values together and dividing by two.