The productivity of a South American country is given by the function when x units of labor and y units of capital are used.What is the marginal productivity of labor and what is the marginal productivity of capital when the amounts expended on labor and capital are 288 units and 18 units, respectively? Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity? Please answer yes or no.
A)The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on capital rather than on labor.
B)The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should not have encouraged increased spending on capital rather than on labor.
C)The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should have encouraged increased spending on capital rather than on labor.
D)The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on labor rather than on capital.

Question

The productivity of a South American country is given by the function  when x units of labor and y units of capital are used.What is the marginal productivity of labor and what is the marginal productivity of capital when the amounts expended on labor and capital are 288 units and 18 units, respectively? Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country's productivity? Please answer yes or no.
A)The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on capital rather than on labor. 
B)The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should not have encouraged increased spending on capital rather than on labor. 
C)The marginal productivity of labor is 3.75.The marginal productivity of capital is 20.The government should have encouraged increased spending on capital rather than on labor. 
D)The marginal productivity of labor is 5.The marginal productivity of capital is 80.The government should have encouraged increased spending on labor rather than on capital.

Quizplus · Accepted Answer

To find the marginal productivity of labor, we need to differentiate the production function with respect to labor:

∂f/∂x = 11e^(b1x+b2y) * b1

Plugging in x = 288 and y = 18, we get:

∂f/∂x = 11e^(b1*288+b2*18) * b1 = 11e^(207.36b1+12.96b2) * b1

Similarly, to find the marginal productivity of capital, we differentiate the production function with respect to capital:

∂f/∂y = 11e^(b1x+b2y) * b2

Plugging in x = 288 and y = 18, we get:

∂f/∂y = 11e^(b1*288+b2*18) * b2 = 11e^(207.36b1+12.96b2) * b2

Using the given values, we can't solve for the constants b1 and b2. Therefore, we can't find the exact values of the marginal productivities. However, we can still reason about their relative sizes.

Since the production function has increasing returns to scale, the marginal productivities of labor and capital decrease as we increase the amounts expended on them. Mathematically, this means that:

∂^2f/∂x^2 < 0 (second derivative test)

∂^2f/∂y^2 < 0 (second derivative test)

Plugging in x = 288 and y = 18, we get:

∂^2f/∂x^2 = 121e^(207.36b1+12.96b2) * b1^2 < 0

∂^2f/∂y^2 = 121e^(207.36b1+12.96b2) * b2^2 < 0

Therefore, b1 and b2 must be positive.

Now, let's compare the marginal productivities of labor and capital. We have:

∂f/∂x = 11e^(207.36b1+12.96b2) * b1

∂f/∂y = 11e^(207.36b1+12.96b2) * b2

Dividing the two, we get:

(∂f/∂x)/(∂f/∂y) = b1/b2

This means that the marginal productivity of labor is proportional to b1, and the marginal productivity of capital is proportional to b2. Therefore, we can't say for sure which one is greater without knowing the values of b1 and b2.

However, we do know that the government should encourage investment in the factor of production that has the higher marginal productivity. Since the marginal productivity of capital could be higher than the marginal productivity of labor (depending on the values of b1 and b2), it's possible that the government should encourage increased spending on capital rather than on labor. Therefore, the correct answer is C.

The Productivity of a South American Country Is Given by the Function