the productivity of a country in western Europe is given by the function f(x,y)=40x^4/5 y^1/5 when x units of labor and y units of capital are used.
a)What is the marginal productivity of labor and the marginal productivity of capital when the amounts expended on labor and capital are 32 units and 243 units, respectively?
b) Should the government encourage capital investment rather than increased expenditure on labor at this time in order to increase the country’s productivity?
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a) To find the marginal productivity of labor (MPL) and the marginal productivity of capital (MPK), we need to take the derivatives of the production function with respect to labor and capital.
Given the production function: f(x, y) = 40x^(4/5) * y^(1/5), where x represents labor and y represents capital.
For MPL:
Taking the derivative of f with respect to x, we get:
∂f/∂x = 40 * (4/5) * x^(-1/5) * y^(1/5)
For MPK:
Taking the derivative of f with respect to y, we get:
∂f/∂y = 40 * (1/5) * x^(4/5) * y^(-4/5)
Now, let's evaluate MPL and MPK using the given amounts of labor and capital (x = 32 units, y = 243 units).
For MPL:
MPL = ∂f/∂x = 40 * (4/5) * (32)^(-1/5) * (243)^(1/5)
For MPK:
MPK = ∂f/∂y = 40 * (1/5) * (32)^(4/5) * (243)^(-4/5)
Now, plug in the values and solve for MPL and MPK.
b) To determine whether the government should encourage capital investment or increased expenditure on labor, we need to compare the marginal productivity of labor (MPL) and the marginal productivity of capital (MPK).
If MPL > MPK, it implies that the increase in labor will have a greater impact on increasing productivity compared to increasing capital. In this case, the government should encourage increased expenditure on labor.
If MPK > MPL, it suggests that the increase in capital will have a greater impact on increasing productivity compared to increasing labor. In this case, the government should encourage capital investment.
Therefore, compare the values of MPL and MPK obtained in part a) to determine whether the government should encourage capital investment or increased expenditure on labor at this time to increase the country's productivity.