No problem, I understand now. So, if we have y=[0.2*(K^-1) + 0.8*(L^-1)]^-1, this can be simplified to y=[0.2/K + 0.8/L]^-1.
Now, if we have wk = wL = 1, meaning the wage rate for capital (wk) is equal to the wage rate for labor (wL), we can substitute these values into the production function.
Substituting wk = wL = 1 into y=[0.2/K + 0.8/L]^-1, we get y=[0.2/1 + 0.8/1]^-1 = [1 + 1]^-1 = 1/2.
So, the production function y will be equal to 1/2 when wk = wL = 1.
Now, moving on to the second part of the question:
We have two production functions:
1) y=[0.2k^-1 + 0.8L^-1]^-1
2) z=K^(1/3)*L^(2/3)
We are asked to determine which of these two production functions, y or z, is more demanding of capital relative to labor.
To do this, we need to compare the marginal products of capital and labor in both functions.
In function 1, the marginal product of capital (MPK) is the derivative of y with respect to K, and the marginal product of labor (MPL) is the derivative of y with respect to L.
Differentiating y=[0.2k^-1 + 0.8L^-1]^-1 with respect to K and L, we get:
MPK = -0.2/(K^2), and MPL = -0.8/(L^2).
In function 2, the marginal product of capital (MPK) is the partial derivative of z with respect to K, and the marginal product of labor (MPL) is the partial derivative of z with respect to L.
Taking the partial derivatives of z=K^(1/3)*L^(2/3) with respect to K and L, we get:
MPK = 1/3 * K^(-2/3) * L^(2/3), and MPL = 2/3 * K^(1/3) * L^(-1/3).
To compare the marginal products of capital and labor, we need to find the ratio of MPL to MPK.
For function 1: MPL/MPK = (-0.8/(L^2)) / (-0.2/(K^2)) = 4 * (K^2/L^2).
For function 2: MPL/MPK = (2/3 * K^(1/3) * L^(-1/3)) / (1/3 * K^(-2/3) * L^(2/3)) = 2 * (K/L).
Now, to determine where the ratio of wL/wk changes, we can equate the two ratios obtained above and solve for K/L:
4 * (K^2/L^2) = 2 * (K/L)
2 * (K^2/L^2) = K/L
2 * (K/L)^2 - (K/L) = 0
Hence, we have a quadratic equation in (K/L). Solving this equation will give us the values of (K/L) at which the ratio of wL/wk changes.
I hope this helps! Let me know if you have any further questions.