Question concerning production function - Cobb douglas. Given is:

y=(0,2K-1 + 0,8L-1)-1
If wk = wL =1, what will k and L be?
Now suppose again:
y=(0,2k-1 + 0,8L-1)-1 and
z=K1/3*L2/3. Is product y or z always more demanding to capitals (in case of labour)? if not, find where % of wL/wk is changing in price.
Notes: -1 after K and L is square and also -1 after ()and 1/3 and 2/3

Im sorry, but I don't understand your notation. You say -1 means "square". However, what does 0,2K-1 mean?
Is it "2 times (K-squared)"?

If so, your first equation becomes y=[2*(K)^2 + 8*(L)^2]^2. Here, again, I'm confused. With the outer squared term, you get one beast of a production function, requiring more calculas than I can remember. Without the outer squared term you get a production function that is, unusual but not unheard of in teaching economics, and mathmatically tractable. (Or it could be that I am missing something).

BTW, the traditional cobb-douglas is your z-equation.

Sorry my english but its root not squared, sorry for this problem but I'm still have problem with this math

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.