Cobb-Douglas Production function

Y=60K^1/3 L^2/3

If K=64 L=8 calculate MPL
What is the equilibrium real wage in the economy

To calculate the marginal product of labor (MPL), you need to take the partial derivative of the production function with respect to labor (L), while keeping the capital (K) constant.

Given the Cobb-Douglas production function:

Y = 60K^(1/3) * L^(2/3)

To find MPL, differentiate the production function with respect to L:

MPL = ∂Y/∂L = (∂/∂L) (60K^(1/3) * L^(2/3))

To differentiate the above expression, you can use the power rule of differentiation:

d(x^n)/dx = n*x^(n-1)

Differentiating the first term:

∂/∂L (60K^(1/3)) = 0

Differentiating the second term:

∂/∂L (L^(2/3)) = (2/3) * L^(-1/3)

Now, let's substitute the given values K = 64 and L = 8 into the MPL equation:

MPL = (2/3) * 8^(-1/3)

To simplify the expression:

8^(-1/3) = 1/(8^(1/3)) = 1/2

MPL = (2/3) * (1/2) = 1/3

Therefore, the marginal product of labor (MPL) in this economy is 1/3.

To find the equilibrium real wage in the economy, we need to equate the MPL to the real wage (w).

MPL = w

w = 1/3

Hence, the equilibrium real wage in the economy is 1/3.