Marginal Product of Capital

The marginal product of capital tells how many additional products the firm makes with a tiny bit more capital. $$MP_K = \frac{d q \left( K, L \right)}{d K}$$

Example

In the cookie factory, the production function is

$$ q \left( K, L \right) = K L^2 $$

The marginal product of capital is

$$ MP_K = \frac{d q \left( K, L \right)}{d K} = L^2 $$

Question

In the cookie factory, the production function is `q \left( K, L \right) = K^{13} L^{77}`.

What is the marginal product of capital?

$$ MP_K = \frac{d q \left( K, L \right)}{dK} = \frac{d K^{13} L^{77}}{dK} = 13 K^{13 - 1} L^{77} = 13 K^{12} L^{77} $$