Example
The cookie factory's production function is `q \left( c, s \right) = K L^2`. The marginal rate of technical substitution is
$$ MRTS = - \frac{MP_L\left( K, L\right)}{MP_K \left( K, L \right)} = - \frac{2 K L}{L^2} = -\frac{2 K}{L} $$Question
The production function is `q (K, L) = K^{37} L^{79}`.
Calculate the marginal rate of technical substitution in function of K and L.
The marginal product of labor is
$$
MP_L = \frac{dq(K, L)}{dL} = 79 K^{37} L^{79 - 1} = 79 K^{37} L^{78}
$$
The marginal product of capital is
$$
MP_K = \frac{dq(K,L)}{dK} = 37 K^{37 - 1} L^{79} = 37 K^{36} L^{79}
$$
Therefore, the marginal rate of technical substitution is
$$
MRTS = - \frac{MP_L}{MP_K} = - \frac{79 K^37 L^{78}}{37 K^36 L^79} = - \frac{79 K}{37 L}
$$