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^{18} L^{41}`.
Calculate the marginal rate of technical substitution in function of K and L.
The marginal product of labor is
$$
MP_L = \frac{dq(K, L)}{dL} = 41 K^{18} L^{41 - 1} = 41 K^{18} L^{40}
$$
The marginal product of capital is
$$
MP_K = \frac{dq(K,L)}{dK} = 18 K^{18 - 1} L^{41} = 18 K^{17} L^{41}
$$
Therefore, the marginal rate of technical substitution is
$$
MRTS = - \frac{MP_L}{MP_K} = - \frac{41 K^18 L^{40}}{18 K^17 L^41} = - \frac{41 K}{18 L}
$$