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^{22} L^{24}`.
Calculate the marginal rate of technical substitution in function of K and L.
The marginal product of labor is
$$
MP_L = \frac{dq(K, L)}{dL} = 24 K^{22} L^{24 - 1} = 24 K^{22} L^{23}
$$
The marginal product of capital is
$$
MP_K = \frac{dq(K,L)}{dK} = 22 K^{22 - 1} L^{24} = 22 K^{21} L^{24}
$$
Therefore, the marginal rate of technical substitution is
$$
MRTS = - \frac{MP_L}{MP_K} = - \frac{24 K^22 L^{23}}{22 K^21 L^24} = - \frac{24 K}{22 L}
$$