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^{72} L^{48}`.
Calculate the marginal rate of technical substitution in function of K and L.
The marginal product of labor is
$$
MP_L = \frac{dq(K, L)}{dL} = 48 K^{72} L^{48 - 1} = 48 K^{72} L^{47}
$$
The marginal product of capital is
$$
MP_K = \frac{dq(K,L)}{dK} = 72 K^{72 - 1} L^{48} = 72 K^{71} L^{48}
$$
Therefore, the marginal rate of technical substitution is
$$
MRTS = - \frac{MP_L}{MP_K} = - \frac{48 K^72 L^{47}}{72 K^71 L^48} = - \frac{48 K}{72 L}
$$