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^{48} L^{57}`.
Calculate the marginal rate of technical substitution in function of K and L.
The marginal product of labor is
$$
MP_L = \frac{dq(K, L)}{dL} = 57 K^{48} L^{57 - 1} = 57 K^{48} L^{56}
$$
The marginal product of capital is
$$
MP_K = \frac{dq(K,L)}{dK} = 48 K^{48 - 1} L^{57} = 48 K^{47} L^{57}
$$
Therefore, the marginal rate of technical substitution is
$$
MRTS = - \frac{MP_L}{MP_K} = - \frac{57 K^48 L^{56}}{48 K^47 L^57} = - \frac{57 K}{48 L}
$$