Marginal Rate of Technical Substitution

The marginal rate of technical substitution (MRTS) tells how much capital is needed to replace a bit of labor. $$MRTS = - \frac{MP_L\left( K, L\right)}{MP_K \left( K, L \right)}$$

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^{100} L^{25}`.

Calculate the marginal rate of technical substitution in function of K and L.

The marginal product of labor is $$ MP_L = \frac{dq(K, L)}{dL} = 25 K^{100} L^{25 - 1} = 25 K^{100} L^{24} $$ The marginal product of capital is $$ MP_K = \frac{dq(K,L)}{dK} = 100 K^{100 - 1} L^{25} = 100 K^{99} L^{25} $$ Therefore, the marginal rate of technical substitution is $$ MRTS = - \frac{MP_L}{MP_K} = - \frac{25 K^100 L^{24}}{100 K^99 L^25} = - \frac{25 K}{100 L} $$