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^{2} L^{89}`.

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

The marginal product of labor is $$ MP_L = \frac{dq(K, L)}{dL} = 89 K^{2} L^{89 - 1} = 89 K^{2} L^{88} $$ The marginal product of capital is $$ MP_K = \frac{dq(K,L)}{dK} = 2 K^{2 - 1} L^{89} = 2 K^{1} L^{89} $$ Therefore, the marginal rate of technical substitution is $$ MRTS = - \frac{MP_L}{MP_K} = - \frac{89 K^2 L^{88}}{2 K^1 L^89} = - \frac{89 K}{2 L} $$