Question

You are given the production function: Q(K,L)=10KαLβ

What is the marginal rate of technical substitution (MRTS)? Simplify fully

Answer 1

Solution:

Marginal Rate of Technical Substitution (MRTS) is the rate at which a firm can substitute capital with labor. MRTS equals the slope of an isoquant, which is a curve that represents the different combinations of multiple factors of production, that is labor and capital that yields the same total production. MRTS tells us the rate at which a producer is willing to switch one input (factor of production) with another.

MRTS is calculated as follows:

MRTS = $\frac{Change\; in \;Capital}{Change\; in\; Labor} = \frac{MP_{K} }{MP_{L} }$

MRTS(_K,L) =$\frac{MP_{K} }{MP_{L} }$

MP_L = $\frac{δQ}{δL} = 10βK^{α}L ^{β−1} ​$

MP_K = $\frac{δQ}{δK} = 10αK^{α−1}L ^β ​$

MRTS = $\frac{10βK^{α}L ^{β−1} }{10αK^{α−1}L ^β } = \frac{βK}{αL }$

MRTS =$\frac{βK}{αL }$