Question

What returns to scale does this production function have? Q = L + K Q = 50LK; • w = per unit cost of labor; • r = per unit cost of capital Use MPL ... (1) & Q = 50LK .... (2) to find out mathematical expressions of L*, K*and TC(Q,w,r) = wŁ* + rK*

Answer 1

Answer:-

Given That:-

The returns of Scale this production function have

Q=L+K

Q=50LK:

$\cdot w =$ per unlit cost of labor

$\cdot r=$per unit cost of capital

Use
MPKr
& $Q=50LK....(2)$

we have to find out mathematical expressions of
L.K*
and $TC(Q,w,r)=wL^{*}+rK^{*}$

Q=L+K

multiply L & K by $\lambda$

$=\lambda (L+K)$

$=\lambda (L+K) = \lambda Q$

power of $\lambda$ is 1 which means it has constant returns to scale .

$Q=50LK$

$mp_{L} = \frac{\partial Q}{\partial L} = 50K$

$mp_{K} = \frac{\partial Q}{\partial K} = 50L$

$MRS = \frac{50K}{50L} = \frac{K}{L}$

$MRS = \frac{w}{r}$

$\frac{K}{L} = \frac{w}{r}\Rightarrow K = \frac{wL}{r}$

$Q=50L( \frac{wL}{r})=50L^{2}.\frac{w}{r}$

$\frac{rQ}{50w} = L^{2}$

$L^{*}=(\frac{rQ}{50w})^{\frac{1}{2}}$

$K^{*}=\frac{w}{r}(\frac{rQ}{50w})^{\frac{1}{2}}$   $=(\frac{wQ}{50r})^{\frac{1}{2}}$

$=w(\frac{rQ}{50w})^{\frac{1}{2}}+r(\frac{rQ}{50r})^{\frac{1}{2}}$

$=(\frac{wrQ}{50})^{\frac{1}{2}}+r(\frac{wrQ}{50})^{\frac{1}{2}}$

$=2(\frac{wrQ}{50})^{\frac{1}{2}}$