Question

Consider a short-run production function 𝑦𝑦=𝑓𝑓(𝐿𝐿)= 6𝐿2⁄3 where 𝐿 refers to labour. Let the price of output be expressed by 𝑝 and the price of labour by 𝑤.

Find the factor demand for labour and the amount of output it will produce. Show all calculations.

How many units of labour will the firm hire, if p = R3 and w = R6?

Using your answer in (1.2), how much output will it produce?

If no other costs are incurred, how much total profits will it earn?

Answer 1

Given the production function

$Q= f(L,K)=L^\frac {1}{3}K^\frac {2}{3}$

conditional input demand function for labor, L and capital, K is the solution to the cost minimization problem of the firm:

min $L,K$

s.t

$wL+rKL^\frac {1}{3}K^\frac {2}{3}>= y$

where y is the target level of output and w and r are the prices of inputs labor and capital respectively.

Solution to the above cost minimization satisfy the following condition:

1. slope of isocost curve = slope of isoquant i.e. $wr= K^\frac {2}{3}L$
2. $wr= K\frac {2}{3}$
3. output = target i.e. $L^\frac {1}{3}K^\frac {2}{3}=y$
4. $L^\frac {1}{3}K^\frac {2}{3} =y$

solving the above system of equations gives us the conditional input demands:

$L(w,r,y)= (2wr)^\frac {2}{3}y$

$K(w,r,y) =(2wr)^\frac {2}{3}y$

$\frac {18}{15}L^\frac{5}{3}+L=0$

$R3=6L^\frac {2}{3}R6$