Answer to Question #171962 in Macroeconomics for Rameen Hamood Mughal

Question

A firm uses skilled (S) and unskilled (U) labour to produce output (Q). The firm’s production function is given by

   $f(U,S)= sqrt(2U+4S)$

where U denotes unskilled and S denotes skilled labour. Assume U ≥ 0 and S ≥ 0. Let w_u = 6 and w_s = 8 denote the prices of unskilled labour and skilled labour respectively.

a. Suppose (U,S) = (8,5). How much output does the firm produce? What is the total cost of producing that level of output?

b. Let Q₀ and C₀ respectively denote the value of output and total costs obtained in part (a). Firm is not particularly interested in cost-cutting but they are averse to cost-increase. Find out the maximum level of output (Q*) that can be produced with a spending cap of  C₀  and recommend the levels of unskilled (U∗) and skilled labour (S∗) necessary to produce Q∗. Find Q∗, U∗, and S∗.

Answer 1

a.To maximize output at these costs, you can use the direct equal cost, or isocost

$C = P \times U+ P \times S$

Substituting the original data, we get the equation:

6U+8S=8.5

$U=\frac{8.5-8S}{6}$

Let's substitute this value into the given equation

$Q=\sqrt{2\times(\frac{8.5-8S}{6})+4S}=\sqrt{(\frac{8.5-8S}{3})+4S}=\sqrt{(\frac{8.5+4S}{3})}$

The aggregate product function reaches a maximum when the marginal product function is zero

$Q'=\frac{2\sqrt{6}}{3\sqrt{8S+17}}$

The marginal product is the first derivative of the aggregate product by a variable factor

 $\frac{2\sqrt{6}}{3\sqrt{8S+17}}=0$

the corresponding equation has no solution

b.6U+8S=0

$U=\frac{-8S}{6}=\frac{-4S}{3}$

$Q=\sqrt{2\times(\frac{-4S}{3})+4S}=\sqrt{(\frac{-8S}{3})+4S}=\sqrt{(\frac{-4S}{3})}$

$Q'=\frac{S}{\sqrt{3}\times (-S)^{1/5}}$

$\frac{S}{\sqrt{3}\times (-S)^{1/5}}=0$

S=0

U=0

Q=0