Question

Question 2: Firms Consider a firm that produces output Y from capital K and labour N using the production iechoolopy Y KNdThe f's capital endowcnt is piven as K 50 Labour is hired to maximize profits. At a wage rate w, the firm's labour costs are wN The firm's profit (as a function of N is therefore 1. Find the firm's labour demand function by maximizing profits and solving the fist order condition for the wage rate w. 2. Plot the labour demand curve (Nd on the horizontal axis, w on the vertical axis). 3. What wage rate would the firm be willing to pay if it were to hire Nd-8 units of labour? (Use your labour demand function!) What would the firm's profit be in this case? (Use the profit function!) 4.

Answer 1

a)

Given

$\pi(N^{^{d}}) =50^{{1/2}}(N^{^{d}})^{1/2}-wN^{d}$

Differentiate with respect to N^d , we get

$\partial \pi(N^{^{d}})/\partial N^{d}=50^{1/2}*1/2*(N^{d})^{-1/2}-w$

Put  $\partial \pi(N^{^{d}})/\partial N^{d}=0$ for profit maximization

$50^{1/2}*1/2*(N^{d})^{-1/2}-w=0$

$50^{1/2}*1/2-(N^{d})^{1/2}*w=0$

$50^{1/2}*1/2=(N^{d})^{1/2}*w$

Squaring both sides

$\frac{50}{4}=N^{d}*w^{2}$

$N^{d}=\frac{25}{2w^{2}}$

Labor demand function is given by

$N^{d}=\frac{25}{2w^{2}}$

b)

w	Nd=12.5/w2
0.5	50.00
1	12.50
1.5	5.56
2	3.13
2.5	2.00
3	1.39
3.5	1.02
4	0.78
4.5	0.62
5	0.50
5.5	0.41
6	0.35

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c)

Given N^d =8, We have derived in part(b) that

$N^{d}=\frac{25}{2w^{2}}$

$8=\frac{25}{2w^{2}}$

$16w^{2}=25$

$w=\frac{5}{4}=1.25$

d)

Firm's Profit in this case is given by

$\pi(N^{^{d}}) =50^{{1/2}}(N^{^{d}})^{1/2}-wN^{d}$

$\pi(N^{^{d}}) =50^{{1/2}}(8)^{1/2}-1.25*8=20-10=10$