Question

Suppose that you have the following production function: Y=9K0.5N0.5. With this production function the marginal product of labor is MPN=4.5K0.5N-0.5 (hint: firms pay workers MPN so this equals w). The capital stock is K=25. The labor supply curve is NS=100[(1-t)]w]2 , where w is the real wage, t is the tax on income, and hence (1-t)w is the after-tax real wage rate.

1. a) Graphically draw (a rough sketch is fine) of the labor market and production function. Show graphically the initial equilibrium and show graphically how would expect w, N, and Y to change if there is an increase in the tax on income.

2. b) Assume that the tax rate on income, t, is equal to zero. Find the equation of the labor demand curve. Calculate the equilibrium levels of the real wage, employment, and level of output. What is the after-tax wage income of workers?

3. c) Repeat part (b) under the assumption that the tax on labor income equals 0.5.

4. d) Suppose that the government institutes a minimum wage of w=$2.5. Using your sketch from part (a) show

   graphically how the minimum wage impacts the model when the tax on income is zero and when it equals

   0.5.

5. e) Numerically, if the government institutes the minimum wage from part (d) and the tax rate on labor equals

   zero what is the equilibrium employment and level of output? As a group, would the introduction of a minimum wage increase the total income of workers?

Answer 1

Y= 9K^0.5L^0.5 Equation 1

MPN=4.5K^0.5N^(-0.5) Equation 2

K=25

labor supply curve is NS=100[(1-t)]w]^2 (Equation 3), where w is the real wage, t is the tax on income, and hence (1-t)w is the after-tax real wage rate.

As MPN = w

4.5K^0.5N^(-0.5)= w

4.5(25)^0.5N^-0.5= w

N^-0.5= w/ 22.5

N= (22.5/w)^0.5 (it is a labour demand curve) Equation 4

For equilibrium, N=NS

By solving for w, w= (2.25/1-t)^0.5= 1.5/(1-t)^0.5   Equation 5

Put value of w into equation of labour demand curve, N= 225(1-t)   Equation 6

Put value of N and k into Y, Y= 675(1-t)^0.5   Equation 7

a)

Increase in tax, cause labour supply to decrease which cause wage rate to increase from w to w' and quantity of labour to decrease from N to N, It also cause production to decrease as now to keep same amount of labour producer need to pay more so it cause fall in its production. (As shown in graphs below).

b) If t=0, labour demand equation= (22.5/w)^0.5

From equation 5, w=1.5/(1-t)^0.5= 1.5/(1-0)^0.5= 1.5= equilibrium wage

From equation 6, N= 225(1-t)= 225(1-0)=225= equilibrium Labour

From equation 7, Y= 675(1-t)^0.5 = Y= 675(1-0)^0.5= 675= equilibrium production.

c) if t= 0.5,

From equation 5, w=1.5/(1-t)^0.5= 1.5/(1-0.5)^0.5= 2.1213= equilibrium wage

From equation 6, N= 225(1-t)= 225(1-0.5)=112.5= equilibrium Labour

From equation 7, Y= 675(1-t)^0.5 = Y= 675(1-0.5)^0.5= 477.297= equilibrium production.

d) Minimum wage will cause situation in which supply of labour is more than its demand.

Case1: If t=0 then labour supply is N3 which is more than labour demand N1.

Case2: If t=0.5 then labour supply is N2 which is more than labour demand N1.