Question

^{Suppose that the production function is Y = 9 60.500.5 With this production function, the marginal product of labor is MPN = 4.5 K.5/10.5 The labor supply curve is NS = 105 x[(1 – t)xw], where w is the real wage rate, t is the tax rate on labor income, and hence (1 – t) xw is the after-tax real wage rate. The capital stock is K = 25. Assume that the tax rate on labor income, t, equals zero The equilibrium level of the real wage is (Round your answer to two decimal places). The equilibrium level of employment is (Round your answer to one decimal place). Now, suppose the tax rate is changed to t = 0.2. Now the equilibrium level of the real wage is (round your answer to two decimal places). With the tax rate now at t= 0.2, the equilibrium level of employment is (round your answer to one decimal place).}

Answer 1

Labor demand equation; MPN = w

where w is real wage rate

=> MPN = w

=> 4.5 (K^0.5/N^0.5) = w

Put K = 25

=> 4.5 (25^0.5 / N^0.5) = w

=> 4.5(5/ N^0.5) = w

=> N^0.5 = (4.5 * 5 / w)

=> N^0.5 = 22.5 / w

Squaring both sides;

=> (N^0.5)² = (22.5/w)²

=> N = 506.25 / w²

Labor demand function: Nd = (506.25 / w²)

Labor supply function: Ns = 105 [(1-t) * w]²

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Case 1: Tax rate is zero i.e., t = 0

At labor market equilibrium; Nd = Ns = N

=>  (506.25 / w²) = 105 [(1-t) * w]²

put t = 0

=> (506.25 / w²) = 105 [(1-0) * w]².

=> (506.25 / w²) = 105 w².

=> 506.25 /105 = w² * w²

=> (506.25 / 105) = w⁴

=> (w⁴)^1/4 = (506.25 / 105)^1/4

=> w = (506.25 / 105)^1/4 ------------ Eq(1)

=> w = 1.48

AnswerHence, the equilibrium real wage is 1.48

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At equilibrium. Nd = Ns = N

=> N = 105 [(1-t)w]²

Since t = 0

=> N = 105 (w)²

Put equation (1)

=> N = 105 [(506.25 / 105)^1/4 ]²

=> N = 105 (506.25 / 105)^2/4

=> N = 105 (506.25 / 105)^1/2

=> N = 230.6

Answer: Hence, the equilibrium level of employment is 230.6

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Case 2:

Tax rate is 0.2 i.e., t = 0.2

At labor market equilibrium; Nd = Ns = N

=>  (506.25 / w²) = 105 [(1-t) * w]²

put t = 0.2

=> (506.25 / w²) = 105 [(1-0.2) * w]².

=> (506.25 / w²) = 105 [(0.8 * w]².

=> (506.25 / w²) = 105 * 0.8 w².

=> (506.25 / w²) = 84 w².

=> 506.25 /84 = w² * w²

=> (506.25 / 84) = w⁴

=> (w⁴)^1/4 = (506.25 / 84)^1/4

=> w = (506.25 / 84)^1/4 ------------ Eq(2)

=> w = 1.57

Answer :Hence,With the tax rate of 0.2,  the equilibrium real wage is 1.57

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At equilibrium. Nd = Ns = N

=> N = 105 [(1-t)w]²

Since t = 0.2

=> N = 105 [(1-0.2)w]²

=> N = 84(w)²

=> N = 84[(506.25 / 84)^1/4 ]²

=> N = 84 (506.25 / 84)^2/4

=> N = 84 (506.25 / 84)^1/2

=> N = 206.2

Answer: Hence, With tha tax rate of 0.2, the equilibrium level of employment is 206.2

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