Suppose that the production function is y= 9k^0.5 N^0.5. With this production function, the marginal product of labor is MPN= 4.5K^0.5 N^-0.5. The capital stock is K= 25. The labor supply curve is NS= 100[(1-t)w]^2, where w is the real wage rate, t is the tax rate on labor income, and hence (1-t)w is the after-tax real wage rate. Required:
a. Assume that the tax rate on labor income, t, equals zero. Find the equation of the labor demand curve. Calculate the equilibrium levels of the real wage and employ- ment, the level of full-employment output, and the total after-tax wage income of workers.
b. Repeat part (a) under the assumption that the tax rate on labor income, t, equals 0.6.

A) i)  w/P = MPN , ( NS ) = 100[ (1-t) w]^2

   ii) w = 1.5 ,  N = 225,  

   iii)  y  =  675 ,      

   iv) 337.5

B) i) ( NS ) =  100[(1-0.6)w]^2

   ii)  w = 2.372 , N = 90

   iii) y = 426.91

   iv)  85.839

Explanation:

Given data :

Production function ( y ) =  9k^0.5 N^0.5

MPN = 4.5k^0.5N^-0.5

capital stock ( K ) = 25

labor supply curve ( NS ) = 100[ (1-t) w]^2

assume P = 1

a) Determine

i) equation of labor demand curve =  w/P = MPN

where; w = 22.5 N^-0.5 , N=506.25/(w^2)

labor supply curve ( NS ) = 100[ (1-t) w]^2

ii) equilibrium levels of real wage and employment

506.25/(w^2) = 100[(1-t)w]^2   ( equilibrium condition )

w ( equilibrium level of real wage ) = 1.5

equilibrium level of employment  = 100[(1-t)w]^2 ; where t = 0 , w = 1.5

                                                  = 100 ( 1 * 1.5 )^2

                                               N = 225

iii) level of full-employment  y = 9k^0.5 N^0.5 ; where N = 225 , k = 25

                                                =  9(25)^0.5 * (225)^0.5

                                               y  = 675

iv) Total after-tax wage income of workers

     =  w*N = ( 225 * 1.5 ) = 337.5

B) assuming t = 0.6

i) equation of labor demand curve

labor supply curve ( NS ) =  100[(1-0.6)w]^2  = 16 w^2

ii) equilibrium levels ; 16w^2 = 506.25/(w^2).

w( equilibrium real wage ) = 2.372

Equilibrium employment ( N )=  16 * ( 2.372 )^2 =90

iii) level of full employment y = 9k^0.5 * 90^0.5

                                                = 9(25)^0.5 * 90^0.5 = 426.91

iv) Total after tax wage/income of workers

 =  (1-0.6)*2.372*90 = 85.839