Question

Question 1: Given the following utility function: (U=Utility, l=leisure, c=consumption) U = 2l + 3c and production function: (Y=Output, N or Ns=Labour or Labour Supply) Y = 30N1/2 If h = 100 and G =10 (h=Hours of labour, G=Government spending). Find the equilibrium levels of the real wage (w), consumption (c), leisure (l), and output (Y). Question 2: (Continuting from question 1) a, Find the relationship between total tax revenue and the tax rate if G = tWN. (G=Government spending, t=tax, W=wage, N=Labour) b, Find the impact of a change in the production function to: Y = 45N1/2

Answer 1

1).

Consider the given problem here the production function is given by, “Y=30*N^1/2”,

=> MPN = (30/2)*N^(-1/2), let’s assume “w” be the real wage here => the demand curve for “N” is given by, “w = MPN”,=> “w = 15/N^1/2”.

Now, the utility function of the consumer is, “U = 2*L+3*C, => the absolute slope of the utility function is “2/3”. Now the budget constraint of the consumer is given by, “C + w*L = w*h, => the absolute slope of the consumer is, “w”.

So, if “MRS = 2/3 < w”, => the person will decide to devote total time to work and will devote nothing to “L”, => N=h=100 and L=0.

So, here the labor supply function is given by, “Ns=h=100, if w > 2/3” otherwise “Ns = 0, if w < 2/3”.

Now, the demand for labor is given by, “w = 15/N^(1/2)”, at N=h=100, “w=15/10 = 3/2 > 2/3.

So, which implied that the “labor demand” will cut the “supply of labor” on the upper segment.

Consider the following fig.

So, given the “Nd” and “Ns” the equilibrium “w” is “3/2 > 2/3”, => the person will devote totally to work and nothing to “L=leisure”. So, “L=0”, “C=w*h=3/2*100=150”.

So, “Y=30*N^1/2=30*100^1/2=30*10=300.

=> the real wage “w=3/2”, consumption “C=w*h=150”, leisure “L=0” and output “Y=300”.

2/a.

Now, let’s assume that “t” be the tax rate, => the “total tax Revenue” is given by, “T = t*(w*N), where “w*N” be the total labor income. So, if “G=t*w*N, => the tax revenue is exactly equal to the “govt. expenditure”.

=> Tax Revenue = G = t*w*N.

2/b.

Now, let’s assume that the production function change to, “Y = 45*N^(1/2)”, => MPN = (45/2)*N^(-1/2).

So, given the production function the demand for labor is given by, “w = MPN = 22.5/N^1/2. So, if we compare these 2 demand function we can see that “MPN” has been increased, => the demand curve will shift upward, => given the labor supply, “Ns=100”, the equilibrium real wage will further increase.

So, at N=100, w = 22.5/10 = 2.25 > 3/2”. So, at the real wage will increase, => the consumption will also increase to “C=2.25*100=225” and the output production will be, “Y=45*10 =450.

=> So, as the “production function change”, => “C” and “Y” both change but “L=leisure” will remain same to “zero”.