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 = 30N^1/2

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 = 45N^1/2

Answer 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 = 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”.

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 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 labour supply, “Ns=100”, the real wage further will 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”.