Question

1. Consider the representative consumer’s problem as follows. The representative consumer maximizes utility by choosing the amount of consumption good C and the amount of leisure l . The consumer has h units of time available for leisure l and for working Ns , that is, h = l+Ns . Government imposes a proportional tax on the consumer’s wage income. The consumer’s after-tax wage income is then (1−t )w(h −l ), where 0 < t < 1 is the tax rate and w is real wage rate. The consumer takes w as given. In addition, the consumer earns profits π from owning the representative firm. There is no lump-sum tax, thus, T = 0. (a) (2 points) Do one of the following: (1) write down the equation for the budget constraint facing the representative consumer; or (2) draw the budget constraint on the graph, the slope and intercepts must be clearly labeled. (b) Suppose that the marginal rate of substitution of leisure for consumption,MRSl ,C , is given by MRSl ,C = C bl where b > 0 is also given. Solve for the consumer’s optimal choices of consumption C and l . (c) Using the above results, obtain the labor supply function of the consumer.

2. The representative firm maximizes profits by choosing the optimal labor input Nd , its production function displays constant returns to scale. Suppose that the marginal product of labor is given by MPN = az(Nd )a−1, in which 0 < a < 1 and capital K has been set to one. The firm takes as given the real wage rate w. Solve for the firm’s optimal choice of Nd .

Answer #1

Answer :-a) It is given in the question that -

Consumption = C , N^{s}= Work(w) , Leisure = L , Assumed
Price of consumption = 1

So, ATQ = Total time = **h = l +
N ^{s}**

**1. (Income
contraint) **After tax inco0me =
**(1 -t)w(h-l) , 0<t<1**

**2. (Time
constraint ) = h = l + N ^{s}**

^{ {}N^{S}= h -
l}

c) Labor supply = N^{S}= h - l

= h - (1-t)h/a+1-h

= {1- (1-t)/a+1-h}h

2. It is given that, MPN = az(Nd )a−1, in which 0 < a < 1

and, K = 1,

As per question,

Income wage = MP_{N}= Work/price of consumption

= az(N^{d} )^{a−1}= W

= az/W = N^{1-a}

N^{d}= (az/w)^{1/(1-b)}

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