Question

# 1. Consider the representative consumer’s problem as follows. The representative consumer maximizes utility by choosing the...

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

Consumption = C , Ns= Work(w) , Leisure = L , Assumed Price of consumption = 1

So, ATQ = Total time = h = l + Ns

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

2. (Time constraint ) = h = l + Ns

{NS= h - l}

c) Labor supply = NS= 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 = MPN= Work/price of consumption

= az(Nd )a−1= W

= az/W = N1-a

Nd= (az/w)1/(1-b)

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