Question

A firm has a production function Y = (K^0.3)*(N^0.7) and the firm uses 1 unit of capital for production, that is, K = 1. Additionally, suppose that the market wage is w = 0.35. Suppose that the government imposes a producer tax. That is, the firm pays t units of consumption goods to the government for each unit of output it produces, so the firm is left with (1-t)Y units of output as revenue.

Question: Determine the effect of this tax on the firm’s demand for labor, assuming that t = 0.2. That is

a) Calculate the optimal number of workers the firm will hire. Round your answer to the closest integer.

b) Compare your answer in a) to a situation where the firm does not have to pay a production tax (t = 0), is the firm hiring more or less workers? Briefly explain why you think this happens.

c) Calculate the profits of the firm using your results from a).

Answer 1

a).

Consider the given problem here the production function is “Y = K^0.3*N^0.7, where K=1”.

=> Y = N^0.7, be the production function for “K=1”. The profit function of the producer is given below.

=> A1 = (1-t)*Y – W*N = (1-t)*N^0.7 – W*N.

Here the FOC of the producer is “dA/dN = 0”.

=> (1-t)*0.7*N^(-0.3) – W = 0, => (1-t)*0.7*N^(-0.3) = W, => N^0.3 = {(1-t)*0.7}/W.

=> N = [0.7*(1-t)/W]^10/3, where “W=0.35” and “(1-t)=0.8”.

=> N = [0.7*0.8/0.35]^10/3 = {1.6}^10/3 = 4.79 = 5, => N*1 = 5.

Here if the tax rate is “0.2” and the wage rate is “0.35”, => the optimum labor demand is “N*1=5”.

b).

Given the production function the optimum level demand is “N* = [0.7*(1-t)/W]^10/3”, where “t=0” and “W=0.35”.

=> N*2 = [0.7/0.35]^10/3 = 10.07 = 10, => N*2 = 10 > N*1 = 5.

So as the tax rate become zero the producer increase the demand for labor and the production.

Here tax rate decreases the marginal revenue of the producer given the cost, => it provides the producer an extra incentive to decreases its production an demand less labor.

c).

The profit of the producer under “part –a”.

=> A1 = (1-t)*Y – W*N = (1-0.2)*N^0.7 – 0.35*N, where “N=5”.

=> A1 = 0.8*5^0.7 – 0.35*5 = 0.72.