Question

Question 3 6 pts Imagine the consumer's utility function is: U = Vevo and that the consumer must pay a proportional profit tax, t, on any profits she makes. Therefore, the consumer pays taxes tp, where P are the profits she receives. There are no other taxes. 1. Solve for the two equations that define the consumer's optimum. (2 pts) 2. What is the equation for consumption demanded by the consumer in terms of exogenous variables and w? (2 pts) 3. What happens to consumption as the wage increases? Explain the income and substitution effects in this case. (2 pts)

Answer 1

a).

Consider the given problem here the utility function of the consumer is, => U = L^0.5*C^0.5. Now, here the consumer get wage “W” for each unit of labor supplied, => the profit of the consumer is “W*N = P”. So, the budget constraint of the consumer is given by.

=> C = (1-t)*P, where “P=W*N=W*(T-L)”, where “N=labor supply”, “L=leisure” and “T=time available”.

=> C = (1-t)*W*(T-L) = W(1-t)*(T-L) = W(1-t)*T - W(1-t)*L.

=> C + W(1-t)*L = W(1-t)*T, be the budget line.

So, the lagrang function is given by.

=> V = L^0.5*C^0.5 + c*[W(1-t)*T – W(1-t)*L - C], where “c=lagrang multiplier”.

So, the FOC for utility maximization require “dV/dC = dV/dL = 0”.

=> dV/dC = 0.5*L^0.5*C^(-0.5) + c*(- 1) = 0, => 0.5*L^0.5*C^(-0.5) = c, ……………(1).

=> dV/dL = 0, => 0.5*L^(-0.5)*C^0.5 + c*[–W(1-t)] = 0.

=> 0.5*L^(-0.5)*C^0.5 = c*W(1-t), ……………………..(2).

So, these are the required two equations for consumer maximization.

b).

Here by (1) divided by (2) we have the following conditions.

=> L^0.5*C^(-0.5) / L^(-0.5)*C^0.5 = 1/ W(1-t).

=> L/C = 1/W(1-t), => C = W(1-t)*L,……………………..(3).

The budget line is given by, => C +W(1-t)*L = W(1-t)*T, => W(1-t)*L +W(1-t)*L = W(1-t)*T.

=> 2W*(1-t)*L = W(1-t)*T, => L = T/2, => N = T-L=T-T/2 = T/2, => N=T/2.

So, the level of consumption demanded is “C = W(1-t)*N = W(1-t)*T/2, => C = W(1-t)*T/2.

c).

Here the consumption function is directly related to wage rate, => as the wage increases the level of consumption increases. Here as the wage increases the “SE=substitution Effect” induce the consumer to reduce leisure and increase the working hours that also increase the consumption. On the other hand the “IE=income effect” induce the consumer to increase leisure and reduce working that reduce consumption. Now, the SE is stronger than the IE on consumption, => consumption increases as wage increases.