Question

A worker receives a wage rate w and has L hours of leisure every day (the total endowment of hours is 24 hours per day). The government taxes his income at the constant rate T. The worker spends all his income.

1. Write a budget constraint of this individual and plot it.

2. Display graphically what is the optimal consumption-leisure choice for this worker.

3. Imagine that the government increases the tax rate to T 0 . What is the new budget constraint? Display on the same picture. In the new optimum is the consumption higher? Explain the answer in terms of wealth and substitution effects.

Answer 1

1).

Consider the given problem here “C = Consumption”, “L = Leisure”, “H = Labor hours”, “W = Nominal Wage” and “A = total time endowment”. So, the budget line is given below.

=> C = W*(1-t)*H, where “t” be the tax rate and “H=A-L”.

=> C = W*(1-t)*[A - L] = W*(1-t)*A - W*(1-t)*L, => W*(1-t)*L + C = W*(1-t)*A, be the equation of a budget line. In the following fig”A1B1” shows the budget line.

U

2).

Now, let’s assume that the consumer have normal convex types utility function. So, here given the budget line “A1B1” the equilibrium point is “E1” where the budget line creates the tangency condition to the utility function “U1”, => the optimal consumption and leisure are “C1” and “L1” respectively.

3).

Now, let’s assume that the government decreases the tax rate, => the budget line gets flatter, => the new budget line is “A2B1”. So, the new equilibrium point is “E2” and the new optimal consumption and leisure are “C2 < C1” and “L2 < L1” respectively.

Now, to decompose the “total effect” into “wealth” and “substitution” effect we have to provide some income so that the consumer gets the initial level of utility “U1”. So, here the budget line will shift upward to “A3B3” and the new equilibrium is “E3”. So, the movement from “L1” to “L3” measure the SE which induce the consumer to work less and consume less with the new after tax wage. Now, “L3” to “L2” measure the “wealth effect” which induce the consumer to increase work. Here the SE is much stronger than the “wealth effect”, => the equilibrium “L” increases from “L1” to “L2” and the “H” and “C” decreases. So, here the equilibrium consumption decreases as tax increases.