Laffer curve In the 1980s, president Reagan based his tax and spending policies on supply side eonomics. The idea behind supply side economics is the marginal tax rate is so high it discourages work. Cutting the tax rate would end up increasing tax revenue. We develop a simple model of this idea to determine the restrictions on the utility function required to generate a Laffer curve. Let T denote the tax rate, w is the real wage rate and n the labor supply. The tax revenue is T=wnT where wn is labor income, which is the tax base. For convinience assume w is constant. There is no reason for this assumption to be true, but we impose it to focus on the restrictions on the utility function to generate the Laffer curve. As the tax rate T increases, workers substitute towards leisure and away from consumption. Hence as T rises, wn falls and tax revenues falls for high enough tax rates. Let U,V satisfy the standard assumptions. The model is static and households are endowed with one unit of time. A representative household solves max[U(c)+V(1-n)] subject to wn(1-T) ≥c. a) Derive the first-order conditions and show the solution is apair of functions c(w,T),n(w,T). b) Determine the impact of an increase in T on the labor supply decision. Show the answer depends on the sign of U"(c)c+U'(c) c) Suppose U"(c)c+U'(c) is monotone. This means for all positive consumption, U"(c)c+U'(c) is always increasing in c, constant, or decreasing. In which of the three cases will the increase in tax rate result in lower tax revenue? to simplify the problem you can assume U(c)=(c1-γ)/(1-γ) d) Explain how the income and substitution effects impact your answer. e) Drop the assumption w is constant. Assume now that output is produced as y=Anα where A>0 is constant and 0<α<1. Households receive labor income and dividend income D where all profits are distributed as dividend. The household's budget is wn(1-T)+D ≥c in equilibrium c=y i. solve the household's problem. assume that there is a firm that manages the production process, hiring labor and paying out profits as dividends. write out the firm's problem. ii. Now vary the tax parameter T and determine what happens to equilibrium labor, consumption and tax revenue as T varies. Write out the equilibrium conditions. Assume the tax reenue disappears and doesn't impact the goods market or provide any additional utility to households. iii. In the first part of the question you were told to treat w as fixed and determine the impact of changes in T on the labor supply and tax revenue. In the second part of the question you incorporated the impact of a change in the tax wage on rate. Explain why a prediction of the impact of an increase in the marginal tax rate based on the assumption the wage rate is constant is an example of Lucas Critique-that some variables are held constant which actually functions of underlying parameters.
Laffer curve
In the 1980s, president Reagan based his tax and spending policies on supply side eonomics. The idea behind supply side economics is the marginal tax rate is so high it discourages work. Cutting the tax rate would end up increasing tax revenue. We develop a simple model of this idea to determine the restrictions on the utility function required to generate a Laffer curve. Let T denote the tax rate, w is the real wage rate and n the labor supply. The tax revenue is
T=wnT
where wn is labor income, which is the tax base. For convinience assume w is constant. There is no reason for this assumption to be true, but we impose it to focus on the restrictions on the utility function to generate the Laffer curve. As the tax rate T increases, workers substitute towards leisure and away from consumption. Hence as T rises, wn falls and tax revenues falls for high enough tax rates.
Let U,V satisfy the standard assumptions. The model is static and households are endowed with one unit of time. A representative household solves
max[U(c)+V(1-n)]
subject to
wn(1-T) ≥c.
a) Derive the first-order conditions and show the solution is apair of functions
c(w,T),n(w,T).
b) Determine the impact of an increase in T on the labor supply decision. Show the answer depends on the sign of
U"(c)c+U'(c)
c) Suppose U"(c)c+U'(c) is monotone. This means for all positive consumption, U"(c)c+U'(c) is always increasing in c, constant, or decreasing. In which of the three cases will the increase in tax rate result in lower tax revenue? to simplify the problem you can assume
U(c)=(c1-γ)/(1-γ)
d) Explain how the income and substitution effects impact your answer.
e) Drop the assumption w is constant. Assume now that output is produced as y=Anα
where A>0 is constant and 0<α<1. Households receive labor income and dividend income D where all profits are distributed as dividend. The household's budget is
wn(1-T)+D ≥c
in equilibrium c=y
i. solve the household's problem. assume that there is a firm that manages the production process, hiring labor and paying out profits as dividends. write out the firm's problem.
ii. Now vary the tax parameter T and determine what happens to equilibrium labor, consumption and tax revenue as T varies. Write out the equilibrium conditions. Assume the tax reenue disappears and doesn't impact the goods market or provide any additional utility to households.
iii. In the first part of the question you were told to treat w as fixed and determine the impact of changes in T on the labor supply and tax revenue. In the second part of the question you incorporated the impact of a change in the tax wage on rate. Explain why a prediction of the impact of an increase in the marginal tax rate based on the assumption the wage rate is constant is an example of Lucas Critique-that some variables are held constant which actually functions of underlying parameters.
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