Consider an infinitely-lived household which has a time endowment of one unit in each period. The time unit is separable and can be divided between working time nt and leisure 1-nt. The instantaneous utility function is increasing and concave in consumption, Ct, and leisure. The household enters a period with asset holdings at. In each period, the household chooses consumption ct, asset holdings at+1 and allocates her time. The total return in period t + 1 from the investment in an asset in period t is 1+r. The household receives a constant wage w per time unit spend with labor. Labor is subject to a proportional income tax, 7, where 0 ≤ T ≤ 1. Also, each household has to pay a lump-sum tax of T20 in each period. (1.1) Write down the flow budget constraint of the household for period t. (1.2) Write down the maximization problem of the household in form of the Lagrangean. (1.3) Derive the optimality conditions of the household. (1.4) Assume that the household's preferences can be described by the following utility function: nt 1-o 1+0 Use the optimality conditions from part c) to derive a formulation for labor supply. U (Ct, nt) = -σ 1 0 1+6 (1.5) Explain shortly how a reduction of the proportional income tax 7 would affect labor supply. What would be the response to a reduction in the lump-sum tax T? (1.6) Derive the elasticity of wealth. Do the income tax and the lump-sum tax affect the elasticity?
Consider an infinitely-lived household which has a time endowment of one unit in each period. The time unit is separable and can be divided between working time nt and leisure 1-nt. The instantaneous utility function is increasing and concave in consumption, Ct, and leisure. The household enters a period with asset holdings at. In each period, the household chooses consumption ct, asset holdings at+1 and allocates her time. The total return in period t + 1 from the investment in an asset in period t is 1+r. The household receives a constant wage w per time unit spend with labor. Labor is subject to a proportional income tax, 7, where 0 ≤ T ≤ 1. Also, each household has to pay a lump-sum tax of T20 in each period. (1.1) Write down the flow budget constraint of the household for period t. (1.2) Write down the maximization problem of the household in form of the Lagrangean. (1.3) Derive the optimality conditions of the household. (1.4) Assume that the household's preferences can be described by the following utility function: nt 1-o 1+0 Use the optimality conditions from part c) to derive a formulation for labor supply. U (Ct, nt) = -σ 1 0 1+6 (1.5) Explain shortly how a reduction of the proportional income tax 7 would affect labor supply. What would be the response to a reduction in the lump-sum tax T? (1.6) Derive the elasticity of wealth. Do the income tax and the lump-sum tax affect the elasticity?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Hi, could you help me solve 1-1.3.?
Transcribed Image Text:Consider an infinitely-lived household which has a time endowment of one unit in each
period. The time unit is separable and can be divided between working time nt and leisure
1 nt. The instantaneous utility function is increasing and concave in consumption, Ct,
and leisure.
The household enters a period with asset holdings at. In each period, the household
chooses consumption ct, asset holdings at+1 and allocates her time. The total return in
period t + 1 from the investment in an asset in period t is 1 + r. The household receives
a constant wage w per time unit spend with labor. Labor is subject to a proportional
income tax, 7, where 0 ≤ T ≤ 1. Also, each household has to pay a lump-sum tax of
T20 in each period.
(1.1) Write down the flow budget constraint of the household for period t.
(1.2) Write down the maximization problem of the household in form of the Lagrangean.
(1.3) Derive the optimality conditions of the household.
(1.4) Assume that the household's preferences can be described by the following utility
function:
nt
1 - 0
1+0
Use the optimality conditions from part c) to derive a formulation for labor supply.
U (Ct, nt)
=
-σ
1
0
1+6
(1.5) Explain shortly how a reduction of the proportional income tax 7 would affect labor
supply. What would be the response to a reduction in the lump-sum tax T?
(1.6) Derive the elasticity of wealth. Do the income tax and the lump-sum tax affect the
elasticity?
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