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EC 502 Labor Income Taxes

创建日期：2024-05-16 15:37:06

所属分类：数学mathematics

标签： EC 502

Labor Income Taxes

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EC 502: Problem Set 4

1. Labor Income Taxes
Consider an RBC economy similar to the one described in Lecture 16 with two periods, no
uncertainty, and fixed capital. The one difference is the introduction of a labor income tax.
Households
Households have a utility function in each period given by U(Ct, Lt), where Ct is consump-
tion in period t and Lt is leisure in period t. In each period, there is a total time budget of T
units, so the implied labor is given by Nt = T − Lt. Households may save the amount S1 in
period 1 with a given interest rate r, starting from an initial wealth level Y0. Labor in period
t earns wages at rate Wt, but the labor income is taxed at marginal rate τNt ≥ 0 in period
t, so that the households only receive the effective wage (1 − τNt)Wt for each unit of labor
supplied. The government rebates the taxes back to the households in lump-sum form with
transfers Qt = τNtWtNt in each period. The utility maximization problem is given by
max
S1,L1,L2
U(C1, L1) + βU(C2, L2)
C1 = Y0 + (1− τN1)W1(T − L1)− S1 +Q1
C2 = (1 + r)S1 + (1− τN2)W2(T − L2) +Q2
Firms
Firms have a production function given by Yt = AtKαN1−αt for t = 1, 2. K is the fixed capital
stock in this economy, and there is no investment. Firms face the wage rate Wt and solve
the static profit maximization problem
max
N
AtK
αN1−α −WtN.
The resulting optimal choice of labor for firms, Nt, represents labor demand in period t.
General Equilibrium
General equilibrium in this economy is a set of prices and quantities Wt, r, Ct, Yt, Lt, and
Nt such that
• Households optimize their utility as laid out above given Wt and r
• Firms maximize profits as laid out above given Wt
• Markets Clear
Labor Markets Clear : Nt = T − Lt
Savings Market Clears: S1 = 0
Resource Constraints Hold : Yt = Ct
1
(a) Derive the household intratemporal optimality conditions for the optimal leisure choices
Lt in periods t = 1, 2. These are also known as the labor supply curves in each pe-
riod. These optimality conditions are simply the first-order conditions of the household
objective with respect to L1 and L2 after substitution of the budget constraints:
U [Y0 + (1− τN1)W1(T − L1)− S1 +Q1, L1]+βU [(1 + r)S1 + (1− τN2)W2(T − L2) +Q2, L2] .
(b) Using 1) the resource constraint, 2) the production function, and 3) labor market clear-
ing, write the household labor supply curve as an equation with Wt on the left hand
side and some function of At, Nt, and K on the right hand side. You should expect an
equation similar to one appearing on p5 in Lecture 16. However, on the right hand side
of the equation, τNt will now also appear.
(c) Derive the firm’s intratemporal optimality conditions for labor Nt in each period t = 1, 2.
These are also known as the labor demand curves.
(d) For period t, set the household labor supply curve equal to the firm labor demand curve
and eliminate Wt from this equation to derive a unified labor market equilibrium condi-
tion. Again, you should expect an equation similar to one appearing on p5 in Lecture
16, but τNt will appear.
(e) Assume that household preferences are given by
U(C,L) = log (C − v(T − L))
where v(N) = N
1+φ
1+φ with φ > 0. Using this particular functional form for preferences,
simplify the unified labor market equilibrium condition from part (d).
(f) With the firm labor demand expression on the left hand side, and the household labor
supply expression on the right hand side, plot the labor market equilibrium condition as
a function of Nt, i.e. generate a graphical depiction of labor market equilibrium analo-
gous to Figure 2 in Lecture 16.
(g) If τNt increases, how does the labor market equilibrium diagram change?
(h) If τNt increases, does Nt increase or decrease? Justify your answer.
(i) If τNt increases, does Yt increase or decrease? Justify your answer.
(j) If τNt increases, does Ct increase or decrease? Justify your answer.
(k) If τNt increases, does Wt increase or decrease? Justify your answer.

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