1. Labor Income Taxes
Consider an RBC economy similar to the one described in the illustrated below pdf attached with two periods, no uncertainty, and ?xed 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 consumption 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 U(C1;L1) + β U(C2;L2)
S1;L1;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 = AtKα Nt1-α for t = 1, 2. K is the ?xed capital stock in this economy, and there is no investment. Firms face the wage rate Wt and solve the static pro?t maximization problem
max AtKα Nt1-α - WtN
N
The resulting optimal choice of labor for ?rms, 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 pro?ts as laid out above given Wt
Markets Clear
Labor Markets Clear : Nt = T - Lt
Savings Market Clears: S1 = 0
Resource Constraints Hold: Yt = Ct
(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 period. These optimality conditions are simply the ?rst-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] + β U [(1 + r)S1 + (1 - τN2)W2(T - L2) + Q2, L2]
(b) Using 1) the resource constraint, 2) the production function, and 3) labor market clearing, 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, Nt 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 ?rm labor demand curve and eliminate Wt from this equation to derive a uni?ed labor market equilibrium condition. Again, you should expect an equation similar to one appearing
(e) Assume that household preferences are given by
U(C, L) = log (C - v(T - L))
where v(N) = (N1+Φ)/ (1+Φ) with > 0. Using this particular functional form for preferences, simplify the uni?ed labor market equilibrium condition from part (d).
(f) With the ?rm 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 analogous to Figure in lecture .
(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 τYt 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.
Attachment:- lecture_161.pdf