Macroeconomics: Overlapping Generation model and lump-sum taxes

[Charlotte87]

Homework Statement

Suppose an individual born at time $t$ maximizes life-time utility

\begin{equation*}
\max \ln(c_{1,t}) + \frac{1}{1+\rho}\ln(c_{2,t+1}), \; \rho>0
\end{equation*}

subject to the budget constraints in periods t and t+1, respectively

\begin{eqnarray}
c_{1,t} + s_{t} &=& w_{t} - \tau \nonumber \\
c_{2,t+1} &=& (1+r_{t+1})s_{t} \nonumber
\end{eqnarray}

where c_1,t is consumption of young individuals at time t and cSUB]2,t+1[/SUB] is consumption of old individuals at time t+1. Savings of the young S_t earn an interest rate r_t+1 and w_t is the wage rate at time t. The government initially balances its budget by financing constant spending per capita g by lump-sum taxes τ. The representative firm maximizes profits

\begin{equation*}
\max_{K_{t},L_{t}} K_{t}^{\alpha}L_{t}^{1-\alpha} - r_{t}K_{t} - w_{t}L_{t}, \; 0<\alpha<1
\end{equation*}

where K_t is the aggregate capital stock and L_tthe labor input. Firms are perfectly competitive and take factor prices r_t and w_t as given. Capital does not depreciate and the labor force grows at rate n>0, that is L_t = (1+n)L_t-1.

(a) Derive the Euler equation governing optimal consumption of the consumer born at time t, and solve for c_1,t and s_t as functions of after-tax wages w_t-τ. Provide a brief intuitive explanation.

(b) State the condition for goods market equilibrium. Combine this condition with the solution for savings from part (a) and the representative firm's optimal factor demands for capital and labor, to derive the equation of motion of the capital stock per capita k = K/L of the form
\begin{equation*}
k_{t+1} = ak_{t}^{b} + d
\end{equation*}
where a,b and d are constant coefficients.

(c) Consider the following fiscal experiment. Assuming constant government spending g>0, suppose that lump-sum taxes are cut by Δτ in period t, financed by an increase in lump-sum taxes (1+r_t+1)Δτ/(1+n) in period t+1. The government balances its budget from period t+2 onwards.

Starting from steady state k^*, derive the effect of this fiscal experiment on k_t+1 and give a brief intuitive explanation.

The Attempt at a Solution

I have solved a and b, but gets problem with c.

a)
Euler:
\begin{equation}
\frac{1+r_{t+1}}{1+\rho} = \frac{c_{2,t+1}}{c_{1,t}}
\end{equation}

\begin{equation}
c_{1,t} = (w_{t}-\tau)\times\frac{1+\rho}{2+\rho}
\end{equation}

\begin{equation}
s_{t} = \frac{1}{2+\rho}(w_{t}-\tau)
\end{equation}

b)
\begin{eqnarray}
K_{t+1} &=& L_{t}\frac{1}{2+\rho}\left((1-\alpha)k_{t}^{\alpha}-\tau\right) \nonumber \\
k_{t+1} &=& \frac{1}{1+n}\frac{1}{1+\rho}\left((1-\alpha)k_{t}^{\alpha}-\tau\right) \nonumber \\
&=& \frac{1}{1+n}\frac{1}{1+\rho}(1-\alpha)k_{t}^{\alpha} -\frac{1}{1+n}\frac{1}{1+\rho}\tau \nonumber \\
&=& ak_{t}^{b} + d
\end{eqnarray}

where

\begin{eqnarray}
a &=& \frac{1}{1+n}\frac{1}{1+\rho}(1-\alpha) \nonumber \\
b &=& \alpha \nonumber \\
c &=& -\frac{1}{1+n}\frac{1}{1+\rho} \times \tau \nonumber
\end{eqnarray}


Can anyone give me any clues as for c?

 

[nate808]

c) Starting from steady state k*, the fiscal experiment results in a decrease in lump-sum taxes in period t, which increases the after-tax wages wt-τ. This increase in after-tax wages leads to an increase in consumption c1,t and savings s,t in period t. In period t+1, the increase in lump-sum taxes (1+rt+1)Δτ/(1+n) results in a decrease in after-tax wages, which leads to a decrease in consumption c2,t+1 and an increase in savings s,t+1. This increase in savings leads to an increase in the capital stock per capita k in period t+1. The equation of motion for k can be written as:

\begin{equation}
k_{t+1} = ak_{t}^{b} + d + \Delta k
\end{equation}

where Δk is the change in the capital stock per capita due to the fiscal experiment. This change in k can be calculated as:

\begin{equation}
\Delta k = \frac{1}{1+n}\frac{1}{1+\rho}\left((1-\alpha)k_{t}^{\alpha}-\tau\right) - \frac{1}{1+n}\frac{1}{1+\rho}\left((1-\alpha)k_{t}^{\alpha}-\tau+\frac{(1+rt+1)\Delta\tau}{1+n}\right)
\end{equation}

This results in an increase in the capital stock per capita k in period t+1, which can be explained intuitively as the increase in savings in period t and t+1 leads to an increase in investment, which in turn increases the capital stock per capita. This increase in the capital stock per capita leads to an increase in output and consumption in the following periods.