# Macroeconomics: Overlapping Generation model and lump-sum taxes

1. Mar 14, 2013

### Charlotte87

1. The problem statement, all variables and given/known data

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 c1,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 St earn an interest rate rt+1 and wt 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 Kt is the aggregate capital stock and Ltthe labor input. Firms are perfectly competitive and take factor prices rt and wt as given. Capital does not depreciate and the labor force grows at rate n>0, that is Lt = (1+n)Lt-1.

(a) Derive the Euler equation governing optimal consumption of the consumer born at time t, and solve for c1,t and st as functions of after-tax wages wt-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+rt+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 kt+1 and give a brief intuitive explanation.

3. The attempt at a solution
I have solved a and b, but gets problem with c.

a)
Euler:

\frac{1+r_{t+1}}{1+\rho} = \frac{c_{2,t+1}}{c_{1,t}}

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

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

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?