Difference equation in OLG framework

Question

This question is also regarding Aiyagari and Gertler (1985). We have the governments flow budget contraint:

$\frac{B_{t-1}}{p_{t}} + \bar{g}w = \tau_{y}(t)+\tau_{o}(t) + \frac{M_{t}-M_{t-1}}{p_{t}} + \frac{B_{t}}{(1+i_{t})p_{t}}$

Each period the government consumes the fraction $\bar{g}$ of the total endowment. $\bar{g}$ is i.i.d with mean $\bar{g}$. The government's expenditures are financed by lump-sum taxes, money and one-period discount bonds. $M_{t}$ and $B_{t}$ are the supplies of money and nominal bonds at the end of period $t$.

$\tau_{i}(t)$ is the lump-sum tax where $i$ is either the old or young generation.

The authors then describe a fiscal policy rule:

Assume taxes are levied only to meet the obligation on the debt. Further, let $(1 - \delta)$ be the fraction of the obligation which is backed by direct taxation, where $0 \leq \delta \leq 1$. That is, the present value of the stream of direct tax levies equals $(1 - \delta)$ times $\frac{B_{t-1}}{p_{t}}$

In and of itself I don't have any problem understanding this. However, they then define a time-stationary tax policy which satisfies:

$\tau_{y}(t)+\tau_{o}(t) = (1-\delta)[\frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}}-\frac{B_{t}-B_{t-1}}{p_{t}}]$ (2.8)

They describe this as:

The policy requires that, each period, tax levies equal $(1 - \delta)$ times the difference between the present value of the current interest obligation on the debt and a term which corrects for the adjustment in the value of the obligation.

This I don't quite understand but I re-arranged the term inside the brackets and obtained $[\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}}]$

This makes somewhat more sense to me because I interpret it as: Each period taxes have to back $(1-\delta)$ the value of the outstanding real debt minus the present value of the future debt (I don't quite understand why the present value of next periods debt is subtracted...)

The next part is in regards to difference equations (I think). They define the expected present value of discounted taxes as $T_{t}$ where $T_{t}$ must satisfy:

$T_{t} = \tau_{y}(t)+\tau_{o}(t) + \frac{E_{t}(T_{t+1})}{(1+i_{t})E_{t}(p_{t}/p_{t+1})}$

I understand what the equation is saying but since I have barely done any dynamic programming at university yet I can't seem to solve it. The authors say:

In view of (2.8) it is clear that $T_{t} = (1-\delta) \frac{B_{t-1}}{p_{t}}$.

I understand why this must be true (given the definition of the fiscal policy) but I cannot show this from the difference equation above.

I know one method of solving difference equations is to 'iterate forward' but I don't think I am doing it correctly.

So my question is regarding how to solve equations like the one above. And what should I read in order to get a better grasp at difference equations. I have heard Sargent and Ljungqvist is a good place to start. And some clarification regarding what the time-stationary policy means intuitively, would be greatly appreciated.

Answer 1

The following approach seems to work in this case:

1. look for the stead state formula of $T_t$. You can do this by taking (2.8) and the formula for $T_t$, and combine them. Then, get rid of all $t$ index:

$$ T = \frac{1-\delta}{p}\left(B - \frac{B}{1+i}\right) + \frac{T}{1+i}$$

Then, rearrange until you get $T$, the steady state formula for $T_j$. You should get:

$$ T= (1-\delta)\frac{B}{p} $$

2. Guess what the time indexes are, and confirm they are correct. To do this, lag the equation for $T_t$ (so you get rid of expectations), and replace $T_{t-1}$ and $T_t$ with your guess. If the equality holds, your guess is correct. If not, try again. For example, let us assume $T_t= (1-\delta)\frac{B_{t-1}}{p_t} $ (the correct one).

Then, the equation is:

$$ (1-\delta) \frac{B_{t-2}}{p_{t-1}} = \frac{1-\delta}{p_{t-1}}\left[B_{t-2} - \frac{B_{t-1}}{1+i_{t-1}}\right] + \frac{T_t p_t}{(1+i_{t-1})p_{t-1}} $$

After simplifying and rearranging you get:

$$T_t= (1-\delta)\frac{B_{t-1}}{p_t} $$

which is correct.

Now, I don't know how general this method is. This is not my area of knowledge. But at least it works here.