<p>Take your second equation, move it forward one period, and rearrange. You get:</p>

<p>$$ B_t = \frac{p_{t+1} s_{t+1} + \Delta M_{t+1}}{R_{t+1}}  + \frac{B_{t+1}}{R_{t+1}} $$</p>

<p>Then, define the <em>nominal primary deficit</em> as $D_{t+1} = -(p_{t+1} s_{t+1} + \Delta M_{t+1})$. The above transforms into:</p>

<p>$$ B_t = -\frac{D_{t+1}}{R_{t+1}} + \frac{B_{t+1}}{R_{t+1}} $$</p>

<p>which has the format you are looking for. To find the infinite-period intertemporal budget constraint, simply proceed with forward substitution.</p>

<p>The notation I've used (regarding the deficit) is taken from <a href="http://press.princeton.edu/titles/9743.html" rel="nofollow noreferrer">Wickens</a>, Chapter 5. The above is roughly equal to equation (5.11) in the book. There, the author has a more general expression, which includes inflation, GDP growth, and debt-to-GDP ratio. Your example is a simplification of his. I suggest you go through <a href="http://press.princeton.edu/chapters/c9743.pdf" rel="nofollow noreferrer">Section 5.4</a> of the book, which it gives a very detailed analysis of the Government budget constraint (it also gives the solution you are looking for, in case you want to compare).</p>

<hr>

<p>To find the final expression you are looking for (after edit), take your second equation and replace $B_t$ with the infinite-period intertemporal budget constraint:</p>

<p>$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B^*_{t} + M_{t}}{p_{t}} + s_{t}$$ </p>

<p>where $B^*_t$ is the result of the forward substitution exercise.</p>

CC BY-SA 3.0
2017-08-15T13:23:13.770Z
2017-08-17T07:14:23.607Z
2017-08-17T07:14:23.607Z
luchonacho
<p>I am trying to learn how to solve difference equations in order to derive intertemporal budget constraints. Consider the government's flow budget constraint:</p>

<p>$$\frac{B_{t}-B_{t-1}}{p_{t}} + \frac{M_{t}-M_{t-1}}{p_{t}} + s_{t} -i_{t-1}\frac{B_{t-1}}{p_{t}} = 0$$ </p>

<p>where $s_{t}$ is real taxes net of government consumption. I re-write the above as</p>

<p>$$\frac{R_{t}B_{t-1} + M_{t-1}}{p_{t}} = \frac{B_{t} + M_{t}}{p_{t}} + s_{t}$$ </p>

<p>where $R_{t}$ is the gross nominal interest rate.</p>

<p>Now I am quite lost. I think what is confusing me is that I have $t-1$ <strong>and</strong> $t$ terms on the left-hand side. The examples I have tried before are usually on the form $a_{t} = a_{t+1} + b_{t} +(...)$ which I then solve "forward" by substituting for $a_{t+1}$ over and over again until I see a pattern developing. But in this case I can't really re-write the LHS in a proper manner. </p>

<p>Any thoughts on how I should proceed? And if there are any books that go through difference equations I would be very interested in some names as I am self teaching myself so far.</p>

<p>EDIT: Thanks to @luchonacho I have obtained the following:</p>

<p>$B_{t} = \frac{B_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} + \frac{D_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} ...\frac{D_{t+1}}{R_{t+1}}$ since we are in an inifite period world I write this as"</p>

<p>$B_{t} = \lim_{T\to\infty}  \frac{B_{t+T}}{R_{t+1}R_{t+2}...R_{t+T}} + \sum\limits_{j=1}^{\infty} \frac{D_{t+j}}{\prod_{s=1}^{j} R_{t+s}}$</p>

<p>I might have messed up the summation/product notation but this is one of the expressions I have seen for the intertemporal constraint. However, another common way I have seen it written is as follows:</p>

<p><a href="https://i.sstatic.net/1MzQt.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1MzQt.png" alt="enter image description here"></a></p>

<p>Ignore the right-hand side as it assumes a functional form of the utility function. I am interested in obtaining an expression with both $B_{t-1}$ and $M_{t-1}$ on the LHS. Is there a way I can go from the expression I obtained above to the one with both $B_{t-1}, M_{t-1}$? I am interested in deriving this because I want an expression for the valuation of all government liabilities. </p>

<macroeconomics><difference-equations>
Deriving intertemporal budget constraint from flow constraint
user11767
# Deriving intertemporal budget constraint from flow constraint

(Asked by: username_0 on 2017-08-12)

I am trying to learn how to solve difference equations in order to derive intertemporal budget constraints. Consider the government's flow budget constraint:

$$\frac{B\_{t}-B\_{t-1}}{p\_{t}} + \frac{M\_{t}-M\_{t-1}}{p\_{t}} + s\_{t} -i\_{t-1}\frac{B\_{t-1}}{p\_{t}} = 0$$

where $s\_{t}$ is real taxes net of government consumption. I re-write the above as

$$\frac{R\_{t}B\_{t-1} + M\_{t-1}}{p\_{t}} = \frac{B\_{t} + M\_{t}}{p\_{t}} + s\_{t}$$

where $R\_{t}$ is the gross nominal interest rate.

Now I am quite lost. I think what is confusing me is that I have $t-1$ **and** $t$ terms on the left-hand side. The examples I have tried before are usually on the form $a\_{t} = a\_{t+1} + b\_{t} +(...)$ which I then solve "forward" by substituting for $a\_{t+1}$ over and over again until I see a pattern developing. But in this case I can't really re-write the LHS in a proper manner.

Any thoughts on how I should proceed? And if there are any books that go through difference equations I would be very interested in some names as I am self teaching myself so far.

EDIT: Thanks to @luchonacho I have obtained the following:

$B\_{t} = \frac{B\_{t+T}}{R\_{t+1}R\_{t+2}...R\_{t+T}} + \frac{D\_{t+T}}{R\_{t+1}R\_{t+2}...R\_{t+T}} ...\frac{D\_{t+1}}{R\_{t+1}}$ since we are in an inifite period world I write this as"

$B\_{t} = \lim\_{T\to\infty} \frac{B\_{t+T}}{R\_{t+1}R\_{t+2}...R\_{t+T}} + \sum\limits\_{j=1}^{\infty} \frac{D\_{t+j}}{\prod\_{s=1}^{j} R\_{t+s}}$

I might have messed up the summation/product notation but this is one of the expressions I have seen for the intertemporal constraint. However, another common way I have seen it written is as follows:

[![enter image description here](https://i.sstatic.net/1MzQt.png)](https://i.sstatic.net/1MzQt.png)

Ignore the right-hand side as it assumes a functional form of the utility function. I am interested in obtaining an expression with both $B\_{t-1}$ and $M\_{t-1}$ on the LHS. Is there a way I can go from the expression I obtained above to the one with both $B\_{t-1}, M\_{t-1}$? I am interested in deriving this because I want an expression for the valuation of all government liabilities.

**Tags:** macroeconomics, difference-equations

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## 1 answers

### Answer by username_1 on 2017-08-15. Score: 2 | Accepted answer

Take your second equation, move it forward one period, and rearrange. You get:

$$ B\_t = \frac{p\_{t+1} s\_{t+1} + \Delta M\_{t+1}}{R\_{t+1}} + \frac{B\_{t+1}}{R\_{t+1}} $$

Then, define the *nominal primary deficit* as $D\_{t+1} = -(p\_{t+1} s\_{t+1} + \Delta M\_{t+1})$. The above transforms into:

$$ B\_t = -\frac{D\_{t+1}}{R\_{t+1}} + \frac{B\_{t+1}}{R\_{t+1}} $$

which has the format you are looking for. To find the infinite-period intertemporal budget constraint, simply proceed with forward substitution.

The notation I've used (regarding the deficit) is taken from [Wickens](http://press.princeton.edu/titles/9743.html), Chapter 5. The above is roughly equal to equation (5.11) in the book. There, the author has a more general expression, which includes inflation, GDP growth, and debt-to-GDP ratio. Your example is a simplification of his. I suggest you go through [Section 5.4](http://press.princeton.edu/chapters/c9743.pdf) of the book, which it gives a very detailed analysis of the Government budget constraint (it also gives the solution you are looking for, in case you want to compare).

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To find the final expression you are looking for (after edit), take your second equation and replace $B\_t$ with the infinite-period intertemporal budget constraint:

$$\frac{R\_{t}B\_{t-1} + M\_{t-1}}{p\_{t}} = \frac{B^\*\_{t} + M\_{t}}{p\_{t}} + s\_{t}$$

where $B^\*\_t$ is the result of the forward substitution exercise.


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