<p>The interest component is already included. It was just their way of writing out the budget constraint that confused me. If anyone is interested:</p>

<p>$\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}} = \frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}} - \frac{B_{t}}{p_{t}} + \frac{B_{t-1}}{p_{t}}$. This is simply the interest on existing debt and the face value of existing debt (on the expenditure side). And real bond issuance on the income side. Simple rearranging and you have it.</p>

CC BY-SA 4.0
2018-05-17T23:08:29.360Z
Dearc @BenBernke I have a question on the similar topic. If you have time please look at my question. You solved very good so I’m asking you thanks. https://economics.stackexchange.com/questions/22026/market-clearing-condition-with-walras-law
studentp
CC BY-SA 4.0
2018-05-19T10:47:30.997Z
Hi, I am super busy with finals right now so I won't be able to answer your question right away. But I'll do my best to get back to you in the next couple of days.
user11767
CC BY-SA 4.0
2018-05-19T22:58:57.253Z
thank you for your reply. No worries. No need right now. Good luck in your exams!!:)
studentp
CC BY-SA 3.0
2017-07-18T22:53:48.350Z
2017-07-18T22:53:48.350Z
user11767
<p>Consider a two period OLG model where each young agent recieves an endowment of $w$ units of the single commodity good in the 1st period of his/her life, and nothing in the second period. </p>

<p>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 $t$. The government's budget constraint is then:</p>

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

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

<p>My only problem with this budget constraint is the $\frac{B_{t-1}}{p_{t}}$ component. I understand that it is the current real obligation but I don't understand why there is no interest component. I am self-studying OLG models so my reasoning might be flawed but here is how I see this constraint:</p>

<p>The RHS is the government's revenue in period $t$, which consists of tax revenue from the young and the old, seigniorage and new bond issuance. The LHS is the expenditure of the government in period $t$. It can spend the income on consumption ($\bar{g}w$), to pay off the existing debt ( $\frac{B_{t-1}}{p_{t}}$) but what about the interest on the existing debt?</p>

<p>I am used to seeing government constraints in the form of:</p>

<p>$g_{t} + r_{t-1}b_{t-1} = t_{t}+(b_{t}-b_{t-1}) + h_{t} - \frac{h_{t-1}}{1 + \pi _{t}}$</p>

<p>This is almost identical to the author's budget constraint only that it includes an interest component. What am I missing here?</p>

<p>For reference, the budget constraint was obtained from <a href="http://www.sciencedirect.com/science/article/pii/0304393285900042" rel="nofollow noreferrer">Aiyagari and Gertler (1985)</a></p>

<government-debt><financial-constraints><olg>
OLG - Government Budget Constraint
user11767
# OLG - Government Budget Constraint

(Asked by: username_0 on 2017-07-18)

Consider a two period OLG model where each young agent recieves an endowment of $w$ units of the single commodity good in the 1st period of his/her life, and nothing in the second period.

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 $t$. The government's budget constraint is then:

$\frac{B\_{t-1}}{p\_{t}} + \bar{g}w = \tau\_{y}(t)+\tau\_{o}(t) + \frac{M\_{t}-M\_{t-1}}{p\_{t}} + \frac{B\_{t}}{(1+i\_{t})p\_{t}}$

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

My only problem with this budget constraint is the $\frac{B\_{t-1}}{p\_{t}}$ component. I understand that it is the current real obligation but I don't understand why there is no interest component. I am self-studying OLG models so my reasoning might be flawed but here is how I see this constraint:

The RHS is the government's revenue in period $t$, which consists of tax revenue from the young and the old, seigniorage and new bond issuance. The LHS is the expenditure of the government in period $t$. It can spend the income on consumption ($\bar{g}w$), to pay off the existing debt ( $\frac{B\_{t-1}}{p\_{t}}$) but what about the interest on the existing debt?

I am used to seeing government constraints in the form of:

$g\_{t} + r\_{t-1}b\_{t-1} = t\_{t}+(b\_{t}-b\_{t-1}) + h\_{t} - \frac{h\_{t-1}}{1 + \pi \_{t}}$

This is almost identical to the author's budget constraint only that it includes an interest component. What am I missing here?

For reference, the budget constraint was obtained from [Aiyagari and Gertler (1985)](http://www.sciencedirect.com/science/article/pii/0304393285900042)

**Tags:** government-debt, financial-constraints, olg

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## 1 answers

### Answer by username_0 on 2017-07-18. Score: 3 | Accepted answer

The interest component is already included. It was just their way of writing out the budget constraint that confused me. If anyone is interested:

$\frac{B\_{t-1}}{p\_{t}} - \frac{B\_{t}}{(1+i\_{t})p\_{t}} = \frac{i\_{t}}{(1+i\_{t})}\frac{B\_{t}}{p\_{t}} - \frac{B\_{t}}{p\_{t}} + \frac{B\_{t-1}}{p\_{t}}$. This is simply the interest on existing debt and the face value of existing debt (on the expenditure side). And real bond issuance on the income side. Simple rearranging and you have it.

> **Comments:**
> **username_1**: Dearc @BenBernke I have a question on the similar topic. If you have time please look at my question. You solved very good so I’m asking you thanks. https://economics.stackexchange.com/questions/22026/market-clearing-condition-with-walras-law  
> **username_0**: Hi, I am super busy with finals right now so I won't be able to answer your question right away. But I'll do my best to get back to you in the next couple of days.  
> **username_1**: thank you for your reply. No worries. No need right now. Good luck in your exams!!:)  

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