# OLG - Government Budget Constraint

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)

$\frac{B_{t-1}}{p_{t}} - \frac{B_{t}}{(1+i_{t})p_{t}} = \frac{i_{t}}{(1+i_{t})}\frac{B_{t}}{p_{t}} - \frac{B_{t}}{p_{t}} + \frac{B_{t-1}}{p_{t}}$. 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.