One-period model: the price level (i.e. CPI inflation) is equal to prior government debt* divided by current government surplus** -> if you increase debt, you increase inflation. if you increase surplus, you lower inflation
Two-period model: the price level is equal to prior gov't debt divided by the sum of gov't surplus with the expected future surplus divided by the real interest rate
i.e. CPI price = prior debt / (gov't surplus + (expected future gov't surplus / real interest rate))
When the government runs a deficit, it can...
1. borrow more. it most promise a higher future surplus in order to prevent inflation. otherwise the assets that it borrows would lose their value
2. inflate the debt away; increase inflation. the value of the dollars redeemed by prior debtors falls by exactly the fall in surplus
3. unwillingly experience lower expected future surplus, making inflation even larger
In post-war historical data, deficits are not strongly correlated with inflation. This implies that the government does not routinely inflate away the debt and instead promises larger future surplus (e.g. through higher taxes).
*in this case, the debt is measured in number of bonds, not dollars
**aka deficit. government deficit becomes a surplus during tax season
For in-depth notes that include equations and lists, see Notes on "Fiscal Theory of the Price Level" by John Cochrane