Fiscal Policy & Price Inflation
Traditionally the view is that it depends on whether or not the central bank also tightens monetary policy. If a central bank is monetizing deficits, and responds to a decline in the deficit by monetizing less, then fiscal austerity will reduce inflation. If on the other hand the central bank tries to inflate more to compensate for the expected drag on growth caused by fiscal action, then it could increase price inflation.
This view is largely corrects, but in a way it begs the question of what happens given a certain monetary policy. Or in other words, what happens if monetary policy is not changed at all?
In general, the effect will be to reduce price inflation. The reason for that is that a lower deficit will reduce direct spending on consumer goods, something which will lower their price. Meanwhile, by reducing demand for loanable funds, fiscal austerity will lower interest rates and raise asset prices and the prices of capital goods.
However, this outcome presupposes two things. First of all, that fiscal action do not have significant negative supply effects. If the effects on supply is greater than the effects on demand, then price inflation will not go down but in fact increase.
If on the other hand fiscal austerity have a positive supply effect, then the inflation reducing effects from lower demand will be reinforced further.
The main example of a supply-reducing fiscal austerity actions is higher marginal tax rates (including increases in consumption taxes). Supply increasing fiscal austerity action includes for example reduced unemployment benefits and a higher retirement age.
Another thing which it depends on is to what extent the lower interest rates caused by fiscal austerity will cause the currency to depreciate or not. Generally this will be the effect, but it is uncertain just how great the effect will be. If the currency depreciation is great enough, then inflation might in fact increase as a result of fiscal austerity.
In the case of the euro area (or any other currency union, or group of countries with fixed exchange rates), relative inflation within the euro area will always be reduced (at least assuming that there isn't significant supply reducing effects caused by tax increases) as a result of fiscal austerity since there won't be any nominal exchange rate movements within the euro area. Because the euro will probably depreciate as a result of fiscal austerity, the net effect on inflation in countries tightening fiscal policy will be uncertain, but it is certain that it will raise inflation in countries that don't tighten fiscal policy.