Inflation: prices going up

Inflation is defined as a persistent increase in the average price level for a sustained period of time (usually measured yearly).  As prices around people go up, they can lose purchasing power (assuming their income isn't tied to inflation through a cost-of-living adjustment), they can start to feel uneasy and possibly get violent (or, at least strike), and they can switch from saving to spending if interest rates aren't properly adjusted.  Countries with higher inflation see less demand for their products as consumers switch to countries with lower rates of inflation.

There are two main types of inflation besides simply increasing the money supply: demand-pull and cost-push.  The graph above shows demand-pull inflation.  Here, macroeconomic equilibrium starts at average price level APL1 and output level Y1.  Then, the aggregate demand curve shifts out from AD1 to AD2 because of some sort of demand shock (i.e. confidence soars, interest rates fall, the FTSE goes up and stays up).  This leads to a new equilibrium with higher output level Y2 (meaning less unemployment) and a higher average price level, inflation, to APL2.  This is not necessarily bad and quite common over time.

The graph above shows cost-push inflation.  Here, macroeconomic equilibrium starts at average price level APL1 and output level Y1.  Then, the short-run aggregate supply curve shifts in from SRAS1 to SRAS2 because of some sort of supply-side shock (i.e. oil prices soar, natural disaster).  This leads to a new equilibrium with a lower output level Y2 (meaning more unemployment) and a higher average price level, inflation, to APL2.  This is usually bad.

Evaluation - How can these two types of inflation be treated?  Don't forget the battle...Keynesians vs. neo-Classicals!  Why is it hard to measure inflation?