If it weren't for the Zero Lower Bound on nominal interest rates, there would be no macroeconomic role for fiscal policy in New Keynesian models. Monetary policy alone could and therefore should be used to hit the macroeconomic target, because this leaves fiscal policy free to try to hit its many microeconomic targets as best it can. (There are perfectly good microeconomic reasons why fiscal policy should be countercyclical, but that's another story.)
But if a big drop in demand causes the natural rate of interest to fall too low, relative to the inflation target, the ZLB maybe be a binding constraint on monetary policy in New Keynesian models, because the central bank cannot drop the actual real rate of interest down to the natural rate. So you might want a countercyclical fiscal policy, that raises the natural rate when private demand is low and the natural rate is low, and lowers the natural rate when private demand is high and the natural rate is high, to keep the economy away from the ZLB. Let's assume you do.
What would you need to do to government spending in a New Keynesian model to raise or lower the natural rate of interest, and to mitigate those fluctuations in the natural rate? The answer is not what you probably think it is.
In New Keynesian models, with a New Keynesian IS curve, the natural rate of interest is a negative function of the rate of change of government spending. That's because the rate of change of private consumption (and investment) spending is a positive function of the rate of interest. That's what the consumption-Euler equation says.
(The natural rate of interest r* is the rate of interest that equilibrates saving and investment at the natural ("potential") level of output Y*. Since C+I+G=Y, and with Y at Y*, G rising over time means C+I falling over time, which means a lower level of r*.)
Draw a nice neat sine wave to illustrate what the natural rate of interest would look like over time in the absence of countercyclical fiscal policy. The hills are booms, when private demand is high and the natural rate is high. The valleys are recessions, when private demand is low and the natural rate is low.
Suppose we want a countercyclical fiscal policy to help smooth out those fluctuations in the natural rate, to help prevent us hitting the ZLB. What would countercyclical fiscal policy look like, in the same picture?
In an Old Keynesian model, we would draw a sine wave for the level of government spending, that was the exact mirror-image of the original sine wave. G would be high when private demand is low, and G would be low when private demand is high. Anyone who has taught basic macro has probably drawn that picture, and talked about the two waves cancelling out, and then gone on to talk about problems with lags etc.
In a New Keynesian model, we would draw a sine wave for the rate of change of government spending, that was exactly in-phase with the original sine wave. The rate of change of G would be low when private demand is low, and the rate of change of G would be high when private demand is high. I bet nobody has ever drawn that picture for their students.
If we use calculus and integrate that New Keynesian picture over time, we can figure out what the New Keynesian picture looks like for the level of G. G would be at a peak at exactly that point between the end of one boom and the beginning of the next recession. And G would be at a trough at exactly that point between the end of one recession and the beginning of the next boom. G would be rising most steeply at the peak of the boom, and falling most steeply at the trough of the recession. The sine-wave for the level of G would be exactly a quarter period out of phase with the original sine wave. I bet nobody has ever drawn that picture for their students either.
Now suppose you had fiscal policy run by a bunch of nutters, who didn't understand macroeconomics at all. The nutters think that booms are a good time to be increasing government spending, "because we can afford it"; and recessions would be a good time to be decreasing government spending, "because we can't afford it". What would it look like? Might it look a lot like optimal New Keynesian fiscal policy?
Just who are you calling "Austerians" now?
(I made this point before, in an old post I now can't find [update: J.V. Dubois found it, thanks]; but I have made it much clearer this time.)