Affects of Fiscal & Monetary Policies
The affects of different stimulus policies can come with different side-effects depending on the type of policy that the government pursues, but it depends on which part of the AS curve is relevant at the time of the policy implementation.
In the short-run Keynesian case, the effects of both fiscal policy expansion and monetary policy expansion are much the same and just produce a boost to aggregate demand which increases economic output without any impact on the price level.
Things are a little more complicated in the classical long-run case.
Fiscal policy with classical aggregate supply
In the classical model, a fiscal expansion such as an increase in government spending or a tax cut, would result in a new aggregate demand curve that intersects aggregate supply at a higher level. Since the supply curve is vertical in the classical long run model, all that results from the extra demand is a higher price level.
Strict adherents of the model insist that even in the short-run there can be no temporary unsustainable increases in output, but this is highly debatable.
For now we simply take the model at face value and recognize that it predicts no change in output and only higher prices as a result of fiscal expansion.
The important implication here is that, after an increase in government spending, that spending must have come at the expense of some other spending i.e. private sector spending. This is a process known as 'crowding out', and in the classical model it is complete. In other words, every penny of extra government spending results in an equal amount of reduced private sector spending.
Monetary policy with classical aggregate supply
An attempt to increase economic output via expansionary monetary policy in a classical aggregate supply model is easy to refute. To see this, imagine that the government increases the money supply. This will lead to a higher aggregate demand curve just as with a fiscal boost, and just as before the only effect is to increase prices.
Now, as prices increase the purchasing power of the money supply decreases. This decrease in purchasing power will continue until the increased money supply has the same purchasing power as the initial money supply.
So, all that has happened is that the increased money supply was eroded in value until it equaled the purchasing power of the original money supply. The aggregate demand increase was effectively decreased back to its original curve.
I've kept controversy out of the conceptual analysis above, because no one really disagrees with the concepts. Where there is disagreement relates on the practical application of economic policy based on the concepts discussed.
Aggregate Supply shifts can happen
You should keep in mind that the AD-AS model was developed with the assumption that aggregate supply is usually very stable. From this assumption the New-Keynesian economists tend to analyse too many contractions as being in need of stimulus.
In the diagram you can see that when supply does contract (as in the 1970s when oil prices quadrupled, and as with the current COVID-19 pandemic that has shut down huge sections of our economies and compromised global supply chains), the result leads to both decreases in economic output (and therefore more unemployment) as well as a higher price level (that may lead to persistent inflation).
If the government were to push for a stimulus package under these circumstances (as they unanimously are throughout the western world) then the predicted result is an even higher price level as output is boosted, and this can create a serious threat of very high inflation that quickly gets out of control.
Deeper analysis is often overlooked
Whilst we may all observe a movement in prices and economic output, there is a great deal of disagreement about what the causes of that movement were, and what the long-term consequences of it are.
Additionally, even if the unthinkable occurred and every economist agreed about the causes & consequences of a movement, there will still be huge disagreement about the correct course of corrective action and the overall size and relative balance of corrective stimulus packages, i.e. the correct balance of fiscal and monetary actions.
Furthermore, it may be that the correct course of action is to do nothing at all. History has provided a great deal of evidence that corrective economic policies can actually be destabilizing rather than corrective, because of all the time lags involved in observing a problem, implementing a solution, and the solution taking effect make corrective policies virtually impossible to calculate in most cases.
Finally, what if a decline in output is a good thing - this possibility is never considered. It is perfectly possible that a decline is desirable if the economy had been overheating prior to that decline. But even if there was no overheating, a decline may have resulted simply because consumers have an increased desire to save more of their earnings.
Whilst an increase in the rate of saving would come with some initial extra unemployment (because output levels depend more on spending rather than saving), it would also imply a higher investment rate, more growth, and ultimately a higher level of economic output i.e. an expansion of the entire aggregate supply curve.
In summary, observing and understanding the concepts behind the aggregate supply model is not controversial, but drawing practical conclusions from it is a different matter, and more often than not leads to strong disagreement.
My personal preference, as usual, is to side with the Austrian school of economics and take a hands-off approach to these things rather than trusting the incompetent hand of government to put together a helpful policy response. I do, however, believe that an enhanced set of automatic stabilizers in the economy are desirable, and that both a stable banking system as well as sensible supply-side economic policies would complement a hands-off approach to fiscal/monetary stimulus.