The answer we normally give, when teaching intro macro, is: "It depends on price stickiness; if prices are very flexible it will be short, and if prices are very sticky it will be long."
A better answer would be: "It depends on monetary policy; if monetary policy is very good it will be short, and if monetary policy is very bad it will last forever."
I think I am on the same page here as Simon Wren-Lewis. And I agree with Simon that this point matters not just for students of intro macro.
Many intro macro texts illustrate the distinction using the AD-AS framework. Put the price level (P) on the vertical axis, and real output (Y) on the horizontal axis. Draw a downward-sloping AD curve, a vertical LRAS curve, and an upward-sloping (or horizontal) SRAS curve. Start out in long run equilibrium with all three curves crossing at the same point. The economy is at point A.
Now suppose a shock shifts the AD curve to the left. In the short run the economy goes to point B, and there is a recession. But eventually, prices (including wages) adjust, the SRAS curve slowly shifts down/right, and the economy eventually returns to (a new) long run equilibrium at point C. And the time it takes for prices to adjust determines how long it takes for the economy to get to point C.
We then say it might be better for the central bank to respond to the shock by loosening monetary policy and shifting the AD curve back to the original red curve, so the economy returns to long run equilibrium at point A more quickly than it would get to long run equilibrium at point C by itself. We then talk about lags in monetary policy, and compare the speed of the central bank's response to the speed of price adjustment. And we discuss the difficulties the central bank may face in identifying shocks and responding appropriately, and discuss rules vs discretion, etc.
It all makes for a nice little essay question on a final exam.
But it does set up a false dichotomy between using the discretionary actions of the central bank vs relying on the "natural" self-equilibrating properties of the economy.
Suppose the central bank "did nothing" in response to the shock. How long would it take for the economy to get from point B to point C, and return to long run equilibrium? That depends on what we mean by "doing nothing". And it could mean almost anything.
Suppose, for example, "doing nothing" means "holding the nominal interest rate constant". To a first approximation, we get a vertical AD curve under that assumption. (More realistically, if we add in debt-deflation effects, or falling prices causing expected deflation and a higher real interest rate, the AD curve would likely slope the wrong way.) With a vertical AD curve, a falling price level would not help at all in returning the economy to a point on the LRAS curve. (With the AD curve sloping the wrong way, a falling price level would just make matters worse.)
Holding the nominal interest rate constant in the face of shocks to AD is just one example of a stupid monetary policy. With a sufficiently stupid monetary policy, the short run would last forever (or until the monetary system collapsed), regardless of how quickly prices adjust.
It makes no sense to say that the length of the short run depends on the degree of price stickiness, without mentioning monetary policy.
But once we recognise that the length of the short run depends on the quality of monetary policy, do we really add anything by saying that it depends on the degree of price stickiness too? It's not just the slope of the AD curve that matters. We cannot even say whether a given shock will shift the AD curve (and how much it will shift the AD curve, and which direction it will shift the AD curve) without specifying the monetary policy being followed. (For example, in the simple Mundell-Fleming model an increase in world interest rates would cause the AD curve to shift left if the central bank holds the exchange rate fixed and shift right if the central bank holds the money supply fixed.)
I think it is more useful to say that the length of the short run depends on the central bank's monetary policy target, and on how quickly the central bank recognises and responds to shocks that might cause it to miss its target.
I think I am agreeing with Simon on that main point.
If the central bank has an inappropriate target, then the degree of price flexibility might matter too (but more price flexibility might make things worse, for some targets). (And flexibility of relative prices, in the face of real shocks, would matter for relative resource allocation even if the central bank had a good macroeconomic target.)
(I think Simon would disagree with me on the usefulness of the AD-AS framework in illustrating this point. I think the AD-AS framwork is useful, because it lets me talk about AD curves sloping the wrong way. And it is also useful if we replace the price level with the inflation rate on the vertical axis, because then the AD curve will slope the wrong way, if the central bank holds the nominal interest rate fixed, and expected inflation equals actual inflation.)