Monetary policy matters (mainly) because of nominal rigidities.
The simplest story of nominal rigidities is that (some) prices (or wages) are sticky. Firms want to change prices when a shock hits, but there is some cost that makes it hard for them to do so. The object of monetary policy should then be to target the stickiest prices. Monetary policy should respond to shocks to try to ensure that firms don't need to change the prices that are costly for them to change. Adjust monetary policy to keep the equilibrium value of the stickiest prices equal to the actual value. This prevents disequilibrium in the markets for those goods with the stickiest prices.
An alternative story of nominal rigidities is sticky coordination. Each individual firm can adjust its price easily, but one firm's equilibrium price depends on the prices it expects other firms to set. And it is difficult for all firms to coordinate a change in all their prices. Each firm does not know whether the other firms will change their prices, and each waits for the others to go first. (My hunch is that "sticky expectation" stories of nominal rigidity are, at root, really stories of sticky coordination.)
There is a very small cost to each of us in changing our watches for daylight saving time. But we need someone (like the government) to help us coordinate so we all change our watches at the same time. When there is a currency reform, and one new peso is worth 100 old pesos, firms have little difficulty changing prices, because that price change is coordinated. These two examples illustrate the sticky coordination story.
If the only shocks hitting an economy were Aggregate Demand (nominal, velocity) shocks, it doesn't matter for monetary policy which one of these two stories is correct. Monetary policy should try to exactly offset those AD shocks, so that firms don't need to change prices, and don't need to change their expectations of the prices that other firms will set. And if all shocks were AD shocks, it also doesn't matter whether monetary policy targets inflation or Nominal GDP. To see this graphically, if the Aggregate Supply curve never shifts, it doesn't matter whether the central bank makes the AD curve horizontal (inflation targeting) or a downward-sloping rectangular hyperbola (NGDP targeting). All that does matter is that the central bank tries to ensure that the AD curve never shifts, by offsetting all shocks to AD.
When there are AS shocks, it does matter which one of the two stories is correct. And it does matter whether monetary policy targets inflation or NGDP.
If the source of nominal rigidity is sticky prices, then monetary policy should target inflation. (If the sticky prices were sticky nominal wages, then monetary policy should target wage inflation.) If some price doesn't want to change, then adjust monetary policy in response to all shocks so that the equilibrium value of that price doesn't change, so the sticky price is always at the equilibrium level despite being sticky.
But suppose there are AS shocks and the source of nominal rigidity is sticky coordination rather than sticky prices?
Lets start with a partial equilibrium experiment. Suppose a "good" supply shock hits an individual firm (or industry). The individual firm, seeing its cost curve has shifted down, will want to increase output and cut its price.
Now let's switch to the general equilibrium experiment in a monetary economy. Suppose a "good" supply shock hits all firms at the same time. What happens? The answer depends on whether monetary policy is targeting inflation or NGDP.
Under inflation targeting, all firms should increase output, but keep their prices constant. Under NGDP targeting, all firms should increase output, and cut their prices.
Do you see the difference? Under inflation targeting, the general equilibrium experiment gives very different results to the partial equilibrium experiment. Under NGDP targeting, the general equilibrium experiment gives qualitatively the same results as the partial equilibrium experiment.
The whole idea behind sticky coordination is that people can solve partial equilibrium problems a lot more easily than they can solve general equilibrium problems. When a shock hits, each individual can solve for how his own reaction function should shift in response to that shock. But he can't easily solve for the new Nash equilibrium point, because that requires him to figure out how every individual's reaction function has shifted, solve for the new intersection point of all those reaction functions, assume everyone else will do the same, and expect everyone else will move to the new Nash equilibrium too. That's hard.
If its hard for people to solve general equilibrium experiments, monetary policy should try to ensure they don't have to solve general equilibrium experiments. Instead, monetary policy should try to ensure that the general equilibrium solution is as close as possible to the partial equilibrium solution, which individuals have a better chance of solving.
Under NGDP targeting, the general equilibrium solution to supply shocks is at least qualitatively similar to the partial equilibrium solution. To parallel the idea of targeting the stickiest price, monetary policy should try to make sure that people don't have to do those things that are hard for them to do, and will do badly or not at all.
(This post is inspired by a comment Bill Woolsey left here many months ago, and is a response to the Governor of the Bank of Canada's recent defence of inflation targeting over NGDP targeting, and is my attempt to get my head around the idea of "good deflation". I have ignored the important distinction between inflation vs price level targeting, and the parallel distinction between NGDP growth rate vs NGDP level path targeting.)