But it introduces another problem: if an inflation shock takes the price level and thus NGDP above the target NGDP path, then the Fed will have to take sharp tightening action which would cause real GDP to fall much more than with inflation targetting and most likely result in abandoning the NGDP target.
Market Monetarists treat nominal GDP as something that really exists--the flow of money expenditures on current output. Critics treat nominal GDP as the product of real output and the price level. "If an inflation shock takes the price level and thus GDP above the NGDP path, then the Fed will have to take sharp tightening action."
Given the growth path of the flow of money expenditures on output, an inflationary shock pushing the price level above its trend growth path will reduce what firms and households will buy and so what firms can sell and produce. While the price level rises to a higher growth path, real output falls to a lower growth path. Nominal GDP remains on target.
If this "inflationary shock" pushed real output below capacity, the resulting surpluses would put downward pressure on prices--automatically reversing the "inflationary shock." Prices would grow more slowly and fall back to their previous growth path and real output would rise again to the growth path of capacity. On the other hand, if the reason for the inflationary shock was a reduction in capacity, then the price level remains on the higher growth path and real output remains on the lower growth path with capacity. Still, nominal expenditures and nominal GDP remain on the target growth path.
On the other hand, suppose nominal GDP is nothing that really exists. It is just the product of the price level and real output. For example, suppose the policy interest rate determines real output in the next period, and then the output gap causes inflation the period after. Of course, we assume some stochastic process that causes inflation shocks during the current period too. Given real output, which was determined by last period's policy rate and its own idiosyncratic shocks, and multiply by the price level, which is already determined by last periods output gap, then multiply and an inflationary shock pushes the product--nominal GDP--above target. How to get the nominal GDP back down? A higher policy rate reduces real output next period, which lowers nominal GDP. And then the period after, the resulting output gap will get the price level back down too. (And will all of this manipulation of the policy rate lead to cycling? Will it be explosive?)
Now, how does this all work? Why does a lower policy rate cause output to change next period? Could it be that it directly causes money expenditures on output to change and firms respond to changes in sales? What happens to the amount that firms can sell next period if the great random number generator in the sky mandates that they charge higher prices? Can they really sell the same amount of output at higher prices? Does the quantity of money rise? Does the demand to hold real money balances fall? Is it that people borrow more? At what interest rate? The last period's interest rate that is supposedly determining output? Or is it the current period's interest rate--the one that will be determining output next period?
If the purpose of a model is to trace out how a given policy rate will impact nominal expenditure, real output, and the inflation rate over time, then perhaps skipping the nominal expenditure step, and going from real output (gaps) in the next period and the inflation rate in the subsequent period will do. But the flow of nominal expenditure on output is the true causal factor in the process. Right? Or are there people who really believe that the policy rate determines output next "period" and output (gaps) determine inflation the following period?
In my view, there is no such thing as an "inflation shock." There are supply shocks where the supply of some particular good or service changes. If the supply of a good decreases, the price rises and the quantity falls. As a matter of arithmetic, the price level rises and real output falls. A price level target requires a monetary contraction to push the price level back down.
An inflation target does nothing to reverse this periods random shock to the price level. But, in the real world, where supply shocks are not produced by stochastic processes but rather reductions in the supply of some particular good, then an inflation target implies a monetary contraction to prevent a developing supply shock from generating inflation. (To pull an example out of thin air, rising oil prices cause worries about inflation expectations becoming unanchored, deterring a decrease in policy rates.)
With a nominal GDP target, the results are more complicated. If the demand for the particular good whose supply decreased is unit elastic, then the increase in the price of the good is inversely proportional to the decrease in the quantity. The arithmetic increase in the price level is inversely proportional to the arithmetic decrease in real output. There is no tendency for nominal GDP to move away from target. Given flow of expenditures in the economy, there is no change in spending in the rest of the economy. There is no need for prices or output to change in other markets.
But if the demand for the good whose supply decreased is inelastic, then the increase in the price of that good is more than proportional to the decrease in quantity. Arithmetically, the price level rises more than in proportion to the decrease in real output. This tends to raise nominal GDP, perhaps pushing it above target. However, if the flow of nominal expenditure on output is already determined, then nominal GDP doesn't rise above target. What happens is that spending rises in the market where supply decreased and falls in the rest of the economy. This decrease in demand tends to depress prices and output in other markets. That is how nominal GDP remains on target despite the sharp increase in the price of the good whose supply decreased.
If demand for the good with the decrease in supply is elastic, then the analysis is reversed. The increase in price is less than in proportion to the decrease in quantity. The arithmetic increase in the price level is less than in proportion to the arithmetic decrease in real output. Nominal GDP tends to fall below target. To the degree that aggregate spending is already determined, then spending on the good whose supply decreased goes down, and spending in the rest of the economy expands. The increase in demand in the rest of the economy tends to raise prices and output for those goods. That is how nominal GDP remains on target despite the sharp decrease in the quantity of the good whose supply has decreased.
In my view, nominal GDP targeting is imperfect. However, the other alternatives are worse. But thinking about random "inflation shocks" in a simple new Keynesian model where the policy rate determines next period's output and this period's output gap determines inflation is highly misleading. It is just as misleading as thinking about random shocks to this periods real output and imagining that inflation must be generated in subsequent periods to return nominal GDP to target.