Sumner's version of the expectations trap is that an expansionary monetary policy can eventually raise money expenditures. Firms expecting higher future sales of their products will purchase more capital goods now. Households expecting their employers to sell more in the future will be more assured of their future employment and purchase more consumer goods now.
The increase in present expenditures will raise the demands for capital goods and consumer goods now, resulting in higher production and higher prices. The expectations trap arises if the central bank is expected to respond to the rising prices with a contractionary policy. That will be expected to eventually reduce money expenditures and those expectations will result in lower current purchases of capital goods and consumer goods. But that is exactly the same time that the initial expansionary monetary policy was supposed to expand money expenditures! And so, firms and households will not expect the increase in money expenditures, and so current investment and consumption will not expand after all.
Sumner argues that this will apply to fiscal policy. The expansionary fiscal policy will be expected to increase nominal expenditure. The expected future sales (to the government and to added government employees,) results in an immediate increase expenditures on capital goods by firms and purchases by households who are more secure in their jobs. That growing demand will raise production and prices immediately.
How it applies to fiscal policy is clear. If the central bank responds to the rising prices by a restrictive monetary policy, then it will eventually result in less nominal expenditures. The immediately contractionary impact of expectations of decrease in future money expenditures will offset the expansionary impact of the fiscal policy. And so, there will be no immediate expansion of investment and consumption.
Sumner's argument is reasonable. Krugman has responded. The government will actually employ people regardless of expectations. Sumner's counter argument is that the private sector will contract to offset that effect due to expectations of the central bank's future contractionary actions.
First, part of the reason why Sumner and his critics appear to talk past one another is the monetary transmission mechanism. Sumner argues that sooner or later, an excess supply of money results in higher monetary expenditures. The new Keynesians assume that monetary policy can only increase real expenditures through a decrease in the real interest rate. Fiscal policy, on the other hand, can expand real expenditure without decreasing the real interest rate.
From a new Keynesian perspective, the normal way that monetary policy impacts real interest rates is by changing short and safe nominal interest rates--the federal funds rate. Once "the" nominal interest rate is zero, real interest rates can only fall through a higher expected inflation rate. If the central bank is truly committed to never increasing the inflation rate, this possibility is out of bounds as well. If no one believes that the central bank will allow higher inflation, the real interest rate cannot fall if the nominal interest rate is already at zero. And so, real interest rates cannot fall. And so monetary policy cannot expand real expenditures.
Fiscal policy doesn't have this problem because it expands real expenditures without lowering the real interest rate. From a Wicksellian perspective, the problem is that the natural interest rate has turned more negative than the inverse of the target for the inflation rate. (For example, less than minus 2 percent.) The central bank lowers the nominal market interest to zero, making the real market interest rate equal to the negative of the expected inflation rate. It remains above the natural interest rate by assumption. Savings is greater than investment, and so total real expenditures is less than the productive capacity of the economy.
Fiscal policy is a decrease in national saving (ignoring additional private saving to fund future tax liabilities--Ricardian equivalence.) That increases the natural interest rate, and saving moves closer to investment. As saving moves nearer to investment, real expenditures moves closer to the productive capacity of the economy. Once the natural interest rate rises to the negative of the target inflation rate, real expenditures equal the productive capacity of the economy. Fiscal policy does not require a credible increase in the target for the inflation rate.
Sumner sometimes says, but it is often in the background, that the only reason worth worrying about for the natural interest rate to be negative is money expenditures are below their trend growth path. In his view, because of sticky wages, a decrease in nominal expenditures below its trend growth path results in depressed real output for an extended period of time. And expectations that this will occur, or even persist, can result in a negative natural interest rate. If money expenditures are expected to return to their trend growth path, then real output will recover. Expectations that real output will recover will raise the natural interest rate. In other words, it is not essential for the real market interest rate to turn negative--and certainly not more negative than the inverse of the target for the inflation rate.
One possible scenario would be for real output to grow only slowly from its depressed levels as money expenditures grow, and so inflation will be temporarily higher. The real market interest rate would turn very negative motivating the growth in money expenditures. But that isn't the only possibility, and it is actually the less desirable one. The better scenario would be for real output to grow rapidly and inflation to rise only moderately, if at all. The rapid growth of real output raises the natural interest rate, so that it becomes less negative and so is no longer less than minus 2 percent, or whatever is the current inflation target.
Krugman's argument needs to be that even if moneyl expenditures were expected to remain growing at trend, the natural interest rate would remain more negative than the expected inflation rate. The only way to get the market interest rate that negative is to raise the expected growth path of nominal expenditures and so the expected inflation rate. Central banks won't do that, and so, fiscal policy is the only option. This will reduce national saving and raise the natural interest rate consistent with real expenditures growing with the productive capacity of the economy and money expenditures growing at trend. Only fiscal policy can raise real expenditures while the inflation rate remains on target.
Of course, there isn't a single interest rate, and there are very few nominal interest rates that are near zero. If a central bank is restricted to purchasing only those securities with near zero interest rates, then much of the new Keynesian argument follows. However, the Federal Reserve is buying all sorts of assets, and all of them have nominal interest rates well above zero. And while it might be necessary to purchase much larger quantities of those assets if the problem is something other than low money expenditures, Sumner is almost certainly correct that the depressed level of real output and expectations of a very slow recovery is currently depressing "the" natural interest rate.
In other words, if the Fed committed to purchase whatever quantity of securities with currently positive nominal yields to return money expenditures back to trend, then only limited purchases would actually be needed. There would be no need for higher expected inflation and lower real interest rates. Perhaps there would be higher expected inflation and lower real interest rates, but it is more likely that rapid real growth in output would raise the natural interest rates, with only a modest increase in expected inflation and reduction in the real interest rates on short term government debt.
Now, if nominal expenditures return to the long term trend growth path, and real expenditures remain depressed, then either expected inflation must rise, or the Fed would need to hold a much larger portfolio of long term, and risky assets. Or, of course, the more radical reforms of moving away from basing the financial system on zero interest, no nominal risk, hand-to-hand currency could be considered.