Because otherwise their models won't work.

And yet the canonical versions of their models, which don't even have money, cannot have a Pigou effect.

And so New Keynesians are guilty of the Old Keynesian accusation of "just assuming full employment".

Here's why: In Old Keynesian models, if Aggregate Demand is too low, a cut in real interest rates will (usually) increase Aggregate Demand to get the economy to full employment. But in New Keynesian models a cut in real interest rates may merely redistribute Aggregate Demand from future to present. Because in New Keynesian models it is only the ratio of present to future Aggregate Demand that depends on the real rate of interest. The overall level of Aggregate Demand, present plus future, is indeterminate, unless you just assume the economy will eventually get back to full employment.

In the olden days, when the Old Keynesians were young, the young Keynesians had a sure-fire way of demolishing an unprepared "classical" economist. "But your model just assumes full employment!" was how they attacked. (Several decades later, "But your model just assumes sticky prices/ignores the Lucas Critique" played a similar role for different young macroeconomists.)

A well-prepared "classical" economist needed some sort of answer ready to defend himself against the young Keynesian accusation.

He might say that if there were an excess supply of goods then the price level would fall, and that fall in the price level would increase the real money supply, and that would cause an excess supply of money, which would mean an increased demand for goods, which would get the economy back to full employment.

Or he might invoke Keynes himself, and say that if there were an excess supply of goods then the price level would fall, and that fall in the price level would increase the real money supply, and that would cause an excess supply of money, which would mean an excess demand for bonds, which would cause bond prices to rise, which would mean interest rates fall, which would cause desired investment and maybe desired consumption to rise, which would mean an increased demand for goods, which would get the economy back to full employment.

Only a really desperate classical macroeconomist would invoke the Pigou effect to defend his assumption of full employment. It was a defence of last resort, to be used only if the young keynesians said "But what if there's a liquidity trap so interest rates can't fall, or what if interest rates do fall but desired investment and consumption are perfectly interest-inelastic?" He would then invoke the Pigou effect, and say that an increase in the real money supply would increase real wealth which would increase desired consumption.

Now let's take one of those vicious young Old Keynesians from more than half a century ago, put him in a time-machine, and wake him up in a seminar where a Woodfordian New Keynesian is presenting his very simple model.

It's a very simple model, with no investment, government expenditure or taxes, or net exports. The only source of Aggregate Demand is consumption. "OK", says the young Old Keynesian, "so let's see the consumption function". And the New Keynesian writes down an Euler equation in which the ratio of today's consumption to tomorrow's consumption is a negative function of the real rate of interest. "OK", says the young Old Keynesian, "that looks a bit strange to me, but I will take your word for it".

There is next a long and confused discussion in which the young Old Keynesian eventually learns that the new name for "full employment" seems to be "the natural rate of unemployment".

Then the New Keynesian says: "The economy will always be at the natural rate of unemployment provided the central bank continuously adjusts the real rate of interest in response to shocks to keep the real interest rate at the right level".

The young Old Keynesian sticks up his hand: "Hang on" he says "I can see that keeping the real interest rate at the right level is a necessary condition for keeping the economy always at full employment in your model, but it's not a sufficient condition. What happens if the real interest rate is always at the right level but consumption is always 50% below full employment output? The ratios between today's and tomorrow's consumption would still be exactly the same, so your Euler equation consumption function thingy is still satisfied."

The young Old Keynesian then makes his old accusation: "Your model just assumes full employment!"

How could the New Keynesian defend his model?

He cannot say that if consumption was too low the central bank would just cut the real interest rate. Because the young Old Keynesian would reply: "But you have already assumed the real rate of interest was at the right level. And in any case, even if the central bank did cut the real interest rate, all you know is that the ratio of today's to tomorrow's consumption will increase, and that might equally well happen by tomorrow's consumption falling with today's consumption staying the same, which doesn't help us get back to full employment today, and simply takes us even further away from full employment tomorrow. Cutting real interest rates only gets us to full employment today if you just assume we're at full employment tomorrow. I repeat my accusation: your model just assumes full employment!"

And he's right.

The young Old Keynesian then decides to say something kind, as well as uncharacteristic. "You know" he says "you could make your model work by adding a Pigou effect to get you to full employment. And unlike those stupid classical economists, your model only needs a really tiny Pigou effect to make it work. Because provided the central bank gets the right rate of interest there is nothing to prevent your model being at full employment; it's just that there's nothing to prevent it not being there either. Maybe if the central bank got the interest rate wrong in a big way, then you would need a big Pigou effect to offset it, like in Samuelson 1958 where M/P has to be really big because people want to save so much and can't invest, and only the really big Pigou effect keeps Samuelson's model at full employment. Trouble is, your model doesn't have 'M' in it, so you would need to change that, if you wanted to add a Pigou effect ."

But it's too late for this kind and uncharacteristic suggestion. The New Keynesian already knows that "M" doesn't matter and that only "r" does. He's off thinking about transversality conditions.