If I produce a car, and sell it, and undertake no obligation to service that car (by changing the oil), and undertake no obligation to buy back that car, and if nobody can sue me if that car goes wrong, then that car I have produced and sold is not one of my liabilities. That's why we don't list "cars sold and now owned by the public" under the liabilities section of a car producer's balance sheet (though we might list service obligations or lease buyback obligations or legal liabilities if the cars turn out to explode on impact).

If I discover a way to produce cars for free and sell them at a good price (maybe because I have a monopoly on producing cars) we still don't list the cars I have produced and sold as my liabilities.

Even if I give some of those cars away for free we still don't list the cars I have produced and given away as my liabilities. Every car I produce for free and give away is added to the wealth of the lucky person who gets it, but isn't a liability to me, so it's net wealth for the economy as a whole.

If something causes the cars I have previously produced for free and sold or given away to go up in value (because they suddenly start performing better) the wealth of the owners of those cars goes up, but my liabilities do not go up, so that is an increase in the net wealth of the economy as a whole.

It's exactly the same if we replace "cars" with "$20 notes". Provided the central bank that produces those notes undertakes no obligation to service them (by paying interest) or buy them back (replacing torn notes with a costlessly-produced new note doesn't count). That's why, under those conditions, we should not list "notes sold and now held by the public" under the liabilities section of the central bank's balance sheet. (Yes, I know lots of us economists do just that, but we are generally wrong to do just that; see my old post.)

Central bank money (or any money produced by a producer who earns monopoly profits from producing money) is net wealth. A permanent increase in the stock of that money, at the existing price level, increases net wealth. Call that "the money-wealth effect". A permanent fall in the price level, with no permanent change in the stock of that money, increases net wealth. That last thing is called "the Pigou effect".

Under those conditions central bank money is net wealth even if Ricardian Equivalence holds for government bonds so bonds are not net wealth. If those same conditions held for bonds too (they might or might not) then government bonds would be net wealth too (and Ricardian Equivalence would be false). For example, if a government could sell bonds at a rate of interest below the permanent nominal growth rate of the economy (this is generally true for money but may or may not be true for bonds) then the government could sell bonds, issue new bonds to pay the interest on the old bonds, roll over the bonds forever, and the debt/GDP ratio would still fall over time, so the government need never increase future taxes.

Paul says that in his model he found no role for a money-wealth or Pigou effect, because the Euler equation pinned down current consumption as a function of future consumption and the real rate of interest, and future consumption is pinned down by future full-employment output, when the economy eventually returns to full employment. But that does not mean there is no money-wealth or Pigou effect in his model. (The Euler equation basically says that the ratio between the marginal utility of current and future consumption equals one plus the real interest rate.)

The money-wealth or Pigou effect means an increase in net wealth. Consumption-smoothers (like in Paul's model), seeing an increase in net wealth, will plan to spread their increased consumption over all current and future periods, to satisfy the Euler equation. They plan to consume more than their income from production. But agents with rational expectations (as in Paul's model) will figure out that if all agents plan to do this (as they will in a representative agent model like Paul's) then, in each period, either production will rise to match increased planned consumption, or else the price level or real interest rate will rise to reduce planned consumption to match production. So the variables in the Euler equation will change because of the money-wealth or Pigou effect. In Paul's model, since production can rise in the current period, but not in any future period (when it's back at full-employment), what happens when there is an increase in net wealth is that the curent price level stays the same and production increases, the future price level rises for all future periods, which means the current real interest rate falls (nominal interest rate stays constant because it's at the ZLB), and all future real interest rates stay the same.

In other words: Paul's dichotomy between the money-wealth or Pigou effect and the Euler equation is a false dichotomy. The money-wealth or Pigou effect tells us that planned consumption will increase sometime, but the Euler equation tells us when that increase will happen, and the money-wealth or Pigou effect, plus the rest of the model, tells us what happens to the variables in the Euler equation. In Paul's model it will happen in the current period, which is exactly when you want it to happen, because his economy is in recession in the current period.

(In fact, if the infinite-horizon Euler equation is true, since it tells us that the real rate of interest only affects the ratio between current and future consumption, cuts in real interest rates are totally useless as a cure for permanent deficiences in aggregate demand. But that's a whole other story, that creates a massive difference between Old and New Keynesians. The Neo-Wicksellian/New Keynesians just assume that people expect the economy will eventually return to full employment, but have no mechanism whatsoever to justify that expectation, even if we ignore the effect of deflation and the ZLB. It's the New Keynesians who really really need a Pigou effect to make their models work, but it's definitely not in their Neo-Wicksellian models.)

Here's the long but relevant quote from Paul:

" Looking at Japan in 1998, my gut reaction was similar to those of
today’s market monetarists: I was sure that the Bank of Japan could
reflate the economy if it were only willing to try. IS-LM said no, but I
thought this had to be missing something, basically the Pigou effect:
surely if the BoJ just printed enough money, it would burn a hole in
peoples’ pockets, and reflation would follow.

But what I did was a little different from what the MMs have done
this time around: I set out to prove my instincts right with a little
model, a minimal thing that included actual intertemporal decisions
instead of using the quasi-static IS-LM framework. [If you have no idea
what I'm talking about, you have only yourself to blame — I warned you
in the headline]. And to my considerable surprise, the model told me the
opposite of my preconception: there was no Pigou effect.
Consumption was tied down in the current period by the Euler equation,
so if you couldn’t move the real interest rate, nothing happened.

One way to say this — which Waldmann sort of says — is that even a
helicopter drop of money has no effect in a world of Ricardian
equivalence, since you know that the government will eventually have to
tax the windfall away. Of course, you can invoke various kinds of
imperfection to soften this result, but in that case it depends very
much who gets the windfall and who pays the taxes, and we’re basically
talking about fiscal rather than monetary policy. And it remains true
that monetary expansion carried out through open-market operations does
nothing at all.

In the simple model, the only channel through which money can operate
when you’re against the zero lower bound is by changing expectations of
future inflation. And that’s hard to do."

Whether the Pigou effect is a strong enough force to matter is another question. Go read my previous post.

Now, please can I go to bed?

Update: I am remiss in not HTing Ashok Rao, both for his post, and for reminding me of my old post.