The Pigou effect is that a lower price level, including lower prices of resources like nominal wages, will increase the real value of money balances. This will increase real wealth. The greater real wealth will result in greater real consumption expenditures. And so, a lower price level leads to increased real expenditure on currently-produced goods and services.
From a Wicksellian framing, the increase in real wealth reduces saving supply and so raises the natural interest rate. If the market interest rate were above the natural interest rate, this would tend to bring them together. But rather than the market interest rate adjusting downward to an unchanged natural interest rate, it would be the natural interest rate rising up to the market interest rate.
The point of the Pigou effect was to show that there was a long run process that would generate a level of real expenditure on output equal to the productive capacity of the economy. This effect would occur even if the nominal (and real) interest rate could not fall because of a liquidity trap or else consumption and investment were perfectly inelastic with regards to the interest rate. (A vertical savings supply curve could still shift to the left, and match investment.)
Patinkin emphasized that this effect only works with "outside money," and not "inside money." Inside money is the most common type of money. These days it is mostly made up of various sorts of bank deposits. While inside money is an asset to the person holding it, it is a liability to the issuer. That is directly the banks or other financial institution that issued the money.
When the price level goes down, those holding the inside money are wealthier, but those who issued it have larger real liabilities and so are poorer. The banks and other financial institutions have assets, typically loans and bonds. When the price level falls, the real value of those financial assets rise. But those are liabilities to the households and firms that borrowed the money by obtaining the loans or issuing the bonds. And so the lower price level raises their real liabilities, making them poorer.
For the inside money, the lower price level raises the real wealth of those households and firms holding money and lowers the real wealth of those households and firms that borrowed from the banks or other financial institutions that issued the money. There is no impact on net real wealth and so no impact on real consumption. There is no Pigou Effect for inside money.
When Pigou made this argument, it was in the background of a gold standard. With a gold standard, gold coins and gold reserves are outside money. They are an asset to those who own them and they are a liability to no one. A lower price level raises the real value of the monetary gold, this increases real wealth, and it should increase real consumption. Of course, gold as a proportion of total wealth was small, so it would seem that a large decrease in the price level would be necessary to substantially increase total wealth much, and so significantly increase real consumption.
Now that the gold standard is long gone, how does the Pigou effect apply to fiat money? In my view, it depends on the monetary regime. If the fiat money regime is purely irresponsible, so that the government prints money and spends it based upon fiscal needs, then the Pigou effect applies. This framing is realistic sometimes--for example Zimbabwe.
If the fiat regime is based upon a quantity of base money rule, with the level of base money being on some growth path, including remaining fixed, then the Pigou effect applies. However, I don't think this framing is very realistic.
Modern paper money evolved from paper money redeemable in gold. Generally, it was issued by private banks, though one such bank often developed into a central bank. The reason was that the government granted it a monopoly on the issue of hand-to-hand currency in return for a share of the benefits from borrowing at a zero nominal interest rate. As these central banks were gradually nationalized, the hand-to-hand currency and balances private banks held at the central bank became effectively a type of government debt. As long as the gold standard was maintained, it was a liability of the government.
Of course, the gold standard is long gone. But no central bank has adopted any type of base money rule. According to the Fed's dual mandate, it is supposed to be promoting a stable price level. And this, of course, was the goal of many economists who favored leaving the gold standard. Rather than keep paper money fixed to gold, it should be fixed to a composite commodity. This is, in effect, stabilizing some price index.
If there is a real commitment to stabilizing the price level, then paper money is a liability of the issuer just as it is if it is redeemable into gold. What the redemption obligation requires is that the quantity of paper money be adjusted according to the demand to hold it. Those holding the paper money are lending to the central bank. The central bank can only borrow the amount the lenders are willing to lend. With a gold standard, when those holding paper money choose to hold less, they could redeem the paper money for gold, but they can also spend the paper money on whatever it is they want to hold instead. Through a variety of avenues, this tends to raise the demand for gold, which the central bank must supply while reducing the quantity of paper money. A rule requiring a stable price level has the exact same effect, though without the enforcement mechanism allowed by the redemption obligation.
While the dual mandate never changed, during the heyday of Keynesian demand management, there was an effort to reduce unemployment at the cost of higher inflation. And then there was a period when central banks sought to blame rising inflation rates on various supply-side factors while using monetary policy to stabilize interest rates and output. Finally, central banks came to their senses and brought inflation under control, but rather than a stable price level, they ended up with a flexible inflation target.
If the central bank had a real commitment to a growth path for the price level, then base money would be a liability, just as it was under the gold standard. The more flexibility that is added, along with the forgiveness of errors inherent in an inflation target, the more nebulous is the nature of this liability. Still, if the demand for base money should fall substantially, the central bank would need to withdraw it from circulation to keep inflation from rising above target. This means that the money that it borrows by issuing base money is limited by the amount of lending in that form that is desired by those who choose to hold base money.
What does this have to do with the Pigou effect? If central bank paper money is a liability, then it is a type of inside money. There is no Pigou effect. A lower price level would increase the real wealth of those holding paper currency or balances at the central bank, but it would increase the real liabilities of the central bank.
If the central bank is nationalized, then this is an increase in the real liabilities of the government. Given the custom of treating the balance sheets of the central bank separate from the rest of the government, then the central bank has assets to match its liabilities. And so, while the real value of its liabilities rise, so does the real value of its assets. The central bank is no poorer.
To the degree that the central bank makes loans to the private sector, say to banks, or else holds privately-issued bonds, then the lower price level raises the real value of those private liabilities and so reduces the real wealth of those who borrowed from the central bank. There is no Pigou effect.
Of course, many central banks hold government bonds. Until recently, it was approximately true that the Federal Reserve only held government bonds. Back in their formative days, the entire point of providing central banks a monopoly on issuing currency was for the government to share in the benefit of borrowing at a zero nominal interest rate, and that could occur by having the central bank make loans to the government at low interest rates.
To the degree that the central bank holds government bonds, or else the central bank's balance sheet is consolidated with the rest of the government, then the lower price level increases the real wealth of those holding base money while increasing the real national debt. This is where Ricardian Equivalence creates an issue. If the real value of the national debt should rise, then this represents increased future real tax obligations. And so, while the increase in the real value of base money increases the real wealth and consumption of those holding the money, it will reduce the real wealth of taxpayers, and so they will consume less and save more to be able to pay the higher taxes in the future.
While I tend to be skeptical of Ricardian Equivalence, I must admit that the impact of a lower price level on public finance is likely to be immediate and direct. The lower price level decreases the nominal tax revenues of the government and the nominal expenditures it must make to maintain current services. However, the nominal interest and principal payments it makes on the national debt remain the same. This requires an immediate increase in taxes or reduction in real government expenditures on goods and services.
Of course, a price level target implies that any decrease in the price level is temporary. The entire scenario for the Pigou effect just doesn't apply. The lower price level doesn't permanently increase real balances or real wealth. That effect of the lower price level is temporary and will disappear once the price level recovers. Of course, with a price level target, the fact that real balances are temporarily high (and the prices of goods and services temporarily low) provides a very strong incentive to increase current real expenditures. Buy now when prices are temporarily low. With a Wicksellian framing, when the price level recovers, the inflation rate will be higher and so real market interest rates are lower, bringing it into equilibrium to the natural interest rate, even if the nominal market interest rate doesn't fall.
However, with an inflation target, a lower price level (or growth path of the price level,) will be allowed to persist. The change in the price level is permanent. Real base money balances and the real wealth of those holding them are permanently higher, (or on a permanently higher growth path.) But nominal government revenues are on a permanently lower growth path, and the nominal national debt is unchanged, creating an immediately apparent public finance crunch. With perfect Ricardian equivalence, this will exactly offset the Pigou effect.
Unlike with a price level target, there is no effect of prices being temporarily lower, because inflation targeting keeps prices from recovering. But because inflation targeting, like price level targeting, does make central bank/government base money into a liability, this permanent change in real balances and the real wealth of those holding real balances does not create a Pigou effect.
P.S. I favor privatizing hand-to-hand currency, mutual fund type reserve balances, and a very small national debt. Not much room for a Pigou effect in my "ideal" institutional framework.