Suppose the natural interest rate is negative. The level of the real interest rate necessary for saving to equal investment given a level of real income equal to the productive capacity of the economy is less than zero.
Further suppose that the monetary order is based upon gold. The dollar price of gold is fixed by definition. The nominal interest rate can be no more negative than the storage cost of gold. Potential lenders will simply hold onto gold rather than lend at any nominal interest rate lower than that.
How does the economy return to equilibrium? Assuming the natural interest rate is below the near-zero nominal bound for the nominal interest rate, how can the real market interest rate turn negative? Or is there some market process that will raise the natural interest rate back up?
Similar to the scenario where there is no gold standard and the real interest rate is negative, the shift from negative yield assets to gold, raises the demand for gold. The nominal and relative price of gold rises enough to clear the market.
With a gold standard, the relative price of gold can still rise. Further, it can rise to a point where it is expected to fall at a rate equal to the (negative) real interest rate. However, it is more conventional to describe this as a decrease in the price level to a point so low that it is expected to rise again from that low level. The rate of recovery in the price level implies expected inflation. This expected inflation reduces the real market interest rate. This process can make the real market interest rate sufficiently negative so that it keeps saving and investment equal given a level of real output equal to the productive capacity of the economy. Given this negative real interest rate, real consumption and real investment together add up to enough real expenditure to purchase everything firms can produce.
This process requires that there is some expectation of a "normal" price level. The decrease in the current price level must be understood to be temporary, and so the lower the price level falls, the more rapid or more persistent the expectation of future inflation. (If the nominal price of gold were free to rise, market clearing based upon expectations of future declines in the price of gold similarly requires an expectation of the long run market clearing price of gold.)
One important characteristic of the gold standard involves risk. Only in the very special situation where "the" natural interest rate is negative and the price path of gold is certain does the price of gold immediately rise and then fall along that path at the natural interest rate. In a more realistic scenario, where it is only safe and short financial assets that have market clearing yields that are negative, and gold is a speculative commodity with a highly uncertain price, then the price of gold may rise to a higher level, but then rise at a slower rate resulting in a slightly lower real yield, making it less competitive with other risky investments.
With a gold standard, the relative price of gold may be subject to uncertain fluctuations, but leaving aside bonds indexed to the price level, short and safe financial assets (government bonds, government insured deposits) will be subject to the same uncertainties. An unexpected inflation will impose losses on those holding government bonds tied to gold just as it imposes losses on those holding gold. So, rather than gold appreciating (or depreciating) like risky financial assets, it must perform like the least risky assets, and so, by assumption, depreciate at the negative natural interest rate that applies to short and safe assets.
The process by which the price level falls to a level such that it is expected to rise again to some kind of long run equilibrium value of the price level is temporary by its very nature. If the natural interest rate is temporarily negative, then this process creates a sufficiently negative real market interest rate to return to equilibrium. If the problem is some sort of perverse expectations, so that expectations of low real income and output in the future lead to increased saving and reduced investment, and so a perversely low natural interest rate, then this process should solve the problem. The price level falls to a point where its expected future increases generate the inflation necessary to push the real market rate down to that natural rate. Real income and output recover, the perverse expectations are falsified, saving and investment and the natural interest rate return to normal.
However, if there is some more persistent problem, where even with expectations of real income and output consistent with reality, preferences and realistic investment opportunities result in a negative natural interest rate, this process is more than a bit paradoxical. The price level must fall to a point where it is expected to rise at a rate equal to the negative natural interest rate forever? In other words, in secular stagnation scenarios, a sharp deflation with the expectation that the price level will sooner or later rise back to normal doesn't really help.
Fortunately, there is a second process at work that increases the natural interest rate. If there is no gold standard, and negative real interest rates increase the demand for gold, the resulting increase in the price of gold results in capital gains for those already owning gold. This increase in wealth results in an increase in consumption, which is equivalent to a decrease in saving. The decrease in the supply of saving raises the natural interest rate. With the natural interest rate below zero by assumption, it becomes less negative. At a sufficiently high price of gold, the natural interest rate rises above the storage cost of gold, and the economy returns to equilibrium.
With a gold standard, this is called the "Pigou effect." At a lower price level, the real value of gold holdings rise. Gold being wealth, this is an increase in real wealth. As wealth rises, consumption rises as well. That is a decrease in the supply of saving. The natural interest rate rises. At a sufficiently low price level, the natural interest rate will rise enough so that it is no longer lower than the cost of storing gold. The real market interest rate matches the natural interest rate, saving equals investment, and real expenditure equals the productive capacity of the economy.
Sometimes the "Pigou effect" is confused with the real balance effect. They are related. At a lower price level, real money balances are higher, and when real money balances are greater than the demand to hold them, then the excess money balances will be spent on something, resulting either directly or indirectly in an expansion in the real demand for output.
However, the "real balance effect" includes the possibility that excess money balances will be spent on financial assets, like bonds, raising their prices and reducing their yields. The lower real interest rate, then, results in an expansion in real expenditure--either consumption, investment or both.
The "Pigou effect," strictly speaking, is a different pathway that works through increased real wealth, increased consumption, and reduced saving rather than lower interest rates. And while that effect should exist given the proper institutional framework, (some money is of the outside variety and not a debt to anyone as is gold with a gold standard,) it is of special interest when nominal interest rates have hit the zero nominal bound, so any "excess" real balances spent on financial assets cannot reduce their nominal yields.
Interestingly enough, if "the" interest rate assumption is loosened, so that it is only short and safe financial assets that have reached the zero nominal bound, and longer and riskier financial assets still have positive nominal yields, then the real balance effect can still impact those yields and so result in increased consumption and investment. Considering a lower price level, a constant real quantity of financial assets is a lower nominal quantity. At a lower price level, some of the excess real money balances (or, for that matter, the real holdings of zero nominal yield short and safe assets) should be shifted to longer and riskier assets, lowering their real yields. The nominal quantity of those securities then, would fall by less than the price level, resulting in a higher real quantity, funding increased real purchases of consumer and capital goods.
Outside of a gold standard, the parallel process would be that at a sufficiently high price of gold, some of those holding gold will switch to other risky financial assets--say equities or long term corporate bonds. This reduces their expected yields and so provide funding opportunities for firms or households to purchase capital or consumer goods.
And finally, without a gold standard, a higher demand for gold and price of gold stimulates gold mining. With a gold standard, the higher demand for gold leads to a higher relative price of gold and a lower price level. This includes the prices of resources like labor. Given lower wages and lower prices of mining equipment, gold mining becomes more profitable and expands.
With a gold standard, any increase in the relative price of gold involves a lower price level. Rather than specialized gold traders buying and selling gold, perhaps on an organized exchange, people specialized in producing and selling other products must add to all of their peculiar concerns specific to their markets a further concern--what is happening with the supply and demand for gold. And while demands for dental work or jewelry or technology in the mining industry would all be included, the emphasis in this post has been "macro" concerns. How much does the relative price of gold need to rise (the price level need to fall) so that expected future depreciation (inflation) makes holding gold no more attractive than holding other sorts of financials assets? How much must the price level fall to expand real wealth enough to expand consumption or the demand for risky financial assets enough to bring recovery?
And, of course, leaving aside indexing contracts for inflation, all of these changes in the relative price of gold--in the price level--disrupt all existing contracts. Unless anticipated, deflation shifts wealth from debtor to creditor, and inflation does the opposite. All contracts become partly speculations on the supply and demand for gold..
However, there is a further complication. All the analysis above assumes that the gold standard will be maintained continuously. Until restrictions were imposed by the government, free banks generally had suspension clauses that allowed them to stop redeeming in gold when necessary. Further, in the U.S., anyway, it was not at all uncommon that the ban on suspension clauses was ignored, and banks suspended gold payments anyway. Of course, "necessary" from the point of view of the banks was when there was a large increase in the demand for gold. In other words, what would happen if the natural interest rate turns negative.
How does the suspension of gold redeemability impact a scenario with a negative natural interest rate?