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“to the extent that the broader measure of money declines, this reflects an excess demand for base money — caused either by an increase in demand or reduction in supply. To the extent that the broader measure of money increases, this reflects either a decline in the demand for base money or increase in the supply of base money”

Could you explain this more? What is the connection between the demand to hold inside money and the demand to hold base money? If the quantity of inside money is determined endogenously by the real demand for it and the costs of supplying it, nominal income need not change to eliminate excesses or deficiencies. So why would any increase or decrease in the quantity of inside money not simply be associated with an equilibrium quantity of inside money, given the price level and quantity of outside money? And how would changes in the supply and demand for inside money be connected to the supply and demand for base money in a multiplier fashion? For instance M2 could increase independently of the base if the public shifted out of currency into deposits, as banks respond to deposit demand M2 would increase and…ok wait I think I’m understanding your statement now. Increases in that broad money aggregate reflect a declining demand for base money as the public shifts out of currency. Hmm, I guess you’d need a pretty broad aggregate to make sure you’re not just capturing shifts between different non base exchange media. Everyone says the multiplier is dead and aggregates aren’t useful, but I think I see the value here. Would be interested to read more you’ve written about this.

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I'm not sure that I have ever written about this in a comprehensive way. My general view is that inside money is supplied competitively and so the supply is essentially demand-determined. (To the extent it is not, that would be due to things like financial regulation and other frictions that might impede supply from adjusting to meet demand.) So changes in the demand for inside money need not necessarily lead to changes in nominal income/spending.

The general way that I think about this issue is as follows. Suppose that I am the central banker. I supply base money. I want to make sure that I'm not supplying too much or not enough base money. What can I look at? My argument would be that you could look at the broader money aggregates. If broader money aggregates are growing at a faster rate than normal, this suggests that there is an excess supply of base money. This could be because I have supplied too much base money or it could be that there has been a decline in the demand for base money. From my perspective, however, that is irrelevant. It provides a signal that I need to tighten policy.

The reason that people don't seem to like the money multiplier is that the typical analysis is quite crude. Take, for example, the M1 multiplier. The money multiplier is simply of function of the currency-to-deposit ratio and the reserve-to-deposit ratio. People (correctly!) complain that it is unreasonable to think that these ratios remain constant over time. However, they are wrong to conclude from that observation that the money multiplier is useless. Instead, they should think about these ratios as being endogenously determined to reflect the preferences for base money. When the currency-to-deposit ratio or the reserve-to-deposit ratio increases, this reflects an increase in the demand for base money. The money multiplier declines. All else equal, this means that the broader measure of the money supply declines as well. The central banker who sees this should then conclude from observing the broader aggregate that there is an excess demand for base money and take a more expansionary stance.

David Beckworth and I wrote a paper about 12 years ago in which we tried to use this basic framework for thinking about what we called "Great Spending Crashes" that you can find here: https://www.degruyter.com/document/doi/10.1515/1935-1690.2380/pdf?srsltid=AfmBOorxbC6pXFNo9O2yr7kmL9iv_Cscqiv9OljcU2NIun3bHcwnPlQF

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