Do we need a bubble?

There is something economists have known since 1958 that we don't talk about much, except in private, like in economics journals that nobody else reads. It's a bit too weird.

There are two sorts of world. In a normal world, the equilibrium rate of interest is above the growth rate of the economy. In a weird world, the equilibrium rate of interest is below the growth rate of the economy. What's weird about a weird world is that it needs a bubble, a Ponzi scheme, a chain-letter swindle, for the economy to work well.

Maybe the world we live in is a weird world. And it needs a bubble to work well. And the economy keeps trying to create a bubble. And when it succeeds the economy does work well. But bubbles are unstable and eventually burst. Then it works badly again. Until the next bubble.

The 1958 paper was Samuelson's "An Exact Consumption-loan Model of Interest with or without the Social Contrivance of Money". Here's a simplified version.

Each cohort of people lives for two periods. The population, and everything else, stays the same. When young they work and produce goods. When old they cannot work and cannot produce goods. So they want to save when young so they can consume when old.

If no goods can be stored, they cannot save, and the old starve. They would like to lend when young, even at very negative interest rates, but there is nobody willing to borrow from them. The next cohort, who could pay them back, haven't been born yet.

A chain letter swindle solves the problem. Each old person sends a letter to one young person, saying "Please give me half your goods, and then you can send this same letter to one young person when you are old". Each young person now saves half the goods he produces, by buying one letter. And he sells the letter when he is old for the same amount of goods. The rate of interest is now zero, as is the growth rate of the economy.

Every cohort is now better off than without the chain letter, because they don't starve when old. But the first cohort to invent the chain letter swindle is much better off, because they consume all their production when young, plus half a young person's production when old. And if any cohort breaks the chain, the preceding cohort is much worse off, because they consume half their production when young, and nothing when old.

Samuelson called the chain letter "money". But it could be an unfunded government pension plan. Or gold could serve the same purpose, even if gold were intrinsically useless except as a store of value. Whatever it is, it's a bubble, that has a market value in excess of any intrinsic value.

The value of the bubble would be equal to half the GDP per period. So, if the "period" is about 20 years (if retired people live for about 20 years), the bubble would need to be worth about 10 years of GDP. That's a very big bubble to make the economy work well. It could be bigger still if people wanted to save for emergencies, or for their kids, as well as for retirement.

Introducing capital alleviates the problem, but may not always solve it. Without capital, the natural rate of interest in the model is minus 100% per period, because the produced goods rot before the next period. Investing in real capital can have a positive rate of return, and so can raise the natural rate of interest above zero. If the marginal return on capital, in equilibrium, is greater than the growth rate, then the world returns to normal. People will save by investing in real capital, and the economy doesn't need a bubble. But if the marginal return on capital, after an allowance for risk, is less than the growth rate of the economy, it still needs a bubble. A chain letter, or bubble, can grow forever at the growth rate of the economy, as long as people believe in it and don't break the chain. And it can pay a rate of interest equal to the growth rate. So people save partly in real capital, and partly in the bubble, and both pay the same risk-adjusted rate of return, equal to the growth rate. (The marginal rate of return on capital rises as the capital stock falls when people hold less capital and more bubble.)

Land may also alleviate the problem, in the same way that capital does. But land is different from capital because they aren't making it any more. That means that land may also serve as the bubble asset, with the price of land made up of a fundamental component equal to the present value of expected rents, plus a bubble component, with the bubble component growing at the growth rate of the economy and yielding capital gains. And house price bubbles are really just bubbles in the price of the land on which the house sits, because the house itself is just reproducible capital.

Governments can create the bubble, with a national debt they never pay the interest on, but just rollover from one year to the next. A Ponzi scheme. If spending and tax revenues grow at the growth rate of the economy, and that growth rate is above the rate of interest, they can do this forever. The long run government budget constraint just doesn't add up, literally.

Bubbles can solve the problem of living in a weird world. But the trouble with bubbles is that they burst. They only stay up because people believe in them. And if people stop believing in them, or think the next generation of fools will stop believing in them, they burst. Or, remembering that it is very profitable to start a chain letter, every bubble is vulnerable to the next bubble that someone invents to take its place in people's savings.

When a bubble is big enough to satisfy people's desire to save, the economy works well. But when it bursts, people often resort to saving in the bubble asset par excellence, money. But money is the medium of exchange. And because the medium of exchange is typically the unit of account as well, and prices are sticky, the real value of the money bubble cannot rise quickly enough to fill the space left by the bursting of the other bubble. So there's an excess demand for the medium of exchange. And that causes a recession. Until the next bubble comes along. And that happens as soon as the growth rate picks up enough to exceed the rate of interest.

Comments

[kevin quinn]

I must be violating the taboo on keeping it in-house, because I always describe this alternate world to my undergrad money and banking students. Now I can give them a link to a much more cogent exposition than they get in my lecture - courtesy of Nick! When Samuelson died, I wrote on some blog or another that this was my favorite Samuelson article and one of my favorite economics articles, period. Tirole's equilibrium bubble paper is another. But I wonder whether the typical economist thinks that this stuff can be safely consigned to the dustbin, since Barro showed us how to rationalize the infinitely-lived representative agent with altruistic dynasties. I don't think you can get these equilibrium Ponzi schemes in Barro's world. It's funny that much of the alternate world exhibits a kind of "long-run" Keynesian flavor: you need continually rolled-over government borrowing to prevent sub-optimality, e.g: and the model in effect starts from the Keynesian dictum that "in the long run we're all dead." Finally I wonder whether part of the impetus for Barro's work may not have been to close the Pandora's Box that Samuelson, Gale and Diamond had opened?

[James Oswald (azmyth)]

"But if the marginal return on capital, after an allowance for risk, is less than the growth rate of the economy, it still needs a bubble."

Only if you want the old to consume as much as the young in each period. Suppose capital earns 0% and the economy grows by 50% between the two periods from 100 units of production to 150.

Period 1: Cohort 1 invests 50, consumes 50.
Period 2: Cohort 2 (new young) invests 25, consumes 75 and spends 50 buying Cohort 1's capital.
Cohort 1 sells their capital for 50 and consumes 50.

This could continue indefinately with the old consuming 2/3rds of what the young do. Another bubbleless alternative is to impose a tax on the young and give that money to the old, similar to U.S. social security.

[Nick Rowe]

Kevin: I am really pleased to see your comment. I was worried that I might be totally out to lunch (again) with this post. Knowing that you know your stuff, and have spotted past errors I've made, gives me more confidence.

Lots of economists are familiar with the Samuelson paper, but I think none of us know quite what to do with it. I never thought it was a very good model of money, interpreted as a medium of exchange. But I think it makes a much better model of bubbles/Ponzi schemes/chain letters.

But what do finance theorists make of the model, I wonder? How many of them are aware of it? My guess is that most finance people are partial equilibrium guys, who don't have a general equilibrium theory of the rate of interest (Graham and Dodds, for example, just take it as given), so won't be able to see the problem. But I may be wrong on that.

"I don't think you can get these equilibrium Ponzi schemes in Barro's world." You might be right. It would be much harder. But I'm not sure if it's impossible. And I think I agree that much of the attraction of thinking about the Barro world is that it lets us avoid thinking about the Samuelson case, closing up that nasty Pandora's box, and doing normal finance theory, where bubbles are an aberration, rather than an integral part of the proper functioning of the economy.

James: If capital earns a 0% return, then it's equivalent to a costless storage technology. But in your example, with a 0% return on capital, and a 50% per period growth rate, a Ponzi scheme could promise a 50% return, and all cohorts would do better by not investing anything in capital, and relying totally on Ponzi-finance for their retirement.

" Another bubbleless alternative is to impose a tax on the young and give that money to the old, similar to U.S. social security."

But that is a bubble. It's an unfunded Ponzi-financed chain letter swindle; and that's formally equivalent to a bubble. The only difference is that it's run by the government. But if someone sets up a big enough private bubble in competition, the market rate of interest will rise to equal the growth rate, and it may burst the government bubble.

[Rogue]

Nick, in this theory, what keeps the bubble growing, besides people's belief? (I'm assuming the bubble needs to keep growing each time it passes to the next generation, otherwise, rates go back below natural rate?)

And what enables the next generation to keep affording an ever larger bubble? (Or do they only pay it off completely once they sell to the next person?) If this is the case, then the last person to sell is the actual loser, because he won't be paid in full?

[Nick Rowe]

Rogue: In the simplest model I started out with, with no population growth and no growth in income per person, the bubble doesn't grow. It doesn't need to grow. People are happy to be able to earn 0% (real) on their savings, compared to minus 100% real by storing perishable goods, or starving. (Assume the utility function is U=ln(C), so marginal utility is 1/C, so the real rate of interest people would be willing to accept is determined by 1+r = Cold/Cyoung).

But if population grows at rate g, or if income per person grows at rate g, then the bubble can grow at rate g too, and pay an interest rate of g, and each person pays the same fraction of his income to buy the chain letter. So, in principle the bubble can last forever. If the bubble grew faster than the economy, then eventually they wouldn't be able to afford to buy the chain letter from the previous generation.

[Rogue]

This seems benign then, compared to the 'bubble' we have actually been seeing. It's like the inverse of the parent foregoing personal consumption in order to raise and send a child to school, so the child gets a job that enables him to raise his own child and send him to school.

[Nick Rowe]

Rogue: The results are benign (the old don't starve!). Every cohort is better off, provided the bubble doesn't burst. And there is nothing in the arithmetic that says the bubble can't last forever, as long as it doesn't grow faster than the economy forever, and it does need to if r is less than g. But if the bubble ever collapses, the results for those left holding the worthless chain letters are worse than if they had never bought them.

[Rajiv Sethi]

Interesting post... It's been a while since I read Barro's 1974 paper, but my recollection is that he simply assumes intertemporal budget balance, thus ruling out this kind of phenomenon a priori. Your post (and Samuelson's paper) make clear why one really ought not to make this kind of assumption.

[Nick Rowe]

Rajiv: Thanks! That's my recollection of Barro too.

I don't know why some other economist/blogger hasn't already made the point I make here (unless I missed seeing it). I confess that I, like most of the profession(?), am uncomfortable thinking about the case where r is less than g, and so intertemporal budgets don't balance. It was commenter RSJ who really forced me to confront this issue, and write this post, by citing reams of data showing that the assumption that r greater than g really didn't hold, at least for some government bonds. Of course, it depends on the asset. For currency, we know that r= -inflation so is less than g. And as we move to less liquid and riskier assets, we know that r is greater than g. So it's not obvious what assumption best works for the economy as a whole. I have ducked this issue.

[Jon]

"Until the next bubble comes along. And that happens as soon as the growth rate picks up enough to exceed the rate of interest."

I see that as a restatement of the Austrian business cycle narrative.
* The banking system extends credit lowering the prevailing rate of interest
* A bubble forms
* "something" causes the rate of interest to rise again
* The bubble bursts

What's fascinating to me is that the Austrians had a good explanation for why the bubble tends to form in the capital goods sector--but reused the same argument for why the bubble would form at all. Its this second point that gets attacked the most--but that's precisely where Samuelson's narrative enters.

[Nick Rowe]

Jon: interesting comparison. But I think it's very different from the Austrian narrative. Here's mine:

* the interest rate equals the natural rate, which is low
* because the interest rate is lower than the growth rate, a bubble forms
* the bubble is net wealth, so desired national savings (in the national income accounting sense) falls
* the fall in desired savings causes the interest rate to increase (I think I would say that the "natural rate" increases with it, though there's a semantic issue there)
* the interest rate rises until it equals the growth rate
* the bubble is now on the margin of sustainability OR
* a competing bubble appears OR
* a sunspot causes people to lose faith in the bubble
* the bubble bursts
* desired savings rises
* the interest rate falls
* there's an excess demand for money
* that causes a recession
* output stops falling
* the interest rate is below the growth rate
* repeat.

[Lord]

Government is the bubble everyone can believe in, even those that dislike it. If the real rate of return is zero though, then do all assets lose their value as well since they generate no income? Does sheer inertia hold them in place should they become valuable in some possible future? Sounds reminiscent of Kondratieff cycles alternating between real growth and bubbles, for each a season.

[scott sumner]

Nick, Interesting post. Here are a few observations:

1. Years ago I did a paper arging that, under the gold standard, when there was a drop in the MEC, nominal and real interest rates fell, this reduced the opportunity cost of owning gold, demand for gold rose, and you had deflation and recession. Of course this problem doesn't occur under fiat money if the central bank is targeting NGDP. But they may not be targeting NGDP, so it also may be a problem under fiat money.

2. It occurs to me that the marginal cost of financing increases in the government debt may be above the growth rate of NGDP even if the average cost of financing government debt is below the rate of growth in GDP. This would imply the government could earn a certain amount of seignorage from providing a risk free and very liquid asset, but that the amount of seignorage might be very limited.

[Peter T]

Samuelson's interesting thought experiment does not need money - a barter exchange does just as well. For instance, middle medieval period English peasants typically married in their mid-twenties, and took over the family arable. The mid-50s aged parents "retired" to a cottage at the end of the garden. They key is, as noted, the inter-generational or other temporal exchange.

If you assume money is just a medium of exchange, the arithmetic works out. Seems to me, though, money is several other things as well - it's also a claim on anticipated earnings and a commodity in it's own right. So there could be "promise" bubbles, "money" bubbles, or "claim" bubbles, with interest rates a partial control on some types.

[Nick Rowe]

Lord: "Government is the bubble everyone can believe in, even those that dislike it."
Did you mean that government itself is a bubble? I agree. All social institutions are generally bubbles. They would stop existing if people stopped believing in them. That's why different societies can have different social institutions. Intrinsically, what we call "the government" is just a bunch of people no different from anyone else. It's only our belief that they are more powerful that makes them more powerful.

Or did you mean that governments have a comparative advantage in creating bubble assets, like unfunded pension plans, or fiat money? Maybe.

"If the real rate of return is zero though, then do all assets lose their value as well since they generate no income?"
People need to save for their retirement, or as insurance against lean years ahead. And if 0% rate of return is the best they can get, they will save at 0%, or even negative rates of return. In fact, wasn't this the norm, historically? A hoard of gold coins, yielding a negative return after storage and guarding costs.

Scott: Thanks!
1. But that is very much the standard ISLM prediction, for example, if the stock of money is fixed exogenously, as it would be under the gold standard. But if money is the medium of exchange, people will hold it even if it is rate of return dominated by other assets. The sort of increased demand for money you would get if the MEC fell would be small relative to GDP, if money were used only as medium of exchange. A competent central bank could handle it by printing more. But if there were a large bubble needed to resolve the r less than g problem, it could be large relative to GDP. And if it burst, and people switched to hoarding money instead, it could require a very large increase in the supply of money to satisfy that extra demand. And it would need to be "outside" money, like helicopter money, so that it is net wealth. So the central bank might need to helicopter money equal to (say) 100% of annual GDP to satisfy the demand for a bubble asset. And then vacuum it back in a year or two, when people switch to another bubble asset. Not easy.

2. That's certainly true for money, as medium of exchange. There's a downward-sloping demand curve. But I think I see your more general point. It might *appear* that we live in a Samuelsonian world, where r less than g, but it might not be true at the margin. I'm not sure.

[Lord]

Both really. It is difficult to conceive of life without government; if it didn't exist we would have to invent it. No one has any doubt about the lender of last resort, except Greece, et al, perhaps. One has to wonder what living in Somalia is really like. The ancients hoped and believed their gold would be valuable in the future but how many times was it dashed against famine and pestilence. Some beliefs may be more believable, some more enduring, and some just the lack of any better alternative.

[Doc Merlin]

The wierd world is the world we live in, because we have a central bank that targets nominal interest rates in the short term and prices in the long term. This leads to vertical AD in the short term and horizontal AD in the long term.

[RebelEconomist]

A thought-provoking post, Nick, although the thoughts that it provokes for me are not much to do with bubbles or money. As you acknowledge, there is always land, and this is the basic capital stock that the old possess. To the extent that they can maintain some degree of control over the use of this capital, the old can survive when their productivity falls by trading land gradually for consumption. As Peter T writes, there is no need for money. Now, since the young may be more productive than the old, it makes sense for all concerned for the old to relinquish the capital stock earlier than they need to in order to consume, and make a contract with the young to receive a certain flow of consumption items in the future. The internal rate of return implied by this contract is "the interest rate" (a summary description of a term structure), and will depend on the relative bargaining power of the old and young, which in turn depends on (the expected value of) variables such as population growth, productivity changes, ability to assert one's rights and instinctive inter-generational altruism. Money provides a convenient way of expressing this contract (although not a good way of securitising it, so in my opinion, OLG models do not add much to the understanding of money), so any change in the size of the stock of money will also affect the nominal interest rate.

Fiat money is no more a bubble than any other type of contract - they all disappear if law and order breaks down. I would understand a bubble to be an increase in value of some asset that derives from nothing more than previous increases in value.

[Nick Rowe]

Peter T: Our comments crossed.

"Samuelson's interesting thought experiment does not need money - a barter exchange does just as well."

I basically agree. Many economists have used Samuelson's model as a model of a medium of exchange, but I don't think this is helpful. You get a velocity of circulation of about once every 25 years if you interpret it literally. Plus there's the problem that the medium of exchange is still used even when other assets yield a much higher rate of return (witness Zimbabwe, etc.). It's a model of a demand for a stock of savings that cannot be satisfied by any real capital investment, and that can create a demand for savings in the form of Ponzi-finance/chain letters/bubbles. And the model works even if people use barter within the period.

But it's nevertheless interesting to ask what would happen if they did use a medium of exchange within the period, and that same medium of exchange happened to become the bubble asset (perhaps after some other bubble asset collapsed). The result, given sticky prices, would be a sudden jump in the demand for the medium of exchange, and this would cause a recession. And that sounds very much like what just happened in 2008. The IS curve shifts left as the old bubble collapses, and the LM curve becomes flat as the demand for money as a new bubble asset rises.

Lord: agreed.

Doc: That's weird too, but nowhere near as weird as a world that needs bubbles.

Rogue: start with the simple model I introduced above, with no storeable goods. Then introduce a very small amount of land, 1 acre. That very small amount of land could become extremely valuable. (A very high price/rent ratio). Add a second acre and the value of land per acre would almost halve. Add some gold, or a chain letter, and the value of land could collapse. The land is valued because it is the only store of value, not because it pays an annual rent.

Here's another way to say the same thing: Take the Present Value of all the rents on land and capital, discounted at the growth rate of the economy, g. That's the "real supply of assets". Now find the "real demand for assets" (the desired stock of savings) if the rate of interest were equal to g. If there's an excess demand for real assets at g, then we are in the weird Samuelsonian world. A bubble/Ponzi scheme/chain letter will and should arise to fill the excess demand for assets. The more I think about it, the more I think that's actually the world we live in.

[azmyth]

I was confused at first by your use of the word bubble. Taxes are not a bubble in a conventional sense - an asset that is overvalued, but they are a bubble in the sense of a social convention that could deteriorate. Roweian bubbles are anything that is not investment that brings purchasing power into the future. Money will fulfill this role if it is not continually inflated: MV=PY, so if Y goes up, P will decline and a stock of money will earn the growth rate of the economy return.

If the return from the "bubble" is higher than productive investment, people will invest in the bubble. Unfortunately that would reduce the stock of investments and reduce the real productive capacity of the economy. Return on investments is not a fixed percent. As more is invested, people invest in more marginal projects driving the return down. The return for bubbles should equal the return for investments in the long run, but the higher the bubble return is, the fewer real investments are undertaken.

[Nick Rowe]

azmyth: In the Solow Growth Model, for example, if the marginal return on investment is less than the growth rate of the economy (in steady state) then the economy is "dynamically inefficient" it is possible to reduce investment, and increase both current and steady state aggregate consumption for all periods in the future. There can be too much investment.

In the Samuelson model, I could accept everything you say about bubbles driving down the amount of real investment, but this could be a good thing. Real investment might be less efficient than a bubble at satisfying the desire to save. Or, more accurately, satisfying 100% of the desire to save with real investment could be less efficient than a mix of real investment plus bubble. If the marginal return on real investment, assuming that real investment satisfied 100% of the desire to save, were less than the growth rate, then having a bubble satisfy some of the desire to save would be efficient.

[RebelEconomist]

Nick, in your reply to rogue, you seem to be using the growth rate of the economy as some kind of benchmark interest rate. Why?

[Nick Rowe]

Rebel: Suppose a chain letter is growing at rate c per year. Every year, you send out 1+c letters for every 1 letter you received last year. Then you earn an interest rate of c. Now suppose income is growing at rate g. If c is greater than g, the chain letter must eventually collapse, because the young generation would not have enough income to pay for all the chain letters. But if c is less than or equal to g, the chain letter would never collapse, because the young generation would pay either a constant or declining share of their income to the previous generation.

So, an interest rate equal to the growth rate of the economy is the highest sustainable interest rate a chain letter can pay. Same thing for a bubble asset or unfunded pension plan.

I've assumed the growth rate is constant and exogenous. I can't figure it out otherwise.

[Nick Rowe]

Or rather, if c is less than or equal to g, the chain letter *need* never collapse. It might, and probably would, if people just stopped believing in it and broke the chain. But it doesn't *have to* collapse.

[RebelEconomist]

Ah, I see, Nick. I would not describe that as the interest rate. It is more the inter-generational rate of change of the consumption of the old, so it is not surprising that it cannot, at least in steady state, outstip the growth of production. I would say that the interest rate is the price that the old pay to the young (or the other way round, depending on relative bargaining power) when they hand over their capital stock in return for a pension. One reason it makes sense to strike that bargain is that the old own the capital stock, but can generate less return from it than the young.

[Nick Rowe]

Rebel: I am drawing a picture of the supply and demand for bubbles, and am going to incorporate it in a new post, that I hope will make things clearer.

[Adam P]

Oh Rebel, surely it's obvious that an inter-generational RATE of change of consumption is an interest rate.

On the other hand, the PRICE that the old pay to the young (or vice versa) sounds like a price, not a rate.

[kevin quinn]

Here's a footnote on the non-weird version of the Samuelson model. So suppose that we have two-period lives but with the bulk of the perishable endowment coming in the second period of life. The young want to borrow from the old, but of course the old won't lend, since they will be gone before they are ever repaid. The interest rate has to be high enough to make people content with consuming their endowments. Let's make this "autarkic" interest rate above g. Now we have what Gale called a "classical", as opposed to a Samuelson economy. In this economy, unlike the Samuelson economy, laissez-faire is efficient. BUT consider that if saving is increasing in the interest rate, then if the government were to tax just a small amount from the initial old and lend it to the young and from then on simply roll over that lending each period, the interest rate would fall and we eventually reach a steady state with r=g. The consumption bundle chosen when r=g is the best stationary consumption choice - a bundle offering more utility must be in feasible. In particular it offers more utility than the laissez-faire bundle. So everyone but the initial old are better off with this solution. The harm to the initial old makes means that the move is not a Pareto move. But that harm can be made arbitrarily small, so that the LF solution, while efficient, is nothing to write home about. So even in the non-weird case, Samuelson's model, I would argue, makes the case for pushing the economy into the golden rule outcome.

[Nick Rowe]

Kevin: I hadn't thought of that! Weird on both sides! My first thought was "would we need a negative bubble?". My second thought was "that can't happen in practice". My third thought was "it's an artefact of a 2-period model, and would disappear in continuous time". I'm not sure, but I'm sticking to the third thought for now. The young borrow from the middle-aged.

[RebelEconomist]

Adam P,

The inter-generational rate of change of the size of the consumption of the old (call it pensions for short) is of course associated with the interest rate struck between young and old. However, in a steady-state, non-growing and non-inflating economy, the size of pensions may not change but there will still be such an interest rate. Perhaps you misread what I wrote?

I thought you work in financial markets? If so, you surely understand that the price of a bundle of future cashflows implies an interest rate. In my day, the relevant Bloomberg function was YA. Try it.

[Scott Sumner]

Nick, I basically agree with your reply. I should add that the gold research was attempting to explain the Gibson paradox, which occurred under the gold standard but not under fiat money. Regarding the necessity of big swings in the amount of base money, I can see the argument, and it might occur. But I doubt it actually would occur if the Fed targeted NGDP at 5%.

[Too Much Fed]

scott sumner said: "1. Years ago I did a paper arging that, under the gold standard, when there was a drop in the MEC, nominal and real interest rates fell, this reduced the opportunity cost of owning gold, demand for gold rose, and you had deflation and recession. Of course this problem doesn't occur under fiat money if the central bank is targeting NGDP. But they may not be targeting NGDP, so it also may be a problem under fiat money."

Can the gold part happen if the fed and the rich are trying to price inflate with "demand deposits" from private currency denominated debt and for various reasons, no more private currency denominated debt is being "produced" or is even being "defaulted" on?

[Too Much Fed]

Nick's post said:

"Scott: Thanks!
1. But that is very much the standard ISLM prediction, for example, if the stock of money is fixed exogenously, as it would be under the gold standard."

Maybe my definition(s) is/are different here, but why is the stock of money (emphasis money) fixed under the gold standard?