Coloured sunspots and shells

This post is a sketch of a model of secular stagnation; and of bubbles that burst and get replaced by different bubbles. I don't formalise the model mathematically, because I don't have a comparative (or absolute) advantage at that sort of thing. But I think it could be formalised fairly easily.

Start with a model where the equilibrium interest rate is below the growth rate of the economy. For example, an overlapping generations model, where the young produce consumption goods, and want to save for their old age, but real investment opportunities either don't exist, or else yield a very low rate of return.

Now suppose there are some intrinsically useless shells that the young can pick up from the beach and store. The shells come in lots of different colours. There is a fixed stock of shells of each colour on the beach.

Every period there is a sunspot that is observed by all. The sunspots come in lots of different colours too, just like the shells. Every period there is a small probability that the sunspot will change colour. The new colour is determined randomly. The sunspots are intrinsically irrelevant events, just like the shells are intrinsically useless assets.

The equilibrium (or one of many possible equilibria) looks like this:

If the sunspot suddenly changes colour to red, the red shells become valuable and all other shells become worthless. Young agents spend part of their time collecting red shells from the beach and storing them. Next period, when that cohort of young agents become old, and if the sunspot is again red, they sell those red shells to the next cohort of young agents, who again sell them to the next cohort of young the following period, and so on.

If the sunspot suddenly changes colour to green, the red shells become worthless, and are thrown away on the beach. The old agents who held those red shells are impoverished. The young agents collect green shells from the beach and store them, and the cycle continues.

We need to assume there are strictly positive costs of storing shells, otherwise the red shells would not become worthless when the sunspot turns green, because agents would buy them and hold them, waiting for the next red sunspot.

We need to assume that the probability of the sunspot changing colour is small, and the cost of storing shells is small, otherwise it would not be profitable for the young to collect and store red shells when they see a red sunspot.

In the limit, as the probability of the sunspot changing colour approaches zero, and as the cost of storing shells approaches zero, the rate of return on holding shells approaches the growth rate of the economy.

We can have an equilibrium in which every individual is behaving rationally, and where intrinsically worthless assets are valuable, but where bubbles burst and get replaced by a new bubble, and all agents know that bubbles eventually burst.

If you are an old agent when the sunspot changes colour, you are worse off than you would be without a bubble. Because you bought red shells when you were young, but they are now worthless. Otherwise, all agents are better off with a bubble. Because they earn a higher rate of return on their saving.

If it is costly to collect shells from the beach, that activity will be measured as "investment", and so investment will be inefficiently high whenever the sunspot changes colour (thanks to Steve Williamson for that point).

One solution might be for the government to issue one trill perpetuity (promising to pay an annual dividend of one trillionth of NGDP forever), maybe broken into a million bits to help liquidity.

But I wonder why intrinsically useful land does not dominate intrinsically useless shells as an infinitely-lived asset. Perhaps land is less liquid?

[I wrote this post because Steve Williamson asked what Paul Krugman might be saying. So I tried to model it. It might be something like this. Or it might not.]

Comments

[Jason]

This model reminds me of the emerald city in the Wiz
http://www.youtube.com/watch?v=ktaknlftNWY

One possible way to account for secular stagnation:
http://informationtransfereconomics.blogspot.com/2013/11/secular-stagnation-and-eu.html

Maybe these sunspot changes are accelerating inflation episodes:
http://informationtransfereconomics.blogspot.com/2013/09/exit-through-hyperinflation.html

[Kevin Donoghue]

From Krugman's opinion piece today:

You may or may not have heard that there’s a big debate among economists about whether we face “secular stagnation.” What’s that? Well, one way to describe it is as a situation in which the amount people want to save exceeds the volume of investments worth making.

When that’s true, you have one of two outcomes. If investors are being cautious and prudent, we are collectively, in effect, trying to spend less than our income, and since my spending is your income and your spending is my income, the result is a persistent slump.

Alternatively, flailing investors — frustrated by low returns and desperate for yield — can delude themselves, pouring money into ill-conceived projects, be they subprime lending or capital flows to emerging markets. This can boost the economy for a while, but eventually investors face reality, the money dries up and pain follows.

I don't think that will be to Stephen Williamson's taste, but it doesn't seem terribly obscure.

[The Market Fiscalist]

"Well, one way to describe it is as a situation in which the amount people want to save exceeds the volume of investments worth making. "

Is he just thinking that people want to save a higher % of their income than borrowers want to borrow even at 0% interest rates ? And faced with 0% returns on their savings savers who need to increase their returns will be more vulnerable to Ponzi-type schemes that generate bubbles. Faced with very low returns it may even be rational to take greater risks (if you have savings worth $10000 and you need $500 to pay for your heart medications you are rational to take a gamble by putting your savings into a risky bond that pays 5% than a safe bond that pays 2%.

Its easy to think of scenarios that drive high savings and low investment. An aging and shrinking population will want to save for its retirement , but investment may be risky because the market is getting smaller and no-one knows for sure what all those old people will want to buy in 10 years time when they start spending their savings.

[Donald A. Coffin]

What's implicit in Krugman's analysis (and I'm surprised no one has yet pointed it out) is this: For there to be too few investments worth making, the internal rate of return on (marginal) investments must be less that the available real rate of interest. At least I think that's right. So the (potential) supply of funds must intersect the demand for investments at a negative real interest rate. Since a zero real interest rate is always available, the marginal investments are not worth undertaking. So this is not in any way different from the argument Krugman (and others) have been making for a while--that the (current) equilibrium real rate of interest is negative.

[Nick Rowe]

Kevin: (quoting Paul): "Well, one way to describe it is as a situation in which the amount people want to save exceeds the volume of investments worth making."

Which begs the question: at what rate of interest?

Why is zero% real a magic number?

But a rate of interest less than the growth rate is a magic number, because Ponzi schemes become sustainable (bubbles can be rational). That's what I'm modelling here.

[Benoit Essiambre]

"But I wonder why intrinsically useful land does not dominate intrinsically useless shells as an infinitely-lived asset. Perhaps land is less liquid?"

Well at some point land gets too expensive and people realise that there is a wave of landowners who are getting old and land might end up in a slow motion fire sale when they downgrade or die.

Even if you look at the long term scenario and ignore the liquidity issues of land and ignore taxes, you have to take into account the fact that on this earth, we have to obey laws of thermodynamics and most, if not all things with intrinsic value degrade and need maintenance or storage or protection from vandalism and theft.

It is not clear to me that rent is enough to offset these costs. It may be that the natural real rate of return on land when we are not in an exceptional economic boom could be well into the negative.

For a very concrete example, take natureconservancy.ca where their goal is to buy actual land and leave it in its natural state. They do have to setup large trust funds and get donations to pay the maintenance costs of the land they own.

Could it be that central banks, by refusing to create sufficiently high inflation and refusing to allow real interest rates to go far enough in the negatives, are giving an unfair advantage to money compared to owners of degrading land and other investors?

[Ralph Musgrave]

Sumner’s “secular stanation” is complete rubbish. The argument is that there’s a shortage of investment opportunities, ergo not much investment spending, ergo we’re condemned to “a persistent slump” as Krugman puts it in his version of the SS theory here:

http://www.nytimes.com/2014/01/31/opinion/krugman-talking-troubled-turkey.html?_r=0

Well . . .doh . . .if there’s not enough investment spending, the boost household or public spending. And if Summers or Krugman can’t work out how to do that, they should go ask some children in the nearest school playground.

[Kevin Donoghue]

"Which begs the question [that rasping noise you hear is the grinding of philosophy teachers' teeth] at what rate of interest?"

The prevailing real rate, which is above the optimum (which may be negative). A lower real rate would require higher inflation, which central bankers won't tolerate. Models with a rate of interest less than the growth rate may be interesting, but it only confuses matters to suggest that's what Krugman is describing. He's talking about an IS curve which lies to the left of potential Y for any r the central bank will deliver.

I'd be happy with your final paragraph if it read: "I wrote this post because Steve Williamson asked what Paul Krugman might be saying. I created a model which definitely isn't what he's saying, but it's more Steve's kind of thing. Anyway let's talk about that."

Which is fine, but I don't have anything at all to say about such models.

[Nick Rowe]

Benoit: but land pays rent to the owner, and shells don't.

Ralph: in my model, aggregate household spending is always high enough. The problem is too much spending by the young, and too little by the old. Shells solve that problem, except the colour keeps changing.

Kevin: I'm not sure if that is what PK is talking about. Is he saying "the economy needs a higher inflation target" or "the economy needs a bubble"? The second is harder, politically, for him to say. But the second fits with the idea of "Alternatively, flailing investors — frustrated by low returns and desperate for yield — can delude themselves, pouring money into ill-conceived projects, be they subprime lending or capital flows to emerging markets. This can boost the economy for a while, but eventually investors face reality, the money dries up and pain follows."

[Kevin Donoghue]

Is [Krugman] saying "the economy needs a higher inflation target" or "the economy needs a bubble"?

He undoubtedly believes that the US and the Eurozone need less austere policies and he'd welcome higher inflation. But in the blogpost which Stephen Williamson refers to and in the opinion piece which followed, he's taking it as given that policy isn't going to change in his preferred direction. So what will investors do? They will gamble. He's not quite saying the world economy needs bubbles; but, as things things are actually shaping up, bubbles are what we're likely to get.

In IS-LM terms: the IS curve is too far to the left when investment prospects are viewed realistically, but it shifts to the right during property booms and suchlike, temporarily bringing periods of high employment.

I really can't see anything he's said which is inconsistent with that reading.

[JW Mason]

With respect to land vs money, recall that Keynes raised the same question in chapter 17 of the General Theory:

It may be that in certain historic environments the possession of land has been characterised by a high liquidity-premium in the minds of owners of wealth; and since land resembles money in that its elasticities of production and substitution may be very low,[17.8] it is conceivable that there have been occasions in history in which the desire to hold land has played the same rôle in keeping up the rate of interest at too high a level which money has played in recent times.
It is difficult to trace this influence quantitatively owing to the absence of a forward price for land in terms of itself which is strictly comparable with the rate of interest on a money debt.
We have, however, something which has, at times, been closely analogous, in the shape of high rates of interest on mortgages.[17.9]
The high rates of interest from mortgages on land, often exceeding the probable net yield from cultivating the land, have been a familiar feature of many agricultural economies.
Usury laws have been directed primarily against encumbrances of this character.
And rightly so.
For in earlier social organisation where long-term bonds in the modern sense were non-existent, the competition of a high interest-rate on mortgages may well have had the same effect in retarding the growth of wealth from current investment in newly produced capital-assets, as high interest rates on long-term debts have had in more recent times.
That the world after several millennia of steady individual saving, is so poor as it is in accumulated capital-assets, is to be explained, in my opinion, neither by the improvident propensities of mankind, nor even by the destruction of war, but by the high liquidity-premiums formerly attaching to the ownership of land and now attaching to money.

[JW Mason]

(Everything except the first sentence is JMK, not me. I hope that's clear.)

[Avon Barksdale]

Does any serious economist pay attention to Krugman any more?

[Benoit Essiambre]

"but land pays rent to the owner, and shells don't."

No that is what I am saying. Land's maintenance cost might well be more than the rent it generates. Net profit on most land might well be in the negative and this doesn't even count taxes. Shells could be very cheap to keep and maintain. Especially if they were virtual Bitshells that exist only in computers.

The other day I was trying to think what would be a good, safe store of value for someone who want's to save in something more tangible than stocks and bonds.

The best thing I could think of was stockpiling nonperishable household supplies in leftover storage space. If you really had spare unused safe and dry storage space and you bought things you knew you were going to need in the next few years, (had no transaction costs to sell it at the end and not enough time for your stock to become obsolete) this would give you near real 0% return (nominal ~2%!). But there are a lot of ifs and you would still be subject to risks of your stock burning down, drowning in a basement flood or getting stolen. On the other hand you might get price benefits from buying in discounted bulk.

However, not everyone has unused storage. I live in an apartment so if I were to stockpile household supplies I would have to pay for storage. It comes down to the laws of thermodynamics. Things degrade and value naturally drops with time, real returns on assets tends to be negative in the universe. On earth the only thing we can depend on to offset this degradation is energy that comes directly or indirectly from sun. The land metaphor works partly because actual land is bathed in sunlight everyday which enables it to generate value. But the concentrated form of sun, oil, is getting sparse and we are left with rather low margin ways to exploit the other forms. I know some farmers and only by working insane hours do they get to stay profitable.

[Herbert]

@Nick: "But I wonder why intrinsically useful land does not dominate intrinsically useless shells as an infinitely-lived asset."

A coherent explanation emerges if you take account of risks:

1. Land and real capital are risky investments.

2. Money (in the sense of currency and overnight deposits) is safe.

3. Hence, land and real capital must yield an interest above the Federal Funds Rate or other short-term interest rates.

4. The upshot is that in order to assess dynamic efficiency, or the possibility of bubbles, Ponzi-games, or shell-sunspots, one must look at the risky interest rates. The safe interest rates are irrelevant. This is known for long, cf. Bohn (1995), for instance.

5. Because the risky interest rates (corporate bond yields, equity yields, mortgage interest rates) are all well above the growth rate, at least in the U.S., Canada and presumably all OECD member states, there is no secular stagnation or "savings glut".

6. Krugman-Summers mix up risky and safe interest rates.

[Benoit Essiambre]

I guess what I am trying to say is that we should get rid of the assumption that we can easily find intrinsically valuable safe assets that have positive real returns. The natural return on assets is often in the negative and might only have been positive during the exceptional boom time of the 20th century.

Prior to the 20th century people had to spend, if not large sums of money, large amount of time to keep land profitable and live off of it. The possibility to extract rent just by owning the land is not at all a given. Especially not without having people work on it in near slave conditions.

[Avon Barksdale]

First, stop with the IS-LM stuff and Keynesian arguments. Arguing this way is not too different than the state of medicine in the medieval times. Should we talk about the virtues of bloodletting too?

I don't understand the purpose of this thought experiment where all asset prices are set exogenously and we start from a dynamically inefficient equilibrium. It's a big so what. It's a pure betting market and the optimal strategy is simple to work out. In fact, the setup is entirely trivial: Dynamically inefficient equilibria lead to inefficient investment - got it.

I suspect that this post is looking to understand bubbles. Before trying to understand “bubbles” in the economy, let's try to define what a bubble is. How can we separate rationally high prices from (and I am guessing at what people mean here) from “irrationally” high prices? How can we measure a price above fundamentals? (Remember any discussion about a bubble is a test of market efficiency, which is always a joint test of market returns with a model of the pricing kernel.) Is it even possible to forecast a negative return in any reliable sense? If the best you can do is create some ex-post story about why prices went down, then you don't have an operational definition of a bubble - you just have some possible explanation of ex-post price movements (a.k.a. elevator economics).

For what it's worth, there are no such thing as “bubbles”. There are slowly time varying risk premia almost certainly connected to macroeconomic conditions, which is entirely rational. The last thing we want is the government to try to temper asset markets because experts claim to “know” the fundamental hold-to-maturity price. If we want to understand the origin of the US stagnation, look no further than to Casey Mulligan's research. When the poorest people in the US have marginal tax rates that exceed 100%, yeah, stagnation is in your future.

[Lord]

Avon: better to remain silent and be thought a fool than to open one's mouth and remove all doubt

[Nick Rowe]

Lord: manners.

[Min]

"We can have an equilibrium in which every individual is behaving rationally, and where intrinsically worthless assets are valuable, but where bubbles burst and get replaced by a new bubble, and all agents know that bubbles eventually burst."

"If the sunspot suddenly changes colour to red, the red shells become valuable and all other shells become worthless."

That's rational?

"If it is costly to collect shells from the beach, that activity will be measured as "investment"

That's rational?

"If the sunspot suddenly changes colour to green, the red shells become worthless, and are thrown away on the beach."

Hmmm. What does that correspond to in your analogy? Losing a war of conquest? A nationwide natural disaster?

And where are the bubbles? IIRC, bubbles are easy to produce in the classroom, with naive traders in a simple market game. Bubbles are endogenous, n'est-ce pas? Sunspots that change color are not.

[Min]

Krugman: "Alternatively, flailing investors — frustrated by low returns and desperate for yield — can delude themselves, pouring money into ill-conceived projects, be they subprime lending or capital flows to emerging markets. This can boost the economy for a while, but eventually investors face reality, the money dries up and pain follows."

Nick Rowe: "But the second (that the economy needs a bubble) fits with (Krugman's statement above)."

Is Krugman saying that the economy needs a bubble, or that recent (and not so recent) history is likely to repeat itself, leading to bubbles? What he described fits the lead up to the recent financial crisis as well as the lead up to the Great Depression.

[Nick Rowe]

Min: suppose there are multiple equilibria. Everyone drives on the right, and everyone drives on the left. Both are rational, if everyone else is doing the same. A sunspot happens, and everyone switches side of the road. Is that rational? Well, it's not irrational.

Here is Wiki on sunspot equilibria.

Avon: those bits of paper in your pocket look like a bubble asset to me. They are intrinsically worthless, and yet people (and governments) treat them as valuable only because everyone else thinks they are valuable. Why can't there be multiple equilibria? Nobody is irrational.

Benoit: I thought that good quality agricultural land in (say) England had paid positive rents to the landlord for centuries.

Herbert: you may be right. But would it be possible to introduce land into my model, and still have a demand for shells? Because my shells are risky too.

JW: Interesting quote. If people get some sort of utility from owning land, then the equilibrium stock of capital will be lower than it otherwise would be (at least, in an OLG model). The "Junker Fallacy" is not a fallacy. But land then helps solve the problem of individuals' desired "saving" exceeding investment opportunities.

Kevin: "So what will investors do? They will gamble. He's not quite saying the world economy needs bubbles; but, as things things are actually shaping up, bubbles are what we're likely to get."

It is not obvious to me (though I have read of others making the same claim) why low interest rates will cause people to make irrational(?) gambles in their "search for yield". If that is what PK is arguing, I would like to see that assumption made explicit.

[Kevin Donoghue]

"...flailing investors — frustrated by low returns and desperate for yield — can delude themselves, pouring money into ill-conceived projects...."

That looks to me like an explicit assumption.

[Nick Rowe]

Kevin: OK. I've got a model here explaining *why* low yields would cause people to pour "money" (i.e. resources) into risky projects, that would appear to be ill-conceived from hindsight.

[Kevin Donoghue]

Nick, I get that. You're responding to Stephen Williamson's demand for a "coherent economic argument", where "coherent" is a Williamsonian term of art. If pressed, I'm sure Krugman could come up with some model exploiting sunspots, or principal-agent problems or whatever, giving him the same result he can get more readily from IS-LM. But in this instance he didn't bother and frankly there's no reason why he should.

[Nick Rowe]

Donald: "Since a zero real interest rate is always available, the marginal investments are not worth undertaking."

What is the zero real interest rate that is always available? Is it buying newly-produced consumption goods and storing them? Some are not storable. But that is nevertheless an investment project.

[Nick Rowe]

Kevin: I think there is a reason. For example, if you have some sort of model, you can see what sort of policy changes would and would not fix the underlying problem. For example, in my model, the government issuing one trill perpetuity would solve the underlying problem. But I don't think you can see that from ISLM.

[Nick Rowe]

Or, is it low nominal yields or low real yields? If it's nominal, increasing the inflation target could be the solution.

Suppose that PK is now writing a post, saying either "Yes! Nick Rowe's model is exactly what I had in mind!" or "Nick Rowe's model is totally different to what I had in mind! Because..." Wouldn't we learn something?

[Kevin Donoghue]

Nick,
I'd like to think he's writing a post saying: "Seriously guys, I wrote about this back around the turn of the century. Just translate my parable of the hunters into game theory."

Imagine the situation back in what ev-psych types call the Ancestral Adaptive Environment. Suppose that two tribes - the Clan of the Cave Bear and its neighbor, the Clan of the Cave Bull - live in close proximity, but traditionally follow different hunting strategies. The Cave Bears tend to hunt rabbits - a safe strategy, since you can pretty sure of finding a rabbit every day, but one with a limited upside, since a rabbit is only a rabbit. The Cave Bulls, on the other hand, go after mammoths - risky, since you never know when or if you'll find one, but potentially very rewarding, since mammoths are, well, mammoth.

http://www.pkarchive.org/theory/iceage.html

[Avon Barksdale]

Nick: “those bits of paper in your pocket look like a bubble asset to me. They are intrinsically worthless, and yet people (and governments) treat them as valuable only because everyone else thinks they are valuable. Why can't there be multiple equilibria? Nobody is irrational.”

Money itself is not a “bubble”, at least not in the sense that most people mean when they say “bubble”. If you mean that the demand for the shortest term debt that pays no interest (i.e., money) seems very high relative to the short end of the yield curve, then yes, the demand for money is quite high. This does not make it a bubble, it reflects the one thing that is special about money: it is the only thing that the government accepts to pay taxes. If everyone needs dollars to pay the government, people will very quickly use dollars to make just about all transactions to solve coincidence problems. The high demand for money relative to the short end of the yield curve is ultimately built from need to pay taxes.

As far as low interest rates and risky investing goes, that is in completely the wrong order. Low interest rates are almost always associated with bad economic times, recessions and the like. In such circumstances, the risk premium is high. Holding any asset with even a little bit of risk will enjoy higher expected returns – you don't need to gamble on some wild growth stock to see a high expected return during recessions, just hold the index. Again, it is utterly impossible to understand anything about asset pricing without thinking about the time varying nature of risk premia.

[Frank Restly]

Nick,

"One solution might be for the government to issue one trill perpetuity (promising to pay an annual dividend of one trillionth of NGDP forever), maybe broken into a million bits to help liquidity."

If you believe that government policy should be counter cyclical instead of pro-cyclical then the annual dividend should be one trillionth of the output gap, not of nominal GDP.

[Lord]

Nick: My point.

If everyone thought it would persist forever, land would dominate too well, ceasing to be traded. It would be transferred through inheritance and marriage. Otherwise, the land bubble may have already burst, leaving them seeking alternatives. Those believing in persistence would want to permanently acquire it while those expecting it to not would seek the most liquid item awaiting what would replace it.

[Nick Rowe]

Avon: Mr Ponzi borrows at rate of interest r, spends the proceeds on consumption, and borrows more to pay the interest. Most people would say that is a Ponzi scheme/chain letter swindle/bubble, and that it must eventually end in tears, and that only people who were irrational or misinformed, or who expected that future people would be irrational or misinformed, would lend to Mr Ponzi. But if r is less than the growth rate of the economy, that need not be the case. Mr Ponzi's long-run budget constrain need not add up. It is a profitable business for Mr Ponzi.

The Bank of Canada borrows by issuing currency and promises to pay minus 2% real interest. It is a profitable business for the Bank of Canada.

"The high demand for money relative to the short end of the yield curve is ultimately built from need to pay taxes."

I used to have to argue theoretically that chartalist theories of money did not work. Now I just point to Bitcoin.

[Nick Rowe]

Totally off-topic: I just checked the spam filter again. I found one of Kevin's comments in there last night, and retrieved it.

But what is really weird: the last spam comment in the filter is 6 days old. Until a week or so ago, we used to get several spam comments per hour trapped in the filter!

Where has all the spam gone???

[Avon Barksdale]

If you are talking about your model, yes, it starts at a dynamical inefficient equilibrium and stays there. It's a toy and it illustrates an obvious point. In the real world, we have risk premia. People might see the buying opportunity of the century (say like borrowing to buy to buy stocks in early 2009), but they refuse because they are more worried about losing their jobs (or house, or if an American, health care). Again, the real problem we have is the giant government subsidy on debt. The "mis-pricing" occurs because those who are taking the risks are not going to be the ones who bear the consequences. But, whatever, it gets votes. I find it funny when people say that shareholders only care about short term gains. They then turn around and trust politicians to do what is in the long term best interest - a group of people who would risk destroying the country's financial system through subsidies if it got them an additional seat in the House for just four more years.

[JP Koning]

"But I wonder why intrinsically useful land does not dominate intrinsically useless shells as an infinitely-lived asset. Perhaps land is less liquid?"

I was wondering this too. Why would the first generation ever start to collect red shells for future sale if there is no guarantee that the next generation will pay a positive price for those shells? Absent some sort of intergenerational coercive mechanism that guarantees to force future generations to accept shells, and given that land already has a positive price, it only makes sense to leave the shells on the ground and focus on amassing property.

But let's say that shells were somehow kickstarted into having a positive price. Let's say they are just as liquid as land. I'd think that the expected capital appreciation on the shells would have to be quite large in order to compensate for both their lack of a non-pecuniary return and the fact that they could be worthless come the next sunspot. Exactly like bitcoin, actually, since anyone holding it expects triple digit returns.

[Benoit Essiambre]

"Benoit: I thought that good quality agricultural land in (say) England had paid positive rents to the landlord for centuries."

This has to be true at least for some land else food wouldn't be produced and the human race could not survive. However...

"Herbert: you may be right. But would it be possible to introduce land into my model, and still have a demand for shells? Because my shells are risky too."

Farmland has some risk of being on a downward productivity curve. For example because of factors such as peak cheap oil and peak cheap fertiliser.

Even if you can find good plots with positive rent, it could be a shrinking positive rent which would mean a capital loss when it's going to be time to sell the land and a total return possibly still in the negatives. Given the potential for rent declines, it might not be that difficult to reach a price for land where shells are still, for some periods of time, a better risk adjusted return. However, I'm not sure this argument would cover your rather long time period of infinity if we maintain that land keeps generating positive rents for that long.

But since land productivity falling under unity makes it impossible to survive, I guess the question could become: Is the hoarding of shells rational under non apocalyptic scenarios or even: Is rational hoarding of shells a predictor of the apocalypse?

[Min]

Benoit Essiambre: " Is rational hoarding of shells a predictor of the apocalypse?"

Well, come the apocalypse, all shells will be worthless, of whatever color, so it would be irrational to hoard shells in preparation for the apocalypse. (In fact, the Bible pretty well says as much. :))

[notsneaky]

"We need to assume there are strictly positive costs of storing shells, otherwise the red shells would not become worthless when the sunspot turns green, because agents would buy them and hold them, waiting for the next red sunspot."

Actually isn't it the overlapping generations assumption that is needed here? If I live for many periods, even with strictly positive costs of storing shells, there's (might be) some positive balance of red shells I want to hold when the sunspot is green in anticipation of it changing. Equating marginal cost of storage with marginal expected future benefit, with marginal future expected cost of picking up new shells (the fact that there's two conditions here means one might be slack, so maybe 0). Depends somewhat on the form the storage costs take, fixed, proportional, convex, etc.

Other way. I live for two periods (and don't care about my descendants). Result depends a bit on sequence of events. In your wording shell picking and storage comes after sunspot occurs, I think. Suppose it's green. Assuming no shocks to costs of picking shells, I know that I'll be able to pick red shells up if the sunspot changes tomorrow. Even if costs of storage are zero. So yes, no point in picking any now, no point in storing them.

So it's really that I live for two periods that makes it work, not the cost of storage.

(if shells have to be picked before sunspot materializes it's a bit different. Last period sunspot is green and probability of transition is low. I go out and pick mostly green shells, with a few red shells thrown in because I'm risk averse. Then a red sunspot happens. I have a lot of green shells on hand. Up to some point where marginal cost of storage is equal to marginal cost of picking green shells all over again, I'll keep some around)

[Nick Rowe]

notsneaky: "Assuming no shocks to costs of picking shells, I know that I'll be able to pick red shells up if the sunspot changes tomorrow."

We can't have an equilibrium where a good has zero price today, has a positive probability of a positive price in future, and has zero storage costs.

[notsneaky]

We can't? Well, you're almost right. Say I can produce a trinket anytime I want with one hour of labor. That one hour of labor could be used instead to watch tv, which I value at T. Today, nobody wants trinkets for consumption. Tomorrow, there's a chance q that someone will want trinkets for consumption and their price will be p>T. It's costless to store a trinket. We all discount the future B<1.

If I produce the trinket today, store it myself and plan to sell it tomorrow my expected profit is (-T+Bpq). Say that's negative. If I just wait, see if the price is positive (>T) tomorrow, and produce or watch TV accordingly then my expected profit today is 0 with probability (1-q) and B(-T+p)>0 with probability q (possibly 1).

There's no consumption demand for trinkets today but there is storage demand. If p=0, others will want to buy the trinkets I produce, hold it till next period and realize the pure expected profit Bpq. But, that means the maximum they'd be willing to pay for a trinket today is Bpq. And that's just not going to make it worth it for me to produce trinkets.

I guess you could say that p(today)=Bpq and no trade, no production is an equilibrium. But so is p(today)=0, if by equilibrium we mean "everyone makes optimal choices, given other's optimal choices" rather than "no excess demand".

Make trinkets perfectly divisible, throw some convexity in there and you're right.

[Nick Rowe]

notsneaky: but people have already collected the red shells, when there was a red sunspot. The costs of picking them up off the beach are a sunk cost. If the sunspot turns green, they won't throw away the red shells, if there are no storage costs.

BTW, I was (implicitly) assuming a fixed number of red shells on the beach, so that when the sunspot turns red, all the red shells get picked up off the beach.

[notsneaky]

That's why I said it depends on sequence of events (within a period). The way I understood your wording was that first we observe sunspot, then we pick shells for today and, possibly tomorrow, so no sunk costs.

Fixed number of shells - so a zero (constant?) marginal cost of picking shells, then an infinite one at the last shell?

The broader point (and this is all sort of pedantic) is that as soon as you allow for a slightly longer horizon, people will want to hold onto the "currently worthless" shells in anticipation of the flip of the sunspot. I think that's the main thing what would make formalizing this model a pain.

[notsneaky]

Actually, here's the devil that's sitting in the details. What do you mean by "storage is costly"?

The usual way to model "costly storage" is by opportunity cost and depreciation. I have some red shells today. I could buy a consumption good with them today. Or I could store them and buy some consumption good with them tomorrow. But if I store 10 shells today, I end up with only 8 tomorrow, 2 of them break. But the relative price of the consumption good can change between today and tomorrow. So storing shells could be worth it or not, given the rate of inflation. But if they are useless today, then I might as well store'em since the opportunity cost is zero. That's not what you mean here.

What you seem to mean here is that there's a "real cost" to storing shells. As in, in order to store shells I have to rent a storage locker or something. To put my shells in. Another words, I have to give up some consumption good today to store my shells for tomorrow. But that introduces another relative price into the model - the cost of storage, which is endogenous and we got to keep track of it. Can't draw any conclusions until we can characterize how that behaves. Ok, suppose the price of a locker is positive (storage costs are not zero, as you assume). The maximum willingness to pay for a locker is then the expected future benefit of storing shells. But you're also assuming that's zero (people throw away red shells when the sun is green). Doesn't work. If the price of storage is positive then that means demand for storage is non-zero. Hence it's worth it (up to a point) to store some shells.

So the only equilibrium with non-zero price of storage is the one where some storage (of the "currently worthless" shells) actually happens.

That's why you actually *do* have to formalize this stuff, comparative advantage or not.

[notsneaky]

or to put it another way, the *real* interest rate matters. You can't assume it doesn't exist or ignore it because you want the number of shells to have real effects.

[Nick Rowe]

notsneaky: "As in, in order to store shells I have to rent a storage locker or something."

Yep. The storage costs must be a function of the physical quantity of shells owned, not their value. It takes part of your time, that could have been used to produce consumption goods.

[notsneaky]

Ok, but then people hold on to some "currently worthless" shells (unless possibly that cost is a chunky fixed quantity and even then you might get multiple equlibria). The value of these will result from the probability that the sunspot might change next period, but also from the fact that next period young people will want some "wrong color" shells, as long as their price is low enough, because the sunspot can change color after that. And another reason the next period young people will want "wrong color" shells is that the young people from the period after that will also want to have some "wrong color" shells. Etc. So what will matter is the overall expected distribution of sunspots (something to do with "ergodicity"?)

I don't know if the fact that people hold on to both kinds of shells is enough to kill the bubbles or maybe give rise to the possibility of two bubbles at once. But the dynamics here are lot more complicated and messy. I guess a direct way to get at it is to simply assume that a green sunspot means not only that red shells cannot be used for exchange but also that they cannot be stored. But that's when the model becomes strangely contrived.

[Nick Rowe]

notsneaky: I think: If there are no storage costs, and people hold onto the wrong-coloured shells, the total value of all the shells is determinate (unless it's zero), but the value of each colour is indeterminate, provided the sum of those values adds up right. Interesting model. With storage costs, people only value one colour of shell, because they want to maximise value per storage cost.

[notsneaky]

I think that's only true if you got linearity everywhere; no risk aversion and consumption over time perfect substitutes, constant returns to scale (output linear in labor). Otherwise they're gonna diversify their portfolio of shells and there will be a relative price each period between the two shells. So if one shell is bubbling (in terms of the consumption good) what is happening to that inter-shell price?