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Perhaps hyperinflation is the eq of infinity?

immortal investors?

Pardon me, I probably do not understand all of the question, but if an asset has an infinite price (or any price above the amount even the wealthiest consumer would be willing to pay), is it not impossible to exchange? Thus, assuming only one consumer can rent the ocean front land at one time, that would put the wealthiest consumer in a pseudo-monopsony position, where the price he pays is equal to the greatest amount which the second wealthiest consumer would be willing to pay to rent the land.

If we allow infinite accuracy, then beach property owners will increasing sub-divide the lots, creating a new production lines selling cottages, then hotel rooms, then cots on the beach. The rent to income ratio is restored.
If we do not allow infinite accuracy, then we reach a point in which the transaction cost of a single trade of land exceeds any gain, the supply/demand curves hit a noise floor. The noise floor is the minimum variance in the Stiglitz paper on EMH.

The assumption that r is equal to g is tripping up your computations. I guess if you consider that there can be no risk premium for lending over the growth rate, then everything, theoretically, can reach up to the sky.

The sun will one day go supernova and the earth will be destroyed.

Well before that, the continents will shift and your ocean front will be a mountain top.

Maybe people will stop liking the beach.

Or perhaps capitalism will end and all beach front property will be nationalzed without compensation.

Or perhaps couples will have less than two children. (I think this has to do with your assumption of constant growth.) Anyway, thinking more about beaches, with a much smaller population, the number of people wanting beach front homes...

Of course, that just means that the price should be really, really high, but not infinite.

And I realize beaches are just an example, but changing technology and preferences (and political risk) could all reduce the value of just about anything.

Applying the Gaia Economic Pie Slice Traders Hypothesis (GEPSTH) http://tinyurl.com/yzdvpm3
one person by chance acquires the most absolutely desirable small piece of the Gaia pie.

Trading proceeds, back and forth, and as the bidding goes up, more and more of the remaining Gaia pie slices are traded to acquire Gaia prima-terre (GP-T). Eventually, one person, Dr Evil has accumulated all of the remaining pie slices and trades them for GP-T. But since Gaia is finite, the trading ends, and his unsatiable appetite for growth is stymied.

So, with Mini-me he creates a black hole machine, places it on GP-T and sucks up the infinite remaining universe.

What happens inside the blackhole, only Stephen Hawkings knows. :)

I think Ron and Bill have the right answers. All the other objections I can get around by tweaking the assumptions slightly.

Don: Regardless of inflation, my model says that the price of one acre of ocean-front, relative to the price of the produced consumption good, should be infinite.

Ryan: people might exchange one acre of south-facing ocean front for two acres of north-facing ocean front (if one person wanted more sun and the other wanted more space). People would offer to buy ocean front by offering millions of produced goods in exchange, but no owner of ocean front would ever accept the offer. It would be like having two sets of goods: C-goods and L-goods. C-goods would traded for other C-goods, and L-goods would be traded for other L-goods, but anyone who offered to sell L-goods for C-goods would always face an excess demand, at any finite price. Sort of like a bifurcated economy.

Matt: maybe. But maybe people just want ocean-front, and they want their own little 10 metres of ocean-front all to themselves, and living in an ocean-front high-rise just isn't the same. (In a sense it's like a positional good).

Rogue: yes, if there were a small risk premium, then the equilibrium price would be finite again. But then I could just increase the income elasticity of demand a little, to make rents grow a little faster, to make the growth rate of rents as high (or higher) than r + the risk premium. Unless....the risk premium is increasing in the price???? Which it might. Have to think about that. It's not implausible.

Just-visiting: I could just assume away monopoly over ocean-front. Or assume the monopolist has to kids/heirs, and 4 grandkids/heirs, etc., so any monopoly gets broken up over time (without primogeniture).

Ron: Suppose investors were mortal, but cared for their kids, so bequeathed their ocean-front to the kids, then it's exactly as if each person is immortal. But my model here becomes very fragile with respect to slight changes in the assumptions. If there is just one owner of ocean-front who knows he will die without heirs, he will sell at a finite price, so we get an equilibrium finite price.

Bill: if there were a constant probability d of total destruction, then it's just like a risk-premium, and I can tweak the assumptions about income elasticity to overcome it, so rents grow as fast as r+d. But if you know that d must increase over time, so that the sun must go supernova before a certain date, then d increases without limit over time, so we get a finite equilibrium price again.

So I think Ron and Bill win this one (maybe Rogue too?). But, could you explain to a Vancouver home-buyer that the sun's eventually going supernova is the ultimate reason why he shouldn't pay that high a price?

Math boy to the rescue.

Dividing by zero doesn't give infinity, it gives an undefined value. It could be very high, it could be very low. It is undefined. Dividing by almost zero does approaches infinity, but is still finite.

Also, I'm not that familiar in economics, but in accounting, present values are usually based on a finite life. For an investor in real estate, that finite life would be his own life (plus, possibly the lives of those he cares about...I'd say 4 generations at best). So no matter how fast or how high rents rise, the total rents are going to be finite, so the present value of those rents is also finite.

Neil: On the maths (not my strong suit). The PV of rents is (where R is today's rent):

PV= R + R(1+g)/(1+r) + R(1+g)^2/(1+r)^2 + ..etc.

The series does not converge to a finite value, unless g is less than r.

If each investor cares about his kids, and doesn't care about his grandkids, but he knows his kids will care about their kids, etc., then we get effectively infinitely-lived agents. That's what Barro showed, when he resurrected the Ricardian equivalence proposition.

Other than no one having the infinite resources to buy it, and not needing them since they only need more than the next, there is not just one such resource. Unless the economy is static, some parts of it will always be growing faster than g and other parts slower at a given time. These trees won't grow to the heavens, but while growing, can grow faster than such a resource, and as long as you time your investments, switching from past to future winners, you can do better. Once you are wealthy beyond your desires, retreating to such safe and stable investments can be much less risky. How long it would last before heirs squander it is another matter.

But, could you explain to a Vancouver home-buyer that the sun's eventually going supernova is the ultimate reason why he shouldn't pay that high a price?

No, just point out that when the Olympics are over, all of those networks and corporate sponsors that paid hundreds of millions to broadcast/sponsor the Games are going to start dumping their condos that they purchased in the runup to house their guests/staff/technical crews/media.

Of course, they'll be sitting on the corporate books at cost (say 2004 prices) so breaking even is probably good enough for the accountants. Make a few grand, good idea, but not their main line of business, so not really concerned about timing of disposal. Instead of renting in an uncertain and tight market, years away, pick up a few condos and guarantee you have a place to stay.

(btw, I was using GEPSTH to prove there is no infinite price, which I gather your reply was also suggesting)

And if you are wealthy enough to buy it, you are probably wealthy enough to occupy it, consuming its rental value.

your post explains latest mega-acquisition of Burlington Northern railroad by Warren Buffett pretty well (he paid a very large, but finite multiple of earnings for the railroad). The only reason he was able to complete the transaction is because market frictions and imperfections sometimes briefly cause r to be much larger than g. Congratulations, you have a great proof that EMH is seriously wrong! You have also proved that there are permanent anti-bubbles in many assets.

Let's combine Ron, Bill, Nick's and my answers. The risk premium could account for the possibility of the sun becoming supernova, but if the investors were immortal anyway, we go back to infinite value. I wonder what types of houses immortals would prefer during a supernova? Plenty of skylight? :)


What about estate taxes?

While an interesting hypothesis,Perhaps, even though mathematicical models suggest something, good ole
human psychology will screw up the best of models?
Consider:This Time is Different-800 years of finacial cycles (Carmen M. Reinhart and Kenneth Rogoff)
Cheers and a good conversation!


The fundamental problem is that different people have different time preferences and those preferences change with prices. You implicitly assume that everyone involved has infinite length time preferences. This is fine for some problems, but fails miserably sometimes.

People with less than infinite time preferences and non-constant discount rates will rationally shift to other goods. This means that the price will be far less than infinite for the housing good.

Second objection:
You also create a situation with an infinite future price, and then say that EMH holds so the current price must also be infinite, this doesn't make sense, because you can't treat infinite prices that way. Let say that I have an object that is worth so much to me that I would not be willing to trade it for any amount of money or goods, this object then has infinite value to me. It doesn't have infinite price, however; it has no price. Whenever we get an answer that has infinite price that just means that no one is selling, therefore there is no market for the good, so EMH doesn't hold.

The constant discounting with infinite time preference makes the problem odd.

There are other mistakes too, such as the fact that supply curves change with time (they usually become more elastic over longer time ranges). We can make more ocean front property for example (just look at what dubai is doing), it just takes a really long time to do. We can also create similar goods to compete with the fixed supply good, which become more attractive as the price of the fixed supply good rises.

J G Ballard experienced this and wrote about it, Steven Spielberg made it into a movie. In "Empire of the Sun" Ballard says that he learned that a man would do anything for a potatoe. I'd call that a working definition of an infinite price.

Nick: I think income and price elasticity can not be assumed to be 1 through out - due to maximum income constraints.

Nick: "The PV of rents is (where R is today's rent):

"PV= R + R(1+g)/(1+r) + R(1+g)^2/(1+r)^2 + ..etc.

"The series does not converge to a finite value, unless g is less than r."

1) Isn't this a form of the St. Petersburg Paradox?

2) Is the proper rent 0? ;) One penny? ;)

3) As others have pointed out, there are a number of implausible assumptions.

4) It has been a long time since I read about the St. Petersburg Paradox, but IIRC you can argue that it is not theoretically correct to use expected values.

Ron: "What about estate taxes?"

My first thought was that estate taxes would reduce the after-tax rate of return to owning ocean-front land below the growth rate of rents, but I could offset this by assuming that the income elasticity of demand is greater than one. My second thought was that I have no idea how to model the government budget constraint if the government puts a tax on an asset with an infinite price. Presumably if estate taxes are 10%, when you die you have to give the government 10% of your land. OK, but now what does the government do with that land? (What does it spend it on?) If it doesn't spend it, the government asymptotically ends up owning all the land. Dunno.

Doc: There has always been a "problem" in infinite horizon models if different people have different rates of time preference proper. [Terminology: "rate of time preference *proper*" means the what the MRS between present and future consumption would be if present and future consumption were equal]. The most patient person always ends up owning everything. We can fix that if the rate of time preference varies with wealth or the scale of consumption (non-homothetic preferences).

My simplest assumption about preferences would be:

U=log(C)+log(L) where C is the produced consumption good and L is living on ocean-front land. The marginal utilities are then:
dU/dC = 1/C and dU/dL = 1/L

If R is real rents on land, and C is numeraire, then R=C/L, and since L is constant, and C is growing at rate g, then R must be growing at rate g.

With zero rate of time preference proper, the real rate of interest will be determined by (1+r)= C(1)/C(0), so r=g too.

If I assumed positive rate of time preference, I think I could still get r=g by making the marginal utility of C diminish more slowly than in the logarithmic utility case. Plus, by making the MU of land diminish more slowly than the MU of C, I could make R rise faster than g, and so still get R growing at rate r, or faster than r, if need be.

In fact, can't I just assume U=C^a.L^b and choose a and b to get whatever I want, even if there is positive time preference proper?

(Any competent econ PhD student familiar with consumption-Euler equations could answer that question, but my math/theory isn't good enough.)

Min: My logarithmic U() function assumed above is exactly the one originally used to resolve the St Petersburg Paradox. I am assuming agents maximise intertemporal utility, not intertemporal consumption.

Ritwik: "Nick: I think income and price elasticity can not be assumed to be 1 through out - due to maximum income constraints."

I'm not 100% sure, but I don't think you are right there. My log U() function does the job. Don't worry about this violating the budget constraint; remember, somebody must own the land, and collect rents on it, so people in aggregate certainly can afford to rent it.


(1+r) = (1+p).MU(Cthisyear)/MU(Cnextyear) where p is the rate of time preference proper. So I can easily rig the utility function, and hence the marginal utility function, to get (1+r)=Cnextyear/Cthisyear, even when there is time preference.

This is what you get when you ignore general equilibrium effects. In a GE version of your model, prices (including the interest rate r) will adjust so that this cannot happen in equilibrium. If it does happen, then somebody is not optimizing in your model. In the example "January 31, 2010 at 06:47 AM", your agent ends up with infinite utility, so you do not have a well-defined maximization problem.

Read Santos and Woodford on the (im)possibility of bubbles. It is closely related (although it is a tough read).

pinus: What GE effects am I ignoring?

Start with a representative agent 2-period model. The agent maximises the sum of this period's plus next period's utility, where the per period utility function is U=log(C)+log(L). Assume L1=L2 (Land is in fixed supply). Assume C falls as manna from heaven, and C2=(1+g)C1.

That problem is certainly well-defined.

Now extend to 3 periods, 4 periods,....and take the limit as the number of periods becomes infinite.

Or does what happen in the limit not equal what happens at the limit?

Or is your objection that with zero time preference proper, an infinitely-lived agent has infinite lifetime utility? If so, I can get around that problem easily, by introducing time preference, but changing the U() function so that the MRS between L and C grows faster than C, so rents grow faster than C, and at the same rate as growth in C plus time preference.

For r=g, you're going to need a discount rate of 1, so of course everything explodes.

Noah: sure it explodes. That was my point. What (if anything) prevents fundamental values of some assets becoming infinite?

(And I don't need a subjective discount rate of 1. I can tweak the utility function to get rents growing as fast as r, even if r exceeds the growth rate of C because of time preference. Just make the MU of C diminish at a different rate than the MU of L.)

Much of the discussion here is beyond my reach, and the supernova idea is whimsical, but if I could interject some more down to earth issues...

Ocean front property. If you were an insurance agency offering coverage for ocean front property, I imagine you'd be following the IPCC forecasts fairly closely for predictions of rising sea levels, and the effects of more violent storms on ocean front property. Over time, your ocean front property may become ocean property, or become retaining wall/dyke front property. IPCC = perfect information? Climate change bearish insurance agencies (increasing astronomic rates) and Denier bullish insurance agencies (hold or moderately rising rates)?

Another factor- offshore wind farms. It seems to me there was a planned offshore windfarm off Nantucket Sound (a number of miles out) that was cancelled due to political pressure by ocean front owners, including some of the Kennedys. Off shore is where some of the most consistent and strongest winds (best suitable in other words) for wind turbines exixts. Over time, political interventions of this nature will be unsuccessful as the public priorities intensify, and the political elite's influence declines (recent passing of "The Lion").

So, the whole game can change dramatically with the sun just doing the status quo thing.

"What (if anything) prevents fundamental values of some assets becoming infinite?"
Some factors in rough order of importance:

Political risk premium
Technological risk premium (after X years virtual reality will reduce the value of coastal land)
Every potential buyer has budget constraints and limited access to leverage - this increases r in equilibrium
Information costs

Can the answer be a practical one to do with how production and consumption functions work, rather than a theoretical one? Two examples:

1. It is hard to find commodities that are truly in fixed supply. We once thought ocean-front land was one of them, until its price became high enough to make someone invent Dubai. Perhaps in practice, people will develop substitutes for any sufficiently valuable commodity.

2. Could it be that in the real world, your hypotheses on discount rates and elasticities would never hold? For example, would price and income elasticity always remain at 1 as people become much richer in real terms, and thus can afford a greater number of alternative leisure options?

My instinct is there's a third constraint, which is to do with the return on capital - would you buy an asset with a permanent return of only 0.5%, 0.1% or 0.01% (today's short-term money market rates notwithstanding)? But I'm not sure about that one. You might have a capital appreciation goal if you expect next year's investor to be one year closer to infinity than you are. A variation of this argument is to look at how much real wealth is actually available in the world, and ask what share of it could conceivably go to buying this kind of asset today. Of course that argument ignores some valid questions based on credit creation.

Perhaps we could take away some of the practical objections by turning the question into: how much would you pay to acquire a perpetual 0.1% share of world GDP? This is a similar question, but strips away some of the distractions to get to (what I think is) the heart of it. It does feel like the answer would be 'a finite amount' but I'm not sure why. I have been getting more convinced that failures in backward induction are the answer to a number of asset pricing puzzles. The finite lifetime objection mentioned above is related to this.

Incidentally, to make the question a little more rigorous and avoid nit-picking from mathematicians, you can state it in terms of "can you say why there should be any finite upper bound on asset prices?" It's the same thing really, but it stops pedants complaining about division by zero.

Leigh: "Incidentally, to make the question a little more rigorous and avoid nit-picking from mathematicians, you can state it in terms of "can you say why there should be any finite upper bound on asset prices?" It's the same thing really, but it stops pedants complaining about division by zero."

Good idea. That's how I want to think of it. Only I would re-state it as: "Can you say why there MUST be any finite upper bound on the price of ANY asset?"

Now one can think of many reasons that might lower the price of any asset (like, in your example, the possibility of increasing the supply of ocean front). But can we say definitively that those reasons MUST impose a finite upper bound? Or could they be offset by other forces that would tend to increase demand? And the same thing about my assumptions about discount rates and elasticities: I can't think of any reason why they *cannot* hold.

Normally, you see, we say that the value of any asset must have an upper bound, because it can only be worth the PV of its earnings. But if that PV could be unbounded, that can't work.

Look at it another way. We never observe infinite asset prices (or do we?). Why is that? Theory says that asset prices (absent bubbles) must be worth the PV of their earnings. And I can't think of any good theoretical reason (absent the sun going supernova, and "weird" reasons like that), that would tell us the PV of earnings must always have a finite upper bound.

That leaves 3 options:
1. It's just a fluke that we have never had the conditions that would make asset prices infinite.
2. The value of assets is actually determined by the fact that the sun will go supernova, and other weird stuff.
3. Theory is wrong. Something else is preventing asset prices going to infinity.

Great thought problem. However, I must quibble with the assumption: "Start with an economy with technology and total factor productivity growing at rate g."

But not all production factors can grow in productivity. The most basic sample is a piece of wood furniture. The wood is the wood. Its there or it isn't. Wood as a production factor cannot grow at 'g'. Its growth is necessarily zero. I think a similar argument applies to coastal real-estate.

Therefore, I think you are begging the question. You assume that coastal real-estate is a fixed good and assume 'g'. Then conclude that there is a singularity. But of course.

I once ran a regression on california real estate prices, comparing some coastal and inland cities, across time, using census data, and found the following:

1. price to rent ratios have been secularly increasing, albeit cyclically, from 1930-2007 -- due to longer and longer loan terms and falling interest rates. http://1.bp.blogspot.com/_fevQMK7kLEI/Sp9XkbA2_eI/AAAAAAAAABc/-WQW0GCxy7A/s1600-h/Price_to_rent_Cities.png

2. Monthly mortgage burden ratios have been relatively constant, and don't show much correlation with ownership rates, coastal/inland, city income, or city gini index.

Jon: yes, I started with a produced consumption good, then added land later. I should have said that the produced consumption good is growing at rate g, and the non-produced good at rate 0.

RSJ: That is an amazing graph you link to.

Nick, thanks for the response. I didn't just mean time preference but rather variance in time preference within the set of buyers and sellers (and the variance over time too). As long as there is a variance, the problem won't blow up. The problem only blows up when the time preferences and utility functions and such are all identical within the population.

"We never observe infinite asset prices "

Sure we do. Any time an owner refuses to sell for any reason, thats an infinite price.
We have many examples:

1. Religious martyrs who refuse to recant their beliefs are an example.
2. People who refuse to move from their homes when they are being confiscated (and then try to kill the police)

What is unique about these circumstances, is that the 'goods' are unique, so we can't have variance in the time preferences of the sellers (because there is just one). And due to the person's desires, we don't really have much variance in time preference over time, either.

Saddle path solutions are ubiquitous in models with an infiniate horizon. Generally, there are two saddle paths: one is 'convergent' and the other 'divergent'. In order to be on the convergent path and so avoid the case in which prices shoot off to infinity or zero, initial conditions must be just right; i.e., the initial price of the asset must be set to satisfy the so-called 'transversality condition', along with arbitrage conditions. This need for 'vision at a distance' is a feature of the solution to all dynamic programming problems solved by backwards induction and has been well understood by economists for some time, starting perhaps with Dorfman Samuelson and Solow in their book Linear Programming and Economic Analaysis. See pages 321-322. It was again brought to the attention of economists by Frank Hahn in his 1966 QJE paper. Curiously, the finance literature appears somehow to ignore the difficulty, I suspect, by simply assuming the existence of a risk fre rate of return to which everything can be tied.

I will take my shot at option 3: the theory is wrong. My intuition is that in all of the cases in which the asset price is infinity, it is also the case that the agents in the economy have infinite lifetime utility. In that case, our standard general equilibrium approach to asset pricing would seem to break down. For instance, it's not clear to me that the consumption Euler equation has to hold in this case. If I have infinite lifetime utility, who cares if I am optimizing intertemporally?

To take a more extreme example, suppose I were an agent in this economy and someone offered to sell me an acre of beachfront land for one penny, even though we all agree the asset price "should" be infinity. If I buy the land, my lifetime utility is infinity. If I don't buy the land, my lifetime utility is still infinity. So in some sense, the agents in this economy don't really care about asset prices at all, even when they are infinite.

Doc: in your two examples, it's not just that the asset is unique. The seller's supply price might be infinite, but the demand price would be finite. In my example, both the supply price and the demand price of ocean-front land should be infinite.

Harvey, and econ_grad_student:

Here's one of the puzzling things. In my model, if there were no market in ocean-front land, so you could rent it, but were not allowed to sell your endowment or buy someone else's endowment, there would be no difficulty in solving for the equilibrium real interest rate on the produced consumer good, and real rental rate on ocean-front land. It's only when we consider a market to buy (rather than rent) ocean-front land that we hit a problem.

And I could tweak the model so that an agent's lifetime utility is finite. Just add in time preference. This will also mean that r is greater than g (the growth rate of C), but I can still make Rents grow faster than g (and so as fast as r) by making the income elasticity of demand for ocean-front land greater than one.

The more I think the more I like Nick's model. Infinite demand price in practice means that marginal buyer is prepared to pay any price provided he has access to leverage. I've seen that in many private equity buyouts before the current crisis.


Are you sure you can tweak the model so that the agent's lifetime utility is finite AND the asset price is infinite? A friend and I have been trying and we can't get it to work. Let me try to sketch out why:
Assume the following utility function:
U = sum from t = 0 to infinity of (beta^t)*(u(C) + v(L))
For the asset price to be infinite, we need g > r. From the Euler equation, we know
u'(c) = beta*(1+r)*u'(c') and by assumption c' = (1+g)c, so we can re-arrange to get
1+r = (1/beta)(MU(c)/MU((1+g)c)). We need 1+r<1+g which implies (1/beta)(MU(c)/MU((1+g)c))<1+g.
Re-arrange to get MU(c) < beta*(1+g)MU((1+g)c), and multiply both sides by c to get
c*MU(c) < beta*(1+g)*c*MU((1+g)c)
Note c*MU(c) is a rough approximation to the agent's flow utility from consuming c today, while beta*(1+g)*c*MU((1+g)c) is a rough approximation to today's discounted value of tomorrow's consumption of c. This inequality suggests that the discounted flow utility from consuming c is growing over time, so lifetime utility cannot converge.
I have to think more about the approximation c*MU(c) = flow utility from c today, but I don't think diminishing marginal utility should upset the argument.
I could be totally wrong though!

How close to infinity can we get? Here is an area with prices exceeding $110k to $165k per linear foot.

Nick, I don't quite see the puzzle here. Are you asking whether prices would be infinite if your assumptions held? Or if they would be infinite if your assumptions were roughly true? Or are you asking why ocean front land does not in fact have infinite value?

I'd say:

1. Yes, if your assumptions were true the value would be infinite.

2. If the assumptions were roughly true, but not exactly, the value would rapidly fall from infinity to a few hundred thousand dollars.

3. The assumptions are not exactly true for all sorts of reasons. I was going to mention a point Leigh made about the income elasticity not staying as high as 1.0 as real GDP went to infinity. (Remember that your example assumes that the public expects to see their real income to eventually approach infinity---that's far more mind-blowing than than the infinite price issue.

But there's lots more. If you introduce risk aversion, and uncertainty about future demand for oceanfront land, that knocks the price down by 99.99999999999999%, well by even more, from infinity to a few million dollars at most. And you'd expect the future value to be uncertain, as infinite technological progress implies future devices that perfectly simulate the experience of standing on the beach.

So to summarize, it isn't puzzling at all that land isn't all that valuable, even though it would have infinite value if all your assumptions held.

TheMoneyDemand: I'm trying to think about leverage. If you already held 1 acre of infinitely valuable land, you could get a 50% mortgage to buy a second infinitely valuable acre. And the mortgage would have a premium increasing over time, paid for by the growing rents on the second acre??

1. Be careful using the greater than and less than symbols in comments. One of them trips it into html (or something).
2. I'm an old guy, useless at mathy/theory.
3. g is the growth rate of C (by definition). You need the growth rate of Rents to be equal to (or greater than) r in order to have an infinite value of land. You can have g less than r (which you will get if beta is less than one) and still have infinite price of land, provided growth rate of R is greater than g.
4. In your analysis, land plays no role. You could delete it from the model and not affect your argument. But if you delete land, the model is very standard. There can't be anything wrong with it.
5. Rents = MU of L/MU of C. Since L is constant, if you have utility separable in C and L you need diminishing MU of C to get rents growing. If you want rents growing faster than C, you need MU of C diminishing faster than C is growing

I hope what I have written here makes sense to you. I'm glad you are working this out. ideally, what we need is a utility function in which the MRS between L and C (which equals Rents) growing at a rate equal to r, where 1+r is (1/beta)(MU(C)/MU((1+g)C)).

Scott: I'm not 100% sure what I'm asking. It just seems to me that there are some assets whose rents would be expected to grow over time, and I can't think of any a priori reason why those rents should not grow as fast or faster than the rate of interest. So I can't think of any a priori reason why the fundamental ("natural") prices of those assets should have any finite upper bound. (And at the back of my mind is some sort of monetary model, where the actual price is trying to approach that "natural price", or the actual rate of return on land is trying to approach the natural rate, and it keeps creating a crisis.)

Sure, my assumptions are rough approximations, but it seems they could be wrong in either direction. So asset values could either be lower or *higher* than my toy model predicts.

I just realised, I don't even need the income elasticity of the demand for land be greater than one. I could just make the price elasticity of demand for land less than the price elasticity of demand for C. All I need is something to get rents growing at rate r, when land is in fixed supply. Since L is constant, and C is growing at rate g, I just need a utility function in which the MRS between L and C to be growing at a rate g or greater.

"The seller's supply price might be infinite, but the demand price would be finite. In my example, both the supply price and the demand price of ocean-front land should be infinite."

Only "supply price" can be infinite. Demand price can't be infinite, because people have finite budget constraints.

For one, you don't have actual dynamics. Quite possible "infinity" is an equilibrium for the dynamic system equivalent to what you have described, but that doesn't guarantee that it's unique, nor that it's the one that the system converges to. You should have differential or finite difference equations there – then there could be a meaningful discussion.

Sorry if this seems rude. Shortest way to say it.

syntaxfree: that wasn't rude; an attempt at a constructive critique. But one of us doesn't understand the other (and it may be me). I'm not saying the equilibrium value should *approach* infinity; I'm saying it should already be infinite, from the very beginning. The model has C and R growing at a constant rate g, and everything else should be constant.

Doc: vendor financing? With a stream of mortgage payments of infinite present value? Like the way TheMoneyDemand is thinking.

Nick, thanks for your response. You have a good point about the non-separability of preferences. I think I was on a slightly wrong track above. Let me try one more time to convince you.

The equilibrium value of land at time zero is the sum from zero to infinity of R_t/((1+r)^t), where R_t is the time t rental rate of land and r is the real interest rate. Furthermore R_t = MU(L_t)/MU(C_t) at any point in time. From the consumption Euler equation, we can write MU(C_0) = (beta^t)*((1+r)^t)MU(C_t), which we can re-arrange to get (1+r)^t = MU(C_0)/((beta^t)(MU(C_t))). Plugging these two equalities into the present value equation we get:
PV(L) = sum from t = 0 to infinity of (MU(L_t)/MU(C_t))/(MU(C_0)/((beta^t)(MU(C_t)))), which simplifies to sum from t = 0 to infinity of (beta^t)MU(L_t)/MU(C_0). Following your assumptions let's assume the marginal utility of land grows at some constant rate h, so that MU(L_t) = MU(L_0)(1+h)^t. Then we can write PV(L) = (MU(L_0)/MU(C_0)) times the sum from t = 0 to infinity of (beta*(1+h))^t. So for the price of land to be infinite, it must be the case that beta*(1+h) is greater than or equal to 1. But think about what that implies: from the agent's perspective, the present value of the flow utility from the marginal piece of land is greater for farther out periods than for closer ones. Then the sum of these discounted marginal utilities must be infinite, which in turn implies that the agent's lifetime utility must be infinite.

I'm not sure if this makes much sense in this format, but it I'm pretty convinced that it's true.

econ grad student: And thanks for keeping on working at this, and posting back.

I am with you (I think), except for just one thing: " Following your assumptions let's assume the marginal utility of land grows at some constant rate h".

We want the *ratio* of the MU of land to the MU of C to grow at some rate h. We could also achieve that by having the MU of C *decline* at rate h, and that way we could keep lifetime utility finite.

"Doc: vendor financing? With a stream of mortgage payments of infinite present value? Like the way TheMoneyDemand is thinking."

This results in any profit seeking vendor having to deal with infinity - infinity, nonsense. A vendor can't actually compare the two and say that the buyer is willing to offer more than what it is worth to him.

Nick, intuitively it seems like having MU(C) decline over time might be enough to generate an infinite asset value, but the math says otherwise. Let me state my argument a tiny bit differently, noting that I have not made any assumptions on the form of the utility function (sorry for the repetition).

The equilibrium value of land at time zero is the sum from zero to infinity of R_t/((1+r)^t), where R_t is the time t rental rate of land and r is the real interest rate. Furthermore R_t = MU(L_t)/MU(C_t) at any point in time. From the consumption Euler equation, we can write MU(C_0) = (beta^t)*((1+r)^t)MU(C_t), which we can re-arrange to get (1+r)^t = MU(C_0)/((beta^t)(MU(C_t))). Plugging these two equalities into the present value equation we get:
PV(L) = sum from t = 0 to infinity of (MU(L_t)/MU(C_t))/(MU(C_0)/((beta^t)(MU(C_t)))), which simplifies to sum from t = 0 to infinity of (beta^t)MU(L_t)/MU(C_0). To go one step further, we can write PV(L) = (1/MU(C_0)) times the sum from t = 0 to infinity of (beta^t)*MU(L_t).

There are only two ways this sum can diverge: beta can be greater than or equal to 1 (in which case it is straightforward that lifetime utility is infinite), or MU(L) can be growing over time. Specifically, MU(L) must be growing quickly enough that beta*(1+h) is greater than or equal to one. A declining marginal utility of consumption will not generate an infinite price of land on its own.

econ_grad_student: neat proof! (I think it's right, though I am not 100% convinced that an infinite discounted sum of the flow *marginal* utlity from land implies an infinite discounted sum of the flow *total* utility from C and L.)

Now watch me try to back-track and weasel out of it!

You said in an earlier comment:
"My intuition is that in all of the cases in which the asset price is infinity, it is also the case that the agents in the economy have infinite lifetime utility. In that case, our standard general equilibrium approach to asset pricing would seem to break down. For instance, it's not clear to me that the consumption Euler equation has to hold in this case. If I have infinite lifetime utility, who cares if I am optimizing intertemporally?"

OK, suppose your intuition is right on that first point (that infinite value of land implies infinite lifetime utility). Does your second point follow (that agents won't care about maximisation)?

Suppose you were immortal, and had no time preference. Would you be indifferent between a lifetime utility of (say):
1+1+1+1+.....to infinity
2+2+2+2+....to infinity?

(This reminds me of Borges' story "The Immortal" http://en.wikipedia.org/wiki/The_Immortal . Someone (Socrates?) has fallen down a hole in the ground, and can't get out. But nobody cares to help him, since he will only be down there a finite time, and he's immortal.)

Again, kudos to you for working this one through!

Nick said:
"I'm trying to think about leverage. If you already held 1 acre of infinitely valuable land, you could get a 50% mortgage to buy a second infinitely valuable acre. And the mortgage would have a premium increasing over time, paid for by the growing rents on the second acre??"

Well, there was a move in that direction, as so called PIK loans became popular in the last leveraged buyout cycle. The main feature of such non-amortizing loan is the deferral of (high) interest payments until maturity. Theoretically PIK loans could also have infinite value, as stock warrants are usually issued to PIK lender. So the private equity world really got closer to your model before the crisis, it's really interesting if these new leveraged buyout features will survive the current crisis.

I think there is a way to test if some market participants behave as if some assets should be priced according to your model. For example, it should be possible to check if buyers are more likely to use PIK loans and share warrants when buying companies that are likely to have permanent growth rates that are closer to g.

Thanks Nick! I agree with you about the weak point in my proof. If I stuck at it for a while I think I might be able to convince you it has to be true, but I don't think I can prove it rigorously or anything (I'm not much of a math person either).

As far as my earlier statement, maybe I will back off of it a little, and just say that it seems like our standard optimization toolbox might not work so well when the objective function isn't well-defined. I think there is actually some research on this issue but I can't find it offhand. Thinking about the philosophical aspect too much makes my head hurt - kind of like reading Borges!

Nick, We know that land isn't worth anything like infinity. And I seem to recall James Mill once said that everything true in theory is also true in reality. So in a sense you could interpret your question as asking "which aspects of theory best explain why we never see infinite prices." My hunch is that risk aversion plays a big role.

Here's another question. Can you imagine some sort of security, like a government bond that pays a steady share of NGDP, that should have an infinite value in theory? That might make the problem more tractable, as the problem with land is that there are so many uncertainties about future demand, that we don't really know where to look. But if you constructed a bond with a real interest rate equal to g, then it should have infinite value except for a relatively few complications like risk aversion and default risk.

A few comments back is a remark about the existence to two market types: the flow market for the services of the property and the stock market for the land as an asset. An interesting book by Andre Burgstaller, Property and Prices, (CUP, 1994) considers the coexistence of two such market types in a variety of more or less familiar settings (eleven models in all). One of them is the pure exchange model where an non-produced asset lies behind each given flow (milk and honey fountains spew forth milk and honey). The question of bounded utility is considered at various points in the book.

Scott:"Can you imagine some sort of security, like a government bond that pays a steady share of NGDP, that should have an infinite value in theory?"

I think we've addressed that security here.

"Rogue" is correct. The model you've written down is the Gordon model for valuing equities. Both g and r are supposed to be constants, set equal to their steady-state values for this model to be valid. It can happen that r < g for relatively short periods of time in dynamics, most recently in the the U.S. between about 1974 and 1979, but cannot last. In such a world, it doesn't make sense for r < g, because then people could borrow infinite sums of money at rate r, invest and earn at rate g, and become billionaires; that cannot be an equilibrium. The condition that r > g is known as the dynamic efficiency condition. There is a chapter on it in Blanchard and Fischer (1989). All the other stuff--asset in fixed supply, various elasticities, and so on--are red herrings.

Hi Bob!: " In such a world, it doesn't make sense for r < g, because then people could borrow infinite sums of money at rate r, invest and earn at rate g, and become billionaires; that cannot be an equilibrium."

They can only do that if the asset they invest in has a finite price. And that there cannot be an equilibrium at any finite price for the asset is exactly my point! Could there be an equilibrium at an infinite asset price?

I would be careful about using these models for valuation purposes. First, the discount rate is the overall growth rate of the economy. Second, if a businesses has a larger profit rate, it will re-invest and grow to the point where the growth rate slows. There are no infinities. It's better here to look at the dividend capitalizaiton model, which is just the Cambridge growth model. This says: earnings yield = present dividend + expected dividend growth rate. Now with a no-arbitrage assumption, all assets will have the same earnings yield, so you are really trading off the present dividend versus the growth rate. I.e. high yield stocks will grow more slowly than low yield stocks. The overall growth rate of the economy will be (in equilibrium) the overall yield. And obviously no company will grow faster than the economy permanently, as it would become the economy. And to have a permanent growth rate equal to the economy would require 100% re-investment, or zero dividends. Anything short of that, and you must assume a growth rate less than the economic growth rate, and your series converges. Typically, people assume high initial growth rates, a slower intermediate rate, and a terminal rate of about 2%, since very large mature companies will pay out high yields and grow very slowly.

Now when you are valuing land that cannot be produced or consumed, then these valuations do not apply. I think it's better to separate land (and similar pet-rock type assets) from productive capital, as the price of land is indeed exogenous, since you are really talking about rentier profits (in the original sense of the term). Originally, rents were determined by what you could squeeze out of your serfs, and this was controlled by government to a large a degree. There is no possibility of increasing the stock when the price gets too high, or decreasing the stock when the price is too low. This is for ideal scenarios (e.g. coastal land in populated areas where zoning laws prevent increasing density).

But in that case, you assumption solves the problem. You assume that (exogenously), people are willing to hand over a fixed proportion of their income as rent. Say 1/4. With that assumption, the price of land goes along for the ride with incomes and must compete with the price of capital. So assume that the economy is growing at 6%, then the rate of interest for capital will be 6%, for a P/E multiple of about 17. And if people living in coastal areas make 200,000, then their incomes will also grow at 6%, so you will be agnostic as to securing a slice of land and earning the rentier profits or a slice of capital and earning the competitive return (6%). Therefore you are also willing to pay 1/4*200,000*17 = 850K for coastal land, and no more. 850K spent on productive capital will exactly reproduce the income stream you would have gotten from owning land. Now, if you assume that there will be income inequality (e.g. concentration of wealth), so that the inhabitants of coastal regions will have incomes that grow faster than the economy as a whole (for a period of time), then you would be willing to pay more and would make similar adjustments as to companies that grow quickly in the beginning and more slowly later on. If you want to take into account property taxes and maintenance, then you would pay a bit less, etc.


Bob is on the right track with this not being an equilibrium but then gets the argument all wrong. The borrowing and investing of infinite sums is consistent with an infinite price, as you correctly responded. In fact the infinite borrowing and investing in the asset is how the price would become infinite in the first place!

No, the reason it would not be an equilibrium is that

r = -marginal_utility_if_consumption_growth_rate

and the dividends from the asset are part of consumption. Now, in the usual model there is one consumption good and one asset and so the dividend from the asset IS consumption. Thus, since the marginal utility of consumption grows more slowly than consumption itself (due to declining marginal utility) we always get r < g.


Now, in your case the dividend from the asset may seem to be a small part of consumption and so perhaps r >= g. Actually no.

Why not? Lets assume r >= g and find a contradiction. The infinte price you refer to would be infinite in real terms (not just money terms). Expressed in real terms the infinite willingess to borrow and invest is really just an infinite willingness to FORGOE CONSUMPTION in return for a share in the dividend stream, THUS AS THE ASSET PRICE GOES TO INFINITY CONSUMPTION FROM OTHER SOURCES DISAPPEARS. We are now back to the text book (tree model) case where the dividend is all of consumption and this implies r < g.

Thus, at some point the dividend becomes a large enough part of aggregate consumption that r < g and we find equilibrium at a finite price.

So, you are not correct to say that there is no equilibrium at a finite price.

Oh and btw Nick, your implicit assumptions to get r = g assume that the rate of time-preference in agent's utility funcitons is 0 (ie, discount factor is 1). This is NOT "very standard".

But it doesn't matter, if you have an asset either in fixed supply (like your example) or with constant returns to scale then you actually CAN'T have r >= g.

"Those assumptions about elasticities, interest rates, growth rates, don't seem totally implausible."

Excuse me? You're serious?

In addition to the assumption of no time discounting in utlity, you've got capital not depreciating and apparently labour supplied completely inelasticly regardless the real wage.

In particular, the infinite asset price means nobody wants the non-ocean front land consumption goods, (infinite willingness to forgoe other consumption) which makes the real wage zero. Yet people still work to produce the other consumption goods?

YOU ARE ALSO ASSUMING THAT BUILDING MORE CAPITAL IS IMPOSSIBLE. Otherwise the owners of the ocean front land would want to invest to increase their future consumption, thus driving r below g.

All this is plausible to you?

Actually Nick, I think you can scratch what I just said. All the dynamics are a red-herring, this all coming from the no time discounting assumption.

There is no equilibrium here because the utility of the representative agent is infinite, since there is no maximum there is no intertemporal allocation. All assets have infinite price.

Adam P:

I still think my assumptions, applied to land, are roughly reasonable. No new land can be built, no depreciation, etc.

My assumption about zero time preference is debatable. I've always thought that time preference is a form of irrationality ("imperfect telescopic faculty"). But nevertheless, is it logically impossible for people to have zero time preference? What would happen if they did?

And what about the social planner's version of the problem? Should a social planner discount the utility of future generations? What shadow price, should a planner put on ocean-front land? (Would it be finite?) Or what Pigou tax if (say) there were some current activity that caused an externality that caused the loss of ocean-front land?

Also, what would be the effect of a growing population? Wouldn't this be like a falling supply of ocean-front land, and possibly cause land rents to rise faster than r+time preference?

As you can see, I'm still puzzling this one out. Is it logically impossible to have anything except an equilibrium with a finite value of all assets? Can we conceive of a case where there would be no finite upper bound? If not, what rules it out?


I don't see the confusion here:) The cost of an income stream has to be the cost of replicating that income stream. In your toy economy, you *must* have capital, not just space on the beach. Beach property does not produce anything. If you don't include capital, then there would be no goods in your economy and no production or consumption. Capital always has a finite price, because it itself is produced from capital and labor, and it is used up in the course of production.

So assume that people spend 1/4 of their income on rent, and you can rent your beach house to a factory owner. He will be diverting 1/4 of his income to you, and his income will be the dividends of the factory. So instead just buy 1/4 of his factory and get the same income stream. And it doesn't really matter who rents your house (except for their incomes), because regardless of what they do in the productive economy, or whether they get their income from labor or financial claims, you can purchase a share of whatever claims they hold, or purchase stock in whatever firm they work for, in enough proportion to replicate 1/4 of their income stream, and the cost of doing this must be the cost of the house. Are we agreed?

If so, then the price of the beach house must be equal to 1/4 of the price of the factory. And the price of the factory cannot be infinite. If factories become too expensive relative to the consumption goods they produce, then you can arbitrage by hiring away labor from operating factories to building factories, this decreases the supply of consumption goods and raises their price, and it also increases the supply of factories and lowers their price. In the same way, if the price of factories is too cheap relative to the price of consumption that they produce, then you can bid resources away from the production of factories and towards the production of consumption.

In none of this does time preference enter into the picture, as regardless of the *amount* of money you decide to invest, you have a choice of going into the capital production business (which is equivalent to shorting capital) or consumption production business (which is equivalent to going long capital), and you will arbitrage so that the profits of both industries are the same. So time preferences can affect the share of income devoted to final consumption, but they cannot shift the relative prices of capital and consumption. It is only in a lucas-tree type model where the stock of capital is unchanging, has no cost of production and indeed cannot be produced that high (long run) savings rates correspond to low (long run) yields. If you move away from this model and classify assets into space and capital, then price of space is set by the return on capital and an exogenous rentier factor.

RSJ: "So time preferences can affect the share of income devoted to final consumption, but they cannot shift the relative prices of capital and consumption."


Consider a 2 good model with no time. Here's the equilibrium condition:
MRS between apples and bananas = relative price of apples to bananas = MRT between apples and bananas.

Draw quantity of apples on one axis, quantity of bananas on the other, and an indifference curve and Production possibility frontier meeting at tangency with a budget line. Slope of I-curve is MRS, slope of PPF is MRT, and slope of budget line is relative price.

Only in a very special case of that model would the relative price be determined by MRT. That's when the PPF is a straight line, so its slope, MRT, is a constant. In that case, the supply curve of apples is perfectly elastic as a function of the relative price of apples and bananas. The supply curve (determined by technology) determines relative prices, and the downward-sloping demand curve (determined by preferences) determines the relative quantities of apples and bananas produced at that relative price. In the more general case, where the PPF is curved, MRT depends on quantities, the relative price is determined both by preferences and by technology.

Now take that same 2-good model, but make the first good "apples today" and the second good "apples next year". That's the Irving Fisher diagram. The equilibrium condition now becomes:
Intertemporal MRS = (1+r) = intertemporal MRT

Only in the very special case, where the intertemporal PPF is a straight line, does technology determine the rate of interest independent of preferences.

"Only in the very special case, where the intertemporal PPF is a straight line, does technology determine the rate of interest independent of preferences."

It is *always* a straight line, unless you are assuming deep irrationality on the part of investors in your model. A banana is a banana -- it is tasty. But capital is worth only it's return. So a unit of land that is twice as fertile is worth twice as much and needs to count as 2 units of less fertile land. The units of capital are by definition the return, so the line is straight.

Only if you get your units all screwed up is the line not straight, but screwing up the units assumes deep irrationality on the part of investors (e.g. that they would pay the same amount for more fertile land as for less fertile land, and would use square feet as the unit of measurement of capital, as opposed to output per square inch.).

No, sorry. it's not. That's the whole point of the Arrow-Debreu dated commodities approach. A banana today is a different good from a banana next year. Even if they are physically identical. They are not (generally) perfect substitutes, either in consumption or in production. The intertemporal indifference curve is not (generally) a straight line (MRS between present and future bananas varies because of diminishing MU of bananas). And the intertemporal PPF is not (generally) a staright line either (MRT varies because of diminishing MPK and increasing MC of producing new Investment).

The greater is current investment, the higher is the marginal cost in terms of foregone current consumption per additional capital good produced; and the higher will be the future stock of capital and so the lower will be the future marginal product of capital in terms of future consumption. For both those reasons, the trade-off in production between future consumption vs current consumption worsens as you increase investment. In equilibrium that trade-off must equal 1+r.

And in the limit, when the MC curve for producing new capital goods becomes perfectly inelastic, capital approaches land. In that case (if all assets are land) then the PPF between present and future consumption becomes reverse-L-shaped, so the rate of interest (and price of land) are determined solely by MRS (the slope of the intertemporal Indifference curve), i.e. preferences.


I am being bombarded with bananas, but am trying to talk about capital. Capital is not a commodity! Capital has transformational properties that determine its characteristics, rather than innate properties of commodities.

To assume that the returns on capital will predictably diminish is to require irrationality on the part of capital markets -- you must assume that each drop in the return on capital takes them by surprise, even though it keeps happening over and over again. So you have a choice between having forward looking capital markets and keeping the mathematical convenience of diminishing *aggregate* returns. You cannot have both.

Now the mechanism that keeps aggregate marginal returns constant is that capital is funded with long term debt commitments. Therefore whatever the current return on capital a firm has now, it has because it has committed itself to defending that return over the long term. As soon as the return falls, the firm will decrease its capital stock via voluntary liquidation or bankruptcy.

All firms have in the beginning increasing returns on capital up until the optimal firm size is reached, whereupon they have decreasing returns to capital. Rarely is the optimal size $1. When the firm is still in the increasing returns area, the market cap of the firm is increased far above its tangible value, effectively funding the firm's increase of capital. At some point the returns start to diminish and they fall below the overall required return, at which point any further fall in marginal product forces the company to liquidate a portion of its capital and return it to investors. At the end of the line, mature firms re-invest only enough to maintain their current capital stock.

If you have arbitrage-free forward looking capital markets, then you cannot say that the aggregate marginal product of capital has any predictable tendency to fall. It can have no predictable tendency other than to remain at whatever rate it is now. And what this means is that in the goods market, there will be productive factors left unutilized, if their physical transformation characteristics do not exceed the required (financial) return. As a result, the goods market cannot be perfectly elastic if the financial market is, which means that competitive capital markets require monopolistic competition in the goods market.

Only in the corner case of zero interest rates can you assume either perfect competition for goods or diminishing marginal returns for capital. But this corner case corresponds to a malthusian no-growth economy (e.g. all capital is like land). In that case, military force or heredity will determine who owns capital, as price is undefined.


1. Diminishing returns refers first and foremost to capital goods, rather than capital in the sense of the value of those goods. Given an increasing supply price of capital goods, diminishing returns to capital in the latter sense would be even stronger.

2. Diminishing returns does not refer to what is happening to returns over time. It refers to what would happen to returns, at the margin, if the stock of capital today were higher today than it in fact is, other things equal.

3. Profit-maximising firms always operate at a point where returns to capital are diminishing. Same for labour and land. If marginal revenue product never diminished, the amount of capital (or labour or land) they want to hire would be either zero or infinite.


You aggregate based on relative prices. You do not aggregate pianos and cherries by volume. You aggregate national output by adding up their price, and then discounting by some "general price index". That is also how economic agents interact. Assume each firm only uses a single variety of capital. Then you indeed have (asymptotically) diminishing returns for that firm. But new firms appear to offset the old firms, so that for the aggregate economy, returns do not diminish when the size of the aggregate capital stock is increased -- because the flat-lined industries are not the targets of the capital increase. The new, growing industries are the targets of the capital increase. The composition of the capital basket changes so that firms with lower returns are cheaper and shrink in the representation of a "unit of capital" that the investment dollar buys.

If you look only at the individual firm, then you don't care that rising per unit input costs are someone else's rising per unit output revenue. Neither do you care that rising cost of investment for that firm corresponds to falling cost of investment for the rest of the economy. But in the aggregate, the whole economy is not a single "firm". It is an endless stream of new forms of capital and new commodities. First you pick fruit, then you spin cotton, then you write software. It is not the case that the economy as whole just picks fruit. As the yields from picking fruit begin to decline, an increase in capital for the economy as a whole is not routed to fruit pickers, but to cotton spinners. It goes on and on. So when "K" is some aggregated form of capital, a general increase in "K" has constant return.

2. Yes, it is the partial derivate with respect to "K". But K is a market-weighted basket representing incomparable forms of capital that change with time. So you cannot generalize the dynamics from a specific firm to the economy as a whole.

3. No, because it takes time to increase capital. Let r(C) be the return on capital for a firm. Then suppose r(C) = (.1)C/(1+c^2). Returns increase up until C = 1. Look Ma, no infinities! :) Btw, r(C) will *always* be zero when C is zero, and it will be zero for small values of C, because there are sunk costs before any revenue is realized.

So returns increase from 0 reach some maximum and then decrease thereafter. If the market rate is 3%, then when C = 3, the firm will hit the market return (from above) and it will not expand C any further.

If you were to try to invest in C when the firm is in consol state, the firm would return that capital to you, because it would not be able to deliver the market return. You cannot force a firm to incur a liability that it cannot repay. That means you need to look for some other firm to invest in -- that form of capital is no longer available to be purchased.

What happens with the market price? Suppose that you believe in 5 years, the firm will hit C = 3. That means that in 5 years, the firm will become a consol paying 3% return on $3 of capital. So the market price today is the present value of those consols i.e. (3/(1.03)^5). That means that you, as the investor do not make more or less money by investing in the firm now that at any future point, or between investing in fast and slow growth firms. This is the only way to aggregate capital, and as result, by definition the marginal product is constant.

What "funds" the expansion of the firm is the difference between the market price and C. As C goes up, that difference falls, as does the capacity to invest in the firm. If you assume that investment can occur at *infinite* speed, then you would assume that all firms are market consols paying the same constant yield, and investment requires the creation of new firms.

If you do not assume infinite growth rates of individual firms, then as C approaches 3 (from below), the P/E multiple of the firm falls, and that means that the P/E multiple of the rest of the market increases, (assuming that the market yield is constant). This can only mean that some other firm is experiencing increasing (average) returns, or a new firm has arrived (one that would not have arrived if multiples were lower).

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