Could there ever be conditions under which the equilibrium (real) prices for some assets are infinite? What would happen to an economy as it approached those conditions? Would those prices keep climbing to the skies heavens, then collapsing in waves of fear and panic? I'm trying to figure it out.
Start with an economy with technology and total factor productivity growing at rate g. Assume everything else (population, capital, etc.) is constant over time, so the growth rate of output, income, and consumption is also g. Assume intertemporal consumption preferences such that the equilibrium real rate of interest, r, is the same as g.
So far very standard.
Now add in an asset in fixed supply. Ocean-front land, maybe. There is a demand to rent ocean-front land. Assume that rental demand has a price elasticity, and income elasticity, both equal to one. That means total annual rents on ocean-front land are a constant fraction of real income. Given the fixed supply, that means rents per acre (or per metre of ocean-front) must be rising in real terms at rate g. The present value of the rents on one metre of ocean-front are therefore equal to (today's rent)/(r-g), which is infinite. If investors are rational, and there are no bubbles, either positive or negative, so prices reflect fundamental values, the equilibrium price of ocean-front land should equal the present value of the rents. So the equilibrium price should be infinite.
And if we assume instead that the income elasticity of demand exceeds one, rents per metre will be rising faster than g, and faster than r. So the equilibrium price of ocean-front land should exceed infinity (yes, I know).
Those assumptions about elasticities, interest rates, growth rates, don't seem totally implausible.
What have I got wrong?
Worthwhile Canadian Initiative 
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.
Nick,
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? 🙂
Nick,
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!
Don
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.
Summarize:
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.
Doc:
(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
Depreciation
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.
Nick,
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
cMU(c) < beta(1+g)cMU((1+g)c)
Note cMU(c) is a rough approximation to the agent’s flow utility from consuming c today, while beta(1+g)cMU((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??
Econ_grad_student:
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.