Monday, January 20, 2014

More on Rational Agents and Irrational Markets: A Wonkish Response to Andy Harless

In a comment on my most recent blog post, Andy Harless "[wishes he] had a better intuition for what is going on in [my] model." I took a stab at responding to Andy in the comment section, but my response became so long that I turned it into a post. Here is my answer to Andy. You can find additional comments over at Economist's View where Mark Thoma was kind enough to post an excerpt.

The model Andy is talking about (here) describes an endowment economy with no production. There are two types of people; patient and impatient. Impatient people have a higher rate of time preference than impatient people and, as a consequence, one group will become lenders and the other group will become borrowers. Both types die in any given period, with the same age-independent probability. They are replaced by new people so that the population of each type is stationary and there is an exponential age distribution for each type.

All asset market transactions occur through a zero profit financial intermediary. There are complete annuities markets and when a person dies, his wealth is returned to the financial intermediary. If a person borrows from the intermediary, he is required to take out life insurance.

In the special example in this post, there is no fundamental uncertainty. I will also assume, in this post, that there are only two shocks. Because of these special assumptions, I will need only two assets to complete the markets. These assets are short-term bonds and long-term bonds. The more general case, where long-term bonds and equity are different assets, is covered in the paper.

Short term bonds represent a claim to one unit of the endowment next period. Long term bonds represent a claim to one unit of the endowment in every period. The price of a long term bond is the same as the value of a new born agent’s human wealth because human wealth and long-bonds are claims to the same income streams.

When they are born, lenders sell a portion of their human wealth to borrowers: in return, they buy short-term debt. Lenders start out life as savers and in the early years of their life they consume less than their endowment in every period.

Borrowers, in contrast, sell sell short-term debt to lenders. In the early years of their life they consume more than their endowment in every period.

As lenders age, eventually they reap the benefits of their youthful choices and they begin to spend the interest on their asset portfolios. Lenders have an increasing consumption profile over time.

As borrowers age, eventually they reap the harvest of their youthful indiscretion and they begin to pay back the interest on their debts. Borrowers have a decreasing consumption profile over time.

So far so good. But what about uncertainty?

The trades I described above imply that a lender shorts long-term debt and goes long in short-term debt. A borrower takes the opposite side of these trades.

A long bond issued in period t is a claim to one unit of the endowment next period PLUS a claim to a long bond in period t+1. Just as in Keynes’ beauty contest example, the price of a long bond next period is worth what the market thinks it is worth. In the paper, we assume that there is a complete set of markets that are indexed to an observable ‘sunspot’ variable. That is simply a short cut for bringing ‘animal spirits’ or market sentiment into the pricing equation. The model displays what George Soros calls ‘reflexivity’.

Our model has many equilibria where a long-bond is worth exactly what the market thinks it is worth. If people think that long bonds are worth a lot; they WILL BE worth a lot. In that case, there will be a resource transfer from lenders to the new born agents and the borrowers. If people think that long bonds are worth less; they WILL BE worth less. In that case, there will be a resource transfer from the new born agents and the borrowers to the lenders.

Our model differs from a representative agent economy because, although borrowers and lenders each make trades that obey a transversality condition, there is no analog of these transversality conditions for the market as a whole. It is this feature that distinguishes our work from most other models. The trades that occur in our model are equilibrium trades in the sense in which that term is used in standard DSGE models.  But unlike most existing models; these trades are NOT Pareto optimal. We think that this is a useful way to understand why financial crises are so painful.

Why are equilibria not Pareto optimal? The answer comes from our assumption that demographics limit participation. If the unborn could participate in the asset markets that occur before they are born, they would eliminate the inefficient sunspot fluctuations. Because everyone is assumed to dislike risk, in the absence of prenatal financial markets; everyone is worse off. The model captures a lot of what appears to have happened in the financial markets, within a framework that is very neoclassical in its structure. For me, that is a virtue.


  1. I'm not getting this yet Roger.

    Does each agent get one unit of endowment each period he is alive?

    "In the special example in this post, there is no fundamental uncertainty. I will also assume, in this post, that there are only two shocks."

    Does that mean there are two different types of sunspots? (a sunspot and a moonspot?)

    I don't see why there would be two types of bonds. If there were only one-period bonds in the model, would the effect of sunspots disappear?

    1. The only thing my intuition can come up with (which I think is the same as what Andy Harless is saying) is that substitution and income (wealth) effects of interest rate changes must be exactly cancelling out somehow, so you get multiple equilibria?

      Since the perpetuities live longer than the agents, a fall in perpetuity prices would make the patient agents less wealthy, and the impatient agents more wealthy, which might (given the right preferences) increase demand for goods and reduce demand for perpetuities, so the equilibrium is unstable. But one-period bonds would not have this wealth effect. With exactly the right mix of perpetuities and one-period bonds, the equilibrium would be on the border between stable and unstable, so you get multiple equilibria.

      Yes, I *think* that is what is going on.

    2. With both patent and impatient agents, the equilibrium interest rate will be a weighted average of the rates of time preference of patient and impatient agents. But those weight depend on the relative wealth of the patient and impatient agents. And their relative wealth (if the bonds are longer-lived than the agents) will depend on the rate of interest.

      With exactly the right duration of the bonds, relative to the lives of the agents, and their preferences, an increase in interest rates will have zero effect on the demand for bonds.

      By having a mix of one-period bonds and perpetuities in the model, you can get the right average duration of bonds to make this happen.

      But why would that exact mix of one-period bonds and perpetuities be an *equilibrium* feature of the model?

      (I haven't read your paper. Because, as you remember from the olden days, i never could do math, and always had to try to figure things out my own way!)

    3. I *think*, that if risk-averse agents could *choose* the duration of the bonds they buy and sell, they would choose bonds with the same duration as their own expected lives, in order to eliminate the uncertainty of those wealth effects.

      Plus, those bonds would have exactly the right time-profile of dividends to deliver agents their desired time-profile of consumption. Each agent would sell all his wealth to the bank, then buy a life annuity with exactly the right rising (for patient agents) or falling (for impatient agents) time profile of consumption. This eliminates uncertainty from future interest rates. And I *think* it eliminates sunspot equilibria.

    4. Apologies Nick for my delayed response -- I was traveling and am only now finding time to blog again. Your conjecture is interesting - but not quite right. Duration doesn't matter because the model has a complete set of Arrow securities. Each individual agent faces a single lifetime budget constraint. The multiplicity arises from the fact that newborn agents are unable to insure against the states of nature into which they are born.

  2. Roger, maybe it's just me but your results seem to have some similarity to the rational bubble in Townsend's turnpike model of money and other models assets that have a role of relaxing credit/liquidity constraints. The question then becomes quantitative. What proportion of asset price volatility can be explained under rational expectations, and what proportion comes from things like overextrapolative expectations and other forms of misperceptions(David Laibson and co's papers on natural expectations have a simple model that goes quite far on this). Realistically, both types of explanations play an important role (even though an economic theorist would say the rational bubble story is more elegant in some way- but empirically that's not a sufficient criterion, you have to allow for deviations from RE and ideally come up with models that allow you to quantitatively disentangle the effects).

    1. Daniels: Thats a fair comment. My research agenda involves progressing as far as possible under the rational agent assumption. If at some point, I hit a macro fact that cannot be explained with that assumption, I will branch out and explore non von-Neuman-Morgenstern preferences. I'm not read to go there yet.


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