Hyperbolic Disasters

From an address by Ray Rees

For the RES Newsletter, January 2012, Online Issue 1

In November 2011 the CESifo Prize Fellowship was awarded to Professor Sir Partha Dasgupta, a past-president of the Society at Spatenhaus, Munich. This is an edited version of the address given by Ray Rees to mark the occasion. A mathematical appendix to this article is available from the author on request (ray.rees@lrz.uni-muenchen.de)

Sir Partha Dasgupta has made many important contributions to the literature of economics, in pure theory as well as applied economics and the economics of public policy. One of his most interesting recent contributions is the 2005 AER paper he wrote with Nobel Laureate Eric Maskin on ‘Uncertainty and Hyperbolic Discounting’, and it is the idea in this paper that I want to present to you tonight.

Hyperbolic discounting is a method for expressing how people value sums of money that will be received at some time in the future, for example the payoff to an investment. This method is very different from the standard way we do that in economics, commonly known as ‘exponential discounting’, and to many economists hyperbolic discounting seems to represent irrational behaviour. In fact it’s often associated with addictive or extremely short-sighted behaviour, a strong preference for immediate as compared to future satisfaction. It also gives rise to ‘time inconsistency’, for example, as when I decide in January that I’m going to save up to buy my wife a fur coat for Christmas but then in July spend the money on a ticket for the World Cup.

The contribution of the Dasgupta/Maskin paper is to show that the behaviour modeled by hyperbolic discounting can in fact be rational in some situations. For example, suppose that you have the choice between an investment A, which gives a net return of 1000 euros no later than in 3 months’ time, and an investment B, which returns 1200 euros no later than in 12 months’ time. Suppose also that there is some probability that the payoffs could arise at any time within the next 3 months, so there is a chance that the payoff from B could in fact occur sooner than the payoff from A. The paper shows that the investor could decide initially that she prefers B, because of the chance that it could yield its return sooner than A. But if, say, after, two months, that has not yet happened, it is perfectly rational for her to switch preferences and say that she now prefers investment A, because the chance that B will materialize in the time remaining is now too small.

As you might expect, Dasgupta and Maskin’s proof of this is both extremely elegant and mathematically sophisticated, so to bring the importance of their result home to you I’d like to suggest a simple concrete example. Suppose investment A takes the form of buying shares in a company that installs ash trays on the handlebars of Harley-Davidson motorbikes. And investment B is to buy 5-year Greek bonds. Then Dasgupta and Maskin have proved that it is perfectly rational to invest in a business installing ashtrays on motorbikes in preference to buying Greek bonds. I think that makes their result extremely plausible.
However, to illustrate the possible dangers inherent in this kind of rationality I would like to tell you the following story. Three friends were on holiday on a cruise ship, cruising off the coast of China. One was an investment banker, one an economic theorist, and the third a moral philosopher. All had studied at Cambridge under Professor Dasgupta and indeed the banker, a tall, well-built man, was also a Cambridge rowing blue. In the dead of night the cruise liner was attacked by pirates, but in the darkness the three were able to escape in a small boat. At daybreak they discovered two unfortunate things: first, that there were only 3 litres of water and 3 kilograms of bread in the boat; and second, that there was a small but steady stream of seawater coming through a leak in the boat just below the water line.

The economist suggested the following division of labour: he and the philosopher would bail out the water that was seeping into the boat, which they had to do with their bare, cupped hands. The banker would row the boat in a direction to be decided — he was after all too big to bail. The others readily agreed.

They also agreed to divide the food and water equally between them right away, at the outset — 1 litre of water and 1 kilo of bread each — but since they were all believers in methodological individualism, they agreed that each would have complete control over the way in which he consumed this, i.e. each was free to choose his own consumption path.

The crucial decision was that concerning the direction in which to row. There were two possibilities. Possibility A was that they could row in the direction of the Chinese mainland, which they estimated they could reach in 2 days. The problem was that they didn’t know what would await them there and how they would be able to find a ship to take them home. Possibility B was that they could row in exactly the opposite direction, out to the main shipping lanes, where they would then be certain to be picked up by a passing ship and their rescue would be complete. But this would take them four days. They knew that there was a possibility that if they chose A, a wind might spring up and propel them more quickly to the shore, while if they chose B, there was a chance that they could be picked up at any time by a ship that had deviated from the shipping lanes, and so they might be rescued earlier than expected. They decided unanimously to choose B and to head for the shipping lanes.

I think this fits the setup in the Dasgupta/Maskin paper very well. I should also tell you that the banker was a (quasi-)hyperbolic discounter, the economist an exponential discounter, and the moral philosopher did not believe in discounting the future at all. He believed it was, well, immoral.

Four days later, the boat was found drifting close to the coast by a Chinese fishing junk. The banker and the economist were lying unconscious at the bottom of the boat, the philosopher was madly bailing water out of the boat, and all the food and water was gone.

At this point it occurs to me to ask Professor Dasgupta if he is able to apply his model to explain this empirical outcome. However given that he has already presented a paper this evening it seems fairer to give the story told by the philosopher. I tell it in his own words.

‘Well, we set off toward the shipping lanes and since it may have taken 4 days, I resolved to consume one quarter of my bread and water each day. As we rowed along, I noticed to my dismay that my friend the banker had consumed one-half of his bread and water on the first day and my friend the economist just over one-third of his, but since I am a believer in an individual’s right to decide I said nothing.

‘Toward the end of the second day, though we were making good progress, I noticed that the banker had now consumed just over one-half of his remaining bread and water, and the economist just under one-third of his. But even more worryingly, the boat had turned completely around and was now heading back towards the coast. When I voiced my concern I was told by my economist friend to shut up because I was not behaving rationally, whereas our banker friend was, given his preferences and probability beliefs. I obviously hadn’t read the Dasgupta and Maskin paper.

‘Then, on the third day, disaster struck. The banker again consumed over one-half of his remaining bread and water and the economist more than a quarter of his, but a few hours later the banker suddenly collapsed. He had exhausted his reserves of energy and could no longer row the boat. The economist took over the rowing, but after some hours he too collapsed and couldn’t row any further. The boat was now simply drifting, at the mercy of the currents, while I kept on bailing. On the fourth day we barely had the strength to consume the last of our bread and water, but then, with no bread and water left, we were lucky enough to be sighted by the Chinese fishing junk and rescued.

‘I have to say however that this terrible experience has brought the three of us much closer together and has greatly strengthened our bonds of friendship.’

Shortly afterwards the Wall St. Journal got hold of this story andpublished it under a headline that nicely summarizes it:

"Chinese Junk Bonds Survivors of Bail-Out Rescue after Banker's Collapse"

From issue no. 156, January 2012, pp.12-13

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