To the Editor Journal of Economic Perspectives If Letters were titled, the title for this one might be "Is Some Better Than None? A Coda on Three Notes and Vice Versa". The first puzzle presented in the puzzle section of the first issue of the Journal of Economic Perspectives (Nalebuff, 1987) was as follows: Puzzle 1: Turn Out the Lights In the event of a temporary shortage of electricity, standard practice is to ration all customers equally, that is, to reduce each customer's available electricity by the same proportion. This practice results in the familiar summer brown-outs. Imagine that priorities for electricity could be bought and sold. A market could be set up so that all payments are made as transfers among the users; the money collected for obtaining a high priority would be paid out to those who are willing to take a lower priority. Since the payments are made as transfers, the electric company generates no revenue from the sale of priorities. In the event of a shortage, the person with lowest priority would be cut off first. If this rationing is insufficient, then the next lowest priority would be cut off and so on up the priority scale. It seems intuitive that those who most value uninterrupted electrical supply and those who lease value an uninterrupted supply are both made better off as a result of the marketing of priorities. Hospitals would be willing to pay a premium for an uninterrupted electrical supply while struggling artists would be willing to endure more frequent power outages in return for compensation. But what about the consumers in the middle? Show that under the competitive equilibrium prices for priority, everyone is at least as well off as they were before the market was created.(pp. 186) And the answer given was this: Answer to Puzzle 1 Since all payments are transfers, the average price for a priority level must be zero. What does the average price buy? It buys the average bundle. Given N priority levels, a consumer could buy 1/N of each priority level at a total cost of zero. The interpretation of owning 1/N of a priority level is that 1/N of a consumer's electricity is supplied at that priority. In the event of a shortage which requires, say, a 20 percent rationing, the lowest 20 percent of priorities would be cut off. The consumer with 1/N of each priority would then receive 80 percent of his demands. Note that this outcome is precisely what would have occurred before the market for priorities was established. Since each consumer has the option of costlessly maintaining the status quo, the market equilibrium can only improve everyone's welfare. This result would have been obvious if the question had been posed slightly differently. Imagine that everyone is given equal endowment of 1/N of each of the N priority levels. A market is opened and trade is allowed. In the resulting competitive equilibrium, no one can be worse off than at their initial endowment; they don't have to trade! The trick in this question is to recognize the equivalence between equal endowments and the requirement that aggregate of the payments for priorities be zero.(p. 190) This puzzle might better have appeared in the Anomalies section of JEP, as I shall try to show. Note I: A Solution Editor Barry Nalebuff apparently assumed that after perusing the answer the reader will completely understand the problem. This assumes too much, I believe. If the problem were so obvious, even after being given the answer, it would not constitute enough of a puzzle to inaugurate the section and JEP, and would not be a basis for a seminar at Stanford. This is not to say that I disagree with the answer. Indeed, a fundamental intuition of economists - that to be given a choice is not likely to worsen a person's (economic) situation - supports that answer. But there is considerably more to be said about the matter. And I believe that the crucial issue is whether and when some of a good, and some knowledge, should be considered better than none. This problem is related to a large class of situations wherein the public and/or lawmakers act as if the making of a market is not a good thing, and therefore the activity is prohibited by law or custom; examples include ticket scalping, the sale of human organs, paying someone else to serve in the army in one's place (e.g. in the Civil War), having someone stand in line to receive a valuable that will be rationed by waiting (such as a preferred class of gym lockers, or a place in a magnet school), auctioning off immigration visas (though maybe this case is a bit different), and paying another professor to take one's mandated place in a university graduation ceremony. The existence of these prohibitions suggests that the logic of the answer as given in the article - purporting to show that all are made better off by the existence of a market - is not compelling to many persons. Indeed, the opposite conclusion is so compelling that for twelve years the airline volunteer scheme for oversales was not even considered an imaginable alternative by almost anyone - though one week's experience proved that it was indeed a Pareto improvement (Simon, 1994). Consider the electricity-rationing problem in the specific terms in which it is presented. Is there a rigorous proof of the answer? I doubt whether the answer as given above would constitute a proof that would meet the test of a logician; it is too sketchy for that. And I wonder whether such a proof has been given elsewhere; the article does not refer to one. Indeed, I do not know of a rigorous proof for even the far-simpler analogous problems mentioned above where there is no uncertainty. In the absence of an analytic proof, here is a numerical illustration to support the answer given. First, let us shift from electricity to water; electricity can be rationed either by reducing the voltage or by shutting off service, and the former is more complex; in contrast, water is usually rationed in quantity (or by water hours, which is intended to ration by quantity). And simplify by assuming only two users A and B, and only two levels of supply - 1 unit or none. Then: 1. Assume dollar amounts that A and B would be willing to pay to ensure receiving the unit of water if there is a stated (say, .5) probability of receiving that unit (B receiving the unit if A does not, and vice versa). The dollar amounts must differ between the users; otherwise, a market will not function. 2. Assign the unit of water to the higher bidder, and transfer half the difference between the stated dollar amounts from the higher bidder to the lower bidder. 3. Since both users have attained a situation preferable to the alternative of having a chance of not having the unit of water and no money being transferred, and each has either paid or been paid a more desirable sum than the person bid, each may be said to have been improved economically by the existence of the market. 4. The illustration immediately generalizes to four persons or to any other even-numbered set of persons by imagining each trade taking place between any person below and any person above the mid-point. And it generalizes to odd-numbered sets by imagining that the middle person remains as before, and therefore is not worsened even if not bettered. The rationing puzzle connects with a puzzle in the Summer, 1988, issue: Puzzle 5: Are There Pareto Improvements? Think of a feasible change in the U.S. economy that does not involve payment of compensation and that would be a Pareto improvement. (Farrell, 1988, p. 178) Answer to Puzzle 5 I don't have an answer to this one. Perhaps if there were such a change it would be made right away, so that there can never be an answer (just as there can't be a $20 bill on the sidewalk). More likely, the difficulty of the problem reflects the fact that every change affects people in many different ways (p. 181). If one restricts consideration only to the economic effects (thereby excluding the annoyance of some persons to see a system change that they had resisted on the grounds that it could not work), and excluding second and lower-order effects, then the volunteer auction system for airline oversales has indeed proven to be a Pareto improvement (effects on the railroads and buses being considered second-order effects). Allowing a market in tickets when a person has no ticket for a flight that is sold out and would like to make an offer to a ticket-holder is another such Pareto improvement, I think. Indeed, all the other schemes mentioned above - and many or most creations of new markets where there are none - are likely to be Pareto improvements. (I assume that the phrase "involve payment of compensation" does not exclude such cases.) Note II: Is None Ever Preferable to Some? So we have an understandable argument - even if it does not constitute a proof - why the making of a market improves the persons involved in the puzzle under discussion. And this may be the end of the story for the economist, who may say that this is all s/he is interested in. But economists are in some cases interested in more than the welfare of individuals, as witness the body of work on concentration in income distributions. And for others than economists this is certainly not the end of the story; improving the welfare of all parties does not always constitute a perceived improvement; this is shown by many children preferring that the self and a companion each have two cookies to the self having three cookies and the other having four. The prohibitions of markets referred to earlier also show that there is more to the story. Additionally, it is easy to imagine participants in the water-rationing program regretting after the fact that they made the deals that they did - the purchaser of the unit when there turns out to be no need for rationing, and the seller when rationing (and cessation of service to that person) does occur. Hence some more of a good for some is not necessarily better than no more for anyone. This is one of the points at which economics and psychology diverge. The economist has an "answer" to the problem, as stated in JEP; but the psychologist will not be surprised if the participants or the society reject the market anyway. And indeed, some philosophers might erect a justification for the society doing so. In some cases the justification may be easy to arrive at. Should people be allowed to sell rights that might lead to the ends of their own lives? Should someone be allowed to sell units of electricity in a rationing scheme if the person is on an electrically-operated life-support device, even if the probability of the rationing coming about is small? All this is reminiscent of Duesenberry's famous statement that economics is about how people make choices, and sociology is about why people have no choices to make. Note III: Can Economists Prefer None to Some? Now a story within a story. Economists also sometimes reject some in favor of none. This is the case with respect to some kinds of knowledge. For example, even if there is no analytic proof of the answer to the puzzle at hand, some economists will reject the above argument by illustration as being worse than no argument at all because it is not comprehensive. Two other examples in fields in which I have worked: 1) Simulation of competition in duopolistic and triopolistic markets can produce answers to questions that cannot be produced with analytic game theory or otherwise, e.g. the effect of a change in the cost of capital. 2) Monte Carlo simulation produces answers to problems in probability and statistics that cannot be understood analytically, or where even skilled probabilists get wrong answers when they first resort to analytic methods, e.g. the Monty Hall three-door problem. Yet journals often refuse to publish the simulation work on the grounds that the results do not constitute complete proofs for all possible cases. In other words, the judgment is made that knowledge of some (perhaps most) cases is no more valuable than knowledge of no cases at all. So we return to the question posed at the beginning of this Letter: Is some really better than none in all cases? REFERENCES Farrell, Joseph, "Puzzles: Sylvia, Ice Cream and More", The Journal of Economic Perspectives, Vol 2, #3, Summer, 1988, 175- 182. Nalebuff, Barry, "Economic Puzzles: Noisy Prisoners, Manhattan Locations, and More", in The Journal of Economic Perspectives, Vol. 1, Number 1, Summer 1987, pp. 186 ff. Simon, Julian L., "Origins of the Airline Oversales Auction System", Regulation, 1994, Number 2, pp. 48-52.