> --- "Robert A. Book" <[EMAIL PROTECTED]> > > 5 cars come at 1pm and SIX cars at 2pm. > > During that time, charge just high enough so that all > who want to park, can. The last car in does get a > space, if he is willing to pay. > > > between. But there's another possible outcome -- > > everyone races to be > > the 9th to arrive and get the last "free" space. > > Everybody knows the price will go up at a particular > time. Every car there will pay the price. The spaces > are no longer free.
Maybe I wasn't clear on your assumptions. Are you assuming that at a time X1, they price goes up to $Y per minute, and everybody whose car is on the lot at that time pays $Y per minute until either they leave, or the price goes down at time X2? And that all of X1, X2, and Y are known in advance with certainty? > > I think you are > > (inadvertently) assuming that there is some > > non-price way of allocating spaces taht is superior > > to an allocation with prices. > > No, that is not a correct inference. Well, I didn't think you'd assume that on purpose! ;-) > > Is the marginal consumer the last to get a > > space, or the first to be turned away? > > It is the next one to get a space. Nobody gets turned > away, because when the lot is full, there is a > positive price. Part of the problem here is that the number of spaces has to be an integer. This means that the "marginal" consumer, the "marginal" value, etc., are (strictly speaking) undefined. These concepts require continuity. Let me ask you this: When the lot is full, the price is high enough that nobody wants to pay for a space -- but at some point a car leaves, not because the price has risen, but because somebody want to go home and therefore that person's value for the space has now dropped to zero. So there is one empty space. Are you assuming that (a) the space stays empty for a while and everybody still in the lot pays zero for that time, or (b) that the price drops a bit, and somebody else immediately takes that space? If the answer is (b), then you are implicitly assuming that the parking spaces are discrete, but there is a continuous space of consumers. Which is sort of the opposite of what is normally assumed. (Aumann used a continuous space of consumers in a game theory problem, but I don't think it applies here.) This doesn't make it wrong, but this might be why our usual instincts are not working. In real life, I think uncertainty is an inherent part of the parking problem. Assume that consumers arrive (and perhaps leave) according to a Poisson Process, with a parameter (mean number of consumers per hour) that varies throughout the day. Assume that conditional on arrival, each consumer has a valuation drawn from some distribution (perhaps also varying throughout the day; perhaps the valuations are on average higher when more consumers are arriving per hour). Then you ought to be able to find a price that maximizes the EXPECTED VALUE of consumer surplus plus revenue (i.e., maximizes efficiency), subject to the constraint that the lot never has more cars than spaces. (You could then also calculate the optimal number of parking spaces, based on the price of constructing a space, by maximizing the discounted net present value of CS+Revenue-Costs, perhaps subject to a constraint that the lot is full no more than X% of the time.) If you do this, I expect you will find that the price is sometimes zero (at periods of low demand), but there will probably be times when the price is positive but there is some probability of spaces being available -- and there is some probability of people being turned away. But these probabilities will be minimized. --Robert Book
