An Option Value Problem From Seinfeld

An Option Value Problem from Seinfeld
Avinash Dixit∗ Forthcoming, Economic Inquiry January 26, 2011

Abstract This is a paper about nothing.

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Introduction

In an episode of the sitcom Seinfeld (season 7, episode 9, original air date December 7, 1995), Elaine Benes uses a contraceptive sponge that gets taken off the market. She scours pharmacies in the neighborhood to stock a largesupply, but it is finite. So she must “reevaluate her whole screening process.” Every time she dates a new man, which happens very frequently, she has to consider a new issue: Is he “spongeworthy”? The purpose of this article is to quantify this concept of spongeworthiness. When Elaine uses up a sponge, she is giving up the option to have it available when an even better man comes along. Thereforeusing the sponge amounts to exercising a real option to wait, and spongeworthiness is an option value. It can be calculated using standard option-pricing techniques. However, unlike the standard theory of financial or many real options, there are no complete markets and no replicating portfolios. Stochastic dynamic programming methods must be used.
Department of Economics, Princeton University,Princeton, NJ 08544-1021. Fax 609-258-6419, E-mail dixitak@princeton.edu. I thank Ricardo Guzm´n for his abstract suggestion, and Economic Inquiry co-editor a Yoram Bauman for many expository improvements. JEL Categories: C61, D91, G11.
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The model

Suppose Elaine believes herself to be infinitely lived; this is a good approximation in relation to the number of sponges she has and hertime-discount factor or impatience. She meets a new man every day. Define the quality Q of each man as Elaine’s utility from having sex with him. This is independently and identically distributed, and drawn each day from a distribution which I assume to be uniform over [0,1]. Each day she observes the Q of that day’s date. Actually this is only her estimate formed from observing and closelyquestioning the man (which is what she does in the episode), not the ex post facto outcome. But I assume that she has sufficient experience and expertise to make a very accurate estimate. Having observed Q, she makes her yes/no decision. Elaine’s per-day discount factor is β. All these assumptions are to simplify the calculations; the method is perfectly general and many bells and whistles can be added tothe analysis. I mention a few of them at the end. If sponges were freely available for purchase at a constant price (which is small in relation to the potential value so I will ignore it), then Elaine’s decision would be yes for any quality greater than zero. But when she has a finite stock and cannot buy any more, her optimal decision will be based on a “spongeworthiness threshold” of quality, Qm ,such that her decision will be yes if Q > Qm . The threshold depends on the number m of sponges she has: the fewer sponges left, the higher the threshold needed to justify using up one of them. Let Vm denote Elaine’s expected present value of utility when she has a stock of m sponges. She meets a man and observes his quality Q. If she decides to use one of her sponges, she gets the immediate payoffQ and has continuation value Vm−1 on the second day, which has present value β Vm−1 . If she decides not to, there is no immediate payoff, only the present value of continuation with m sponges, namely β Vm . Therefore her decision rule is Spongeworthy if that is, if Q + β Vm−1 > β Vm Q > Qm ≡ β (Vm − Vm−1 ) , (1)

and the value recursion formula of dynamic programming is Vm = E [ max{ Q + β Vm−1, β Vm } ] . 2 (2)

Although (1) gives an expression for the spongeworthiness threshold, it is not a full solution since the (Vm − Vm−1 ) that appears on its right hand side is endogenous. We must obtain the solution using (2). The expectation on the right hand side of (2) is found by integrating over the range of Q, which splits into two parts, depending on which of the two expressions...