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An Option Value Problem from Seinfeld

This is a paper about nothing. (JEL C61, D91, G11) I. 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 large supply, but it is finite. So she must "re-evaluate 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. Therefore using 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. II. 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 her time 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 closely questioning 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 [beta]. All these assumptions are to simplify the calculations; the method is perfectly general and many bells and whistles can be added to the 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, [Q.sub.m], such that her decision will be yes if Q > [Q.sub.m]. 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 [V.sub.m] 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 payoff Q and has continuation value [V.sub.m - 1] on the second day, which has present value [beta][V.sub.m - 1]. If she decides not to, there is no immediate payoff, only the present value of continuation with m sponges, namely [beta][V.sub.m]. Therefore her decision rule is (1) Spongeworthy if Q + [beta][V.sub.m - 1] > [beta][V.sub.m] that is, if Q > [Q.sub.m] [equivalent to] [beta]([V.sub.m] - [V.sub.m - 1]), and the value recursion formula of dynamic programming is (2) [V.sub.m] = E[max{Q + [beta][V.sub.m - 1], [beta][V.sub.m]}]. Although Equation (1) gives an expression for the spongeworthiness threshold, it is not a full solution because the ([V.sub.m] - [V.sub.m - 1]) that appears on its right-hand side is endogenous. We must obtain the solution using Equation (2). The expectation on the right-hand side of Equation (2) is found by integrating over the range of Q, which splits into two parts, depending on which of the two expressions yields the max. Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. …