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UP OR DOWN? A MALE ECONOMIST’S MANIFESTO ON THE TOILET SEAT ETIQUETTE

JAY P. CHOI∗

This paper develops an economic analysis of the toilet seat etiquette. I investigate whether there is any efficiency justification for the presumption that men should leave the toilet seat down after use. I find that the “down rule” is inefficient unless there is a large asymmetry in the inconvenience costs of shifting the position of the toilet seat across genders. I show that the “selfish” or the “status quo” rule that leaves the toilet seat in the position used dominates the down rule in a wide range of parameter spaces including the case where the inconvenience costs are the same. (JEL D7, H4)

Dear Annie: I read with interest the letters about putting down the toilet seat. I’ve been browbeaten by various women for the past 60 yr about proper seat etiquette, starting with my mother. If I forget to put the seat down even once, my wife reminds me for hours about this life-threatening situation.

I know you said the last column was the final word on the subject, but I hope you’ll reopen the issue. I want to ask women: Who gave you exclusive ownership of the bathroom? If men are nice enough to put the lid down, why can’t you ladies lift it up when you are done? When I suggested this to my wife, she wanted to have me taken out and shot. It’s time to rebel! — Fed Up in Salem, Ore.

Dear Fed Up: What is it about toilet seats that excites people? We received hundreds of letters on this subject and decided the “last word” would have to wait — Kathy Mitchell and Marcy Sugar.1

I. INTRODUCTION

Should the toilet seat be left up or down after use? This is a question that arises when members of the opposite sex share the same toilet. For some reason, this seemingly trivial question elicits passion from all sorts of people. It has become a topic of national debates in popular

*I thank Carl Davidson and Roger Lagunoff for helpful discussions and many colleagues for sharing their experi- ences. I am solely responsible for the views expressed in this paper. Choi: Professor of Economics, Department of Economics,

Michigan State University, East Lansing, MI 48824. Phone 51-353-7281, Fax 517-432-1068, E-mail [email protected]

1. Annie’s Mailbox by Kathy Mitchell and Marcy Sugar, October 29, 2002. Annie’s Mailbox is written by Kathy Mitchell and Marcy Sugar, long time editors of the Ann Landers syndicated column.

syndicated columns by Ann Landers and TV sitcoms such as ABC’s “Home Improvement” and NBC’s “3rd Rock from the Sun.” It is clear that this age-old debate is divided by the gender. Women complain that it should be the man’s responsibility to lower the toilet seat after use. “Leaving the toilet seat up” is often described as a problem, and there is even a toilet seat that goes down automatically after about 2 min, claiming that it has the perfect solution to the problem. Men seem to question why women should be the free-riders all the time. To quote Larry James (2004), a personal relationship counselor, “The most hotly contested battlefield in the gender wars may not necessarily be in the bedroom. It may be the bathroom. The seat-up versus seat-down debate rages on . . .”

Despite high emotions in the debate, sci- entific inquiries into this issue are sparse. In fact, it is not obvious why there should be a presumption that men are expected to leave the toilet seat down after use. Inter- net search generated the following noneco- nomic/scientific reasons for the down rule. First, there is an argument that being considerate to one’s love partner’s needs supports things going well in and out of the bedroom. To quote a phrase in the Internet (available at http://www.celebratelove.com/littlethings.htm), “Foreplay begins with putting the toilet seat down without being asked!” Second, it is not

ABBREVIATIONS BPH: Benign prostatic hyperplasia

303

Economic Inquiry (ISSN 0095-2583) Vol. 49, No. 1, January 2011, 303 – 309

doi:10.1111/j.1465-7295.2009.00277.x Online Early publication March 11, 2010 © 2010 Western Economic Association International

304 ECONOMIC INQUIRY

good Feng-Shi to leave the toilet seat up. Third, a toilet is not the most attractive household appliance. Closing the lid improves its appear- ance and prevents things from falling into the bowl. The last argument, however, proposes not only the seat down but also the lid down.

In this paper, I investigate whether there is any justification for the down rule based on economic efficiency. I find that the down rule is inefficient unless there is large asymmetry in the inconvenience costs of shifting the position of the toilet seat across genders. I show that the “selfish” or the “status quo” rule that leaves the toilet seat in the position used dominates the down rule in a wide range of parameter spaces including the case where the inconvenience costs are the same. The intuition for this result is easy to understand. Imagine a situation in which the aggregate frequency of toilet usage is the same across genders, that is, the probability that any visitor will be male is 1//2. With the down rule, each male visit is associated with lifting the toilet seat up before use and lowering it down after use, with the inconvenience costs being incurred twice. With the selfish rule, in contrast, the inconvenience costs are incurred once and only when the previous visitor is a member of different gender. The worst case under the selfish rule would occur when the sex of the toilet visitor strictly alternates in each usage. Even in this case, the total inconvenience costs would be the same as those under the down rule if the costs are symmetric. If there is any possibility that consecutive users are from the same gender, the selfish rule strictly dominates the down rule because it keeps the option value of not incurring any inconvenience costs in such an event. This logic can be extended to the case of asymmetric aggregate frequency of toilet usage across genders.

The remainder of the paper is organized in the following way. In Section II, I compare three plausible rules for the toilet seat position — up, down, and selfish — on an efficiency criterion. I show that the selfish rule always dominates the other two if the inconvenience costs of changing the toilet seat position are the same across genders. In Section III, I characterize the optimal rule for the toilet seat position. It turns out that the selfish rule is the most efficient rule in a wide range of parameter spaces. I also derive the condition that the down rule can be the most efficient one when the inconvenience costs are asymmetric. Section IV extends the analysis to the case where the inconvenience

costs are heterogeneous even within the same gender. Section V contains concluding remarks.

II. THE BASIC MODEL

I consider the usage of a toilet that is shared by members of the opposite sex.

Assume that the proportion of male to all users of a certain toilet is given by α. Let me assume the frequency of using a toilet by male and female is the same without loss of generality. If one gender uses the toilet more often, this asymmetry can be reflected in α. Thus, the parameter α represents the relative aggregate frequency of male using the toilet.2

I analyze an infinite horizon discrete time framework where the toilet is used once in each period. The discount factor is given by δ. With the assumption about the relative frequency of the toilet usage by each gender, the probability that the user is male in each period is given by α.3 The inconvenience cost of lowering the toilet seat for women is given by cf . The corresponding cost of lifting the toilet seat for men is given by cm. Even though I use the term inconvenience costs, cf and cm can encompass other types of costs such as “unwittingly placing one’s bottom directly on the porcelain” and risk of falling in by sitting down without looking when the seat is up or “leaving sprinkles on the seat” when it is down, respectively.

My goal in this section is to compare the expected aggregate inconvenience costs of three rules — down, up, and selfish — concerning the position of the toilet seat. In this comparative analysis, I abstract from other considerations such as being considerate to members of the opposite sex, aesthetic aspects, the wear costs of the seat hinge, etc.

A. The Down (Female-Friendly) Rule

This is a rule that leaves the position of the seat down after one is done with the bathroom task. In particular, this rule implies that each visit by a male member will be associated with the inconvenience costs of 2 cm, whereas female members will incur no costs.

2. The relative frequency of men going “number 1” ver- sus “number 2” can be also incorporated in the parameter α.

3. Equivalently, I could envision a continuous time model in which the arrival rate is given by a Possion process with the arrival rate being a function of the number of total users. The probability that a particular arrival is male is given by α. I derive essentially the same results with this continuous model.

CHOI: A MALE ECONOMIST’S MANIFESTO ON THE TOILET SEAT ETIQUETTE 305

Let V DOWNm and V DOWN f

denote the value functions with the down rule when the particular user in the current period is male and female, respectively. Then, these value functions satisfy the following recursive relationships.

V DOWN m = −2cm + δ

[ αV

DOWN m(1)

+ (1 − α)V DOWNf ]

V DOWN f = δ

[ αV

DOWN m + (1 − α)V DOWNf

] (2)

By solving these two equations, we can get

V DOWN m = −

( 1 − δα

1 − δ )

(2cm)(3)

V DOWN f = −

δα

1 − δ (2cm)(4)

Because the probability of a particular arrival being male is α, the value function associated with the down rule is:

V DOWN = αV DOWNm + (1 − α)V DOWNf(5)

= − α 1 − δ (2 cm)

B. The Up (Male-Friendly) Rule

This is a rule that leaves the position of the seat up after one is done with the bathroom task. In this case, all the inconvenience costs are incurred by females. The case is a mirror image of the down rule and the value function of this rule can be derived in an analogous way.

Let V UPm and V UP f

denote the value functions when the particular user is male and female, respectively. Then, these value functions satisfy the following relationships.

V UP m = δ

[ αV

U P m + (1 − α)V UPf

] (6)

V UP f = − 2cf(7)

+ δ [ αV

DOWN m + (1 − α)V UPf

] By solving these two equations, I can derive

V UP m = −

δ(1 − α) 1 − δ (2cf )(8)

V UP f = −

( 1 − δ(1 − α)

1 − δ )

(2cf )(9)

Because the probability that a particular arrival is male is α, the value function associated with the down rule is:

V UP = αV UPm + (1 − α)V UPf(10)

= − (1 − α) 1 − δ (2cf )

A comparison of Equations (5) and (10) yields the following proposition.

PROPOSITION 1. The down rule is more effi- cient than the up rule if and only if

cf cm

> α

1−α .

C. The Selfish (Status Quo) Rule

This is a rule that leaves the position of the seat as it was used.

Let V SQm and V SQ f

denote the value func- tions when the particular user is male and female, respectively, under the selfish rule. Then, these value functions satisfy the following relationships.

V SQ m = − (1 − α)cm(11)

+ δ [ αV

SQ m + (1 − α)V SQf

]

V UP f = −αcf + δ

[ αV

SQ m + (1 − α)V SQf

] (12)

By solving these two equations, I get

V SQ m = −

[ (1 − α)(1 − δ(1 − α))

1 − δ cm (13)

+ δα(1 − α) 1 − δ cf

]

V SQ f

= − [ δα(1 − α)

1 − δ cm + α(1 − δα)

1 − δ cf ](14)

Because the probability that a particular arrival is male is α, the value function associated with the down rule is:

V SQ = αV SQm + (1 − α)V SQf(15)

= − α(1 − α) 1 − δ (cm + cf )

Comparisons of Equations (5), (10), and (15) give me the following result. See also Figure 1.

PROPOSITION 2. If the inconvenience costs are the same across genders (cm = cf ), the self- ish rule dominates both the up and down rules.

306 ECONOMIC INQUIRY

FIGURE 1 Comparisons of the Up, Down, and Selfish Rule for the Symmetric Inconvenience Costs

(cm = cf = c)

Per-period average costs

2c2c

a = 1a = 0

Average costs with the selfish rule (2a(1 − a)c)

Average costs with the selfish rule (2a(1-a)c)

Average costs with the Up Rule (2(1 − a)c)

Average costs with the Down Rule (2ac)

a

The intuition for Proposition 2 is easy to understand. With either up or down rule, each member of one gender group has to incur the inconvenience costs two times with each usage. This practice can be obviously inefficient in the event that consecutive users are from the same gender to which the inconvenience costs are attributed. This inefficiency can be avoided by using the selfish rule because the inconvenience costs are incurred only when the consecutive users are from different genders. Even in such an event, the aggregate costs would be the same as those under the up or down rule if the inconvenience costs are the same across genders.

I cannot rule out the optimality of, say, the down rule if the inconvenience costs are asymmetric across genders. My analysis, how- ever, suggests that to justify the down rule on efficiency grounds, the inconvenience costs for

female should be very high relative to those for male. More precisely, the condition for the down rule to dominate the selfish rule is γ = cf

cm >

1+α 1−α .

For instance, if male and female users visit the toilet with the same frequency (α = 1/2), the inconvenience costs for female should be three times higher than the corresponding costs for male to justify the down rule.

Up to now, I have considered only three potential mechanisms. These three rules, how- ever, are not the only rules we can entertain. For instance, I can imagine a rule such that the posi- tion of the seat should be restored to the prior position before use. Alternatively, I can also consider a mutually considerate rule in which male users leave the seat down, whereas female users leave the seat up after use. In the next section, however, I show that all these rules are dominated by one of the three rules I have

CHOI: A MALE ECONOMIST’S MANIFESTO ON THE TOILET SEAT ETIQUETTE 307

considered. Thus, restricting my attention to the three rules does not entail any loss of generality in the analysis.

III. CHARACTERIZATION OF THE OPTIMAL RULE: A MECHANISM DESIGN APPROACH

In the previous section, we compared three simple rules that can be used for the toilet seat position. The task of this section is to derive the most efficient rule among all possible mechanisms. I show that one of the three rules discussed in the previous section is always optimal. Thus, restricting my attention to the three rules does not entail any loss of generality if the only concern is to minimize the aggregate inconvenience costs of toilet users.

The general rule can be considered a col- lection of four numbers (σum, σdm, σuf , σdf ) where σij denotes the probability that the seat be down after use when the position of the seat before use is i and the visitor is j , where i = u, d and j = m, f . The first subscripts u and d denote up and down, respectively, and the sec- ond subscripts m and f denote male and female, respectively. The objective is to search for the best mechanism that minimizes the aggregate inconvenience costs.

In the Appendix, I prove that the position of the seat before one’s use should not count in the optimal rule.

LEMMA. The optimal rule should depend only on the gender of the user, not the position of the seat before one arrives.

With the help of lemma, I can restrict my search for the optimal mechanism to a class of rules that can be written as (σm, σf ), where σm and σf are the probabilities that the toilet seat should be in the down position after usage by a male and a female, respectively.

Let Vm(σm, σf ) and Vf (σm, σf ) be the corresponding present discounted value when a particular user in the current period is male and female, respectively.

Vm(σm, σm) = − [ασm + (1 − α)σf ]cm − σmcm + δ[αVm + (1 − α)Vf ]

Vf (σm, σf ) = −[α(1 − σm) + (1 − α) × (1 − σf )]cf − (1 − σf )cf

+ δ[αVm + (1 − α)Vf ]

Then, the corresponding value function for the rule (σm, σf ) can be written as

V (σm, σf ) = αVm(σm, σf ) + (1− α)Vf (σm, σf ) = [−αM + (1 − α)F ]/(1 − δ),

where M = [ασm + (1 − α)σf ]cm + σmcm and F = [α(1 − σm) + (1 − α)(1 − σf )]cf + (1 − σf )cf .

The search for the optimal mechanism is equivalent to solving

Minσm,σf αM + (1 − α)F.

PROPOSITION 3. The optimal toilet etiquette is given by the following:

Let γ = cf cm

be the relative cost of changing the toilet seat position for male and female. Then, the optimal rule is characterized by two critical values of γ (γ and γ) such that:

(1) The toilet should be down if γ > γ = 1+α 1−α

(2) The toilet should be left as it was used if α

2−α = γ < γ < γ = 1+α1−α (3) The toilet should be up if γ < γ = α2−α

Proof: Because the objective function αM + (1 − α)F is a linear function of σm and σf , I have corner solutions except the knife-edge cases. By differentiating αM + (1 − α)F with respect to σm and σf , the optimal rule is given by:

σm =

⎧⎪⎨ ⎪⎩

1 if γ > 1+α1−α any number between 0 and 1 if γ = 1+α1−α

0 if γ < 1+α1−α

σf =

⎧⎪⎨ ⎪⎩

1 if γ > α2−α any number between 0 and 1 if γ = α2−α

0 if γ < α2−α

Because α2−α = γ < γ < 1+α1−α , we have the desired result. Figure 2 summarizes the opti- mal configuration for different values of α and γ = cf

cm . Q.E.D.

My analysis can be easily extended to the case of time-varying γ and α. These values, for instance, can change depending on the time of the day. The mistake costs of “unwittingly placing one’s bottom directly on the porcelain” or risk of falling in are presumably higher during the nighttime when the light is turned off. If this is the case, the optimal rule could be time-dependent, with the selfish rule during the

308 ECONOMIC INQUIRY

FIGURE 2 The Optimal Rules with Asymmetric Inconvenience Costs

daytime and the down rule during the nighttime. A countervailing argument against the down rule during the nighttime is that nocturia (needing to urinate frequently during the night time) is more common with men due to benign prostatic hyperplasia (BPH).4

IV. HETEROGENEOUS COSTS WITHIN THE SAME GENDER

In the previous sections, I assumed that the inconvenience costs are the same within the same gender. I extend the analysis to the case where different users have different inconve- nience costs even within the same gender. Let me assume that cm and cf are distributed accord- ing to continuous distribution functions G(.) and H (.), respectively on [0, ∞). Let cm and cf

4. I thank Carl Davidson for this observation.

denote the mean values of the inconvenience costs for male and female, respectively:

cm = ∫ ∞

0 cmdG = E(cm),

cf = ∫ ∞

0 cf dH = E(cf )

It is clear that with heterogeneous inconve- nience costs, the optimal rule should be char- acterized with two critical values cm* and cf * such that a male visitor should put the toilet seat down if and only if his cm ≤ cm* and a female visitor should put the toilet seat up if and only if her cf ≤ cf *. Then, σm = G(cm∗) and σf = 1 − H (cf ∗) using our previous nota- tion. The value function with the critical values of cm* and cf * can be written as

V (cm ∗ , cf

∗ ) = − αM + (1 − α)F

1 − δ ,

CHOI: A MALE ECONOMIST’S MANIFESTO ON THE TOILET SEAT ETIQUETTE 309

where M = [αG(cm∗) + (1 − α)(1 − H (cf ∗))] cm + G(cm∗)E(cm |cm ≤ cm∗) and F = [α(1 − G(cm

∗)) + (1 − α)H (cf ∗)]cf + H (cf ∗)E(cf | cf ≤ cf ∗)

With this observation, the optimal critical values for cm* and cf * can be derived by solving

Mincm∗,cf ∗ αM + (1 − α)F The first order condition with respects to cm

∗ and cf

∗ can be written as: cm∗ = −αcm + (1 − α)cf

cf ∗ = αcm − (1 − α)cf (= −cm∗) Because inconvenience costs are nonnegative, the two first order conditions cannot be satisfied as interior solutions simultaneously. Taking the boundary conditions into consideration, I can state the following result.

PROPOSITION 4. With the heterogeneous inconvenience costs, the optimal toilet rule is given by the following:

(1) If cf cm

> α

1−α , women should leave the seat as it was used and a male visitor should put the toilet seat down after use if cm ≤ cm

∗ − αcm + (1 − α)cf and use the selfish rule otherwise.

(2) If cf cm

< α

1−α , men should leave the seat as it was used and a female visitor should put the toilet seat up after use if cf ≤ cf ∗ = αcm − (1 − α)cf and use the selfish rule otherwise.

V. CONCLUDING REMARKS

In this paper, I conducted an economic analy- sis of the most efficient rule concerning the posi- tion of the toilet seat when the objective is solely to minimize the aggregate costs of inconve- nience. This research is in line with an argument that social norms are instruments of collective optimization when the price system is inappli- cable.5 The main result is that unless there is a large degree of asymmetry in the inconvenience costs of changing the seat position across users, the selfish rule is the most efficient one.6 In addi- tion, the selfish rule is incentive-compatible in

5. See Arrow (1971). 6. This result, however, should not be construed as

implying that selfish behavior is optimal in other bathroom etiquettes. My analysis will apply only to the cases where different users have conflicting preferences. For instance, every user prefers clean toilets. In this case, cleaning after use for the next visitor should be the proper etiquette. This is especially so because cleaning one’s own is less unpleasant than the previous user’s.

that it can be self-enforcing without any outside sanctions for violating the rule.

The analysis has applicability to a wide variety of circumstance in which a facility is used by different people and the facility can be customized according to the preference of each user. For instance, a car can be shared by family members who need different configurations of the driver seat. The same logic implies that it would be most efficient to leave the seat position as it was used last if the inconvenience costs of reconfiguring the position are the same across family members. A computer shared by different people is another example.

APPENDIX: PROOF OF LEMMA

To derive the optimal rule, I allow the possibility that a visitor’s decision can depend on the toilet position prior to one’s use. The general rule can be considered a collection of four numbers (σum, σdm, σuf , σdf ) where σij denotes the probability that the seat be down after use when the position of the seat before use is i and the visitor is j , where i = u, d and j = m, f .

Suppose that the visitor in a particular period is male. At the time of his arrival, the toilet seat could be either up or down. Let Vum and Vdm be the value functions when the position of the toilet is up and down at the time of visit, respectively. I also denote corresponding value func- tions by Vuf and Vdf when the visitor is female. Then, I can write

Vum =Maxσum [−cm + δ{αVdm + (1 − α)Vdf }] + (1 − σum)[δ{αVdm + (1 − α)Vdf }]

Vdm =Maxσdm − cm + σdm[ − cm + δ{Vdm + (1 − α)Vdf }] + (1 − σdm)[δ{αVdm + (1 − α)Vdf }]

Comparison of Vum and Vdm reveals that the optimal values for σum and σdm should be the same (σum = σdm), that is, the optimal rule should be forward-looking and not contingent on the position prior to the visit.

Similarly, when the visitor in a particular visit is female, we can write the value functions as

Vuf = Maxσuf − cf + σuf [δ{αVdm + (1 − α)Vdf }] + (1 − σuf )[−cf + δ{αVdm + (1 − α)Vdf }]

Vdf = Maxσdf σdf [δ{αVdm + (1 − α)Vdf }] + (1 − σdf )[−cf + δ{αVdm + (1 − α)Vdf }]

Once again, comparison of Vuf and Vdf reveals that the optimal values for σuf and σdf should be the same (σuf = σdf ).

REFERENCES

Arrow, K. “Political and Economic Evaluation of Social Effects and Externalities,” in Frontiers of Quantitative Economics, edited by M. Intriligator. Amsterdam: North-Holland, 1971, 3 – 25.

James, L. “Put the Toilet Seat DOWN! For Men Only.” 2004 http://www.celebratelove.com/littlethings.htm.

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