Up or Down? A Male Economist s Manifesto on the Toilet Seat Etiquette

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1 Up or Down? A Male Eonoist s Maniesto on the Toilet Seat Etiquette By Jay Pil Choi* Mihigan State University Noveber 2002 Abstrat This paper develops an eonoi analysis o the toilet seat etiquette, that is, whether the toilet seat should be let up or down. I investigate whether there is any eiieny justiiation or the presuption that en should leave the toilet seat down ater use. I ind that the down rule is ineiient unless there is a large degree o asyetry in the inonveniene osts o shiting the position o the toilet seat aross genders. I show that the selish or the status quo rule that leaves the toilet seat in the position used doinates the down rule in a wide range o paraeter spaes inluding the ase where the inonveniene osts are the sae. The analysis an be applied to other shared ailities that an be ustoized to eah user s preerene. JEL Classiiation: D7, H4. Keywords: gender, eonoi eiieny, soial nor. Correspondent: Jay Pil Choi Departent o Eonois, Mihigan State University East Lansing, MI E-ail: * I thank Carl Davidson and Roger Laguno or helpul disussions and any olleagues or sharing their experienes. I a solely responsible or the views expressed in this paper.

2 Dear Annie: I read with interest the letters about putting down the toilet seat. I've been browbeaten by various woen or the past 60 years about proper seat etiquette, starting with y other. I I orget to put the seat down even one, y wie reinds e or hours about this lie-threatening situation. I know you said the last olun was the inal word on the subjet, but I hope you'll reopen the issue. I want to ask woen: Who gave you exlusive ownership o the bathroo? I en are nie enough to put the lid down, why an't you ladies lit it up when you are done? When I suggested this to y wie, she wanted to have e taken out and shot. It's tie to rebel! Fed Up in Sale, Ore. Dear Fed Up: What is it about toilet seats that exites people? We reeived hundreds o letters on this subjet and deided the "last word" would have to wait Kathy Mithell and Mary Sugar Introdution Should the toilet seat be let up or down ater use? This is a question that arises when ebers o the opposite sex share the sae toilet. For soe reason, this seeingly trivial question eliits passion ro all sorts o people. It has beoe a topi o national debates in popular syndiated oluns by Ann Landers and TV sitos suh as ABC s Hoe Iproveent and NBC s 3 rd Rok ro the Sun. It is lear that this age-old debate is divided by the gender. 2 Woen oplain that it should be the an s responsibility to lower the toilet seat ater use. Men see to question why woen should be the ree-riders all the tie. Despite high eotions in the debate, sientii inquiries into this issue are sparse. In at, it is not obvious why there should be a presuption that en are expeted to leave the toilet seat down ater use. 3 In this paper, I investigate whether there is any 1 Annie s Mailbox by Kathy Mithell and Mary Sugar, Otober 29, Annie's Mailbox is written by Kathy Mithell and Mary Sugar, long tie editors o the Ann Landers syndiated olun. 2 To quote Larry Jaes, a personal relationship ounsellor, The ost hotly ontested battleield in the gender wars ay not be neessarily be in the bedroo. It ay be the bathroo. The seat-up vs. seat-down debate rages on 3 Leaving the toilet seat up is oten desribed as a proble and there is even a toilet seat that goes down autoatially ater about two inutes, laiing that it has the peret solution to the proble. 2

3 justiiation or the down rule based on eonoi eiieny. 4 I ind that the down rule is ineiient unless there is large asyetry in the inonveniene osts o shiting the position o the toilet seat aross genders. I show that the selish or the status quo rule that leaves the toilet seat in the position used doinates the down rule in a wide range o paraeter spaes inluding the ase where the inonveniene osts are the sae. The intuition or this result is easy to understand. Iagine a situation in whih the aggregate requeny o toilet usage is the sae aross genders, i.e., the probability that any visitor will be ale is ½. With the down rule, eah ale visit is assoiated with liting the toilet seat up beore use and lowering it down ater use, with the inonveniene osts being inurred twie. With the selish rule, in ontrast, the inonveniene osts are inurred one and only when the previous visitor is a eber o dierent gender. The worst ase under the selish rule would our when the sex o the toilet visitor stritly alternates in eah usage. Even in this ase, the total inonveniene osts would be the sae as those under the down rule i the osts are syetri. I there is any possibility that onseutive users are ro the sae gender, the selish rule stritly doinates the down rule sine it keeps the option value o not inurring any inonveniene osts in suh an event. This logi an be extended to the ase o asyetri aggregate requeny o toilet usage aross genders. The reainder o the paper is organized in the ollowing way. In setion II, I opare three plausible rules or the toilet seat position up, down, and selish on an 4 Internet searh generated the ollowing non-eonoi/sientii reasons or the down rule. First, there is an arguent that being onsiderate to one s love partner s needs supports things going well in and out o the bedroo. To quote a phrase in the internet (available at Foreplay begins with putting the toilet seat down without being asked! Seond, it is not good Feng-Shi to leave the toilet seats up. Third, a toilet is not the ost attrative household appliane. Closing the lid iproves its appearane and prevent s things ro alling into the bowl. The last arguent, however, proposes not only the seat down but also the lid down. 3

4 eiieny riterion. I show that the selish rule always doinates the other two i the inonveniene osts o hanging the toilet seat position are the sae aross genders. In setion III, I haraterize the optial rule or the toilet seat position. It turns out that the selish rule is the ost eiient rule in a wide range o paraeter spaes. I also derive the ondition that the down rule an be the ost eiient one when the inonveniene osts are asyetri. Setion IV extends the analysis to the ase where the inonveniene osts are heterogeneous even within the sae gender. Setion V ontains onluding rearks II. The Basi Model I onsider the usage o a toilet that is shared by ebers o the opposite sex. Assue that the proportion o ale to all users o a ertain toilet is given by. Let e assue the requeny o using a toilet by ale and eale is the sae without loss o generality. I one gender uses the toilet ore oten, this asyetry an be releted in. Thus, the paraeter represents the relative aggregate requeny o ale using the toilet. 5 I analyze an ininite horizon disrete tie raework where the toilet is used one in eah period. The disount ator is given by δ. With the assuption about the relative requeny o the toilet usage by eah gender, the probability that the user is ale in eah period is given by. 6 The inonveniene ost o lowering the toilet seat or woen is given by w. The orresponding ost o liting the toilet seat or en is given 5 The relative requeny o en going nuber 1 vs. nuber 2 an be also inorporated in the paraeter. 6 Equivalently, I ould envision a ontinuous tie odel in whih the arrival rate is given by a Possion proess with the arrival rate being a untion o the nuber o total users. The probability that a partiular arrival is ale is given by. I derive essentially the sae results with this ontinuous odel. 4

5 by. My goal in this setion is to opare the expeted aggregate inonveniene osts o three rules down, up, and selish onerning the position o the toilet seat. 7 In this oparative analysis, I abstrat ro other onsiderations suh as being onsiderate to ebers o the opposite sex, aestheti aspets, the wear osts o the seat hinge, et. The Down (Feale-Friendly)Rule This is a rule that leaves the position o the seat down ater one is done with the bathroo task. In partiular, this rule iplies that eah visit by a ale eber will be assoiated with the inonveniene osts o 2 whereas eale ebers will inur no osts. Let V and V denote the value untions with the down rule when the partiular user in the urrent period is ale and eale, respetively. Then, these value untions satisy the ollowing reursive relationships. V = 2 + δ[v + (1 ) V ] (1) V = δ[v + (1 ) V ] (2) By solving these two equations, we an get δ V = 1 (2 ) (3) 1 δ δ V = (2 ) (4) 1 δ 7 Even though I use the ter inonveniene osts, and an enopass other types o osts suh as unwittingly plaing one s botto diretly on the porelain and risk o alling in by sitting down without looking when the seat is up or leaving sprinkles on the seat when it is down, respetively. 5

6 Sine the probability o a partiular arrival being ale is, the value untion assoiated with the down rule is: V = V + (1 ) V = (2 ) (5) 1 δ The Up (Male-Friendly)Rule This is a rule that leaves the position o the seat up ater one is done with the bathroo task. In this ase, all the inonveniene osts are inurred by eales. The ase is a irror iage o the down rule and the value untion o this rule an be derived in an analogous way. Let V and V denote the value untions when the partiular user is ale and eale, respetively. Then, these value untions satisy the ollowing relationships. V = δ[v + (1 ) V ] (6) V = 2 + δ[v + (1 ) V ] (7) By solving these two equations, I an derive V = δ (1 ) (2 ) (8) 1 δ δ (1 ) V = 1 (2 ) (9) 1 δ Sine the probability that a partiular arrival is ale is, the value untion assoiated with the down rule is: V = V (1 ) + (1 ) V = (2 ) (10) 1 δ A oparison o equations (5) and (10) yields the ollowing proposition. 6

7 Proposition 1. The down rule is ore eiient than the up rule i and only i > 1. The Selish (Status Quo) Rule This is a rule that leaves the position o the seat as it was used. Let SQ V and SQ V denote the value untions when the partiular user is ale and eale, respetively under the selish rule. Then, these value untions satisy the ollowing relationships. SQ V = (1 ) + δ[v SQ + (1 ) V ] (11) SQ V = + δ[v SQ + (1 ) V ] (12) SQ By solving these two equations, I get SQ V = (1 )(1 δ (1 )) δ (1 ) + 1 δ 1 δ SQ δ (1 ) (1 δ ) V = + 1 δ 1 δ (13) (14) Sine the probability that a partiular arrival is ale is, the value untion assoiated with the down rule is: SQ V = V SQ SQ (1 ) + (1 ) V = ( + ) 1 δ (15) Coparisons o equations (5), (10), and (15) give e the ollowing result. See also igure 1. 7

8 Proposition 2. I the inonveniene osts are the sae aross genders ( = 1), the selish rule doinates both the up and down rules. Per-period average osts 2 Average Costs with the Up Rule ( 2(1 ) ) Average Costs with the Down Rule ( 2 ) 2 Average osts with the selish rule ( 2 (1 ) ) = 0 = 1 Average osts with the selish rule ( 2 (1 ) ) Figure1. Coparisons o the Up, Down, and Selish Rule or the Syetri Inonveniene Costs ( = 1= ) 8

9 The intuition or Proposition 2 is easy to understand. With either up or down rule, eah eber o one gender group has to inur the inonveniene osts two ties with eah usage. This pratie an be obviously ineiient in the event that onseutive users are ro the sae gender to whih the inonveniene osts are attributed. This ineiieny an be avoided by using the selish rule sine the inonveniene osts are inurred only when the onseutive users are ro dierent genders. Even in suh a event, the aggregate osts would be the sae as those under the up or down rule i the inonveniene osts are the sae aross genders. I annot rule out the optiality o, say, the down rule i the inonveniene osts are asyetri aross genders. My analysis, however, suggests that to justiy the down rule on eiieny grounds, the inonveniene osts or eale should be very high relative to those or ale. More preisely, the ondition or the down rule to doinate the selish rule is γ = 1+ > 1. For instane, i ale and eale users visit the toilet with the sae requeny ( =1/2), the inonveniene osts or eale should be three ties higher than the orresponding osts or ale to justiy the down rule. Up to now, I have onsidered only three potential ehaniss. These three rules, however, are not the only rules we an entertain. For instane, I an iagine a rule suh that the position o the seat should be restored to the prior position beore use. Alternatively, I an also onsider a utually onsiderate rule in whih ale users leave the seat down whereas eale users leave the seat up ater use. In the next setion, however, I show that all these rules are doinated by one o the three rules I have onsidered. Thus, restriting y attention to the three rules does not entail any loss o generality in the analysis. 9

10 III. Charaterization o the Optial Rule: A Mehanis Design Approah In the previous setion, we opared three siple rules that an be used or the toilet seat position. The task o this setion is to derive the ost eiient rule aong all possible ehaniss. I show that one o the three rules disussed in the previous setion is always optial. Thus, restriting y attention to the three rules does not entail any loss o generality i the only onern is to iniize the aggregate inonveniene osts o toilet users. The general rule an be onsidered a olletion o our nubers (σ u, σ d, σ u, σ d ) where σ ij denotes the probability that the seat be down ater use when the position o the seat beore use is i and the visitor is j, where i =u, d and j =,. The irst subsripts u and d denote up and down, respetively and the seond subsripts and denote ale and eale, respetively. The objetive is to searh or the best ehanis that iniizes the aggregate inonveniene osts. In the Appendix, I prove that the position o the seat beore one s use should not ount in the optial rule. Lea. The optial rule should depend only on the gender o the user, not the position o the seat beore one arrives. With the help o lea, I an restrit y searh or the optial ehanis to a lass o rules that an be written as (σ,σ ), where σ and σ are the probabilities that the toilet seat should be in the down position ater usage by a ale and a eale, respetively. 10

11 Let V (σ,σ ) and V (σ,σ ) be the orresponding present disounted value when a partiular user in the urrent period is ale and eale, respetively. V (σ,σ ) = [σ + (1 )σ ] σ + δ[v + (1 ) V ] V (σ,σ ) = [(1 σ ) + (1 )(1 σ ) ] (1 σ ) + δ[v + (1 ) V ] Then, the orresponding value untion or the rule (σ,σ ) an be written as V(σ,σ ) = V (σ,σ ) + (1 ) V (σ,σ ) = M + ( 1 ) F 1 δ, where M = [σ + (1 )σ ] + σ and F= [(1 σ ) + (1 )(1 σ ) ] + (1 σ ) The searh or the optial ehanis is equivalent to solving Min M + (1 )F σ, σ Proposition 3. The optial toilet etiquette is given by the ollowing: Let γ = be the relative ost o hanging the toilet seat position or ale and eale. Then, the optial rule is haraterized by two ritial values o γ (γ and γ ) suh that: (1)The toilet should be down i γ > γ = 1+ 1 (2) The toilet should be let as it was used i (3) The toilet should be up i γ < γ = 2 =γ <γ <γ =

12 Proo. Sine the objetive untion M + (1 )F is a linear untion o σ and σ, I have orner solutions exept the knie-edge ases. By dierentiating M + (1 )F with respet to σ and σ, the optial rule is given by: σ σ 1 = any nuber between 0 and1 0 1 = any nuber between 0 and i γ > 1-1+ i γ = 1-1+ i γ < 1- i γ > 2 - i γ = 2 - i γ < 2 - Sine 1+ =γ < γ =, we have the desired result. Figure 2 suarizes the 2 1 optial oniguration or dierent values o and γ =. Q.E.D. My analysis an be easily extended to the ase o tie-varying γ and. These values, or instane, an hange depending on the tie o the day. The istake osts o unwittingly plaing one s botto diretly on the porelain or risk o alling in are presuably higher during the nighttie when the light is turned o. I this is the ase, the optial rule ould be tie-dependent, with the selish rule during the daytie and the down rule during the nighttie. 8 8 A ountervailing arguent against the down rule during the nighttie is that noturia (needing to urinate requently during the night tie) is ore oon with en due to BPH (benign prostati hyperplasia). I thank Carl Davidson or this observation. 12

13 γ = RULE 1+ γ = 1 γ = 1 (Syetri ost) SELFISH RULE γ = 2 RULE 0 1 Figure 2. The Optial Rules with Asyetri Inonveniene Costs 13

14 IV. Heterogeneous Costs within the Sae Gender In the previous setions, I assued that the inonveniene osts are the sae within the sae gender. I extend the analysis to the ase where dierent users have dierent inonveniene osts even within the sae gender. Let e assue that and are distributed aording to ontinuous distribution untions G(.) and H(.), respetively on [0, ). Let and denote the ean values o the inonveniene osts or ale and eale, respetively: = dg = E( ), = dh = E( ) 0 0 It is lear that with heterogeneous inonveniene osts, the optial rule should be haraterized with two ritial values * and * suh that a ale visitor should put the toilet seat down i and only i his * and a eale visitor should put the toilet seat up i and only i her *. Then, σ = G( *) and σ = 1 H( *) using our previous notation. The value untion with the ritial values o * and * an be written as V( *, * ) = M + ( 1 ) F 1 δ, where M = [G( *) + (1 )(1 H( *))] + G( *) E *) and F= [(1 ( G( *)) + (1 ) H( *)] + H( *) E( *) With this observation, the optial ritial values or * and * an be derived by solving Min M + (1 )F *, * The irst order ondition with respets to * and * an be written as: * = + (1 ) 14

15 * = (1 ) (= *) Sine inonveniene osts are nonnegative, the two irst order onditions annot be satisied as interior solutions siultaneously. Taking the boundary onditions into onsideration, I an state the ollowing result. Proposition 4. With the heterogeneous inonveniene osts, the optial toilet rule is given by the ollowing: (1) I >, woen should leave the seat as it was used and a ale visitor 1 should put the toilet seat down ater use i * + (1 ) the selish rule otherwise. and use (2) I <, en should leave the seat as it was used and a eale visitor 1 should put the toilet seat up ater use i * = (1 ) selish rule otherwise. and use the V. Conluding Rearks In this paper, I onduted an eonoi analysis o the ost eiient rule onerning the position o the toilet seat when the objetive is solely to iniize the aggregate osts o inonveniene. This researh is in line with an arguent that soial nors are instruents o olletive optiization when the prie syste is in appliable. 9 The ain result is that unless there is a large degree o asyetry in the inonveniene 9 See Arrow (1971). 15

16 osts o hanging the seat position aross users, the selish rule is the ost eiient one. 10 In addition, the selish rule is inentive-opatible in that it an be sel-enoring without any outside santions or violating the rule. The analysis has appliability to a wide variety o irustane in whih a aility is used by dierent people and the aility an be ustoized aording to the preerene o eah user. For instane, a ar an be shared by aily ebers who need dierent onigurations o the driver seat. The sae logi iplies that it would be ost eiient to leave the seat position as it was used last i the inonveniene osts o reoniguring the position is the sae aross aily ebers. A oputer shared by dierent people is another exaple. 10 This result, however, should not be onstrued as iplying that selish behavior is optial in other bathroo etiquettes. My analysis will apply only to the ases where dierent users have onliting preerenes. For instane, every user preers lean toilets. In this ase, leaning ater use or the next visitor should be the proper etiquette. This is espeially so sine leaning one s own is less unpleasant than the previous user s. 16

17 Appendix: Proo o Lea To derive the optial rule, I allow the possibility that a visitor s deision an depend on the toilet position prior to one s use. The general rule an be onsidered a olletion o our nubers (σ u, σ d, σ u, σ d ) where σ ij denotes the probability that the seat be down ater use when the position o the seat beore use is i and the visitor is j, where i =u, d and j =,. Suppose that the visitor in a partiular period is ale. At the tie o his arrival, the toilet seat ould be either up or down. Let V u and V d be the value untions when the position o the toilet is up and down at the tie o visit, respetively. I also denote orresponding value untions by V u and V d when the visitor is eale. Then, I an write V u = V d = Max σ u Max σ d σ u [ + δ{v d + (1 ) V d + σ d [ + δ{v d + (1 ) V d }] + (1 σ u )[ δ{v d + (1 ) V d }] }] + (1 σ d )[ δ{v d + (1 ) V d }] Coparison o V u and V d reveals that the optial values or σ u and σ d should be the sae (σ u = σ d ), that is, the optial rule should be orward-looking and not ontingent on the position prior to the visit. untions as Siilarly, when the visitor in a partiular visit is eale, we an write the value V u = V d = Max σ u Max σ d + σ u [δ{v d + (1 ) V d σ d [δ{v d + (1 ) V d }] + (1 σ u )[ + δ{v d + (1 ) V d }] }] + (1 σ d )[ + δ{v d + (1 ) V d }] One again, oparison o V u and V d reveals that the optial values or σ u and σ d should be the sae (σ u = σ d ). 17

18 Reerenes Arrow, Kenneth, Politial and Eonoi Evaluation o Soial Eets and Externalities, in Intriligator, M., ed., Frontiers o Quantitative Eonois, Asterda: North- Holland, 1971, pp Jaes, Larry, Put the Toilet Seat! For Men Only, available at 18

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