IB(.,0W) (2.9)For this indirect utility function I include the wage variable w9 as it will play an importantrole. For a shirker, however, it is shown in the appendix that the possibility of household tradesdoes not necessarily strictly raise utility. Furthermore the following Lemma shows that, forgiven p\u2019, a shirker wifi trade no more than a non-shirker.Lemma 1 H H*.Proof: See appendix.This Lemma is intuitively obvious. If expending less effort at work, effort at home causes less16t should be noted that ensuring household difference leads to mutually beneficial trades does not require thisassumption, since \u00eb> 0 and the marginal disutility of effort is increasing, mutually beneficial within householdtrades will often exist between a person in a good job and one working a piece rate. At the cost of somecomplexity it is possible to relax this assumption and more accurately determine the reduced limits withinwhich p\u2019 must lie, however this adds nothing to the results. In fact, modelling the household in Becker\u2019s (1991)classic framework (which includes all members of the household maximizing a joint welfare function) ensuresthat a household will always benefit by such a division of labour and will not alter any of the model\u2019s conclusions.21disutility so one wifi not wish to buy more household services. Furthermore, it follows fromLemma 1 that not being able to trade household services internafly, makes a non-shirker atleast as much worse off as a shirker. That is:Lemma 2 pB(w9 pfl) \u2014 iIIB(w9,.) iBO(w9,pn) \u2014Proof: See appendix.I now determine the value of wages which are sufficient to dissuade shirking.2.2.1 Wages sufficient to dissuade shirkingIt is necessary to define an individual\u2019s no shirking constraint for two different situations: 1)the person in a good job has a spouse who is a piece rate worker, 2) the person in a good jobhas a spouse in a good job.In situation (1) the no shirking constraint is the following:BO(n) + p1_s(I) (w,p) (2.10)remembering that S (.) is the indirect utility function of a potential seller of household serviceswhen trades are not available. To simplify notation, the argument for the limit on the amountof H which can be traded is suppressed, since it plays no role. The left hand side is thediscounted lifetime stream of utility obtained when deciding to shirk in the current period.The right hand side is the discounted lifetime stream of utility obtained if one decides to be a22non-shirker. \u2018 Note that there is no uncertainty in these calculations. A non-shirker expects tomaintain employment forever while a shirker is certain to be dismissed in the next period and tonever again receive a good job.\u20198 The first term on the left hand side is the one period benefitto shirking, note that, even if shirking, an individual in a good job stifi has the opportunity totrade with his or her spouse, though such trade may not be worthwhile. The second term isthe discounted infinite stream of utility obtained when employed as a piece rate worker fromthe next period on. In this case, trade within the household is not possible, as one\u2019s spouse isalso a piece rate worker and does not demand one\u2019s services. The right hand side is calculatedallowing for trades, since in this situation the worker has a good job and buys household servicesfrom their piece rate working spouse.In situation (2) the no shirking constraint is given by:iF1S( n\u2019 g \u2018+\u2018 \u2018\u2018\u2018\u2018\u2018) (2.11)i\u2014p\u2014 i\u2014pThis equation is similar to (2.10) except that now within household trading opportunities arereversed. Here, whenever one is in a good job, i.e. the first term on the left hand side andthe term on the right hand side, household trades are unavailable since one\u2019s spouse also hasa good job and wifi not provide household services. However when not in a good job, one cansell household services to one\u2019s spouse and thus benefit from internal trade, i.e. the first termon the left hand side.\u2018TAs is standard in efficiency wage models, if one wishes to shirk, then it is optimal to do so immediately.The no shirking condition then needs to only compare returns to shirking in the current period to those fromdeciding to never shirk.\u201881t wiil be shown that in both the discrimination and non-discrimination equilibria which are consideredfurther on, no one shirks and thus no one is dismissed in any period. Thus if someone were to shirk theycalculate their return to shirking under the expectation that, if dismissed, they will never again receive a goodjob. Note however that given expectations which define a non-zero probability of termination being due toshirking, firms will never hire workers previously dismissed from good jobs. Thus (2.10) would stifi apply evenif the model were extended to allow for exogenous, involuntary turnover as in Shapiro and Stiglitz (1984).23Denote solving (2.10) with equality w and denote ziP solving (2.11) with equality w.As will soon be clear, d and n are intended to denote discrimination and non-discriminationrespectively. From the assumed separability of the utility function and the use of the envelopetheorem, it follows that for given w9, < where is the derivative of \u2018P withrespect to uP, see the appendix for details.\u20199 Thus, since p > 1, an increase in uP serves tomake shirking less likely in both situations since it raises the right hand side in equations (2.10)and (2.11), by more than the left. That is, as would be expected in an efficiency wage model,increasing the wage raises the relative returns to not shirking. The separability can also beshown to imply that the values w and w are unique.2\u00b02.3 Discrimination equilibriumI now establish a result which is essential in proving the existence of a discrimination equilibrium, i.e. an equilibrium in which only men receive good jobs.Proposition 1 w