UNSTABLE RELATIONSHIPS¤ Kenneth Burdett University of Essex Colchester CO4 3SQ, United Kingdom Ryoichi Imai Nagoya University of Commerce and Business Administration Nissin, Komenoki, Sagamine 4-4, Aichi 470-0193, Japan Randall Wright University of Pennsylvania 3718 Locust Walk, Philadelphia, PA 19104 USA 9/9/99 Abstract We analyze models where agents search for partners to form rela- tionships (employment, marriage, coauthor, etc.), and may choose to continue searching for better partners while in relationships. Matched agents are less inclined to search if their match generates more instan- taneous utility, and also if it is more stable. If one partner searches the relationship will be less stable, and so the other is more inclined to search. Thus, instability can be a self-ful¯lling prophecy. This new source of multiplicity can lead to a continuum of steady state equilib- ria. We also show that it tends to lead to too much search, and also to too much unemployment and too much inequality in equilibrium. JEL classi¯cation numbers: C78, D83 Key Words: search, matching, marriage, unemployment ¤ We thank Gadi Barlevy, Dale Mortensen and Tracy Webb for their inputs. The NSF provided ¯nancial support. 1 \To be faithful to one would be cruel to the rest." Don Giovanni 1 Introduction We analyze models where agents search for partners to form relationships (e.g., employment, marriage, or coauthor relationships). While in a relation- ship, an agent may or may not choose to continue to search, at a cost, for a di®erent partner. Matched agents are less inclined to search { or, more inclined to be \faithful" to their current partner { if the match is better in the sense of the instantaneous utility derived from the relationship, and also if the match is more stable. What lends stability to a relationship? If one partner is searching then the relationship is less secure for the other, since he is more likely to be abandoned, and hence he is more inclined to search himself. In this way, endogenous instability can be a self-ful¯lling prophecy. We show these considerations can lead to multiple equilibria, and indeed to a continuum of steady state equilibria, in some cases. We also show that they tend to lead to too much search, too much unemployment, and too much inequality in equilibrium. The multiplicity in the current model is new, and has nothing to do with the standard thick-market e®ects that have been understood in equilibrium search theory since Diamond (1982). The standard thick-market e®ect works as follows: assuming increasing returns in the matching technology, if there is more search then it is easier to meet people, which makes people more inclined to search. This is not what is going on here. To emphasize the dis- tinction, we assume constant returns to scale in the meeting technology, so that your probability of meeting a potential partner is independent of the ag- 2 gregate amount of search, and also make some other assumptions that reduce the strategic interaction between market activity and the private gains from search. This allows us to focus clearly on the strategic interactions within re- lationships, and to show that there can be more unstable relationships simply because people think relationships will be more unstable.1 The rest of the paper can be summarized as follows. Section 2 speci¯es the basic framework, and in particular our assumption that when any two people meet they draw a random variable x that gives the instantaneous utility each will receive if they form a partnership; thus, agents are homogeneous ex ante, although relationships are heterogeneous ex post due to match- speci¯c idiosyncracies. Section 3 considers a relatively simple version where relationships come in two varieties, x1 and x2 > x1 . We characterize the set of equilibria in terms of when agents enter into relationships and when they search. For some parameter values there is a unique equilibrium, while for others there are multiple equilibria. The leading example is the following: agents will accept x1 matches, and may either be: \unfaithful" and continue to search for a better match; or \faithful" and stop searching because they are satis¯ed with x1 , this satisfaction arising because these matches are relatively stable, and this stability arising because agents in x1 matches do not search. Indeed, we show that there can even exist a \perverse" equilibrium where agents prefer x1 over x2 matches, even though x2 > x1 , because they believe 1 In the labor market contect, searching while matched is referred to as on-the-job search, and has been previously studied by Burdett (1977) in a single-agent framework, by Mortensen (1978) in the context of two agents, and by Pissarides (1994) and Webb (1998) in simple general equilibrium models, for example. See Mortensen and Pissarides (1998) for other related references. None of the existing papers looks at the main issue considered here, however, which is the possibility of endogenous instability. 3 the former are more stable, and this belief is correct because agents search in x2 matches and not in x1 matches (a special case occurs when x1 = x2 , so that there is really no fundamental di®erence between matches, but people simply believe that certain relationships will be less stable than others). We also analyze welfare, and derive the result that there tends to be too much search in equilibrium.2 We also show that this leads to excessive unemployment and excessive inequality: compared to the solution to the social planner's problem, as the equilibrium tends to have too much search, it also has too many people unmatched, too many in x2 matches, and too few in x1 matches. We go on in Section 4 to consider the case where the match value x is drawn from a general, rather than a two-point, probability distribution. We consider the class of equilibria where agents choose a reservation match value R, such that they enter relationships i® x ¸ R, and a critical match value Q > R, such that they search while matched i® x · Q. Even within this potentially limited class, it turns out that under reasonable parameteric assumptions we can generate a continuum of equilibria: any value of Q in some interval is consistent with equilibrium. We also derive the distribution of agents across states and distribution of match values across existing re- lationships, and show how these vary across equilibria. In general, we show that equilibria with more search (a higher value of Q) generate more unem- ployment and greater inequality. We interpret our ¯ndings, that there can 2 To be more precise, we show that endogenous instability tends to generate too much search, although there is another e®ect that tends to generate too little search, according to an ex ante welfare criterion, that arises here as in any typical search models where agents are impatient. If the discount rate is not too big, however, this standard impatience e®ect is dominated by the new endogenous instability e®ect, and there will be either too much search or an e±cient amount of search depending on other parameter values. 4 exist a great multiplicity of equilibria, with di®erent search behavior and dif- ferent implications for unemployment, inequality and welfare, to mean that at least theoretically endogenous instability can be a powerful force. 2 The Basic Framework Time is continuous. There is a [0; 1] continuum of in¯nitely-lived agents who are interested in forming (bilateral) partnerships. While unmatched, agents search, and while doing so they receive an instantaneous payo® b, which can be thought of as the utility from being single net of search costs. While searching, they meet other agents according to a Poisson process with arrival rate ®. All agents are homogeneous ex ante but matches are heterogeneous ex post: when any pair meet, they draw a random variable x giving the instantaneous utility that each will receive if they enter into a relationship. Agents are always free to reject a potential partner in favor of continued search { as they presumably would do if they draw a low value of x. The distribution of potential match qualities is given by F (¹) = pr(x · x) and is x ¹ assumed to be exogenous; however, the distribution of match qualities across existing relationships, call it G(x), is endogenous, given that agents accept some values of x and reject others.3 3 One can assume that in each meeting there is a realization of total surplus X, and that agents bargain in such a way that each gets x = X=2 (as would follow from at least some versions of standard bargaining theory). Alternatively, one can assume nontrannsferrable utility. A closely related model with nontrannsferrable utility (but no on-the-job search) is contained in Burdett and Wright (1998), although there it is assumed that when two agents i and j meet they draw a pair (xi ; xj ) representing the utlity each would receive from the match. In that model, xi and xj are independent. The present model can be thought of as a version of that general structure with perfect dependence: xi = xj = x. In any case, nothing we do in this paper depends on whether one interprets utilty as nontrasferrable, or as trasferrable with bargaining dividing the surplus evenly. 5 While in a relationship, agents enjoy the match value x, and also decide whether to continue searching. If they search they pay cost d > 0 and continue to meet other agents at the same rate ®; if they do not search they pay no cost and meet no one new.4 When a matched agent meets someone new, we assume that he ¯rst leaves his current partner and then draws the new value of x associated with the other person. He may then form a relationship with the new person at that value of x or reject the new person and become unmatched, but he cannot go back to his previous partner. One could also analyze the model where agents are allowed to go back to their previous partner after checking out a potential alternative; obviously, in this case they will choose whichever relationship yields the greater x. However, we think that our assumption not only seems realistic, at least in some contexts, it is also the appropriate assumption if one wants to focus on the strategic interactions within relationships. To explain this, consider the opportunities available when you search. First, your arrival rate of meetings ® could depend on the number of other agents that are searching. As this e®ect has been studied extensively in the past, here we assume constant returns to scale in the meeting technology, which implies that ® is a ¯xed constant.5 However, even give a constant ®, if we allow a matched agent who meets someone new to choose between 4 If d = 0, all agents will search, as in some previous analyses of related models, such as Webb (1998) or Burdett and Coles (1999); hence, those models cannot address the key aspect on which we focus { the decision whether to search while matched. 5 Let M (N ) be the total number of meetings in the market per unit time when the number (measure) of agents who are searching is N > 0. The arrival rate for a representa- tive individual is given by ® = M (N )=N , and so as long as M exhibits constant returns, ® does not depend on N . Intuitively, if N is bigger there are more people you might meet, but you are also competing with more people to meet them, and constant returns implies that these e®ects exactly o®set. 6 the new person and his current partner, your e®ective arrival rate will still be endogenous. To see this, observe that when you meet matched agents, if they are allowed to choose, they will only enter a relationship with you if you beat their current value of x, and therefore to know your chances you need to know the endogenous distribution of x within existing matches, G(x). Since our assumption is that a matched agent can either go with you or become unmatched, but cannot stay with his current partner, meeting a matched agent is the same as meeting an unmatched agent. Summarizing, in our model your e®ective arrival rate (if not your payo®) is una®ected by what other agents are doing: you have a constant probability of meeting someone, and every meeting is a draw from F (x) which can either be accepted or rejected in favor of unmatched search. Our assumptions may somewhat discourage search while matched, compared to alternatives such as allowing matched agents the option of staying with their current part- ners, but in any case we will show below that there is too much search in equilibrium, and if anything this result is more striking give these assump- tions. Moreover, they simplify the analysis. One thing that keeps the model relatively tractable is the following: given that you have decided to search while matched, when you meet someone new, there is no additional decision to leave or stay with your current partner (i.e., to stay without observing the new x; once you observe the new x our assumptions do not allow you to stay). This is because all new people look identical ex ante, so if you would not leave with one new person then you would not leave with any, and therefore you would not be engaged in costly search in the ¯rst place. Hence, matched agents engaged in search are always willing to leave their 7 current partners as soon as they meet someone else. For generality, we also assume that all matches are terminated exogenously according to an inde- pendent Poisson process with arrival rate ¾. Whenever you are thrown out of a relationship, either because of an exogenous separation or because your partner meets someone new, you return to unmatched search. You also re- turn to unmatched search when you are in a relationship, meet someone new, and ¯nd them lacking. Thus, in the context of labor markets, the model has well-de¯ned notions of quits and layo®s { voluntary and involuntary sepa- rations, if you will { concepts that are often hard to distinguish in theory. However, we will not belabor this interpretation in the analysis that follows. 3 A Simple Model Suppose that x = x2 with probability ¼ and x = x1 < x2 with probability 1¡¼. Agents can be in one of three states: unmatched, in an x1 relationship, or and in an x2 relationship. The fraction in each state is denoted N0 , N1 , and N2 , respectively, where N0 + N1 + N2 = 1. Let the payo®, or value, function of an agent in each state be denoted V0 , V1 , and V2 . Agents need to choose strategies for deciding when to accept a match and when to search while matched. Let Ai be the probability that a representative agent agrees to enter into an xi match and let Si be the probability that he searches while in an xi match, i = 1; 2. The value functions satisfy the following continuous time dynamic pro- gramming equations: rV0 = b + ®¼A2 (V2 ¡ V0 ) + ®(1 ¡ ¼)A1 (V1 ¡ V0 ) 8 rV1 = x1 + (¾ + S1 ®)(V0 ¡ V1 ) + S1 §1 (1) rV2 = x2 + (¾ + S2 ®)(V0 ¡ V2 ) + S2 §2 ; where §i is the net gain from searching while in an xi match: §1 = ®¼[A2 V2 + (1 ¡ A2 )V0 ¡ V1 ] + ®(1 ¡ ¼)(1 ¡ A1 )(V0 ¡ V1 ) ¡ d §2 = ®¼(1 ¡ A2 )(V0 ¡ V2 ) + ®(1 ¡ ¼)[A1 V1 + (1 ¡ A1 )V0 ¡ V2 ] ¡ d: For example, the third equation in (1) equates the °ow value rV2 to the sum of the instantaneous utility x2 , plus the probability you become unmatched, either because of an exogenous separation or because your partner meets someone new, ¾ + S2 ®, times V0 ¡ V2 , plus S2 times your net gain from search.6 Given (A1 ; A2 ; S1 ; S2 ) one can determine (N0 ; N1 ; N2 ). We will do so below, but it is important to note that we do not need to know (N0 ; N1 ; N2 ) in order to analyze equilibrium strategies, since (N0 ; N1 ; N2 ) does not enter (1). This is a consequence of two assumptions. Due to constant returns in the meeting technology, the arrival rate ® does depend on the number of agents searching, N0 + N1 S1 + N2 S2 . Moreover, even if everyone is searching, so that ® is determined, suppose you are unmatched and meet a matched agent. Under our assumption that he cannot return to his previous partner, this is equivalent to meeting an unmatched agent. If we alternatively assumed that he could return to his previous partner, he will match with you only if the new match is better, and you would have to know (N0 ; N1 ; N2 ) to calculate the probability of this event. 6 One could be more explicit at the cost of additional notation by saying that you choose sj taking as given others choose Sj . Then one would write rV2 = x2 + (¾ + S2 ®)(V0 ¡ V2 ) + s2 §2 , for example. In equilibrium, of course, sj = Sj . 9 A (steady state) equilibrium can now be de¯ned as a list including value functions (V0 ; V1 ; V2 ) satisfying (1), the distribution across states (N0 ; N1 ; N2 ) satisfying conditions given below, and strategies (A1 ; A2 ; S1 ; S2 ) satisfying the following best response conditions: 8 8 > 1 < if ¢j > 0 > 1 < if §j > 0 Aj = > [0; 1] if ¢j = 0 ; Sj = > [0; 1] if §j = 0 (2) : : 0 if ¢j < 0 0 if §j < 0 where ¢j = V1 ¡ V0 is the net gain from accepting an xj match, while §j is the net gain to searching while in an xj match, de¯ned above. Consider for now pure strategy equilibria, given by setting each of the elements of (A1 ; A2 ; S1 ; S2 ) to either 0 or 1. There are sixteen such strategy pro¯les; however, only ¯ve potentially constitute equilibria. First, there is a \degenerate" or type D equilibrium where agents reject all matches: A1 = A2 = 0. Second, there is a \choosy" or type C equilibrium where agents accept x2 but reject x1 , and { naturally, since they would not accept x1 { they do not search in x2 matches: A1 = 0, A2 = 1 and S2 = 0. Third, there is a \faithful" or type F equilibrium where agents accept x1 as well as x2 and do not search in either case: A1 = A2 = 1 and S1 = S2 = 0. Fourth, there is an \unfaithful" or type U equilibrium, in which agents accept both matches but now continue to search in x1 matches: A1 = A2 = 1, S1 = 1 and S2 = 0. Finally, there is a \perverse" or type P equilibrium where agents accept both and, perhaps counter to all intuition (but see below), search in x2 but not x1 matches: A1 = A2 = 1, S1 = 0 and S2 = 1. These are all the potential equilibria, at least given the normalization x2 > x1 (if we reverse the inequality then other cases arise, but they are merely relabellings of what has already been considered). Any other can- 10 didate equilibrium can easily be ruled out. For example, suppose A1 = 0, A2 = 1, and S2 = 1. These strategies imply §2 = ®(1 ¡ ¼)(V0 ¡ V2 ) ¡ dS1 and ¢2 = V2 ¡ V0 ; but the best response conditions require §2 > 0 and ¢2 > 0, which is a contradiction. Intuitively, it cannot be an equilibrium to accept and then spend resources to search in x2 matches since there is no possible gain if you are going to reject x1 matches. The other cases are similar. The next step is to derive parameter values for which each of our ¯ve candidate equilibria exist. Consider ¯rst type D equilibrium, where A1 = A2 = 0, and therefore rV0 = b. We use the unimprovability principle: to check whether strategies constitute an equilibrium, it su±ces to show the payo®s from using these strategies cannot be improved by deviating, in any possible contingency, one time and then reverting to the candidate strategies. Consider the payo® to deviating from A2 = 0 by accepting an x2 match. It cannot be optimal to accept the match and then search, given that you revert to the candidate strategy A1 = A2 = 0 (it would be better to quit immediately). So we can set S2 = 0, which means rV2 = x2 + ¾(V0 ¡ V2 ), and this implies ¢2 is proportional to x2 ¡ b. Hence, ¢2 · 0, and A2 = 1 does not improve your payo®, i® x2 · b. Similarly, A1 = 1 does not improve your payo® i® x1 · b, although this is not binding. Hence, type D equilibrium exists i® x2 · b, shown as region D in Figure 1, which also shows the regions where each of the equilibria exist in (x1 ; x2 ) space (note that only the area above the 45o line is relevant since x2 > x1 ). Now consider type F equilibrium, where A1 = A2 = 1 and S1 = S2 = 0. 11 This implies rV0 = b + ®¼(V2 ¡ V0 ) + ®(1 ¡ ¼)(V1 ¡ V0 ) rV1 = x1 + ¾(V0 ¡ V1 ) rV2 = x2 + ¾(V0 ¡ V2 ): For this to be an equilibrium, we require ¢1 ¸ 0, which holds i® (r + ¾ + ®¼)x1 ¡ (r + ¾)b x2 · y1 = ; ®¼ and ¢2 ¸ 0, which is not binding. We also require §1 · 0, which holds i® (r + ¾)d x2 · y2 = x1 + ; ®¼ and §2 · 0, which is not binding. Hence, type F equilibrium exists in the region F, to the southeast of both lines y1 and y2 , in Figure 1. Intuitively, for this equilibrium to exist, we need x1 to be big enough that agents accept it and x2 to be small enough that agents stop searching once they accept x1 . Consider type U equilibrium, where A1 = A2 = 1, S1 = 1 and S2 = 0. This implies rV0 = b + ®¼(V2 ¡ V0 ) + ®(1 ¡ ¼)(V1 ¡ V0 ) rV1 = x1 + (¾ + ®)(V0 ¡ V1 ) + ®¼(V2 ¡ V1 ) ¡ d rV2 = x2 + ¾(V0 ¡ V2 ): We require ¢1 ¸ 0, which holds i® x1 ¸ b + d, and ¢2 ¸ 0, which is not binding. We also require §1 ¸ 0, which holds i® µ ¶ r+¾+® ® r+¾ ® x2 ¸ y3 = x1 + b+ + d; r + ¾ + 2® r + ¾ + 2® ®¼ r + ¾ + 2® 12 and §2 · 0, which is not binding. The region where type U equilibrium exists is shown as region U in the Figure. Notice that we need x2 to be big enough that agents want to search in x1 matches, and also x1 ¸ b + d, so that agents prefer to search while matched than to search while unmatched. Consider now type P equilibrium, where A1 = A2 = 1, S1 = 0 and S2 = 1. This implies rV0 = b + ®¼(V2 ¡ V0 ) + ®(1 ¡ ¼)(V1 ¡ V0 ) rV1 = x1 + ¾(V0 ¡ V1 ) rV2 = x2 + (¾ + ®)(V0 ¡ V2 ) + ®(1 ¡ ¼)(V1 ¡ V2 ) ¡ d: We require ¢2 ¸ 0, which holds i® x2 ¸ b + d, and ¢1 ¸ 0, which is not binding. Also, we require §2 ¸ 0, which holds i® r + ¾ + 2® ® (r + ¾)(r + ¾ + 2®) + ®2 (1 ¡ ¼) x2 · y4 = x1 ¡ b¡ d; r+¾+® r+¾+® ®(1 ¡ ¼)(r + ¾ + ®) and §1 · 0, which is not binding. See region P in the Figure. In this strange case, agents prefer x1 over x2 matches even though x1 < x2 , because these strategies make x1 matches are more secure. This is only possible if x1 is not too much less than x2 , since agents will only sacri¯ce so much instantaneous utility for security, and also x1 and x2 are large relative to b + d, since it is a high cost of becoming unmatched that makes security important. Finally, consider type C equilibria, where A1 = 0, A2 = 1 and S2 = 0. This implies rV0 = b + ®¼(V2 ¡ V0 ) rV1 = x1 + (¾ + S1 ®)(V0 ¡ V1 ) rV2 = x2 + ¾(V0 ¡ V2 ): 13 Notice we have left S1 in the expression for rV1 . It turns out that, even though no one accepts an x1 match, it matters what they believe about S1 (i.e., it matters what agents believe would happen o® the equilibrium path). It can be shown that a type C equilibrium exists i® x1 · b + d and x2 ¸ y1 , although the beliefs concerning S1 have to be di®erent, depending on parameter values.7 See region C in the Figure. This completes our characterization of existence, with the results in terms of pure strategy equilibria summarized in Figure 1.8 The ¯gure is drawn making no assumptions about parameter values, other than obvious things such as r > 0, ¼ 2 (0; 1), etc. { everything one needs to know for all of the 7 Explicitly, given x1 · b + d and x2 ¸ y1 , there is an equilibium with A1 = 0, A2 = 1, S2 = 0, and: S1 = 1 if x2 ¸ y5 ; S1 = 0 if x2 · y6 ; and S1 2 (0; 1) if x2 2 (y6 ; y5 ), where (r + ¾ + ®¼) ®¼ ¡ (1 ¡ ¼)(r + ¾) (r + ¾ + ®)(r + ¾ + ®¼) y5 = x1 + b+ d ¼(2® + ¾ + r) ®¼(2® + ¾ + r) ®¼(2® + ¾ + r) (r + ¾ + ®¼) (1 ¡ ¼)(r + ¾) (r + ¾)(r + ¾ + ®¼) y6 = x1 ¡ b+ d: ¼(® + ¾ + r) ¼(® + ¾ + r) ®¼(® + ¾ + r) Notice that when x2 2 (y6 ; y5 ) the equilibrium requires agents use mixed strategies. In any case, all of the type C equilibria are observationally equivalent, since no one ever accepts an x1 match in equilibrium. 8 Although we focus mainly on pure strategy equilibria, for completeness, we show that when the type F and type U equilibria coexist one can also construct a mixed-strategy equilibrium as follows. Suppose agents accept x1 matches and are indi®erent between searching and not searching, and respond by searching with probability S1 2 (0; 1) (your partner does not know if you are searching, only the probability S1 ). The payo® to searching and not searching in an x1 match must be the same; hence x1 + (¾ + S1 ®)(V0 ¡ V1 ) + ®¼(V2 ¡ V1 ) ¡ d = x1 + (¾ + S1 ®)(V0 ¡ V1 ): Solving for S1 , we have [®¼(x2 ¡ x1 ) ¡ (r + ¾)d] (¾ + r + ®) S1 = : ®[®¼(b + d ¡ y) + (r + ¾)d] It is easy to show that S1 2 (0; 1), and hence the mixed strategy equilibrium exists, i® x2 2 (y3 ; y2 ), as claimed. 14 qualitative properties (e.g., relative slopes and intercepts) holds generally. As mentioned above, type F and type U equilibria coexist when x2 2 (y3 ; y2 ). In this case, agents in x1 matches may either be \faithful" or \unfaithful" depending on what other agents are doing. If other agents are searching in x1 matches then you prefer to search in x1 matches because these matches are unstable, and therefore, speaking heuristically, you are less inclined to be satis¯ed with your x1 match, and more inclined to desire an x2 match, which is relatively secure as well as generating higher instantaneous utility. Indeed, this e®ect is su±ciently powerful that one can produce a type P equilibrium where agents \perversely" prefer x1 matches, even though x1 < x2 , because they believe these matches are more stable, and hence they search for an x1 match while in x2 matches, thus rationalizing their beliefs. A special case of the type P equilibrium occurs when x1 = x2 (on the 45o line in the Figure), which means that there is really no fundamental di®erence between matches, but people simply believe that certain relationships will be unstable. This has an interpretation in terms of discrimination. For example, suppose all agents are distinguished by one of two identi¯able characteristics, say black and white. It is logically possible for individuals to believe that black-white relationships will be less stable that black-black or white-white relationships, and for this belief to be true in equilibrium, even if color has zero impact on payo®s. As we said earlier, however, notice that for the type P equilibrium to exist we need x1 and x2 similar and su±ciently bigger than b + d. Also notice that whenever the type P equilibrium does exist there also exists another equilibrium, either type U or both type F and type U, depending on x1 and x2 . 15 As mentioned, given strategies one can compute (N0 ; N1 ; N2 ). Taking account of the °ows between states, one can see that, in general: _ N1 = N0 ®(1 ¡ ¼)A1 + N2 S2 ®(1 ¡ ¼)A1 ¡N1 [¾ + S1 ® + S1 ®(1 ¡ ¼)(1 ¡ A1 ) + S1 ®¼] _ N2 = N0 ®¼A2 + N1 S1 ®¼A2 ¡N2 [¾ + S2 ® + S2 ®(1 ¡ ¼) + S2 ®¼(1 ¡ A2 )]: For example, the °ow into x1 matches consists of those who are unmatched, get an x1 draw and accept, it plus those in x2 matches who are searching, get an x1 draw and accept it; the °ow out consists of those in x1 matches whose relationships break up exogenously, those who are abandoned by their partners, those who are searching and get an x1 draw and reject it, plus those who are searching and get an x2 draw (whether they accept or reject it). _ _ Steady state solves N1 = N2 = 0, as well as N0 = 1 ¡ N1 ¡ N2 . Letting NiT denote the fraction of agents in state i in a type T equilibrium, we have: D D D N0 = 1 N1 = 0 N2 = 0 C ¾ C C ®¼ N0 = N1 = 0 N2 = ®¼ + ¾ ®¼ + ¾ F ¾ F ®(1 ¡ ¼) F ®¼ N0 = N1 = N2 = ®+¾ ®+¾ ®+¾ U ¾(¾ + ® + ®¼) U ¾®(1 ¡ ¼) U ®¼(¾ + 2®) N0 = N1 = N2 = ·U ·U ·U P ¾(¾ + 2® ¡ ®¼) P ®(1 ¡ ¼)(¾ + 2®) P ¾®¼ N0 = N1 = N2 = ·P ·P ·P where to save space we have written ·U = (¾ + 2®)(®¼ + ¾) and ·P = (¾ + 2®)[®(1 ¡ ¼) + ¾]. One can easily show the following. In terms if the number of unmatched C P U F P S agents, we have N0 > N0 ; N0 > N0 , with N0 > N0 i® ¼ > 1=2. Intuitively, 16 the type C equilibrium where agents are \choosy" and reject x1 matches maximizes N0 , while the type F equilibrium where agents all matches and never voluntarily leave their partners (since they are \faithful") minimizes N0 . In particular, in the labor market context, one should interpret N0 as unemployment, then the conclusion is that the unemployment rate is lowest when there is no on-the-job-search. In terms if the number of agents in x1 P F U C matches, we have N1 > N1 > N1 > N1 . Thus, the type P equilibrium, where agents \perversely" prefer x1 over x2 matches, maximizes N1 . In terms C U F P of the number of agents in x2 matches, we have N2 = N2 > N2 > N2 . Intuitively, equilibria where agents either accept only x2 matches or accept x1 matches but continue to search will maximize N2 , while naturally the type P equilibrium where agents in x2 matches search for x1 matches minimizes N2 . We now proceed to a discussion of e±ciency, de¯ned in terms of the standard social planner's welfare criterion: W = N0 V0 + N1 V1 + N2 V2 : As x2 < b implies the e±cient outcome is obviously A1 = A2 = 0, let us proceed to the more interesting case of x2 > b. In this case, any e±cient outcome clearly entails A2 = 1 and S2 = 0 (and so the type P equilibrium cannot be e±cient); it is not so clear, however, what the planner will choose for A1 and S1 { i.e., whether he prefers the type C, type F or type U strategies. To determine this we need to derive the closed form for W as a function of 17 A1 and S1 . Inserting Vi and Ni , after some algebra, one can derive:9 f¾®S1 [2¡(1¡¼)A1 ]+¾2 gb+¾®(1¡¼)A1 (x1 ¡S1 d)+®¼(¾+2®S1 )x2 rW = ¾®(1¡¼)(1¡S1 )A1 +(¾+®¼)(¾+2®S1 ) : (3) There are three relevant possibilities for the planner: he can pick the type C strategy, A1 = 0, which yields ®¼x2 + ¾b rW = ; ®¼ + ¾ he can pick the type F strategy, A1 = 1 and S1 = 0, which yields ®(1 ¡ ¼)x1 + ®¼x2 + ¾b rW = ; ®+¾ or he can pick the type U strategy, A1 = 1 and S1 = 1, which yields ®¾(1 ¡ ¼)[x1 ¡ (b + d)] + (¾ + 2®)(®¼x2 + ¾b) rW = : (¾ + 2®)(®¼ + ¾) It is easy to verify that the planner's strategy is A1 = 0 if x1 · b + d and x2 ¸ yA , where ®¼ + ¾ b yA = x1 ¡ ; ®¼ ®¼ it is Ax = 1 and Sx = 0 if x2 · yA and x2 · yS , where ¼(¾ + 2®) + ¾ ¾b ¾(¾ + ®)d yS = x1 ¡ + ; ¼(¾ + 2®) ¼(¾ + 2®) ®¼(¾ + 2®) and it is Ax = 1 and Sx = 1 if x1 ¸ b + d and x2 ¸ yS . In words, the planner wants agents to accept x1 matches i® x1 is su±ciently big, and to search in these matches i® x2 is su±ciently big. We want to compare e±cient and equilibrium outcomes. For this purpose, we ignore the type D equilibrium, which is always e±cient if it exists, and 9 In this derviation we have already inserted A2 = 1 and S2 = 0. Notice that the reduced form for W is a linear combination of the instantaneous utilities in each of the states: b, x1 ¡ S1 d, and x2 . 18 the type P equilibrium, which is always ine±cient, and concentrate on the e±ciency properties of type F , type U and type C equilibria. It facilitates the exposition to begin with the limiting case r ! 0, since in this case the line y1 that divides equilibrium regions C and N coincides with the line yA that divides the planner's choice between C and N { i.e., when r ! 0, the equilibrium choice of A1 is always e±cient, and we can concentrate for now on the e±ciency of S1 . Figure 2 shows a version of Figure 1 drawn with r = 0, and highlights two regions in which the equilibrium di®ers from the planner's choice. In the region labeled 1, a planner would choose the type F strategy but the unique equilibrium is type U. In this case, there is too much search, in the sense that agents always search in x1 matches in equilibrium even though this is ine±cient. In the region labeled 2, the a planner would again choose the type F strategy but there are multiple equilibria, including type F but also including type U. In this case, the may be too much search in the sense that there is an equilibrium where agents search in type x1 matches even though this is ine±cient, although there is also an e±cient equilibrium. In all other regions the equilibrium coincides with the planner's choice. The general con- clusion is that, in the limiting case when r ! 0, there is a tendency towards too much search. The reason is simple: when you search while matched, you take into account your own bene¯ts and costs, but neglect the fact that when you meet someone new you abandon your current partner, and moreover, if the new partner is was already matched, they also abandon their current partner. We can use the steady state results derived above to discuss how the dis- 19 tribution of agents across states di®ers between the equilibrium and e±cient outcomes. When these outcomes do in fact di®er, in regions 1 and 2 of Figure 1, the ine±ciency aries when we are in a type U equilibrium but the e±cient U F out come entails type F strategies. As has been established, N0 > N0 , U F U F N2 > N2 , and N1 < N1 . Hence, the planner prefers fewer unmatched agents, fewer x2 matches, and more x1 matches. In other words, the e±cient outcome entails less inequality than the equilibrium outcome. It seems to be a general property that when there is too much search there will be too much inequality { after all, what agents are searching for is to move up in the income distribution (that is, the x distribution here). It is not hard to understand how excessive search leads to too many people in the upper tail (that is, at x2 ); what is perhaps slightly less obvious is that it also leads to too many people in the lower tail (unmatched), since their excessive search generates excessive separations.10 The e®ect that generates too much search (you neglect the impact on other agents when they are abandoned) is always there; however, there is another e®ect that arises when r > 0 that goes the other way. This other e®ect generates too little search, according to criteria W , because impatient agents are less inclined than the planner to, in the ¯rst place, reject an x1 match, and in the second place, to keep searching while in an x1 match. This is because the gains to ¯nding an x2 match accrue only in the future. Figure 3 generalizes Figure 2 to the case of r > 0, and shows that, in addition to the regions 1 and 2 with too much search, there are three new regions with too little. 10 This logic does not rely on a two-point x distribution; as we shall see, something similar occurs in the generalization of the model in the next section. 20 In the region labeled 3, the planner chooses the type C strategy but the unique equilibrium is type F : impatient agents accept x1 matches and stop searching, while the planner wants them to reject these matches. In the region labeled 4, the planner chooses the type U strategy but the unique equilibrium is type F : impatient agents accept x1 matches and stop searching, while the planner wants them to accept these matches but continue to search. Finally, in the region labeled 5, the planner chooses the type U strategy but there are multiple equilibria, including type U but also including type F. Note that, for a given r, these regions occur only in the range where x1 and x2 are relatively small { for x1 or x2 su±ciently big, the equilibrium must either entail too much search or e±cient search.11 We close this section by reiterating that the source of the multiplicity here has nothing to do with the standard thick-market externality, but results ex- clusively from the strategic interaction between the partners: if other agents are searching then matches are less secure and hence less valuable, and so you are more inclined to search. This can lead to multiple equilibria even without thick-market e®ects. Moreover, our endogenous instability e®ect tends to generate too much search, while standard thick-market e®ects tends to generate too little search. While there are other e®ects discussed in the literature that can also lead to excessive search, the endogenous instability e®ect seems new and di®erent. 11 Equivalently, given any x1 and x2 , we can pick r small enough that there will be either too much search or e±cient search, but not too little search. 21 4 The General Model We now present an extension of the model in the previous section, which is exactly the same, except that we now allow match quality x ¸ 0 to have a general distribution described by the cumulative distribution function F (x). If F is di®erentiable we denote the density by f (x), but we do not require di®erentiability for anything important. We will restrict attention to equilib- ria with the following property: unmatched agents enter into relationships i® they draw a value of x above the reservation match quality R; and matched agents stop searching i® x is above the critical match quality Q. Moreover, we are mainly interested in equilibria with Q > R (since otherwise there is no search while matched). Even given this restrictions to a certain class of outcomes, we will show that there can exist a continuum of equilibria. Generalizing the discussion in the previous section, an equilibrium here is de¯ned as a list including the value functions for agents who are unmatched and agents who are in a relationship with match value x, [V0 ; V (x)], a steady state distribution that can be characterized by the number of unmatched agents and the distribution of match values across existing relationships, [N0 ; G(x)], and strategies (R; Q) satisfying conditions to be given below. Note that for ease of presentation we assume as a tie-breaking rule that agents stop searching while matched i® x > Q (i.e., they search when they are indi®erent); this is not particularly important for anything, but it allows us to write the fraction of matched agents enaged in search as G(Q). To begin the analysis, observe that if you are unmatched your value func- 22 tion satis¯es Z 1 Z 1 rV0 = b + ® max[V (z) ¡ V0 ; 0]dF (z) = b + ® [V (z) ¡ V0 ]dF (z): (4) 0 R where R is the reservation match quality, given by V (R) = V0 . Also, if you are in a relationship with match quality x your value function satis¯es rV (x) = x + (¾ + S®)[V0 ¡ V (x)] + s§(x) (5) where S = 1 if your partner is searching and 0 otherwise, and Z 1 §(x) = ®[V0 ¡ V (x)] + ® [V (z) ¡ V0 ]dF (z) ¡ d R is your expected gain from search.12 In any equilibrium where Q > R, we know that x = R implies s = 1 (agents always search in relationships at the reservation match value), which means Z 1 rV (R) = R + ® [V (z) ¡ V0 ]dF (z) ¡ d: (6) R Comparing (6) and (4), one sees immediately that R = b + d. Thus, un- matched agents will agree to form a relationship i® the instantaneous return net of the cost of continued search, x ¡ d, exceeds b.13 12 Equivalently, Z R Z 1 §(x) = ® [V0 ¡ V (x)]dF (z) + ® [V (z) ¡ V (x)]dF (z) ¡ d: 0 R Notice that we are maintaining the assumption that, if you search while matched, you must leave your current partner when you meet someone new. Hence, when you meet someone new and draw an x below R you will become unmatched. This assumption is made here, as in the previous section, so that one does not have to know N0 or G(x) to solve the individual agent's decision problem. 13 Recall that in the previous section we could have an equilibrium of type F where agents accept matches with x < b + d. This does not contradict the current result that R = b + d, since the current result is predicated on being in an equilibria with Q > R, and in the case in the previous section has Q < R (agents do not search while in realtionships at the reservation match quality). 23 To proceed, we now solve for V (x). First, rewrite (5) as rV (x) = x ¡ sd + (¾ + S® + s®)[V0 ¡ V (x)] + s®I; R1 where I = b+d [V (z) ¡ V0 ]dF (z) does not depend on x. Then insert V0 = (b + ®I)=r into V (x) and rearrange to yield r(x ¡ sd) + (¾ + S® + s®)b + (¾ + S® + s® + sr)®I V (x) = : (7) r(r + ¾ + S® + s®) Moreover, for future reference we note that I can be simpli¯ed as Z 1 Z Q Z 1 0 [1 ¡ F (z)]dz [1 ¡ F (z)]dz I= [1 ¡ F (z)]V (z)dz = + dz b+d b+d r + ¾ + 2® Q r+¾ by integrating by parts and inserting V 0 (x) from (7). This expression gives I = I(Q) as a function of Q, but otherwise it depends only on exogenous variable. Notice that I is a decreasing funcion of Q. It is now useful to express a matched agent's payo® explicitly as a function of x, his search decision s, the search decision of his partner S, and the rule being used by all other agents, which is to search while matched i® x · Q. Letting this payo® be denoted vsS (x; Q), we have from (7) r(x ¡ d) + (¾ + 2®)b + (¾ + 2® + r)®I(Q) v11 (x; Q) = r(r + ¾ + 2®) rx + (¾ + ®)b + (¾ + ®)®I(Q) v01 (x; Q) = r(r + ¾ + ®) r(x ¡ d) + (¾ + ®)b + (¾ + ® + r)®I(Q) v10 (x; Q) = r(r + ¾ + ®) rx + ¾b + ¾®I(Q) v00 (x; Q) = : r(r + ¾) Figure 4 shows the four vij functions. Notice that they are linear in x, with @v00 =@x > @v01 =@x = @v10 =@x > @v11 =@x. Also, as shown in the ¯gure, 24 we have v10 = v11 > v01 ; v00 at x = b + d, which must be true as long as Q > R = b + d.14 Given all agents search i® x · Q, your value function V (x) is given by maxfv01 ,v11 g when your partner is searching and maxfv00 ,v10 g when your partner is not searching { as shown by the thick line in the ¯gure. Given any Q > b + d, it is clear from Figure 4 that there is a unique q0 (Q) > b + d such that v00 = v10 (that is, your best response switches from searching to not searching, given that your partner is not searching), and a unique q1 (Q) > b +d such that v01 = v11 (that is, your best response switches from searching to not searching, given that your partner is searching). So, given Q, your best response to your partner's behavior is to search i® x · q0 when S = 0, and to search i® x · q1 when S = 1. Figure 4 depicts a situation with b + d < q0 (Q) < Q < q1 (Q), in which case it should be clear that it is an equilibrium for all agents to search while matched i® x · Q, and moreover that we have Q > R, as we have been assuming throughout the argument. To see why this constitutes an equilibrium, suppose that everyone searches i® x · Q: if you are in a match with x · Q, then S = 1 and you want to search because x < q1 (Q); while if you are in a match with x > Q, then S = 0 and you do not want to search because x > q0 (Q). The remaining di±culty is to show that the situation depicted in the ¯gure, b + d < q0 (Q) < Q < q1 (Q), can actually arise (which is not obvious, because as Q varies all of the curves shift, and hence so do q0 and q1 ). To 14 This last result has a simple intuitive interpretation: given that you are indi®erent between accepting x = R = b + d and rejecting it to remain unmatched, it does not matter to you if your partner searches, since all their search does is increase your probability of returning to the unmatched state. Since Q > b + d, the relevant value functions at b + d are v10 and v11 , and so we conclude v10 = v11 at b + d. The result v1j = v0j also follows from the fact that searching is better than not searching at b + d, given Q > b + d. 25 this end, we ¯rst equate v0j = v1j and solve explicitly for qj (Q): ®q0 (Q) = ®b ¡ (¾ + ® + r)d + (¾ + 2® + r)®I(Q) ®q1 (Q) = ®b ¡ (¾ + r)d + (¾ + ® + r)®I(Q): As seen in Figure 5, the functions q0 (Q) and q1 (Q) are decreasing { because I(Q) is { and therefore each has at most one ¯xed point. Consider the following mild parameter restriction, which always holds if d is small (as one naturally needs to construct an equilibrium where agents accept matches and continue to search): Z 1 (r + ¾)2 d [1 ¡ F (z)]dz > : (8) b+d ® (r + ¾ + 2®) It is easy to verify that (8) guarantees q0 (b + d) > b + d, and so the ¯xed point of q0 (Q), call it q, satis¯es q > b + d. Also, one can verify that q > b + d guarantees q1 (q) > q, and so the ¯xed point of q1 (Q), call it q, satis¯es q > q. Summarizing, (8) implies b + d < q < q, and so if we choose any Q 2 [q; q] we have b + d < q0 (Q) · Q · q1 (Q), exactly the situation depicted in Figure 4. In other words, there exists a nondegenerate interval [q; q] with the prop- erty that any Q 2 [q; q] constitutes an equilibrium in the class under con- sideration. What lies behind this continuum of steady state equilibria? The answer is the discontinuity at x = Q, as seen in Figure 4. As x crosses the critical value Q there is a discrete jump upward in V (x) because other people stop searching; moreover, you strictly prefer to search when x < Q and strictly prefer not to search when x > Q (because the relevant branches of vij change from vi1 to vi0 when other agents stop searching). If Q1 is an equilibrium and we change Q1 to Q2 in the neighborhood of Q1 , it will be 26 the case that you strictly prefer to search for x < Q1 and strictly prefer not to search for x > Q1 . Hence, if Q1 is a best response to itself, any Q in the neighborhood of Q1 is also a best response to itself. Given Q and F (R) = ', to complete the characterization of an equilib- rium we describe the distribution of agents across states (which we need this to discuss inequality). To begin, as a preliminary step, we take G(x) as given, and observe that the number of unmatched agents evolves according to:15 _ N0 = (1 ¡ N0 )¾ + (1 ¡ N0 )G(Q)(® + ®') ¡ N0 ®(1 ¡ '): _ Setting N0 = 0, we can solve for the steady state value of N0 as a function of G(Q), ¾ + G(Q)®(1 + ') N0 = : (9) ¾ + G(Q)®(1 + ') + ®(1 ¡ ') To derive the distribution of match quality, ¯rst note that G(b + d) = 0. Next, denote the measure of set of agents who are matched with match quality x · x by ¹(¹) = (1 ¡ N0 )G(¹). For x 2 [b + d; Q], this evolves ¹ x x according to ¹(x) = N0 ®[F (x) ¡ '] + (1 ¡ N0 )[G(Q) ¡ G(x)]®[F (x) ¡ '] _ ¡G(x)(1 ¡ N0 )f¾ + ® + ®' + ®[1 ¡ F (x)]g: For x > Q, ¹(x) evolves according to ¹(x) = N0 ®[F (x) ¡ '] ¡ (1 ¡ N0 )[G(x) ¡ G(Q)]¾ _ ¡G(Q)(1 ¡ N0 )f¾ + ® + ®' + ®[1 ¡ F (x)]g: 15 In words, the °ow into the set of unmatched agents is the number of matched agents who su®er an exogenous separation, plus the number of matched and searching agents who are abandoned by their partner or meet someone with match value below b + d, while the °ow out is the number of unmatched agents who meet someone with match value above b + d. 27 As another preliminary step, we can insert (9) into ¹(x) = 0 and solve for _ the steady state G(x), for any x ¸ b + d, as a function of G(Q): 8 > [F (x) ¡ '][¾ + 2®G(Q)] > > > if x 2 [b + d; Q] > > (1 ¡ ')(¾ + 2®) < G(x) = > (10) > ¾[F (x) ¡ '] + 2®G(Q)[1 ¡ F (x)] > > > > : if x > Q ¾(1 ¡ ') Now we can solve for G(Q) by setting x = Q in (10) and rearranging: ¾F (Q) ¡ ¾' G(Q) = : (11) ¾(1 ¡ ') + 2® ¡ 2®F (Q) Then we can substitute (11) into (10) to arrive at the ¯nal expression for the distribution of match values above b + d: 8 > > > ¾F (x) ¡ ¾' > > ¾(1 ¡ ') + 2® ¡ 2®F (Q) if x 2 [b + d; Q] > < G(x) = > (12) > (¾ + 2®)F (x) ¡ ¾' ¡ 2®F (Q) > > > > : if x > Q ¾(1 ¡ ') + 2® ¡ 2®F (Q) Similarly, we can substitute (11) into (9) to arrive at the ¯nal expression for the number of unmatched agents: ¾[¾ + 2® ¡ ®F (Q) + ®'] N0 = (13) (¾ + 2®)[¾ + ® ¡ ®F (Q)] This yields the closed form for the distribution of agents across states, as a function of exogenous variables and Q (which is endogenous but not pinned down by the model, due to the multiplicity of equilibrium). Several remarks are in order concerning this distribution. First, G trans- forms F by truncating it below b + d, scaling it linearly between b + d and Q, and again scaling it linearly (but di®erently) above Q. It is easy to verify that G(x) · F (x) for all x, with strict inequality on the interior of the support of 28 F (i.e., G ¯rst order stochastically dominates F ). Also, G is continuous at x = Q as long as F is, but in any case G has a kink at Q. Thus, if F has a density f , then G has a density g, but it is discontinuous at x = Q: 8 > > > ¾f (x) > > ¾(1 ¡ ') + 2® ¡ 2®F (Q) if x < Q > < g(x) = > (14) > > (¾ + 2®)f (x) > > > : if x > Q ¾(1 ¡ ') + 2® ¡ 2®F (Q) Figure 6 shows the density f , and the transformed density g for two dif- ferent values of the critical match value, Q1 and Q2 > Q1 ; Figure 7 shows the cumulative distribution functions F , G1 and G2 (these were drawn as- suming x was distributed log normally). Concentrating on the densities, one can see from (14) that increasing Q has the following e®ects: g shifts up for x < Q1 and x > Q2 , and shifts down for x 2 (Q1 ; Q2 ). Thus, we can say the following about multiple equilibria in the model: an equilibrium with a higher value of Q leads to more weight in both tails and less weight in the middle of the distribution of x values across existing matches. Additionally, (13) implies @N0 =@Q > 0, and so an equilibrium with a higher value of Q also displays a greater number of unmatched agents. Summing up, across di®erent equilibria, ones with higher values of Q entail more unemployment and more inequality. 5 Conclusion The paper has analyzed a model where agents choose whether to search while in relationships. One key ¯nding is that there can be multiple equilibria, and indeed a continuum of steady state equilibria, generated by endogenous 29 instability. The intuition is basic: when other people are more inclined to search, relationships will be less stable and hence less attractive, and so you will be more inclined to search. We even showed that for certain parameters there is an equilibrium where agents prefer matches that are fundamentally inferior because they believe these matches are more secure, which is true in equilibrium simply because they believe it. We analyzed welfare, and showed that there was a tendency for too much search { intuitively, because agents neglect the cost they impose on their current partners when they meet someone new, plus the cost they impose on the current partners of the matched people they meet. We analyzed the endogenous distribution of agents across states and the endogenous distribution of match values across relationships. A general conclusion is that more instability leads to more unemployment and more inequality.16 How robust is the notion of endogenous instability? Why can't two mar- ried people, for example, simply sit down, talk it over, and agree to be \faith- ful" to each other? There are several issues here. First, recall that there are two distinct ways in which there can be too much search, corresponding to regions 1 and 2 in Figure 2. The ¯rst case is a prisoner's dilemma: your best response is to search regardless of your partner's behavior, and there is no way to credibly promise not to despite the fact that search is ine±cient. The second case is a coordination failure: when both partners say they will not search, not searching is a best response, if they believe each other { but this can be a big if in some relationships. In any event, it is true by de¯nition 16 One is tempted to interpret this logic in terms of American versus European labor markets, as the former are conventionally regarded as having more turnover and more inequality. Pursuing this idea seems interesting, but beyond the scope of the current project. 30 that we can improve on a bad outcome when we have a coordination fail- ure as long as agents just cooperate. However, although we only explicitly analyzed bilateral relationships, in principle the message is meant to apply more generally to n-person organizations, and when n is big, it may not be easy to get the entire coalition together to sit down, talk it over, and agree to cooperate. There are many directions in which to extend this line of research. One branch involves thinking about generalizations of the technical assumptions. For example, if agents are heterogeneous ex ante { some people are more generally desirable than others { or they do not generally agree on match values { when two agents i and j meet they draw a pair (xi ; xj ) where xi need not equal xj { then if utilty is transferrable some interesting bargaining issues may arise. In particular, one may need to think about countero®ers when one's partner meets someone new, or one may need to think about o®ering one's partner a big enough share of the surplus to keep them from searching in the ¯rst place. Also, it would be useful to examine how the results are a®ected if we allow a continuous choice of search intensity, rather than s = 0 or 1. Another branch of future work would be to assess the empirical relevance of the idea. Just how susceptible are various relationships to endogenous instability? It might be interesting to examine this in the context of particular labor markets, such as those in professional sports or in academics, and also in the marriage market. We leave this to future research. 31 References Burdett, Kenneth [1977] \On-the-Job Search," American Economic Re- view. Burdett, Kenneth, and Coles, Melvyn G. [1999] \The Simple Analytics of Match Formation," mimeo. Burdett, Kenneth, and Wright, Randall [1998] \Two-Sided Seach with Non-Transferrable Utility," Review of Economic Dynamics. Diamond, Peter A [1982] \Aggregate Demand Management in Search Equilibrium," Journal of Political Economy. Mortensen, Dale T. [1978] \Speci¯c Capital and Labor Turnover," Bell Journal of Economics. Mortensen, Dale T., and Christopher A. Pissarides [1998] \New Develop- ments in Models of Search in the Labor Market," mimeo. Pissarides, Christopher A. [1994] \Search Unemployment with On-the- Job Search," Review of Economic Studies. Webb, Tracy [1998] \On-the-Job Search," mimeo. 32 x2 y4 U,P y2 U F ,U , P 45 C F,U y3 r b d F y1 D b x1 b d Figure 1: 33 x2 yS y2 U 1 45 C 2 y3 b d F y1 yA D b b d x1 Figure 2: 34 x2 yS y2 1 U 45 C 2 y3 r 5 b d 4 b d y1 3 F yA D b b d x1 Figure 3: 35 v00 v10 v01 v11 x b d q0 Q q1 Figure 4: 36 q1 (b d ) 45 q 0 (b d ) q1 (Q) q 0 (Q ) Q b d q q Figure 5: 37 0.025 0.022 0.02 f( x ) 0.015 g1( x ) g2( x ) 0.01 0.005 0 0 0 50 100 150 200 250 300 0 x 300 Figure 6: 1 0.8 0.6 F( x ) G1( x ) 0.4 G2( x ) 0.2 0 9 2.014 10 0.2 0 50 100 150 200 250 300 0 x 300 Figure 7: 38