A Generalized Model of Commitment

A Generalized Model of Commitment Scott Drewianka∗

Abstract The durability of economic relationships, such as employment or family relationships, is influenced by agents’ willingness to “commit” by making match-specific investments or arrangements. This paper shows that many properties of optimal decisions and comparative statics in such models are quite robust, emerging from the abstract structure of commitment decisions rather than specific assumptions about functional or distributional forms. Recognizing this structure should enable future research to consider a broader set of empirically-relevant complexities and may help to identify promising explanations for variations in commitment. The exercise also clarifies the circumstances under which increased uncertainty increases or decreases commitment. JEL Codes: D800, J120, J600 Keywords: match-specific investments, turnover, comparative statics

∗ Department of Economics, University of Wisconsin-Milwaukee, Bolton Hall 860, 3210 N. Maryland Ave., Milwaukee, WI 53211. Phone: (414) 229-2730; fax: (414) 229-3860; e-mail: [email protected]. This paper has benefitted from comments and suggestions from many people, especially Gary Becker, Boyan Jovanovic, Jean-François Laslier (the editor), Dan Levin, Yoshio Niho, two anonymous referees, and participants of workshops at the Universities of Chicago and Wisconsin—Milwaukee and the Midwest Economics Association. I am grateful for financial support from the University of Chicago and the National Institute for Child Health and Human Development (grant number T32HD07302). Any remaining flaws are solely my responsibility.

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1

Introduction Economists are often interested in the stability of employment and household relationships. However,

the likelihood that those relationships will end, and the ramifications if they do, are typically not exogenous. Rather, individuals make a series of decisions that influence the durability of their relationships, such as choices between legal statuses (e.g., legal marriage versus non-marital cohabitation, or permanent versus temporary employment), investments in match-specific capital, and decisions about the intensity of on-thejob search. In making such decisions, a person effectively decides how intensely to commit to continuing the relationship in the future. In order to emphasize this commonality, this paper shall refer to all such relationship-stabilizing actions as “commitments.” There are substantial and successful literatures on many of these commitment decisions and the factors that influence them. Just to cite a few prominent examples, the list of topics includes the effects of firing costs and other labor market regulations on employment and turnover rates, the causes and consequences of recent changes in family structure patterns, the roles of firm-specific human capital and job changes in the careers of young workers, and any number of questions involving fertility. Considering that all of these issues revolve around a decision about commitment, it is perhaps not surprising that most theoretical models in these literatures build off a common framework, nor that this framework has been thoroughly explored over the past several decades. Nevertheless, economists continue to develop new variations on these models, and it seems fair to say that their work could proceed more quickly were it not for the complexity of the models involved. However, that complexity seems inevitable given that the decision almost necessarily involves uncertainty, fixed costs, decisions over time, and option values. This is not to deny that necessary tools exist, but rather to point out that they are often unwieldy when extending an existing model. For example, anyone who has tried to incorporate alternate (perhaps even seemingly natural) assumptions about probability distributions, discrete choices, or choice sets that depend on past decisions has likely found it challenging (though not necessarily impossible) to work through the model. Researchers have thus often traded generality for tractability, either in the form of strong simplifying assumptions and convenient functional forms, or else by ignoring potentially interesting extensions involving commitments of varying intensity or interactions between different

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commitment decisions. Of course, no one could deny that those simplifications have enabled great progress. Still, it would be desirable to relax them if possible–both to allay any concerns about robustness and to extend the scope of the analysis. This paper will show that most previous findings are actually quite robust. A general model of commitment is developed that can accommodate either discrete or continuous payoffs and costs, any number of commitment mechanisms, variation in the intensity of any type of commitment, choice sets that may depend on past decisions (or put another way, the possibility that present choices may affect one’s future options), and arbitrary probability distributions. Since this structure allows for virtually any reasonable specification, the properties it generates are not the product of a specific application or a particular modeling assumption, but rather a natural consequence of the fundamental structure of commitment decisions. Among other things, this model generates complementarities that place a good deal of structure on agents’ behavior. The intuition for these complementarities is easiest to understand if commitments are thought of as (at least partially) irreversible match-specific investments: such investments create match-specific rents that increase the probability that the match will continue, and anything that makes the match more likely to continue (including both a better intrinsic quality of the match and those commitments) raises the expect return on other match-specific investments. These forces ensure that agents who are (a) better matched, or (b) have made greater commitments in the past will be more likely to continue their matches and will make greater future commitments than other agents, ceteris paribus. However, the paper’s study of comparative statics also reveals a series of perturbations that would decrease the optimal intensity of commitments. To be sure, some previous authors have identified similar features in specific models, but it has not always been clear whether those properties are robust or simply the result of particular functional forms employed in the analysis. In a notable example, Jovanovic (1979b, pp. 1257—58) explicitly discusses some complementarities in the context of his influential model and briefly speculates that they may hold under more general circumstance, but he leaves it as a topic for future research. By posing the question more abstractly, this paper is able to confirm his conjecture. Recognition of this robustness should lighten the burden of investigating some particularly complex models. Indeed, the general model developed here already covers many empirically relevant complexities

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that researchers may wish to consider more fully–for instance, commitment decisions that have implications for one’s future options (e.g., having a second child eliminates the possibility of having only one in the future, but opens the possibility of having a third child). Where it does not, the proofs may still suggest a way to proceed, or at least what could go awry. Moreover, it is hoped that the paper’s more abstract perspective on these decisions will help future scholars to build intuition and to develop useful analogies across applications. The paper also identifies one established result that is perhaps less robust than may have been understood. The previous literature has found that greater uncertainty about future gains encourages the formation of new matches and discourages the dissolution of existing ones (Jovanovic, 1979a). While logically unassailable, that conclusion conflicts with the usual intuition that uncertainty raises the value of flexibility and thus makes commitment less attractive. By allowing a broader set of choices, the model developed below reconciles these two ideas. In short, under typical circumstances greater uncertainty encourages commitment at the extensive margin, but it always reduces the intensity of commitment at the intensive margin. One novel implication is that greater uncertainty about future match quality may provide a promising explanation for the recent growth in low-commitment institutions like cohabitation and temporary employment. The discussion proceeds as follows. Section 2 begins by presenting a representative model with commitment– a simple model of household formation and dissolution– that serves to demonstrate some key ideas and results and to provoke consideration of the many different ways one might model decisions about commitment. Section 3 then presents the general model of commitment decisions that is used to explain the structure of optimal commitment decisions. Section 4 extends the discussion by examining a series of comparative statics that correspond to four broad classes of explanations for variation in commitment intensity, and it discusses some implications. Section 5 concludes with a brief summary and a suggestion for future work.

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A Simple Model with Commitment To begin, let us briefly consider a simple model that one might plausibly use in a particular application–in

this case, household formation. In many ways, it is a simplified version of an elaborate model used by Brien, Lillard, and Stern (forthcoming), although it is instructive that they eliminated one aspect of this model (endogenous fertility) on the grounds that it made the model too complex. At any rate, the immediate

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purpose is not to draw substantive conclusions, but rather to provoke consideration of how sensitive the results are to differences in modelling assumptions, to illustrate the more general model of commitment developed in Section 3, and to introduce some properties that will hold more generally.

A

The Simple Model In this model an individual makes decisions about the status of a relationship over two time periods,

with discount rate β. In order to keep the analysis relatively simple, bargaining between partners will not be modelled explicitly here. This could be interpreted as assuming that the “relationship” is between a person and a non-rational being (e.g., a person deciding how intensely to adopt a new technology), but it could also be that the payoffs defined below are the outcome of a bargaining process that occurs in the background. At any rate, little is lost by this simplification, as Drewianka (2004) finds qualitatively similar effects using a two-sided variant of this model that considers bargaining explicitly. Individuals are initially unmatched and search for partners from a heterogeneous population. In the first period, agents meet potential partners and observe the “quality” of their match π1 ∈ (−∞, ∞). This is the baseline level of additional utility the agent would receive by forming the match. The agent may then decide to remain unmatched and search again in the second period, drawing secondperiod match quality π2 from distribution G. If a match is formed in the second period, the agent then receives benefit π2 in period 2. Thus, since the model ends after the second period and there are no costs to forming a match, an agent who is unmatched after the first period will form a match in the second period if π 2 > 0. The value W of the option to continue searching is thus

W =β

Z

∞

π 2 dG(π2 ).

(1)

0

Alternatively, the agent may decide to form the match. In that case, he or she receives benefit π 1 immediately and chooses a legal form for the match (say, marriage or cohabitation) and how much to invest in match-specific capital (e.g., kids), k ∈ [0, ∞). After these decisions are made, circumstances change and the match-quality evolves, with the second period match quality π 2 ∼ N (π 1 , σ 2 ). After observing π 2 , the matched individual decides whether to maintain the match. A person who remains in a cohabitational match

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receives π2 + ρ(k) (with ρ, ρ0 > 0), and a person who remains in a legal marriage receives π2 + ρ(k) + µ (with µ > 0). On the other hand, an agent may end a cohabitational match in period 2 by paying a penalty that decreases utility by k, but ending a legal marriage costs the individual a penalty of (δ + k) (where δ > 0).1 Since both the legal form of the match and the investment in match-specific capital provide benefits if the match lasts, but impose penalties if the match is dissolved, both factors encourage continuation of the match. Whereas an unmatched agent would remain unmatched in period 2 if π2 < 0, a cohabitational match would end in period 2 only if π2 < −[k + ρ(k)], and a marriage would end only if π 2 < − [k + ρ(k) + µ + δ]. Both the legal form and k may therefore be considered forms of “commitment.” Accordingly, in the first-period a person who has chosen cohabitation (c) and who has decided to have k children has expected utility Z∞

J(π 1 , c, k) = π 1 − βkF (− [k + ρ(k)] , π 1 ) + β

[π2 + ρ(k)] dF (π 2 , π1 ) ,

(2)

−[k+ρ(k)]

where F (π 2 , π 1 ) = Θ [(π2 − π1 ) /σ] and Θ is the standard normal distribution. The first term on the right-hand side represents the payoff from the match in period 1, the second represents the agent’s second period utility in the event that the match ends (which occurs with probability F (− [k + ρ(k)] , π 1 )), while the integral represents the expected period 2 utility in the event that the match remains intact. Likewise, if the person has decided to marry (m) and have k children, expected utility is

J(π 1 , m, k) = π 1 − β(δ + k)F (− [δ + k + µ + ρ(k)] , π 1 ) + β

Z∞

[π 2 + µ + ρ(k)] dF (π 2 , π 1 ) .

(3)

−[δ+k+µ+ρ(k)]

The first-period value function V (π 1 ) is thus the upper envelope of the options defined by (1)—(3):

V (π1 ) =

max {W, max J(π1 , c, k), max J(π1 , m, k)},

h∈{0,c,m}

k≥0

k≥0

(4)

where we have defined h as the legal form of the agent’s first period match, with h = 0 representing the situation in which the agent remains unmatched after the first period.

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This framework characterizes much of the optimal behavior. The following results are indicative of those that will receive attention later in this paper, so formal proofs are left for the more general case (Section 3). 1. The marginal benefits J(m) − J(c) and J(c) − W are increasing in π 1 , so agents sort into single, cohabiting, or married based on π1 . The optimal k is also increasing in π1 and greater for married agents than cohabitors, and when k is larger marriage becomes more attractive relative to cohabitation. All (conditional) value functions J and V are convex in π 1 . See Figure 1. 2. Marriages are less likely to end in period 2 than are cohabitational matches, even conditional on π 1 and k. Those with larger k and/or π 1 are also less likely to separate. 3. An increase in uncertainty σ increases each conditional value function J and the overall value function V , but it raises the attractiveness of cohabitation relative to both marriage and staying single. It also reduces the optimal k and increases the likelihood of separation in period 2–both conditional on relationship status and k and unconditionally. The first two points are probably not surprising, although note that they indicate the two types of commitment (marital status and kids) are complements. The third point flows from the fact that max{−k, π + ρ(k)} is convex, so a mean-preserving increase in risk raises the expected future utility for agents who match in period 1, while W remains constant. However, ∂J/∂σ is itself decreasing in k and relationship status, so the increased risk lowers the relative attractiveness of all but the minimal level of commitment.

B

Comparison with Other Models The structure of decisions in the model above are in many ways strikingly similar to results from the

theoretical literature on careers, where it is also common to find commitment (whether investments in match-specific capital, avoidance of on-the-job search, or just willingness to continue the current match) increasing in match quality and past commitments, certain complementarities, and greater willingness to remain matched when there is greater uncertainty about future match quality. Yet in spite of the similarity of these results, the assumptions used in the model above differ in many ways from those used in that literature. It is standard to consider an infinite horizon, rather than the

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two-period framework above. Some models assume that the current match quality is known with certainty and turnover occurs due to “on-the-job search” (Mortensen, 1978; Jovanovic, 1979b; Pissarides, 1994), while others involve “learning” about the quality of the match through experience (Jovanovic, 1979a; Mortensen, 1988; Weiss and Willis, 1997; Brien Lillard, and Stern, forthcoming)–and neither of those is identical to the assumption above that match quality evolves. The search distribution G is frequently taken to be normal. Comparative statics on uncertainty about future match quality are typically considered only in the context of “learning,” where uncertainty varies because of differences in tenure, rather than cross-sectional or intertemporal differences in predictability–and the effect of uncertainty on decisions at the intensive margin is rarely discussed. The endogeneity (or even existence) of match-specific capital is sometimes ignored, as are costs of making a commitment, switching between different levels of commitment, or ending a match entirely. Very few models allow agents to choose between different match “statuses” (like the choice between cohabitation and marriage in the model above); the choice is usually just to be matched or not to be matched. The point of all this is not to argue that one set of assumptions is better than another, and is certainly not to disparage previous research. Rather, the point is that seemingly very different models nevertheless produce many qualitatively similar effects. This leads one to wonder which results are contingent on specific assumptions and which are properties of commitment generally. The following section answers that question by showing that most, though not quite all, of these properties emerge even in a much more abstract model.

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A General Model of Commitment We now generalize the assumptions above in order to nest as many variations as possible. The only major

qualitative change from the model above is that agents will now face an infinite time horizon.2 In addition to being more closely aligned with the job matching literature, that assumption is interesting economically because at least three time periods are needed to assess any role for possible alternate matches (the G distribution) to influence commitment decisions at the intensive margin. The generalized conditions used here are designed to allow application of Topkis’ (1978) Monotone Maximum Theorem (MMT) to the greatest extent possible. While there has been increasing awareness of Topkis’ result over the past decade, it appears that awareness is still far from complete. Accordingly, Appendix A

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provides a brief introduction to the theorem, the concepts and vocabulary underlying it, and some papers that survey the literature. For now, the main things to recognize are that the MMT characterizes a set of minimal conditions guaranteed to yield decision rules that are monotonically increasing in parameters, and that it is especially useful for analyzing models, like this one, with inherent complementarities. While this approach may appear unnecessarily technical, there are several reasons to prefer it in this model. For one thing, Topkis’ result does not require the objective function to be continuous, differentiable, or concave–all of which are particularly useful here because commitment decisions are often discrete (as seen above) and frequently create convexities in the objective function (as shown below). Furthermore, the conditions underlying the MMT, while perhaps less familiar, are more parsimonious and less restrictive than those underlying a calculus-based approach, and they arguably permit a clearer understanding of the relationship between assumptions and conclusions.

A

Revised Notation and Assumptions Continue to let match quality π be real-valued,3 π ∈ Π ⊆ R, and now denote the level of n different

n types of commitments by I ∈ Φ ⊆ R+ , where Φ is a lattice. A new match has I = 0, and that is also

the new level of commitment for agents who choose to search again. Otherwise, any ongoing match has I 6= 0. For example, the vector of commitments I might include a component that equals 1 if the agent is matched and 0 otherwise. The simple model of Section 2 fits into this framework, with I ≡ (h, k) and Φ ≡ {(0, 0)} ∪ ({c, m} × R+ ), which is indeed a lattice. As before, unmatched agents draw potential matches from distribution G(π 0 ). They may also receive a benefit (or cost) ζ, which may represent an unemployment benefit or a search cost. For matched agents, π evolves according to a Markov process; if the current match quality is π, the distribution of future match quality π 0 is F (π0 , π), where π a ≥ πb implies that F (π 0 , π a ) first-order stochastically dominates F (π 0 , π b ). Agents who remain matched select a new level of commitment I 0 optimally from some set Γ(π, I) ⊆ Φ, where Γ(π, I) is an increasing correspondence that maps each (π, I) to a complete sublattice of Φ, and 0 ∈ Γ(π, I) for all (π, I).4 In other words, agents who make an incremental commitment do not lose previouslyheld options to commit even more, and agents may always end the match (possibly at a very high cost).

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One possibility is simply that Γ(π, I) = Φ for all (π, I), but the more general notation allows for irreversible match-specific investments (e.g., kids), technological or legal restrictions on transitions between levels of commitment, and commitment opportunities that are available only to sufficiently well-matched agents. The payoff to a matched agent (I 6= 0) with quality π who chooses a new level of commitment I 0 is given by the bounded5 function R(π, I, I 0 ), which is strictly increasing in π and at least one component of (I, I 0 ) (otherwise the commitment would never be attractive), (weakly) increasing in all components of (I, I 0 ), (weakly) supermodular in (I, I 0 ), and has non-decreasing differences in π. In addition, some results will require that R be (again, at least weakly) convex in π, but that will be stated when necessary. Finally, let the cost of the switch from I to I 0 be given by the bounded (weakly) submodular function c(I, I 0 ). It might appear that these assumptions introduce convexity and complementarity into the model, as supermodularity is commonly thought of as a generalized form of complementarity. However, it is more accurate to say that the assumptions simply rule out the possibility that opportunities for substitution overwhelm the complementarities that arise naturally in commitment decisions. The assumptions above are all “weak”–that is, they include the commonly used specification in which the net payoff R − c is linear in π and additively separable. Since we have not assumed that R − c is necessarily strictly convex or that it has any strict complementarities, the fact that we will find complementarities and convexities anyway means that those properties are actually generated by the nature of commitment itself, rather than introduced through properties of the net payoff (R − c). Indeed, those properties may result even if the payoff function actually involves substitutes and concavities, as long as those effects were not too strong. Nevertheless, it is attractive to retain the more general assumptions in order to clarify the sources of our conclusions and to allow for, e.g., the multiplicative payoff functions that sometimes appear in established theoretical models. For example, a multiplicative payoff arises in Jovanovic’s (1979b) model because the worker’s payoff (her wage) is the product of her baseline productivity and her match-specific “investment” in time spent working on the job rather than searching for a new match. Moreover, there are actually some good reasons to expect the transition cost function c to be strictly submodular. The marginal cost of making new commitment I 0 may quite reasonably depend on how much commitment I was made previously. For example, a typical adjustment cost functions might depend on

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incremental new “investments” (Ij0 − δ j Ij )–for example, c(I, I 0 ) =

P

θj (Ij0 − δ j Ij )2 , with δ j , θj ≥ 0. This

function is not additively separable, but it is submodular. Similarly, in the model of Section 2, the cost function was submodular, but not additively separable in (I, I 0 ), because the cost was paid only when the match ended: c((m, k), I 0 ) = (δ + k) if I 0 = 0 and 0 otherwise. As a brief final point, note that nothing substantive would change if any of the notation above were a function of time (call it t) or match duration (τ ). The latter can simply be modeled as another type of match-specific investment, although one in which the agent has no choice (except by ending the match). (This is one example in which it is helpful to have a choice set Γ that differs across periods–ignoring other types of commitments, one could write Γ(τ ) = {0, τ + 1}.) Likewise, all of the proofs that follow hold with only minor changes if R, c, G, F , or Γ varied systematically with time. We will avoid the additional terms and subscripts in order to simplify an already-complex notation, but clearly the option of including time or duration effects expands the set of models that fall within the general framework developed here. For example, they permit us to cover models with payoffs or costs that vary with an individual’s age, options that expire after some period, and distributions that depend on the duration of the match. To cite one notable example, it accommodates Jovanovic’s (1979a) learning model, in which F (π τ , π τ −1 ) ∼ N (π τ −1 , σ 2τ ) and σ 2τ decreases in τ . Much of the model’s popularity stems from that decreasing variance, as it leads to the empirically-validated predication that turnover eventually decreases with match duration. ¡ ¢ However, learning imposes substantial restrictions on the rate at which σ 2τ decreases; σ2t − σ 2t−1 /σ 2t−1 = −2/(ω + t) for some positive constant ω. This can cause problems for empirical work if the actual rate at

which turnover decreases is inconsistent with this functional form.6 One implication of the present exercise is thus that a less restrictive form can be used without fear of changing the qualitative behavioral predictions.

B

Value Functions Given these cost and benefit functions, agents seek to maximize the discounted expected value of net

payoffs given current quality π and current commitment I. Let that expected discounted present value be represented by the value function V (π, I), and for I 0 > 0 define conditional value function J(π, I, I 0 ):

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J(π, I, I 0 ) = R(π, I, I 0 ) − c(I, I 0 ) + β

Z

V (π0 , I 0 ) dF (π0 , π).

(5)

π 0 ∈Π

Likewise, define W as the discounted expected value of payoffs for an unmatched agent,

W ≡ζ +β

Z

V (π, 0) dG(π).

(6)

π∈Π

As an aside, note that agents have an opportunity cost of remaining matched only if matches are rival and replaceable. If “remarriage” were prohibited, matched agents would have W = 0, and W = 0 for everyone if committing to one match does not inhibit the ability to form others (e.g., under polygamy). These cases will appear briefly later on, but for the bulk of the paper we define W as in equation (6). The Bellman equation is thus ½ V (π, I) = max W − c(I, 0),

max

I 0 ∈Γ(π,I)\{0}

0

¾

J(π, I, I ) .

(7)

Finally, as this formulation suggests, it is convenient to define J(π, I, 0) ≡ W − c(I, 0).

C

Optimal Commitment Intensity We may now establish our main results about the determinants of agents’ optimal commitment decisions.

All propositions are proved in Appendix B. Proposition 1 The value function V (π, I) and the conditional value function J(π, I, I 0 ) are increasing in π (strictly in the case of J, and strictly for V only over the range where the optimal choice involves maintaining the match), supermodular in I, and they have increasing differences. In addition, both V and J are convex in π if R is at least weakly convex in those arguments. The implications and logic of this result are likely familiar, as they arise frequently in the literature. Agents with better match quality are better off due to both the initial advantage of having a better match and the higher expected benefits that come from their greater optimal level of commitment (as shown below). This is one reason that V and J are convex in π.7 (Another is simply that agents have a choice, maximization

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being a convex operator.) Since those value functions are convex, uncertainty about future match quality is actually desirable, as Mortensen (1988) has noted. Further implications are explored in Section 4. Proposition 1’s results about supermodularity and increasing differences flow from the fact that the technology of commitment creates complementarities–between different types of commitment, between current match quality and future commitment, and between past and future commitments. Again, this is true even if there are no complementarities in the instantaneous payoff function (R − c) itself. The intuition has two (complementary!) pieces that generalize from the job turnover literature. First, anything that raises the probability that a match will remain intact also raises the expected return on all forms of commitment. Second, because they create economic rents only for the current match, commitments do just that: raise the probability that match will continue. The complementarity between different types of commitments has some interesting implications. For one thing, an unexpected increase in one type of commitment encourages other forms of commitment as well– think of weddings prompted by unexpected pregnancies. Another implication is that standardization of a formerly match-specific investment (for example, when other firms adopt a technology that was once unique to a given firm, such as when a patent expires) reduces other dimensions of commitment too (e.g., other specific human capital, relationships with co-workers, compensation that becomes vested over time, and so on)–all of which would further reduce the match specific rents and compound the increase in turnover. Some readers may wish to extend this logic to argue for convexities in any particular type of commitment. This is indeed possible–the commitment extends the expected life of the match and thus raises its own expected return, suggesting a convexity. To see this, let us briefly employ the familiar derivative notation: ¯ ∂2J ∂ 2 J ¯¯ = ¯ ∂Ij0 ∂Ik0 ∂Ij0 ∂Ik0 ¯

π∗

+

∂ 2 J ∂π ∗ . ∂Ij0 ∂π ∗ ∂Ik0

(8)

This expression decomposes the effect into two parts: (a) the direct effect, holding constant the probability that the match will endure, and (b) the indirect effect due to the fact that it is more likely to endure. The latter is necessarily non-negative, as is the former if j 6= k and (R − c) is at least weakly supermodular–that is, as long as the two types of commitment are not substitutes. However, for j = k, the first term is usually taken to be negative. Accordingly, at any level of commitment, it is unclear whether the objective function

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is convex or concave in a given component of I 0 –although of course it would ordinarily be concave at the margin if agents behave optimally. A further consequence of Proposition 1 is that agents make greater new commitments, and are more likely to keep existing commitments, when they have greater match quality and existing commitments. A direct application of the MMT (see Appendix A), the conclusion is stated precisely in the following corollary. Corollary 2 The correspondence of optimal new commitment γ (π, I) ≡ argmaxI 0 ∈Γ(π,I) J(π, I, I 0 ) is an increasing subcomplete sublattice of Φ. Note that since present and future match quality are correlated, matches with better current quality are less likely to end tomorrow, and thus they have greater expected returns to commitment. New matches thus sort into levels of commitment segregated by initial match quality, as we saw in the simple model of Section 2 (Figure 1). Indeed, a similar pattern holds for any current level of commitment I, not just new matches. Likewise, existing commitments increase the likelihood of maintaining the match because they raise the benefits of the match and/or the costs of terminating it. Consequently, conditional on current quality π, agents with more existing commitment make greater additional commitments. Similarly, such agents will maintain a match (choose I 0 > 0) in some cases where less committed agents would not (thus confirming that the concept is meaningful). To put it two other ways, (a) the reservation level of quality need to maintain the match, call it π∗ (I), is decreasing in I (as is the reservation π needed to commit at any particular level), and (b) decisions about commitments are positively history-dependent. A related conclusion is that matches are initially stable. The minimum π needed to form a match (π ∗ (0)) is greater than the level needed to continue that match (π ∗ (I0 ), where I0 > 0 is optimal for a marginally acceptable match), so even if an initially marginal match never improved and additional commitments were not made, the match would not end until π fell by π∗ (0) − π∗ (I0 ). As an aside, it is interesting to note that it is unclear whether that gap is more likely to be bridged by a single large shock or by a series of small shocks. The answer obviously depends on F , but it may also depend on the size of the gap itself, which also varies with R and c. In other words, it may be possible to alter the expected timing of separations, even holding constant the stochastic process generating π, by changing the

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structure of benefits and costs. Accordingly, predictions about the hazard rate at which matches end may not be very robust to alternate specifications. This point is significant in that predictions about changes in that hazard rate over the duration of matches have occasionally been used to test theories against one another. In particular, some have noted that Jovanovic’s two seminal models of employment turnover make different predictions about the hazard rate at which matches end–his model that emphasizes firm-specific human capital (Jovanovic, 1979b) predicts that matches are always more likely to survive the longer they have endured, whereas his model of “job learning” (Jovanovic, 1979a) (in which employers slowly learn about workers’ abilities by observing their performance over several periods) predicts that the separation hazard could initially be increasing in match tenure, although it will eventually decrease. Since the empirical evidence for both employment and households shows that the separation hazard is initially increasing (Farber, 1994; Weiss and Willis, 1997; Brien, Lillard, and Stern, forthcoming), this has sometimes been taken as evidence that learning is an especially salient feature of turnover models. However, the analysis here suggests that nearly any model of turnover could be reconciled with an initially increasing separation hazard simply by introducing a fixed cost of ending a match, at least for some parameter values.8 Thus, the proposed test does not appear capable of effectively distinguishing between competing hypotheses.

4

Comparative Statics Having established a model of commitment intensity decisions, we now address the sources of variation in

real-world commitment decisions by cataloguing and distinguishing several classes of hypotheses that might explain observed patterns and trends. Five perturbations seem most interesting: an increased benefit of being unmatched (an increase in ζ), an improvement in the distribution of potential match quality (G2 firstorder stochastically dominates G1 ), an increase in the diversity of match qualities (a mean-preserving spread of G), an increase in the uncertainty of the future quality of a given match (a mean-preserving spread of F ), and changes in the schedule of marginal benefits or costs associated with commitment. In a labor market context, such changes might correspond respectively to increases in the level of unemployment payments, a

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positive industry-wide productivity shock, increasingly specific job descriptions, greater uncertainty due to more rapid technological change, and changes in laws governing different types of employment relationships. This paper considers the first four classes of hypotheses. The fifth class is also of interest, as it includes models involving “firing costs” in the job market case and “divorce costs” in the household composition case. However, modelling the fifth class requires a more elaborate framework that would be disproportionate with the scope of this paper, and such a model has already been analyzed by Drewianka (2004). In order to make these comparisons, let the distribution G be indexed by two parameters, α and s, where α2 > α1 implies that G(α2 ) first-order stochastically dominates G(α1 ), and s2 > s1 implies that G(s2 ) is a mean-preserving spread of G(s1 ). Likewise, let the conditional distribution of future match quality be indexed by σ, where σ 2 > σ 1 indicates that F (π 0 , π, σ2 ) is a mean-preserving spread of F (π 0 , π, σ 1 ).

A

Monotonic Effects We are now ready to state our first comparative statics result.

Proposition 3 The following changes all (a) decrease the optimal commitment I ∗ (π, I) and (b) increase agents’ private welfare: 1. An increase in the benefit of being unmatched (increase in ζ) 2. A first-order stochastic improvement in the distribution of potential matches (increase in α) 3. A mean preserving spread in the distribution of potential matches (increase in s), if R is at least weakly convex in π. In addition, if R is at least weakly convex in π, greater uncertainty about future match quality (increase in σ) increases private welfare. It may be surprising that decreased commitment is related to increased private welfare in the three numbered cases in Proposition A. Each describes a mechanism that raises the opportunity cost of being matched (W ): the first directly, the second due to improved prospects, and the third due to the the convexity of V in π (again, by Jensen’s inequality, though the formal proof is complicated by the fact that the shape of V varies with s). Raising W makes everyone better off, but the effect is not uniform–unmatched agents

16

receive the benefit immediately and with certainty, while the effect is smaller for matched agents. The risk of separation conditional on (π, I) thus increases, which in turn reduces the marginal net benefit of commitment. The effect is then compounded by the complementarities discussed above. A couple caveats should be noted. First, all of these effects are conditional on (π, I). In the short run, this necessarily means that the turnover rate would increase as formerly marginal matches become no longer worth maintaining. However the long run change in the unconditional turnover rate is ambiguous because the change raises the standards for forming a match in the first place. This dynamic applies to all three perturbations considered here. Furthermore, especially in the case of increasing α, although it may become less attractive to commit in a given set of circumstances, we might nevertheless observe more commitment in the aggregate because the same perturbation changes the distribution of (π, I) in the population. Second, all of the welfare statements refer only to private welfare, not social welfare. The distinction is non-trivial here because societies often behave as if commitments have positive or negative externalities– consider recent debates over family policy, or economic models in which one person’s commitment alters the distribution of potential matches for another person (e.g., by removing a worker from the labor market or a potential spouse from the marriage market (Diamond, 1981; Drewianka, 2003)). In such cases, depending on the nature of the externality, social welfare may either increase or decrease. As for the last part of the proposition, the convexity of V again causes private welfare to rise with σ. However, we cannot immediately predict changes in I ∗ because changes in commitment are not necessarily monotone in this case. We already saw this in Section 2, where an increase in the analogous parameter led to decreased commitment at the intensive margin, but increased commitment at the extensive margin. As we shall show below, such non-monotonicity at the extensive margin is frequently possible because the benefits from the convexity of V accrue only to matched agents. Consequently, while the added risk tends to reduce commitment elsewhere, it may nevertheless reduce agents’ willingness to be unmatched.

B

More Volatile Matches Since the effects of σ are plausibly non-monotonic, the more familiar calculus-based comparative statics

will prove more useful than approach based on the Monotone Maximum Theorem. Thus, let us now make

17

the usual assumptions: compact sets (Π and Γ(π, I)\{0}), functions that are continuous and differentiable almost everywhere (R, c, J, V ), a concave objective function, smooth and differentiable distributions with no mass points (F and G), and so forth. Two additional assumptions will help to clarify what happens to commitment at the intensive margin when match volatility σ rises. One rules out the possibility that greater uncertainty actually lowers the risk of separation: dF [π∗ (I ∗ , σ), π, σ] /dσ > 0.9 The other assures that marginal returns to commitment (∂R/∂I) are not convex in π. Otherwise, commitment could (possibly, not necessarily) become a better gamble under greater uncertainty because the “upside” returns to I exceed the “downside” losses, which seems like another odd case with little practical application. Thus, assume that ∂R/∂I is weakly concave in π (R112 ≤ 0), which is consistent with the most common functional forms (e.g., additively separable or linear in π). Under these assumptions, the following is true: Proposition 4 For a mean-preserving increase in match volatility σ: 1. The optimal commitment intensity I ∗ (π, I) falls at the intensive margin. 2. I ∗ (π, I) may either increase or decrease at the extensive margin, depending on whether agents’ odds of being matched in the future are higher if they stay matched or if they search. 3. However, if R is weakly convex in π, then I ∗ (π, I) necessarily increases at the extensive margin for currently matched agents if subsequent matches are prohibited, and for all agents if there is no opportunity cost to forming a match. In what seems to be the usual case (3), the increased risk reduces commitment at the intensive margin, but increases it at the extensive margin. This may suggest that hypotheses involving increased uncertainty offer promising explanations for the recent rapid growth of institutions like temporary work and cohabitation. Besides its theoretical content, Proposition 4 suggests two ways to distinguish empirically between the effects of match volatility and the other factors. First, if commitment conditional on (π, I) decreased at the intensive margin and increased at the extensive margin, that would square only with increased match volatility (although decreasing commitment at both margins would not necessarily rule out a volatility effect). Alternatively, suppose one can estimate of π and π ∗ and decompose the change in the hazard rate

18

for default into components associated with changing π∗ and an increasing likelihood that π0 < π ∗ . The second component would be non-zero only for an increase in σ (which increases the default hazard even conditional on π, I, and π∗ ). In the other cases discussed above the change is entirely due to changes in π, I, and π ∗ , so the conditional default hazard is constant. To provide a concrete example, suppose that we observe a number of variables that plausibly predict a person’s gain from marriage, but we find that there is a strong secular trend in individuals’ divorce hazard even conditional on their own levels of those variables and the distribution of those variables in the population (In fact, this is the conclusion of Pierret (1995)). Then the logic above means that either (a) we have failed to observe some important variable (which becomes less likely the more variables we consider), or (b) an important role was played by increased uncertainty (or possibly a change in marginal net benefits, the comparative static not considered here).

5

Summary The paper has shown that commitment decisions and their comparative statics are robust to a large set

of possible specifications. This robustness should put to rest some potential objections to existing models, especially the concern that previous results may be contingent on distributional assumptions. It should also enable future researchers to add more extensions and complexities to their models. Models that include multiple types of commitments seem particularly interesting, as the inherent complementarities may generate a multiplier effect. Empirical research may also benefit from these results. For one thing, the complementarities identified above have clear implications for empirical specifications. More importantly, as demonstrated above, the comparative statics results can help to identify hypotheses to explain the considerable variation in commitment levels over time and across people. As a final point, it may be worth noting that other literatures also feature decisions that influence the durability of some sort of relationships–for example, firms’ decisions about supply relationships (e.g., whether to buy on the spot market, write long-term contracts, or vertically integrate), immigrants’ decisions about learning new customs and whether to seek citizenship, and customers’ decisions between buying or renting. While models of these decisions differ from those generalized here, they are related in that they

19

generally involve risk, sunk costs, and match-specific investments. It thus may be fruitful to ask whether those models can be mapped into the abstract framework developed here, or at least one sufficiently similar to permit a comparable analysis. If so, then the results developed here may advance research on those questions as well, and perhaps they might enable further exchange of ideas between researchers in these disparate fields.

Appendix A: Monotone Comparative Statics Topkis’ (1978, 1998) Monotone Maximum Theorem (MMT) characterizes minimal conditions under which decision rules are monotone in parameters. Since that result is used extensively in this paper and since it may be unfamiliar to some readers, this appendix defines some key terms and states the MMT. Longer introductions to this literature, many of which contain useful extensions and applications, have been written by Milgrom and Roberts (1990, 1995), Milgrom and Shannon (1994), Athey (2002), and Amir (2005). Let (X, ≥) be a partially-ordered set. The relevant example in this paper is Rn with the usual partial ordering (x2 ≥ x1 if every component of x2 − x1 is non-negative). For any x1 , x2 ∈ X, let x1 ∨ x2 and x1 ∧ x2 be the least upper bound and greatest lower bound (respectively) of x1 and x2 . In Rn , x1 ∨ x2 (x1 ∧ x2 ) is the vector whose j-th component is the maximum (minimum) of the j-th components of x1 and x2 . A subset S ⊆ X is a lattice if for all s1 , s2 ∈ S, s1 ∨ s2 and s1 ∧ s2 ∈ S. A subset A ⊂ S is a sublattice of S if for all s1 , s2 ∈ A, s1 ∨ s2 and s1 ∧ s2 (defined relative to S) are in A. In addition, A is complete if for every subset C ⊆ A, sup(C) and inf(C) ∈ A. For example, the set {(1, 0), (0, 1)} is not a lattice, but {(0, 0), (0, 1), (1, 1)} is a lattice. The set [0, 1] × [0, 1] is a complete lattice, and (0, 1) × (0, 1) is a sublattice of it that is not complete. A correspondence Γ(x) : X ⇒ S is increasing (in the strong set order) if x2 ≥ x1 and si ∈ Γ(xi ) (i = 1, 2) implies s1 ∨ s2 ∈ Γ(x2 ) and s1 ∧ s2 ∈ Γ(x1 ). A function f (s, x) : S × X −→ R is supermodular in s if for all s1 , s2 , f (s1 ∨ s2 , x) + f (s1 ∧ s2 , x) ≥ f (s1 , x) + f (s2 , x). It has the weaker property of increasing differences in x if s1 ≤ s2 implies that f (s2 , x) − f (s1 , x) is increasing in x. If f were differentiable, both properties would apply if all cross-partial derivatives were non-negative. A function g is submodular if −g is supermodular.

20

We may now state the Monotone Maximum Theorem: if S is a lattice, X is a partially ordered set, Γ(x) is an increasing correspondence, function f (s, x) is supermodular in s and has increasing differences in (s, x), and the correspondence γ(x) ≡ arg max f (s, x), then γ(x) is a sublattice of S and increasing in x. s∈Γ(x)

Appendix B: Proofs Proof of Proposition 1: (Note: this proof uses recursive methods developed by Stokey and Lucas with Prescott (1989) and extended to monotone problems by Hopenhayn and Prescott (1992). This model extends Hopenhayn and Prescott’s model mildly by including an option to exit to search.) Let W0 be a constant and define operator T1 for bounded functions J0 such that

where

Z T1 J0 (π, I, I 0 ; W0 ) ≡ R(π, I, I 0 ) − c(I, I 0 ) + β V0 (π 0 , I 0 , W0 , ) dF (π 0 , π), ½ ¾ 0 V0 (π, I, W0 ) ≡ max W0 − c(I, 0), 0 max 0 J0 (π, I, I ; W0 ) . I ∈Γ(π,I),I 6=0

It is straightforward to verify that T1 satisfies Blackwell’s (1965, p. 232) sufficient conditions, so there exists a unique fixed point for T1 , call it J0∗ (·; W0 ). Recall that (R − c) is strictly increasing in π, supermodular in (I, I 0 ), and has increasing differences–and in one part of the hypothesis, convex in π. All these properties are preserved by maximization, integration over an increasing transition function, and summation, so T1 maps functions J0 having those properties into the (closed) subset of such functions. Thus, J0∗ has those properties, as does the associated V0∗ (π, I, W0 ), except that V0∗ is constant over the range where V0∗ = W0 − c(I, 0). Finally, define a second operator T2 by T2 W0 ≡ ζ + β

R

V0∗ (π, 0, W0 ) dG(π). Blackwell’s conditions apply

again, so T2 is a contraction, and thus it has a unique fixed point. Call it W and define V (π, I) ≡ V0∗ (π, I, W ). Then V is the value function, and it has the properties of the more general function V0∗ (π, I, W0 ) above. Further, J ≡ J0∗ (·; W ), so it too has the properties demonstrated for the more general class of functions J0∗ .¤ Proof of Corollary 2: First we show that the optimal I 0 is 0 only for sufficiently low π and I.

Since

J(π, I, I 0 ) is strictly increasing in π, if J(π0 , I, I 0 ) ≥ W − c(I, 0), then J(π, I, I 0 ) ≥ W − c(I, 0) for all π ≥ π0 .

21

Given π, I 0 , and I2 ≥ I1 , the difference in marginal returns is

J(π, I2 , I 0 ) − [W − c(I2 , 0)] − J(π, I1 , I 0 ) + [W − c(I1 , 0)] ≥ R(π, I2 , I 0 ) − R(π, I1 , I 0 ) +c(I2 , 0) − c(I1 , 0) − c(I2 , I 0 ) + c(I1 , I 0 ).

The right-hand expression is non-negative, since R is increasing in I and c is submodular. Thus, if the optimal I 0 = 0 at (π 0 , I0 ), it will also be optimal for all (π, I) ≤ (π 0 , I0 ). For (π, I) such that

sup I 0 ∈Γ(π,I)\{0}

J(π, I, I 0 ) ≥ W − c(I, 0), it follows that V (π, I) =

sup

J(π, I, I 0 ).

I 0 ∈Γ(π,I)\{0}

We thus seek to maximize a supermodular function over a lattice, and Topkis (1998, Corollary 2.7.1 and Theorem 2.8.1) has shown that the set of solutions (γ(π, I)) is an increasing subcomplete sublattice.¤ Proof of Proposition 3: The proof parallels that of Proposition 1. First consider a rise in ζ. Let W0 (−ζ) be some decreasing function of −ζ. Define operator T1 for bounded functions J0 and function V0 by Z T1 J0 (π, I, I 0 ; W0 (−ζ), −ζ) ≡ R(π, I, I 0 ) − c(I, I 0 ) + β V0 (π, I, W0 (−ζ), −ζ) dF (π 0 , π) ½ ¾ V0 (π, I, W0 (−ζ),−ζ) ≡ max W0 (−ζ)−c(I, 0), 0 max J0 (π, I, I 0; W0 (−ζ),−ζ) . I ∈Γ(π,I)\{0}

T1 is a contraction, so call its fixed point J0∗ . Suppose some function J0 is increasing and convex in π, increasing in W0 , decreasing in −ζ, and supermodular; has increasing differences; and that for all ζ 1 ≥ ζ 2 and for all (π, I, I 0 ), J0 (·, W0 (−ζ 2 ), −ζ 2 ) − J0 (·, W0 (−ζ 1 ), −ζ 1 ) ≥ W0 (−ζ 2 ) − W0 (−ζ 1 ). Then so do T1 J0 , the ½ ¾ fixed point J0∗ , and V0∗ (π, I, W0 (−ζ),−ζ) ≡ max W0 (−ζ) − c(I, 0), 0 max J0∗ (π, I, I 0 ; W0 (−ζ), −ζ) . Further, define operator T2 by T2 W0 (−ζ) ≡ ζ +β

R

I ∈Γ(π,I)\{0}

V0∗ (π, 0, W0 (−ζ)), −ζ)) dG(π).

T2 is also a contraction,

so call its fixed point W (−ζ). Note that W (−ζ) is a decreasing function of −ζ (because ζ is, trivially, and V0∗ (π, 0, W0 (−ζ)), −ζ)) is, from above), verifying the initial claim that W is a decreasing function. Then the value function is V (π, I; −ζ)) ≡ V0∗ (π, I, W (−ζ), −ζ), the conditional value function is J ≡ J0∗ (·; W, −ζ), and both have the properties of the more general functions above. Thus, (a) the value function is increasing in ζ, and (b) by the logic of Corollary 2, the optimal I 0 decreases in ζ.

22

Now consider an increase in α. The proof is the similar, with α in place of ζ, except where we showed that the fixed point of T2 is decreasing. For α1 ≥ α2 , Z

[V0∗ (·,W0 (−α2 ),−α2 )−V0∗ (·,W0 (−α1 ),−α1 )] dG(π 0 ,α1 ) Z Z ∗ 0 +β V0 (·, W0 (−α2 ), −α2 ) dG(π ; α2 ) − β V0∗ (·, W0 (−α2 ), −α2 ) dG(π 0 ; α1 ),

T2 W0 (−α2 ) − T2 W0 (−α1 ) = β

which is ≤ 0 because V0∗ is decreasing in −α and G(π0 ; α1 ) dominates G(π 0 ; α2 ). Now consider an increase in s. Again, the proof is similar, with s taking the place of ζ. We only must show that the fixed point of T2 is a decreasing function. For s2 ≥ s1 , Z

[V0 (·, W0 (−s1 ),−s1 )−V0 (·, W0 (−s2 ),−s2 )] dG(π 0, s2 ) Z Z +β V0 (·, W0 (−s1 ), −s1 ) dG(π0 ; s1 ) − β V0 (·, W0 (−s1 ), −s1 ) dG(π 0 ; s2 ),

T2 W0 (−s1 ) − T2 W0 (−s2 ) = β

which is non-positive because V0 is decreasing in −s and convex in π (by Jensen’s Inequality). Finally, consider an increase in σ. Let W0 (−σ) be some decreasing function of −σ, and define operator T1 and function V0 analogously to the cases above. Again, the unique fixed point of this operator, call it J0∗ (W0 (−σ), −σ), and the associated V0∗ are increasing and convex in π. We now show that J and V are decreasing in −σ. If J0 is decreasing in −σ, so is the function V0 , and thus υ(π, I, −σ) ≡ V0 (π, I, W0 (−σ), −σ) is decreasing in −σ and convex in π. Thus if σ 2 > σ 1 , Z

[υ(·, −σ1 ) − υ(·, −σ2 )] dF (π0 , π, σ 1 ) Z Z +β υ(·, −σ2 ) dF (·, σ 1 ) − β υ(·, −σ 2 ) dF (·, σ 2 ),

T1 J0 (·, −σ 1 ) − T1 J0 (·, −σ 2 ) = β

which is non-positive. We then must show that the fixed point (W ) of the analogue to operator T2 is decreasing in −σ, but this follows immediately from the fact that V0∗ is. The rest of the proof follows.¤ Proof of Proposition 4: Since the components of I move together anyway, treat I as univariate in order to minimize the notational burden. Under the given assumptions, the optimal I ∗ at the intensive margin is

23

determined by J3 (π, I, I ∗ (π, I, σ), σ) ≡ 0, so sign(∂I ∗ /∂σ) =sign(J3σ ). Computing this derivative, 1 J3σ β

=

=

∂ ∂σ

Z

J2 (π 0 , I ∗ , I ∗∗ ) dF

(where I ∗∗ ≡ I ∗ (π 0 , I ∗ , σ))   Z ∂   R2 − c1 dF  −F (π ∗ (I ∗ ), π)c1 (I ∗ , 0) + ∂σ π ∗ (I ∗ )

  · ¸ Z  ∂  −F (π ∗ )c1 (I ∗ , 0) + [1 − F (π∗ )] (R2 − c1 ) | + R12 [1 − F ] dπ0 =  ∂σ  π∗ π∗ · ¸· ¸ Z d = − F (π ∗ , π, σ) (R2 − c1 ) | + c1 (I ∗ , 0) − R12 Fσ dπ0 , dσ π∗ π∗

by the Envelope Theorem and integration by parts. The first term is positive by assumption, and the second is since c has non-increasing differences. Since σ indexes a mean-preserving spread, (Rothschild and Stiglitz, 1970), so

R

π∗

R

Fσ = 0 and

R

Fσ < 0

π∗

R12 Fσ ≤ 0 for R112 ≤ 0. Thus J3σ < 0 at the intensive margin.

At the extensive margin, commitment increases if Jσ (π ∗ (I), I, I ∗ ) > Wσ . Wσ = 0 in the described cases, so there that result follows directly from Proposition 3. Otherwise,

(Jσ − Wσ ) /β =

Z

V

∂ (dF ) + ∂σ

·Z

Vσ (π0 , I ∗ ) dF (π0 , π∗ (I)) −

Z

¸ Vσ (π0 , 0) dG(π 0 ) .

The first term is necessarily positive, but the difference in brackets may be either positive or negative depending on the characteristics of F and G, so the net effect is ambiguous.¤

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References [1] Amir, R., 2005. Supermodularity and Complementarity in Economics: An Elementary Survey. Southern Economic Journal 71, 636—660. [2] Athey, S., 2002. Monotone Comparative Statics Under Uncertainty. Quarterly Journal of Economics 117, 187—223. [3] Blackwell, D., 1965 Discounted Dynamic Programming. Annals of Mathematical Statistics 36, 226—235. [4] Brien, M.J., Lillard, L.A., Stern, S., forthcoming. Cohabitation, Marriage, and Divorce in a Model of Match Quality. International Economic Review. [5] Diamond, P.A., 1981. Mobility Costs, Frictional Unemployment, and Efficiency. Journal of Political Economy 89, 798—812. [6] Drewianka, S., 2003. Estimating Social Effects in Matching Markets: Externalities in Spousal Search. Review of Economics and Statistics 85, 409—423. [7] Drewianka, S., 2004. How Will Reforms of Marital Institutions Influence Marital Commitment? A Theoretical Analysis. Review of Economics of the Household 2, 303—323. [8] Farber, H.S., 1994. The Analysis of Interfirm Worker Mobility. Journal of Labor Economics 12, 554-593. [9] Hopenhayn, H.A., Prescott, E.C., 1992. Stochastic Monotonicity and Stationary Distributions for Dynamic Economies. Econometrica 60, 1387—1406. [10] Jovanovic, B., 1979a. Job Matching and the Theory of Turnover. Journal of Political Economy 87, 972—990. [11] Jovanovic, B., 1979b. Firm-specific Capital and Turnover. Journal of Political Economy 87, 1246—1260. [12] Loughran, D.S., 2002. The Effect of Male Wage Inequality on Female Age at First Marriage. Review of Economics and Statistics 84, 237—250. [13] Milgrom, P., Roberts, J., 1990. The Economics of Modern Manufacturing: Technology, Strategy, and Organization. American Economic Review 80, 511—528. 25

[14] Milgrom, P., Roberts, J., 1995. Complementarities and Fit: Strategy, Structure, and Organizational Change in Manufacturing. Journal of Accounting and Economics 19, 179—208. [15] Milgrom, P., Shannon, C., 1994. Monotone Comparative Statics. Econometrica 62, 157—180. [16] Mortensen, D.T., 1978. Specific Capital and Labor Turnover. Bell Journal of Economics 9, 572—586. [17] Mortensen, D.T., 1988. Wages, Separations, and Job Tenure: On-the-Job Specific Training or Matching? Journal of Labor Economics 6, 445—472. [18] Pierret, C.R., 1995. Models of the Marriage Market: Theory and Evidence. Ph.D. dissertation, Northwestern University, Evanston, IL, USA. [19] Pissarides, C.A., 1994. Search Unemployment with On-the-Job Search. Review of Economic Studies 61, 457—475. [20] Rothschild, M., Stiglitz, J.E., 1970. Increasing Risk I: A Definition. Journal of Economic Theory 2, 225—243. [21] Stokey, N.L., Lucas, R.E., Jr., with Prescott, E.C., 1989. Recursive Methods in Economic Dynamics. Harvard University Press, Cambridge, MA. [22] Topkis, D.M., 1978. Minimizing a Submodular Function on a Lattice. Operations Research 26, 305—321. [23] Topkis, D.M., 1998. Supermodularity and Complementarity. Princeton University Press, Princeton, NJ. [24] Weiss, Y., Willis, R.J., 1997. Match Quality, New Information, and Marital Dissolution. Journal of Labor Economics 15, S293—S329.

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Notes 1. The extra cost of ending the marriage might be an incentive-compatibility mechanism to ensure that the marginal benefit of marriage µ is only received by persons likely to stay married. If there were more than two periods (as in the more general model below), it could also represent a reduction in the value of search insofar as potential new partners dislike persons who have broken previous commitments. 2. The infinite horizon is not critical; all of the results also emerge from a finite-horizon model–indeed, by a similar logic, with proofs proceeding inductively rather than recursively. 3. Nothing would change if π had more than one dimension (π ∈ Rk ), but the notational burden is greatly reduced by keeping it unidimensional. The multi-dimensional case would be interesting, e.g., if the process generating π0 depended on multiple lags of π. 4. Unless modified by the word “strictly,” all of the properties discussed are meant only in the weak sense–for instance, the properties “increasing,” “convex,” and “supermodular” include the cases “constant,” “linear,” and “additively separable.” 5. Boundedness is another departure from the simple model we studied earlier. It is assumed here in order to facilitate use of the sup-norm for our proofs. However, in a finite horizon model, unboundedness would be permissible as long as the value function remains integrable. 6. For example, that restriction caused some problems in an early version of the paper by Brien, Lillard. and Stern (forthcoming). 7. However, the functions may not be convex if R is not. The result in the careers literature thus depends in part on its assumption (unusual in other fields) that workers are risk-neutral over wages. 8. Jovanovic’s (1979b) model does not contain such a cost, but the absence of such a cost is surely a matter of simplicity and algebraic tractability rather than a fundamental feature of his model. 9. There are two ways that could happen. First, if the single-crossing point π c (π) < π ∗ (I) (where F (π c , π, σ 2 ) ≡ F (πc , π, σ 1 )), then all of the increased risk of a bad match would fall in the range in which the match would have ended anyway–which is hardly what we have in mind when discussing greater uncertainty (though see Loughran’s (2002) discussion of a related point). Second, it is conceivable, if unlikely, that the convexity of V could be so strong that π∗ drops dramatically and separation becomes less likely.

27

Cohabit

Stay Single

Marry

Value: V(π,0)

J(m) J(c) V

W

V J(m) J(c) W

W

J(c) J(m) Initial Payoff (π)

Figure 1: Agents with better quality matches make stronger commitments to them. This is one source of the value function’s convexity.

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