Review of Economic Studies (2010) 77, 1138–1163 © 2010 The Review of Economic Studies Limited 0034-6527/10/00411011$02.00 doi: 10.1111/j.1467-937X.2009.00599.x Interdependent Durations BO E HONORE´ Princeton University ´ AUREO DE PAULA University of Pennsylvania First version received February 2008; final version accepted September 2009 (Eds.) This paper studies the identification of a simultaneous equation model involving duration measures It proposes a game theoretic model in which durations are determined by strategic agents In the absence of strategic motives, the model delivers a version of the generalized accelerated failure time model In its most general form, the system resembles a classical simultaneous equation model in which endogenous variables interact with observable and unobservable exogenous components to characterize an economic environment In this paper, the endogenous variables are the individually chosen equilibrium durations Even though a unique solution to the game is not always attainable in this context, the structural elements of the economic system are shown to be semi-parametrically identified We also present a brief discussion of estimation ideas and a set of simulation studies on the model INTRODUCTION This paper investigates the identification of a simultaneous equation model involving durations We present a simple game theoretic setting in which spells are determined by multiple optimizing agents in a strategic way As a special case, our proposed structure delivers the familiar proportional hazard model as well as the generalized accelerated failure time model In a more general setting, the system resembles a classical simultaneous equation model in which endogenous variables interact with each other and with observable and unobservable exogenous components to characterize an economic environment In our case, the endogenous variables are the individually chosen equilibrium durations In this context, a unique solution to the game is not always attainable In spite of that, the structural elements of the economic system are shown to be semi-parametrically point-identified The results presented here have connections to the literatures on simultaneous equations and statistical duration models as well as to the recent research on incomplete econometric models that result from structural (game theoretic) economic models (Berry and Tamer, 2006) The paper also adds to the research on time-varying explanatory variables in duration models In that literature, the time-varying explanatory variable is considered to be “external” (see, for instance, Heckman and Taber, 1994; Hausman and Woutersen, 2006) In an earlier paper, Lancaster (1985) considers a duration model where there is simultaneity with another (nonduration) variable for a single agent In this paper, we focus on simultaneously determined duration outcomes with more than one agent More recently, Abbring and van den Berg (2003) consider a model where a duration outcome depends on a time-varying explanatory variable, another duration variable, and endogeneity arises because an unobserved heterogeneity term 1138 Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 and HONORE´ & DE PAULA INTERDEPENDENT DURATIONS 1139 P(Ti ≤ t|Tj = tj ) = if t < tj Fi (t)(1 − πi (tj )) Fi (t)(1 − πi (tj )) + πi (tj ) otherwise where i = j , Fi (·) is a continuous CDF, and πi (·) is between and In other words, conditional on Tj , Ti has a continuous distribution, except that there is a point mass at Tj One can motivate such a distribution by a model in which three types of events occur The first two “fatal events” lead to terminations of the spells for individuals and 2, respectively, and the third will lead both spells to terminate These “shock” models, introduced by Marshall and Olkin (1967), have been used in industrial reliability and biomedical statistical applications (see, for example, Klein, Keiding, and Kamby, 1989) In these models, the relationship between the durations is driven by the unobservables, but no direct relationship exists between them This is similar to the dependence between two dependent variables in a “seemingly unrelated regressions” framework In economics, it is interesting to consider models in which durations depend on each other in a structural way, allowing for an interpretation of estimated parameters closer to economic theory This is the aim of our paper As such, the difference between Marshall and Olkin’s model and ours is similar to the difference between seemingly unrelated regressions and structural simultaneous equations models To achieve this, we formulate a very simple game theoretic model with complete information where players make decisions about the time at which to switch from one state to another Our analysis bears some resemblance to previous studies in the empirical games literature, such as Bresnahan and Reiss (1991) and, more recently, Tamer (2003) Bresnahan and Reiss (1991), building on the work in Amemiya (1974) and Heckman (1978), analyse a simultaneous game © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 impacts both of the two durations One can think of the contribution of this paper as providing an alternative framework that allows for endogeneity There are many situations in which two or more durations interact with each other Park and Smith (2006), for instance, cite circumstances in which late rushes in market entry occur as some pioneer firm creates a market for a new service or good In our model, the decision by the pioneer is understood as having an impact on the attractiveness of the market to other potential entrants In another related example, Fudenberg and Tirole (1985) examine technology adoption by a set of agents In their setting, the adoption time by one agent affects the other agent’s adoption time in a number of ways Under some circumstances, a “diffusion” equilibrium arises, in which players adopt the new technology sequentially For other parametric configurations, adoption occurs simultaneously and there are many equilibrium times at which this occurs Our model allows for similar results where sequential timing arises under some realizations of our game and simultaneous timing occurs as multiple equilibria for other realizations Peer effects in durations also play a natural role in some empirical examples leading to interdependent durations In Paula (2009), soldiers in the Union Army during the American Civil War tended to desert in groups Mass desertion could be thought of as lowering the costs of desertion, directly and indirectly, as well as reducing the combat capabilities of a military company Another example involves the decision by adolescents to first consume alcohol, drugs, or cigarettes, or to drop out of high school In this case, the timing chosen by one individual could have an effect on the decisions of others in a given reference group Other phenomena that could also be analysed with our model include the decision to retire among couples, the simultaneous bidding on EBay auctions, and the pricing behaviour of competing firms The examples above typically result in a positive probability of concurrent timing Let Ti and Tj denote the duration variables for two individuals i and j , and suppose that we are interested in the distribution of Ti conditional on Tj , P(Ti ≤ t|Tj = tj ) (and vice versa) From a statistical viewpoint, one might specify a reduced-form model for the conditional distributions as 1140 REVIEW OF ECONOMIC STUDIES THE ECONOMIC MODEL The economic model consists of a system of two individuals who interact Information is complete for the individuals Each individual i chooses how long to take part in a certain activity by selecting a termination time Ti ∈ R+ , i = 1, Agents start at an activity that provides a utility flow given by the positive random variable Ki ∈ R+ At any point in time, an individual can choose to switch to an alternative activity that provides him or her with a flow utility See Hougaard (2000) and Frederiksen, Honor´e and Hu (2007) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 with a discrete number of possible actions for each agent A major pitfall in such circumstances is that “when a game has multiple equilibria, there is no longer a unique relation between players’ observed strategies and those predicted by the theory” (Bresnahan and Reiss, 1991) When unobserved components have large enough supports, this situation is pervasive for the class of games they analyse Tamer (2003) characterizes this particular issue as an “incompleteness” in the model and shows that this nuisance does not necessarily preclude point identification of the deep parameters in the model Our model also possesses multiple equilibria and, like Tamer, we also obtain point identification of the main structural features of the model This is possible because certain realizations of the stochastic game we analyse deliver unique equilibrium outcomes with sequential timing choices while multiplicity occurs if and only if spells are concurrent We are then able to obtain point identification using arguments similar to the ones used to obtain identification in mixed proportional hazards models (see, for example, Elbers and Ridder, 1982) Since the econometrician observes outcomes for two agents, our model is a multiple duration model The availability of multiple duration observations for a given unit provides leverage in terms of both identification and subsequent estimation (see Honor´e, 1993; Horowitz and Lee, 2004; Lee, 2003) In the panel duration literature, subsequent spells, such as unemployment durations for workers or time intervals between transactions for assets, are typically observed for a given individual This allows for the introduction of individual-specific effects In this paper, parallel individual spells are recorded for a given game, and some elements in our analysis can be made game-specific, mimicking the role of individual-specific effects in the panel duration literature.1 We use a continuous time setting This is the traditional approach in econometric duration studies and statistical survival analysis Many game theoretic models of timing are also set in continuous time The framework can be understood as the limit of a discrete time game As the frequency of interactions increases, the setting converges to our continuous time framework, which can in turn be seen as an approximation to the discrete time model The exercise is thus in line with the early theoretical analysis by Simon and Stinchcombe (1989), Bergin and MacLeod (1993) and others and with most of the econometric analysis of duration models (e.g Elbers and Ridder, 1982; Heckman and Singer, 1984; Honor´e, 1990; Hahn, 1994; Ridder and Woutersen, 2003; Abbring and van den Berg, 2003) See also van den Berg, 2001 The remainder of the paper proceeds as follows In the next section we present the economic model Section investigates the identification of the many structural components in the model The fourth section discusses extensions and alternative models to our main framework Section briefly discusses estimation strategies and the subsequent section presents simulation exercises to illustrate the consequences of ignoring the endogeneity problem introduced by the interaction or misspecifying the equilibrium selection mechanism We conclude in the last section HONORE´ & DE PAULA INTERDEPENDENT DURATIONS 1141 ti ∞ Ki e −ρs ds + Z (s)ϕ(xi )e 1(s≥Tj ) δ −ρs e ds ti The first-order condition for maximizing this with respect to ti is based on: Ki e −ρti − Z (ti )ϕ(xi )e 1(ti ≥Tj ) δ −ρti e (1) where 1A is an indicator function for the event A This may not be equal to zero for any ti since it is discontinuous at ti = Tj Given the opponent’s strategy, the optimal behaviour of an agent δ in this game consists of monitoring the (undiscounted) marginal utility Ki − Z (t).ϕ(xi ).e (t ≥Tj ) at each moment of time t As long as this quantity is positive, the individual participates in the initial activity, and he or she switches as soon as the marginal utility becomes less than or equal to zero As mentioned previously, the relative flow between the inside and outside activities is the ultimate determinant of an individual’s behaviour As is the case with the familiar random utility model, our model identifies relative utilities For example, suppose that the destination state is retirement, with utility flow given by Z1 (t)ϕ1 (xi ), and that the utility flow in the nonretirement state is Ki Z2 (t)ϕ2 (xi ) (where Ki represents initial health, t is age, and xi is a set of covariates, and we abstract from the interaction term e δ ) This would be observationally equivalent to a model where the utility flow in the current state is Ki and utility in the outside activity is Z (t)ϕ(xi ) with Z (t) ≡ Z1 (t)/Z2 (t) and ϕ(xi ) ≡ ϕ1 (xi )/ϕ2 (xi ) An appropriate concept for optimality in the presence of the interaction represented by δ is that of mutual best responses Consider the optimal Ti of individual i given that individual j has chosen Tj It is clear from (1) that T1 = inf{t1 : K1 − Z (t1 ).ϕ(x1 ).e 1(t1 ≥T2 ) δ < 0} T2 = inf{t2 : K2 − Z (t2 ).ϕ(x2 ).e 1(t2 ≥T1 ) δ (2) < 0} In the absence of interaction (δ = 0), the individual switches at Ti = Z −1 (Ki /ϕ(xi )) or ln Z (Ti ) = − ln ϕ(xi ) + i ≡ln ki which is a semi-parametric generalized accelerated failure time (GAFT) model like the one discussed in Ridder (1990) For example, if Z (t) = λt αi , ϕ(xi ) = exp(xi β) and Ki ∼ exp(1), One could in principle allow for (“external”) time-varying covariates, but these would have to be fully forecastable by the individuals © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 U (t, xi ) where the vector xi denotes a set of covariates.2 This utility flow is incremented by a factor e δ when the other agent switches to the alternative activity We assume that δ ≥ Since only the difference in utilities will ultimately matter for the decision, there is no loss in generality in normalizing the utility flow in the initial activity to be a time-invariant random variable In order to facilitate the link of our study to the analysis of duration models, we adopt a multiplicative specification for U (t, xi ) as Z (t)ϕ(xi ) where Z : R+ → R+ is a strictly increasing, absolutely continuous function such that Z (0) = Assuming an exponential discount rate ρ, individual i ’s utility for taking part in the initial activity until time ti given the other agent’s timing choice Tj is: 1142 REVIEW OF ECONOMIC STUDIES the cumulative distribution function of Ti is given by FTi (t) = P[(Ki e −xi β /λ)1/αi ≤ t] = P(Ki ≤ t αi λe xi β ) = − exp(−t αi λ exp(xi β)) T i = Z −1 (Ki e −δ /ϕ(xi )), i = 1, T i = Z −1 (Ki /ϕ(xi )), i = 1, Because δ > 0, T i < T i , i = 1, If t < T i , then Z (t)ϕ(xi ) − Ki < Z (t)ϕ(xi )e δ − Ki < 0, and as a result agent i would not like to switch activities regardless of the other agent’s action Analogously, if T i < t < T i , then Z (t)ϕ(xi )e δ − Ki > but Z (t)ϕ(xi ) − Ki < 0, and agent i would switch if the other agent switches, but not if the other player does not Finally, if t > T i , then Z (t)ϕ(xi ) − Ki > and the agent is better off switching at a time less than t In region of Figure 1, T1 < T2 and the equilibrium is unique This is because the region is such that K1 /ϕ(x1 ) < K2 e −δ /ϕ(x2 ) and hence T < T Here, for any t less than T , Z (t)ϕ(x2 )e δ − K2 is less than zero and agent has no incentive to switch even if agent has already switched Also, Z (t)ϕ(x1 ) − K1 is less than zero and agent would not switch either Once t > T , then Z (t)ϕ(x1 ) − K1 is strictly greater than and agent will prefer to have switched earlier, no matter what action the second agent might take It is therefore optimal for agent to switch at T1 = T This in turn induces agent to switch at T2 = T > T1 Figure Equilibrium regions © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 and the model corresponds to a proportional hazard duration model with a Weibull baseline hazard When δ > 0, the solution to (2) depends on the realization of (K1 , K2 ) There are five scenarios depicted in Figure To understand the alternative scenarios, we first define T i and T i , i = 1, as the values δ δ that set expression (1) to zero when e (ti ≥Tj ) = e δ and when e (ti ≥Tj ) = 1, respectively: HONORE´ & DE PAULA INTERDEPENDENT DURATIONS 1143 In region 2, T1 = T2 and there are multiple equilibria This region is given by K1 /ϕ(x1 ) > K2 e −δ /ϕ(x2 ) and K2 /ϕ(x2 ) > K1 e −δ /ϕ(x1 ) This implies that T > T and T > T To see that individuals will stop simultaneously and there are many equilibria, let T = max T , T and T = T , T © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Because T > T and T > T , we have that T ≤ T We now consider three cases depending on t’s location relative to T and T For t < T , let j be the agent such that T = T j Since t < T j , individual j would not be willing to switch regardless of the action of the other agent, i Also since t < T i , individual i will not switch either given that individual j does not switch Hence no agent switches when t < T For T ≤ t ≤ T , T i ≤ t ≤ T i for each agent At each point in time in the interval, an agent can therefore no better than the alternative activity if the other agent has already switched Hence, any profile such that T ≤ T1 = T2 ≤ T will be an equilibrium Finally, for T < t, T i is less than t for at least one individual, who then has an incentive to decrease his or her switching time toward T regardless of what the other agent does Hence, simultaneous switching at any t in the interval [T , T ] is an equilibrium Region is similar to region The only difference is that the subscripts have been exchanged In this region, T2 < T1 and the equilibrium is unique The final two cases are when K1 /ϕ(x1 ) = K2 e −δ /ϕ(x2 ) or K1 /ϕ(x1 ) = K2 e −δ /ϕ(x2 ) In these cases, the equilibrium is unique and individuals switch simultaneously Since K1 and K2 are continuous random variables, these regions occur with probability zero and we therefore skip a detailed analysis Regions and also deliver a unique equilibrium In region 2, a simultaneous switch at any t in [T , T ] would be an equilibrium This interval will be degenerate if δ is equal to zero It is also important to note that region can be distinguished from regions and by the econometrician, since this will be used in the identification of the model We end this section with a brief discussion on the multiple equilibria encountered in region In our approach, we are agnostic as to which of these equilibria is selected Some of the solutions in that region may be singled out by different selection criteria nevertheless The Nash solution concept we use is equivalent to that of an open-loop equilibrium (as discussed, for example, in Fudenberg and Tirole, 1991, Section 4.7): one in which individuals condition their strategies on calendar time only and hence commit to this plan of action at the beginning of the game If individuals can react to events as time unfolds, a closed-loop solution concept, which here would be equivalent to subgame perfection, would single out the earliest of the Nash equilibria, in which individuals switch at T Intuitively, an optimal strategy in region contingent on the game history would prescribe switching simultaneously at any time between T and T Faced with an opponent carrying such a (closed-loop) strategy, an individual might as well switch as soon as possible to maximize his or her own utility flow This outcome also corresponds to the Pareto-dominant equilibrium In this case, the equilibria displayed in our analysis would still be Nash, but not necessarily subgame-perfect In selecting one of the multiple equilibria that may arise, the early equilibrium is nevertheless a compelling equilibrium and we give it special consideration in the simulation exercises performed later in the paper Other selection mechanisms may nonetheless point to later equilibria among the many Nash equilibria available Players need to know when to act and so in a coordinated way: to take the initiative a person needs to be confident that he or she will not be acting alone as the switching decision is irreversible This coordination risk may lead to later switching times For this reason, we remain agnostic as to which Nash equilibrium is selected 1144 REVIEW OF ECONOMIC STUDIES IDENTIFICATION In this section we ask what aspects of the model can be identified by the data once one recognizes the endogeneity of choices and abstains from an equilibrium selection rule The proof strategy is similar to that in, for example, Elbers and Ridder (1982) and Heckman and Honor´e (1989) applied to the events T1 < T2 and T1 > T2 Like those papers, we rely crucially on the continuous nature of the durations, and it is not straightforward to generalize our results to the case where one observes discretized versions of the durations The subsequent analysis relies on the following assumptions: Assumption The function Z (·) is differentiable with positive derivative Assumption At least one component of xi , say xik , is such that supp(xik ) contains an open subset of R Assumption The range of ϕ(·) is R+ and it is continuously differentiable with non-zero derivative In Assumption 1, we require that g(0, 0) be bounded away from zero and infinity This assumption is related to assumptions typically used in the mixed proportional hazard/GAFT literature with respect to the distribution of the unobserved heterogeneity component To see this, consider a bivariate mixed proportional hazards model with durations Ti , i = 1, that are independent conditional on observed and unobserved covariates The integrated hazard is given by Z (·)ϕ(xi )θi , i = 1, with Z (·) as the baseline integrated hazard; ϕ(xi ), a function of observed covariates xi ; and θi , a positive unobserved random variable In other words, for this model, at the optimal stopping time and when Ti < Tj : Z (Ti )ϕ(xi ) = K˜ i /θi ≡ Ki , i = 1, where K˜ i follows a unit exponential distribution (independent of x’s and θ ’s) See, for example, Ridder (1990) Let f (·, ·) denote the joint probability density function for (θ1 , θ2 ) Then the joint density for (K1 , K2 ), g(·, ·), is: g(k1 , k2 ) = R+ R+ θ1 θ2 e −k1 −k2 f (θ1 , θ2 )d θ1 d θ2 This gives g(0, 0) = E(θ1 θ2 ), which is positive by assumption Our requirement that it be finite is then essentially the finite mean assumption in the traditional mixed proportional hazards model identification literature Economically, it is clear that the model is observationally equivalent to one in which the same monotone transformation is applied to the utilities in the two activities Since a power transformation would preserve the multiplicative structure assumed here, this means that the model should only be identified up to power transformations Assumption rules out such a transformation, since the transformed K ’s would not have finite, non-zero density at the origin Assumptions 2–4 are stronger than necessary Most importantly, the Appendix shows that for some of the identification results one can allow xi to have a discrete distribution The identification of ϕ(·) uses variation in at least one component of xi © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Assumption K1 and K2 are jointly distributed according to G(·, ·), where G(·, ·) is a continuous cumulative distribution function with full support on R2+ Furthermore, its corresponding probability density function g(·, ·) is bounded away from zero and infinity in a neighbourhood of zero HONORE´ & DE PAULA INTERDEPENDENT DURATIONS 1145 The following results establish that Assumptions 1–4 are sufficient (though not necessary in many cases) for the identification of the different components in the model We begin by analysing ϕ(·) Theorem (Identification of ϕ(·)) Under Assumptions and 2, the function ϕ(·) is identified up to scale if supp(x1 , x2 ) = supp(x1 ) × supp(x2 ) fT1 ,T2 |x1 ,x2 (t1 , t2 |x1 , x2 ) = λ(t1 )ϕ(x1 )λ(t2 )ϕ(x2 )e δ g(Z (t1 )ϕ(x1 ), Z (t2 )ϕ(x2 )e δ ) where t Z (t) = λ(s)ds Given two sets of covariates (x1 , x2 ) and (x1 , x2 ) we obtain that fT1 ,T2 |x1 ,x2 (t1 , t2 |x1 , x2 ) ϕ(x1 )ϕ(x2 )g(Z (t1 )ϕ(x1 ), Z (t2 )ϕ(x2 )e δ ) = lim (t1 ,t2 )→(0,0) fT ,T |x ,x (t1 , t2 |x1 , x2 ) (t1 ,t2 )→(0,0) ϕ(x1 )ϕ(x2 )g(Z (t1 )ϕ(x1 ), Z (t2 )ϕ(x2 )e δ ) 2 lim t1 ln (ϕ(x1 )/ϕ(x2 ))) (5) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Proof Consider the absolutely continuous component of the conditional distribution of (T1 , T2 ), the switching times for the agents, given the covariates x1 , x2 When T1 < T2 , using the fact that T1 = Z −1 (K1 /ϕ(x1 )) and T2 = Z −1 (K2 e −δ /ϕ(x2 )), we can use the Jacobian method to obtain the probability density function for (T1 , T2 ) on the set {(t1 , t2 ) ∈ R2+ : t1 < t2 } It is given by: 1146 REVIEW OF ECONOMIC STUDIES Identification of δ traces the survivor function (and consequently the cumulative distribution function) for the random variable ln K1 − ln K2 − δ = W − 2δ Since this is basically the random variable W displaced by 2δ, this difference is identified as the (horizontal) distance between the two cumulative distribution functions that are identified from the data (the events T1 > T2 and T1 < T2 conditioned on x) Figure (2) illustrates this idea From this argument, the parameter δ is identified In the proof of Theorem 3, Assumptions and are invoked to guarantee the identification of ϕ(·) If this function is identified for other reasons, we can dispense with this assumption Finally, we establish the identification of Z (·) and G(·, ·), the join distribution of K1 and K2 Theorem (Identification of Z (·) and G(·, ·)) Under Assumptions 1–4, the function Z (·) is identified up to scale, and the distribution G(·, ·) is identified up to a scale transformation Proof We first consider identification of Z (·) On the set {(t1 , t2 ) ∈ R2+ : t1 < t2 }, consider the function t1 h(t1 , t2 , x1 , x2 ) = t2 t1 = ∞ ∞ fT1 ,T2 |x1 ,x2 (s1 , s2 |x1 , x2 )ds2 ds1 λ(s1 )ϕ(x1 )λ(s2 )ϕ(x2 )e δ g(Z (s1 )ϕ(x1 ), Z (s2 )ϕ(x2 )e δ )ds2 ds1 t2 Consider the change of variables: ξ1 = Z (s1 )ϕ(x1 ) ξ2 = Z (s2 )e δ ϕ(x2 ) and rewrite h as Z (t1 )ϕ(x1 ) h(t1 , t2 , x1 , x2 ) = ∞ Z (t2 )e δ ϕ(x2 ) g(ξ1 , ξ2 )d ξ1 d ξ2 Then notice that d ln Z (t1 ) ϕ(x1 ) ∂h/∂t1 λ(t1 )ϕ(x1 ) = = ∂h/∂x1k Z (t1 )∂k ϕ(x1 ) dt1 ∂k ϕ(x1 ) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Figure HONORE´ & DE PAULA INTERDEPENDENT DURATIONS 1147 Integrating and exponentiating yields CZ (s)ϕ(x1 )/∂k ϕ(x1 ) The mechanics of the proof suggests that we can also allow Z (·) to depend on i as is the case with ϕ(·), but the characterization of the equilibrium in Section assumes Z (·) to be the same for both individuals As in the previous result, the identification would still hold were the covariates for the two agents identical for a given draw of the game (x1 = x2 = x) The requirement that xi contain a continuously distributed component is not necessary either In the Appendix we present an alternative proof that dispenses with that assumption EXTENSIONS AND ALTERNATIVE MODELS In this section, we discuss results for some variations on the model depicted in Section 4.1 Individual-specific δ As mentioned earlier, in certain problems (such as the interaction between husband and wife) players may be easily labelled In this case, one can consider different δs for different players: δi , i = 1, The previous result would render identification for δ1 + δ2 The following establishes the identification of δ1 − δ2 and hence of δi , i = 1, Theorem (Identification of δi , i = 1, 2) tions 1–4 δi , i = 1, are identified under Assump- Proof The sum δ1 + δ2 is identified according to the arguments in the previous theorem Let k > Then lims→0 λ(s)λ(ks)ϕ(x1 )ϕ(x2 )e δ2 lims→0 fT1 ,T2 |x1 ,x2 (s, ks|x1 , x2 ) = lims→0 fT1 ,T2 |x1 ,x2 (ks, s|x1 , x2 ) lims→0 λ(ks)λ(s)ϕ(x1 )ϕ(x2 )e δ1 = λ(s)λ(ks) ϕ(x1 )ϕ(x2 )e δ2 × lim δ s→0 λ(ks)λ(s) ϕ(x1 )ϕ(x2 )e k >1 =e δ2 −δ1 which identifies δ2 − δ1 This and the previous result identify δi , i = 1, It is also possible to allow δ1 and δ2 to depend on x1 and x2 , respectively.3 In that case the right-hand side of (3) becomes ϕ(x1 )ϕ(x2 )e δ(x2 ) ϕ(x1 )ϕ(x2 )e δ(x2 ) , which again identifies ϕ up to scale (by varying We thank a referee for pointing this out © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 where C is a constant Given the identification of ϕ(·) up to scale, Z (·) is therefore identified up to scale (the constant C ) We next turn to identification of G(·, ·) Note that h defines the cumulative distribution function of (K1 , −K2 ), which can be traced out by varying Z (t1 )ϕ(x1 ) and Z (t2 )e δ ϕ(x2 ) (making sure that t1 < t2 ) Since δ is identified and Z (·) and ϕ(·) are identified up to scale, the distribution of (K1 , −K2 ) is identified up to a scale transformation The distribution of (K1 , K2 ) is therefore identified up to a scale transformation HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1123 5.2 Asymmetric countries G( )=1− k , for ≥ As is well known, the shape parameter k controls the dispersion of , with smaller values of k representing more dispersion It has to be larger than for the variance of productivity to be finite We show in the Appendix how the equilibrium conditions are simplified when productivity is distributed Pareto, and these equations are used to generate our numerical examples One convenient implication of the Pareto assumption is that condition (11) implies δdj + δxj = kfe , and therefore revenue of an average firm in the differentiated sector is independent of labour market frictions and is the same in both countries For the simulations, we also assume that a0A = a0B = a0 , so that labour market frictions in the homogeneous sector are the same in both countries, as a result of which expected income of workers, ω0j , is also the same in both countries, i.e ω0A = ω0B = ω0 In addition, we assume that aA > aB > a0 , so that labour market frictions are larger in the differentiated sectors of both countries than in their homogeneous sectors, and particularly so in country A This implies bA > bB > 1/2 Combining equations (23) and (24), we obtain the following expression for global revenues generated in the differentiated sector: ζ ζ QA + QB = 1+β MA (δdA + δxA ) + MB (δdB + δxB ) = ω0 (NA + NB ) φ2 β ζ ζ Therefore, whenever QA + QB rises, the world-wide allocation of workers to the differentiated sector, NA + NB , must also increase.41 Next note that Proposition establishes that a reduction in trade costs raises Qj in both countries Therefore, the above discussion implies that a 41 Note that this result does not rely on the Pareto assumption Under the Pareto assumption, however, we ζ ζ additionally have QA + QB = kfe (MA + MB )/φ2 , so that the total number of entrants into the differentiated sector must also increase Moreover, in the Appendix we show that in this case ω0 Nj /Mj = βkfe /(1 − β) That is, the number of workers searching for jobs in the differentiated sector relative to the number of firms depends on expected income ω0 , but does not depend on the trade cost or labour market frictions in the differentiated sector © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 We address in this section the impact of trade and labour market frictions on unemployment when the two countries are not symmetric We first discuss some analytical results and then turn to numerical examples to illustrate the key mechanisms and various special cases In our working paper, Helpman and Itskhoki (2007), we provide analytical results for countries that are nearly symmetric, in the sense that they have no labour market frictions in the homogeneous sector and the difference between their labour market frictions in the differentiated sector is very small Under these circumstances, bA > bB implies that: (i) a reduction in a country’s labour market frictions reduces the rate of unemployment in its trade partner, yet it reduces home unemployment if and only if the initial levels of friction in the labour markets are low; and (ii) country B has a lower rate of unemployment if and only if the levels of labour market frictions are low to begin with Evidently, a country’s level of unemployment depends not only on its own labour market frictions but also on those of its trade partner Moreover, lower domestic labour market frictions not guarantee lower unemployment relative to the trade partner, unless the frictions in both labour markets are low As a result, one cannot infer differences in labour market rigidities from observations of unemployment rates Richer results obtain with large labour market frictions, as we show below For our numerical illustrations, we use a Pareto distribution of productivity levels, 1124 REVIEW OF ECONOMIC STUDIES 42 In Figures 3–4 we use the following parameters: m0 = 2v0 = 1, fx = 3, fd = 1, fe = 0.5, k = 2.5, β = 0.75, ζ = 0.5, and L = 0.1 43 In Figures 3–4, country A specializes in the homogeneous good when bA ≥ b ; in Figure 4, country B specializes in the homogeneous good when bA ≤ b o ; in Figure country B specializes in the differentiated good for bA ≥ b © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 reduction in trade costs increases NA + NB In the Appendix we also show that NA /NB declines with reductions in τ when bA > bB This then implies that NB , the number of job-seekers in the differentiated sector of country B , necessarily increases Since a fall in τ does not affect sectoral labour market tightness, we conclude that a reduction in trade costs increases unemployment in country B , which has lower labour market frictions in the differentiated sector The effect on NA and hence on the unemployment rate in country A is ambiguous, as we illustrate below The intuition behind this result is the following Lower trade impediments increase the global size of the differentiated sector, which features increasing returns to scale and love of variety As a result, the country with a more flexible labour market, which has a competitive edge in this sector, becomes more specialized in differentiated products That is, the number of entering firms, employment, and the number of job-seekers in the differentiated sector, all increase in country B This compositional shift leads to a higher rate of unemployment in this country because the sectoral rate of unemployment is higher in the differentiated sector Finally, the reallocation of labour in country A may shift in either direction, depending on how strong the comparative advantage is (see below) Figure depicts the response of unemployment rates to variation in country A’s labour market frictions aA , which changes monotonically with bA ; the rising broken-line curve represents country B and the hump-shaped solid-line curve represents country A.42 Country B has bB = 0.55 > 1/2, and therefore the two countries have the same rate of unemployment when bA = 0.55 As bA rises, country A becomes more rigid This raises initially the rate of unemployment in both countries, but B ’s rate of unemployment remains lower for a while At some point, however, the rate of unemployment reaches a peak in country A, and it falls for further increases in bA As a result, the two rates of unemployment become equal again, after which further increases in rigidity in country A raise the rate of unemployment in country B and reduce it in country A, so that the rate of unemployment is higher in country B thereafter The mechanism that operates here is that, once the labour market frictions become high enough in country A, the contraction of the differentiated-product sector leads to overall lower unemployment in A despite the fact that its sectoral unemployment rate is high When bA is very high, the sectoral unemployment rate is very high, but no individuals search for jobs in this sector, as a result of which there is no unemployment at all This explains the hump in A’s curve Note that in the range in which the rate of unemployment falls in country A, the rate of unemployment keeps rising in country B The reason is that there is no change in market tightness in country B and its differentiated-product sector becomes more competitive the more rigid the labour market becomes in A As a result, the differentiated sector attracts more and more workers in country B , which raises its rate of unemployment The monotonic impact of country A’s labour market rigidities on the unemployment rate in B holds globally, and not only around the symmetric equilibrium.43 Figure is similar to Figure 3, except that now the level of labour market frictions in country B is higher, i.e bB = 0.65 > 1/2, and therefore the two curves intersect at bA = 0.65 Moreover, starting with a symmetric world that has these higher labour market rigidities, increases in bA always raise unemployment in B and reduce unemployment in A As a result, country A has lower unemployment when bA > bB and higher unemployment when bA < bB HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1125 Unemployment as a function of bA when bB is low (bB = 0.55 and τ = 1.1) Figure Unemployment as a function of bA when bB is high (bB = 0.65 and τ = 1.1) © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Figure 1126 REVIEW OF ECONOMIC STUDIES That is, in this case a more rigid country always has a lower unemployment rate when it specializes (incompletely) in the low-unemployment sector A comparison between Figures and demonstrates the importance of the overall level of labour market rigidities for unemployment outcomes When labour market frictions are high, a relatively more flexible country always has a higher rate of unemployment Moreover, the rates of unemployment in the two countries move in opposite directions as labour market frictions change in either of the countries In contrast, when labour market rigidities are low and the difference in labour market frictions across countries is not large, the rate of unemployment is lower in a more flexible country and the rates of unemployment in both countries co-move in response to changes in labour market frictions The next three figures depict variations in unemployment in response to trade frictions, in the form of variable trade costs τ : Figure for the case of low frictions in labour markets, Figure for the case in which frictions are low in country B but high in A, and Figure for the case in which frictions are high in both countries.44 In all three cases, unemployment rises in B and falls in A when trade frictions decline.45 Nevertheless, the rate of unemployment is not necessarily higher in A In particular, unemployment is always higher in A when frictions in labour markets are low in both countries, yet unemployment is always higher in B when 44 In Figures 5–7 we use the following parameters: m0 = 2v0 = 1, fx = 5, fd = 1, fe = 0.5, k = 2.5, β = 0.75, ζ = 0.5, and L = 0.1 45 This pattern is not general As we know, in the symmetric case lower trade impediments raise unemployment in both countries, which is also the case when countries are nearly symmetric We can also provide examples in which the rigid country has a hump in its rate of unemployment as trade frictions vary © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Figure Unemployment as a function of τ when bA and bB are low (bA = 0.6 and bB = 0.56) HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1127 Unemployment as a function of τ when bA is high and bB is low (bA = 0.68 and bB = 0.56) frictions in labour markets are high in both countries In between, when labour market frictions are low in B and high in A, the relative rate of unemployment depends on trade impediments; it is lower in A when the trade frictions are low and lower in B when the trade frictions are high This shows that labour market frictions interact with trade impediments in shaping unemployment FIRING COSTS AND UNEMPLOYMENT BENEFITS Our analysis has focused on search and matching as the main frictions in labour markets, and we used a0j = 2v0j /m0j1+α and aj = 2vj /mj1+α as measures of labour market rigidity Evidently, in this specification rigidity in a sector’s labour market is higher if either it is more costly to post vacancies in this sector or the matching process is less efficient in it We can also incorporate firing costs and unemployment benefits as additional sources of labour market rigidity These labour market policies are widespread and they differ greatly across countries But note that governments can also influence search and matching costs by facilitating the flow of information about job vacancies and about unemployed workers Moreover, in some countries there are government agencies that directly assign unemployed workers to firms, and workers need to try these jobs in order to be eligible for unemployment benefits In other words, government policies can influence not only firing costs and unemployment benefits but also our measures of labour market frictions, a0j and aj , which were analysed above In order to save space, we briefly describe in this section results of a formal analysis conducted in our working paper (Helpman and Itskhoki, 2009a), under the simplifying assumption © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Figure 1128 REVIEW OF ECONOMIC STUDIES that there is full employment in the homogeneous sector This analysis can be extended to allow for labour market frictions in the homogeneous-good sector, as in the earlier sections of the current paper With firing costs and unemployment benefits, (xj , bj ) remains a sufficient statistic for labour market frictions, with bj reinterpreted to represent the overall effective labour cost for a differentiated-sector firm, while the definition of xj does not change; it remains the same measure of labour market tightness in the differentiated sector Importantly, the effects of xj and bj on the equilibrium outcomes described in Sections 3–5 not change, except for the qualification of welfare effects to be discussed below Firing costs operate similarly to matching frictions, yielding a type of equivalence between the hiring and firing costs Specifically, higher firing costs reduce labour market tightness xj and increase the effective labour cost bj Moreover, as long as unemployment benefits are not too high (see below), the effects of firing costs on welfare, trade patterns, productivity, and unemployment in trading economies, are the same as those of matching frictions That is, all the earlier results of this paper extend to the case in which there are positive firing costs in addition to matching frictions Higher unemployment benefits always reduce equilibrium labour market tightness xj , but they may increase or decrease the effective labour cost bj The intuition for this result is that unemployment benefits provide unemployment insurance to the workers on the one hand and a better outside option in the wage bargaining game on the other Because higher unemployment benefits provide better unemployment insurance, workers are willing to search for jobs in a less tight labour market, with a higher sectoral rate of unemployment This effect reduces the cost of hiring for firms On the other side, the better outside option of workers at the wage bargaining © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Figure Unemployment as a function of τ when bA and bB are high (bA = 0.95 and bB = 0.8) HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1129 CONCLUSION We have studied the interdependence of countries that trade homogeneous and differentiated products, and whose labour markets are characterized by search and matching frictions Variation in labour market frictions and the interactions between trade impediments and labour market rigidities generate rich patterns of unemployment For example, lower frictions in a country’s labour markets not ensure lower unemployment, and unemployment and welfare can both rise in response to a policy change Contrary to the complex patterns regarding unemployment, the model yields sharp predictions about welfare In particular, both countries gain from trade Moreover, changes in one country’s labour market frictions can differentially impact welfare of the trade partners For example, reducing a country’s frictions in the labour market of the differentiated sector raises competitiveness of its firms This improves the foreign country’s terms of trade, but also crowds out foreign firms from the differentiated-product sector As a result, welfare rises at home and declines abroad, because the terms-of-trade improvement in the foreign country is outweighed by the decline in the competitiveness of its firms Nevertheless, a common reduction in labour market frictions in the differentiated sectors raises welfare in both countries These results contrast with the implications of models of pure comparative advantage, in which movements in the terms of trade dominate the outcomes.47 We also show that labour market frictions confer comparative advantage, and that differences in these labour market characteristics shape trade flows In particular, the country with relatively lower labour market frictions in the differentiated sector exports differentiated products on net and imports homogeneous goods Moreover, the larger the difference in these relative frictions, the lower the share of intra-industry trade These are testable implications about trade flows and international patterns of specialization In addition, we show that trade raises total factor productivity in the differentiated-product sectors of both countries, while productivity does not change in the homogeneous sector And productivity is higher in the country with relatively lower labour market frictions in the differentiated sector 46 Severance pay affects labour costs similarly to unemployment benefits, except that it has no impact on disposable income 47 See, for example, Brăugemann (2003) and Alessandria and Delacroix (2008) The former examines the support for labour market rigidities in a Ricardian model in which the choice of regime impacts comparative advantage The latter analyses a two-country model with two goods, in which every country specializes in a different product and governments impose firing taxes The authors find that a coordinated elimination of these taxes yields welfare gains for both counties, yet no country on its own has an incentive to it © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 stage improves their bargaining position and increases the effective cost of labour to firms Either of these effects can dominate Therefore, bj may rise or decline in response to higher unemployment benefits When bj decreases, it leads to an expansion of the differentiated sector, which raises welfare But because unemployment benefits need to be financed by (lump-sum) taxes, the additional taxes required to finance higher unemployment benefits reduce disposable income and hurt welfare Therefore, on net welfare may rise or decline, but it definitely rises in response to a small rise in unemployment benefits which reduces bj when the initial level of these benefits is small.46 We also show that firing costs and unemployment benefits notwithstanding, international trade may raise unemployment in both countries The reason is that trade attracts more workers to the differentiated sector without affecting sectoral labour market tightness Therefore, when this sector has the lower labour market tightness, trade increases aggregate unemployment 1130 REVIEW OF ECONOMIC STUDIES An important conclusion from our analysis is that simple one-sector macro models that ignore compositional effects may be inadequate for assessing labour market frictions, and especially so in a world of integrated economies Moreover, a focus on terms-of-trade as the major channel of the international transmission of shocks misses the impact of competitiveness, which can dominate economic outcomes APPENDIX A A.1 An alternative specification with homothetic preferences ζ C = ϑ 1−ζ q0 + (1 − ϑ)1−ζ Q ζ 1/ζ , < ϑ < ζ < β, The ideal price index associated with this consumption bundle is −ζ P = ϑ + (1 − ϑ)P 1−ζ 1−ζ − ζ , where the price of the homogeneous good p0 is again normalized to and P is the price of the differentiated product in terms of the homogeneous good The demand for homogeneous and differentiated goods is given by ϑE q0 = ϑPζ /(1−ζ ) E = −ζ , ϑ + (1 − ϑ)P 1−ζ Q = (1 − ϑ) P P −1 1−ζ E = P −1 (1 − ϑ)P 1−ζ E −ζ , ϑ + (1 − ϑ)P 1−ζ where E is expenditure in units of the homogeneous good Using these demand equations, we derive the indirect utility function V= E E 1−σ P 1−σ Since P is increasing in P , the indirect utility is falling in P for a given EE 1−σ Also Q is decreasing in P Next, the demand level for differentiated varieties is β−ζ D ≡ QP 1−β = (1 − ϑ)P (1−β)(1−ζ ) E −ζ , ϑ + (1 − ϑ)P 1−ζ which increases in P given β > ζ It proves useful to introduce the aggregate revenue variable −ζ R ≡ PQ = D 1−β Q = β (1 − ϑ)P 1−ζ E −ζ , ϑ + (1 − ϑ)P 1−ζ which, like Q and opposite to D, decreases in P Note that with homothetic utility, demands and revenues are linear in income, E , which allows for simple aggregation Specifically, in the expressions above, E can be interpreted as aggregate income equal to E = ω0 (L − N ) + wxN and we normalize L = since under homothetic demand it is without loss of generality Most of the remaining derivation of equilibrium conditions remains unchanged, with D replacing Q −(β−ζ )/(1−β) in the text Specifically, after this substitution the free-entry condition and zero-profit conditions are unchanged, which © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 We consider here an alternative specification of the model, with CRRA-CES preferences instead of quasi-linear preferences used in the main text, leaving the rest of the setup unchanged The expected utility is U = EC1−σ /(1 − σ ), where E is the expectations operator, σ ∈ [0, 1) is the relative risk aversion coefficient, and C is a CES bundle of homogeneous and differentiated goods: HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1131 allows us to solve for equilibrium cutoffs and equilibrium Ds in the same manner as in the text Qualitatively, all the relationships still hold, except that now instead of Q as the sufficient statistic for welfare and demand level it is more convenient to express all aggregate variables as functions of P Additionally, R = PQ replaces Q ζ in the expressions for M and N One block of the equilibrium system that changes is the indifference conditions of workers between sectors which now becomes x0 w01−σ = xw 1−σ bH = β β R= M (δd + δx )/φ2 , 1+β 1+β where the δz s are average revenues per entering firm as defined in the text We still have H = xN Using xb 1−σ = x0 (1/2)1−σ , we have the expression for the number of workers searching for a job in the differentiated sector: N = R β + β b σ x0 (1/2)1−σ Since R is decreasing in b, we have that N decreases in b as before As a result, there are still two opposing effects on ˆ = sign (1 − σ )bˆ + x0 /x − Nˆ The change in the unemployment the unemployment rate when x < x0 : sign {u} rate is again ambiguous: it still falls if the initial labour market friction is low enough and increases otherwise These results are qualitatively the same as those derived in the text under quasi-linear preferences Additionally, we can discuss now ex post inequality When b > b0 , a fall in b increases x and reduces w , which both lead to lower ex post inequality At the same time it increases N , which may increase or reduce the inequality depending on the initial size of the differentiated sector It follows that the comparative statics for inequality are ambiguous in the same way as those for the rate of unemployment For discussion of these and other issues, see Helpman, Itskhoki, and Redding (2009) in a related but different model A.2 Conditions for incomplete specialization We derive here a limit on bA /bB which secures an equilibrium in which both countries are incompletely specialized When this condition is violated, the country with a relatively more rigid labour market (higher b) specializes in the production of the homogeneous good Throughout we assume for concreteness that A is the relatively more rigid country, so that bA /bB ≥ We assume that L is large enough in both countries so that both countries always produce the homogeneous good Following the main text, we analyse only equilibria with xj > dj > , so that not all 48 In the case of σ ≥ 1, we need to introduce unemployment benefits in order to dispense with the family risk-sharing © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 The wage rate in the homogeneous sector is still w0 = b0 = 1/2 and equations characterizing x0 and ω0 = x0 w0 (12) and (13) still hold The wage rate in the differentiated sector is still w = b, where b = ax α is the hiring cost (and similarly b0 ) As a result, when a = a0 , we have b = b0 = 1/2, x = x0 , and w = w0 and a > a0 implies b > b0 = 1/2, x < x0 , and w > w0 In the latter case, there is a risk premium for searching for a job in the differentiated sector so that xw > ω0 = x0 w0 with the size of risk premium depending on risk aversion σ Finally, since all workers are indifferent between searching for a job in the two sectors, we have for every worker EE 1−σ = x0 w01−σ = x0 (1/2)1−σ , which is pinned down by the labour market friction in the homogeneous sector, a0 Therefore, holding a0 constant, the welfare in the economy depends only on the price level, P, which in turn is determined by the price of the differentiated good, P Note that with homothetic preferences and ≤ σ < 1, we have dropped the family interpretation In this case, the structure of demand and indirect utility does not change if the worker becomes unemployed, and aggregation is straightforward.48 As a result, this specification can be used to analyse issues such as the ex post income distribution and winners and losers from policy reforms Without showing the explicit derivation (which follows the same steps as in the text), we provide as an illustration a few comparative statics results for the symmetric open economies with homothetic preferences Specifically, we consider proportional labour market deregulation in the differentiated sector of both countries (i.e a decrease in a ˆ so that, as before, P decreases and Q and R increase as b falls This holding a0 constant) We have Dˆ = β/(1 − β)b, also implies an increase in welfare As before, we can express the total wage bill in the differentiated sector as 1132 REVIEW OF ECONOMIC STUDIES producing firms export and there are also firms that exit As shown in the text, this requires fx > fd which we assume holds Given bA > bB , incomplete specialization implies that there is positive entry of firms in the differentiated sector of country A, i.e MA >0 Equation (23) in the text implies that MA = whenever QA QB δdB ζ ≤ δxB When this condition is satisfied with equality, we also find, using (19), that xB δdB dB −β fd 1−β τ fx 1−β ζ β−ζ = δxB xB )-space, lying between the Lemma Let τ > and bA > bB Then there exists a unique function b(τ ) > 1, with b (τ ) > 0, such that (A1) holds for bA /bB = b(τ ) For bA /bB < b(τ ), there is incomplete specialization in equilibrium so that MA > For bA /bB ≥ b(τ ), country A specializes in the homogeneous good so that MA = Proof Recall that dB is decreasing and −β 1−β xB is increasing in τ This implies that δdB /δxB is increasing in τ Equation (22) implies that τ xB / dB is increasing in τ Next, xB / dB and δdB /δxB are decreasing in bA /bB These considerations, together with (A1), imply that b(τ ) is unique and increasing in τ whenever it is finite.50 Finally, QA /QB is decreasing in bA /bB Therefore, from (23), MA > whenever bA /bB < b(τ ) and MA = whenever bA /bB ≥ b(τ ) Evidently, Lemma implies that there is an upper bound on how different the relative labour market frictions can be in the two countries for complete specialization not to occur in equilibrium As we show in the numerical examples of Section 5.2, a wide range of bA /bB > is consistent with incomplete specialization equilibrium See the working paper version, Helpman and Itskhoki (2007), for the analysis of equilibria with complete specialization A.3 Proof of Lemmas 1–5 and Proposition Proof of Lemma This follows immediately from (22) First note that in equilibria with = 1−β dj < xj implies β δdA δdB − δxA δdB > Indeed, δdj fd > δxj fx xj dj < xj , we have dj Using these inequalities for j = A, B together with (21) implies δdA δdB / δxA δxB > τ 2β/(1−β) > 1, in which case > 0.51 Then an increase in bA /bB reduces dA and xB and increases dB and xA (see (22)) Therefore, bA > bB implies dA < dB and xA > xB since in a symmetric equilibrium these relationships hold with equality Proof of Lemma (Lemma 4) Proof of Lemma This also follows immediately from (22) and the fact that δdj > δxj , which we prove below This follows from (19) and Lemma Note that (19) implies: QA QB When bA > bB , Lemma implies dA < dB β−ζ 1−β = dA dB bB bA β 1−β and hence we have QA < QB 49 In the special case of a Pareto distribution, (A1) is a ray through the origin 50 Note that b(τ ) > by construction, since xB = dB τ β/(1−β) fx /fd when bA = bB 51 This also implies δdj > δxj in at least one country and in both countries in the vicinity of a symmetric equilibrium © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Note that this relationship is an upward-sloping (generally non-linear) curve in ( dB , 45◦ -line and xB = dB τ β/(1−β) fx /fd (i.e the equilibrium condition when bA = bB ).49 We can now prove the following: (A1) HELPMAN & ITSKHOKI LABOUR MARKET RIGIDITIES AND TRADE 1133 Proof of Lemma This follows from (23), the incomplete specialization requirement Mj > and Lemmas and Specifically, when bA > bB , MA > together with (23) imply ζ QB QA δdB > δxB > 1, where the last inequality follows from Lemma Lemma implies that δdA > δdB and δxA < δxB since δzj is a decreasing function of zj (z = d , x and j = A, B ) Therefore, δdA /δxA > δdA /δxA > Proof of Lemma This follows from (23) and Lemmas and Specifically, (23) implies (1 − β)φ2 β ζ ζ δdB + δxA QA − δdA + δxB QB When bA > bB , Lemma implies QA < QB and Lemma implies δdA > δdB > δxB > δxA Therefore, in this case MA < MB Proof of Proposition This follows from Lemmas 1, 4, and and the definition of intra-industry trade When bA > bB , Lemma states that xA > xB and dA < dB , which implies that a larger fraction of firms export in country B : − G( xj ) / − G( dj ) is greater in B In the text we show that exports of differentiated products is equal to Xj = Mj δxj /φ2 When bA > bB , Lemma states that δxB > δxA and Lemma states that MB > MA , which implies XB > XA ; that is country B exports differentiated goods on net Balanced trade implies that is has to import the homogeneous good In the text, the share of intra-industry trade is shown to equal XA /XB = δxA MA / δxB MB Using equation (23), we have: XA = XB δdB δxB δdA δxA ζ QB QA − ζ QB QA −1 From equations (22) and (26) and using Lemma 4, an increase in bA /bB leads to a decrease in δdB /δxB and to increases in δdA /δxA and QB /QA Therefore, an increase in bA /bB reduces XA /XB In Appendix A.5, we prove additionally that under Pareto-distributed productivity the total volume of trade increases in the proportional gap between relative labour market frictions, bA /bB A.4 Derivation of results on productivity for Section 4.3 We first show that ϕzj = ϕ( zj ) is monotonically increasing in ⎡ ϕˆ zj = zj G zj ) ⎣ ( ∞ zj zj dG( ) − ∞ zj zj The log-derivative of ϕ( 1/β zj 1/β dG( zj ) is ⎤ ) ⎦ ˆ zj for z = d , x The term in the square brackets is positive since ⎛ 1/β zj ∞ zj 1/β dG( ) 1−G( zj ) bB ) equals 2XB , where we have XB = φ2−1 MB δxB fd δdA ζ ζ Q − QA fx δxA B = = δdA δdB fd −1 δxA δxB fx2 xA dA k ζ ζ QB − QA xA xB dA dB = k −1 fd fx k xA ζ 2βk fx fd τ 1−β ζ QB − QA dA 2(k −1) −1 As bA increases or bB falls, the denominator remains unchanged while xA / dA and QB increase and QA decreases As a result, the volume of trade unambiguously rises Finally, one can also show that XB decreases in τ Substitute the expression for xA / dA (derived from (19)) in the expression for XB to get βk τ 1−β ζ XB = QA fx fd k −1 2βk τ 1−β fx fd ζ +k QB QA 2(k −1) β−ζ 1−β −1 −1 Now note that XB decreases in τ since QA and QB /QA decrease in τ and QB > QA Proof that NA /NB decreases in τ when bA > bB In the text we show that NA + NB increases as τ falls We show now that when bA > bB , NA /NB decreases as τ falls, which implies that NB necessarily increases Under the Pareto assumption, δdj + δxj = kfe Therefore, equations (23) and (24) imply ω0A NA MA = = ω0B NB MB ζ δdB QA ζ δdA QB ζ − δxB QB ζ − δxA QA 1− = δdA kfe δxB kfe 1+ 1+ QB QA QB QA ζ < 1, ζ −1 where the last inequality comes from Lemma under the assumption that bA > bB Recall that ω0A /ω0B depends only on a0A /a0B and does not depend on τ From Proposition 1, QB /QA increases as τ falls Taking this and the fact that NA < NB into account, it is sufficient to show that d δxB − d δdA = δxB δˆxB − δdA δˆdA > in response to a fall in τ , to establish that NA /NB declines in this case Under Pareto-distributed productivity, k δdA ˆ dA − δxB ˆ xB δxB δˆxB − δdA δˆdA (δdB δxB − δdA δxA )(δdA − δxA ) = =k > 0, −τˆ −τˆ where the second equality comes from (22) and the inequality is obtained by Lemma and the fact that under the Pareto assumption δdA + δxA = δdB + δxB = kfe This proves that NB increases as τ falls when bA > bB Since changes in τ not affect labour market tightness, x0B and xB , the only effect on the unemployment rate uB is through NB , and hence the unemployment rate in the flexible country increases in response to trade liberalization © 2010 The Review of Economic Studies Limited Downloaded from http://restud.oxfordjournals.org/ at Taylor's University College on December 6, 2012 Finally, using the condition for Nj (24) and the free entry condition under the Pareto assumption, we get: 1136 REVIEW OF ECONOMIC STUDIES Acknowledgements We thank Alberto Alesina, Pol Antr`as, Jonathan Eaton, Emmanuel Farhi, Larry Katz, Kala Krishna, 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