Mobility Costs and the Dynamics of Labor Market Adjustments to External Shocks: Theory1 Stephen Cameron, Shubham Chaudhuri Department of Economics Columbia University New York, NY 10027 John McLaren University of Virginia Charlottesville, VA 22903 July, 2001 Abstract. We construct a dynamic, stochastic rational expectations model of labor reallocation that is designed so that its key parame- ters can be estimated for trade policy analysis. A key feature is the presence of time-varying idiosyncratic moving costs faced by work- ers. As a consequence of these shocks: (i) Gross ‡ows exceed net ‡ows (an important feature of empirical labor movements); (ii) the economy features gradual and anticipatory adjustment to aggregate shocks; (iii) wage di¤erentials across locations or industries can per- sist in the steady state; and (iv) the normative implications of policy can be very di¤erent from a model without idiosyncratic shocks, even when the aggregate behaviour of both models is similar. It is shown that the solution to a particular planner’s problem yields a compet- itive equilibrium, thus facilitating the analysis and simulation of the model for policy analysis. 1. Introduction. 1 We are grateful to seminar participants at Koc University, Syracuse University, University College London, the University of Virginia, and the World Bank; to participants of the European Research Workshop in International Trade, July, 2000; and also to Bill Gentry, Ann Harrison, Glenn Hubbard, and Marc Melitz for comments. This project is supported by NSF grant 0080731. The e¤ect of a given change in trade policy is crucially a¤ected by the costs workers may face in adjusting to it. This is espcially true of the distributional e¤ects of the change, but it also extends to the e¢ciency e¤ects. For example, the e¤ects of opening up a sector of the economy previously protected from import competition depend crucially on how easily the workers in that sector can …nd employment in other sectors. If geographic or sectoral mobility costs are high, the e¢ciency bene…ts are thereby reduced and the burden borne by those workers is increased. Analysis of the e¤ect of trade on wages thus always requires the use of some assumption on the degree of labor mobility.2 Further, the e¤ects of immigration into a particular region of the country depend on how ‡uid labor is between that region and others, sand so the literature on labor-market e¤ects of immigration has always required assumptions on the degree of mobility (see Borjas et. al. (1996), Slaughter and Scheve (1999)).3 The cost of labor reallocation is also a crucial issue driving the political econ- omy of trade policy, as emphasized for example in static approaches by Magee (1989) and Irwin (1996), and in dynamic analyses of endogenous trade policy such as Staiger and Tabellini (1999) and McLaren (1997, 1999). This paper proposes a workhorse model of equilibrium labor reallocation that is designed to address these policy questions head-on. It incorporates a number of features that are intended to make the model helpful in analyzing trade policy changes in particular, and to be consistent with the broad empirical features of the adjustment process. It also has the bene…t that its parameters can be esti- mated econometrically, thus providing for more detailed policy analysis through simulation, a project which is being carried on in parallel with the theoretical execise detailed here. The model is an in…nite-horizon dynamic stochastic model with rational ex- pectations, in which from time to time random shocks may hit labor demand either in a sector or in a region of the country (for example, changes in trade policy or terms-of-trade shocks). In response to these shocks, each worker at each 2 For example, speci…c-factors models and the Stolper-Samuelson approach have very di¤erent implications for the relationship between trade and wages, driven entirely by di¤erent assump- tions about mobility costs; and the appropriate time horizon for measuring the labor-market e¤ects of trade also depends on assumptions about mobility costs. See Slaughter (1998) for an extended discussion. 3 For example, the di¤erences between the Hecksher-Ohlin approach, the “factor-proportions analysis” approach, and the “area analysis” approach to the e¤ects of immigration (Borjas et. al., 1996) are entirely driven by di¤erent assumptions about labor mobility. See Slaughter and Scheve (1999) for an extensive discussion. 2 moment may choose whether to remain where she is or to move to another sector or geographic location. If the worker moves, she will pay a cost that has two components: A portion that is the same for all workers making the same move, which is a parameter of the model and is publicly known; and a time-varying idiosyncratic portion. The latter is an extremely important feature of the model, because it generates all of the model’s dynamics and allows for gross ‡ows to exceed net ‡ows. If individual situations can vary, one may …nd large numbers of workers moving in opposite directions at the same time, and this is indeed a prominent feature of the equilribium of the model. This is important because empirically gross ‡ows of workers across geographical locations and industries are substantially larger than net ‡ows. Many authors have proposed theoretical models of the dynamics of factor reallocation in response to a trade or policy shock (a number of the issues are reviewed in Neary (1985). Mussa (1978, 1982) studies the dynamics of adjustment in a trade model, with capital as a quasi-…xed factor bearing convex adjustment costs. In both models, labor is either completely immobile (that is, labor faces in…nite moving costs) or costlessly mobile (faces zero moving costs). The roles of the capital and labor could easily be reversed to consider labor adjustment dynamics. Dixit (1993) studies a similar model with random trade shocks and a …xed cost to each reallocation, and Dixit and Rob (1994) consider …xed labor- adjustment costs in a model with random labor-demand shocks and risk-averse workers. Matsuyama (1992) studies a model whose workers cannot reallocate once they have chosen a sector, so the dynamic adjustment to a trade shock comes entirely through new labor market entrants. Dehejia (1997) studies political- economic implications of the adjustment process in a Mussa-type model. Finally, two important papers are particularly closely related to the model used here. Jovanovic and Mo¢t (1990) o¤er an approach based on a matching model, in which workers disappointed in the job-match with their employers search for a better match, and in each period some fraction of them move across sectors to do so. Topel (1986) studies the dynamics of geographic reallocation of labor using an equilibrium overlapping generations model with idiosyncratic moving costs. Our theoretical model di¤ers from all of the above approaches in two ways. First, we allows gross ‡ows to exceed net ‡ows, which is important given the em- pirical importance of gross ‡ows highlighted above. Jovanovic and Mo¢t (1990) shares this feature, but the other studies mentioned above do not. For this rea- son, idiosyncratic shocks are a key feature of our model. Unlike Jovanovic and Mo¢t (1990), we allow for such shocks to be non-pecuniary in nature (such as 3 job dissatisfaction or personal constraints on geographic location).4 Topel (1986) allows for idiosyncratic moving costs, but constrains gross interregional ‡ows to be equal to net ‡ows.5 Second, our model has been tailor-made to allow for estimation of the moving- cost parameters, a feature shared by none of the other equilibrium models. In examining the model, we …rst study a particular (distorted) planner’s prob- lem in some detail, because it turns out that the planner’s solution is also a market equilibrium. This provides a number of results on the market equilibrium that would be very di¢cult to derive by other means. The key properties include gradual adjustment of the economy to an external shock; anticipatory adjustment of the economy to an anticipated shock; and persistent wage di¤erentials (across sectors or regions of the economy) even in the long run steady state, for reasons that appear to be novel in the literature. In addition, it is shown that if the variance of idiosyncratic shocks is su¢ciently high, the aggregate behaviour of the model will mimic a static model with no labor mobility, even though in fact mobility will be high and the normative features of the equilibrium will be very di¤erent from that of the static model. This highlights the importance of second moments of moving costs (such as the variance of the idiosyncratic shocks) as well as the …rst moment, and points out an advantage of our structural approach over reduced-form econometric approaches. The following section lays out the structure of the model. The subsequent section analyzes the solution to the planner’s problem of the optimal rule for the allocation of labor, and …nds the key Euler condition that charcterizes optimal- ity. The subsequent section shows that this optimal rule is implemented by the decentralized rational expectations equilibrium. The following section elaborates the most important properties of the equilibrium. Finally, we brie‡y discuss a special case of the model that o¤ers a simple form to the equilibrium, a¤ording empirical estimation. 4 In a sense, this actually …ts their data better than their own model, since they …nd that movers on average experience a loss in wages, which is the opposite of what one would expect if the point of moving was to …nd a higher wage. 5 In addition, Topel (1986) requires the number of regions to be large so that asymptotic properties can be used to solve the equilibrium. Our model requires no such assumption. 4 2. The model. Consider a model in which production may occur in any of n ‘cells,’ where a cell is taken to mean a particular industry in a particular place. For example, ‘pharma- ceuticals in New Jersey’ might be one of the cells, as might ‘pharmaceuticals in Delaware’ or ‘food service in New Jersey.’ In each cell there are a large number of competitive employers, and the value of their aggregate output in any period t is given by xi = X i (Li ; st ), where Li denotes the labor used in cell i in period t, and t t t st is a state variable that could capture the e¤ects of policy (such as trade protec- tion, which might raise the price of the output), technology shocks, and the like. Assume that X i is strictly increasing, continuously di¤erentiable and concave in i its …rst argument. Its …rst derivative with respect to labor, denoted X1 , is then a continuous, decreasing function of labor; this is, then, the demand P curve for labor in the cell. Denote the total value of output by xt = X(Lt ; st ) ´ i X i (Li ; st ). t Assume that s follows a …rst-order Markov process on some state space S s . Note that this formulation allows for advance warning of policy changes, for ex- ample. To incorporate this possibility within this framework, the variable s could be a vector with two elements: the …rst, a tari¤ ¿ on imports competing with cell i’s ouput, and the second a variable · that measures the political climate, taking a value of either 0, indicating a protectionist climate, or 1, indicating a liberalizing climate. If · follows a non-degenerate …rst-order Markov process independent of ¿ , and ¿ follows a Markov process such that the distribution of ¿ t+1 conditional on ¿ t and · = 0 stochastically dominates the distribution conditional on · = 1, then a change in · can signal a likely future change in tari¤ policy. The economy’s workers form a continuum of measure L. Each worker at any moment is located in one of the n cells. Denote the number of workers in cell i at the beginning of period t by Li , and the allocation of workers by Lt = [L1 ; : : : ; Ln ]. t P t t This allocation vector must lie in the domain S L ´ fL 2 0. For any number ", t t de…ne: Z µZ 1 Z 1 ¶Y ij i i j j ¡ k k¢ Â(") ´ D ("; Lt ; st )f (" )d" f (" )d" f (" )d" , and ¡1 "j +" k6=i;j Z ÃZ Z "j +" ! Y¡ 1 ¢ »(") ´ Dii ("; Lt ; st )f ("i )d"i f ("j )d"j f("k )d"k . ¡1 ¡1 k6=i;j (In other words, for any number ", Â(") is the fraction of i workers who have "i ¡"j > " and move to j; and »(") is the fraction of i workers who have "i ¡"j < " and remain in i.) Then there exists "ij such that Â("ij ) = »("ij ) = 0 . We will adopt the convention that "ii = 08i, and will denote the matrix of these thresholds as " ´ f"ij gi;j2(1;:::n) . An important note is that "i ¡ "j < "ij does not ensure that the worker goes to j, because it is possible that she will choose a third option. That point is clari…ed by the following proposition, which shows how all of the "ij together fully determine the choices of each worker (to within a set of measure zero). Proposition 3.2. Let the conditions in the previous proposition hold, and sup- pose that we have chosen a set of "ij as described there. Then Dij ("; Lt ; st ) = 1 if and only if j solves: max f"k + "ik g k2f1;:::ng 8 (except possibly on a set of measure zero). Equivalently, Dij ("; Lt ; st ) = 0 if and only if j does not maximize f"k + "ik g, except possibly on a set of measure zero. This allows us to write the planner’s problem in a simple way, as the choice of a function "(L; s) giving the thresholds at each date and state. The realized current-period payo¤ to a given worker in cell i is equal to that worker’s wage, wt , plus ("j ¡ C ij ), if that worker moves to cell j. Conditional on the "ik ’s and on i Q "j , the probability that this worker does move to cell j is k6=j F ("j + "ij ¡ "ik ). For this reason, the realized value of the objective function (3.2) will be: 1 X Efst g1 t=1 ¯ t U (Lt ; st ; "(Lt ; st )), (3.3) t=0 where " à !# n X X Z n 1 Y U (L; s; ") ´ X i (Li ; s) + Li ("j ¡ C ij )f("j ) F ("j ¡ "ij + "ik )d"j . i=1 j=1 ¡1 k6=j (3.4) We can write the gross ‡ows of workers out of sector i as a function of the ij " ’s: Z 1 Y ij i m (" ) = f ("j ) F ("j + "ij ¡ "ik )d"j , (3.5) ¡1 k6=j where "i = ("i1 ; : : : ; "in ). We can write mi¢ ("i ) = (mi1 ("i ); : : : min ("i )). This allows us to write the law of motion as a function of the "ij ’s: X Li ii i t+1 = m (" )Lt + i mki ("k )Lk (3.6) k6=i = L0t m("), (3.7) where m denotes the full matrix of gross ‡ows and a prime on a vector indicates the transpose. The planner, then, maximizes (3.3) subject to (3.6). 9 3.1. The gross ‡ows function. Equation (3.5) de…nes all gross ‡ows out of cell i as a function of "i . It is convenient to de…ne a truncated version of this function, which allows us to state a useful property of the gross ‡ows. First, let x¡k denote the vector made by deleting the k th element of x (if x has fewer than k elements, x¡k = x). After deleting one or more elements of a vector, continue to index the remaining elements in the same way, so, for example, if x 2 de…ne mi¢ : 0. There are no other changes in the economy at any time. This can be incorporated into the model by letting st = t8t > 0, and by letting X 2 (¢; s) have one functional form when s ¸ T and a di¤erent one when s < T . The function is shifted down and ‡atter when s ¸ T compared with the function when s < T . Let L¤ , "¤ and m¤ denote the steady state values for the economy with the tari¤ in place (call this the ‘tari¤-a¤ected steady state’), and suppose that L0 = L¤ . It can be seen quickly that no matter how large T is, the adjustment begins immediately, in the sense that because of the announcement the gross ‡ows even in period 0 are already di¤erent from m¤ . In this two-cell situation, the Euler condition (3.14) becomes: ¡ 2 ¢ "12 + C 12 = ¯Et X1 (L2 ; st+1 ) ¡ X1 (L1 ; st+1 ) + ­("2 ) ¡ ­("1 ) + "12 + C 12 t t+1 1 t+1 t+1 t+1 t+1 (5.1) for movers from cell 1 to 2, and vice versa for movers in the other direction. Given that "21 = ¡"12 ¡ C 12 ¡ C 21 at all times (see (3:9)), we can meaningfully write t+1 t+1 "21 as a function of "12 . Using this in the Euler equation, it is straightforward to t+1 t+1 show that the third, fourth and …fth terms on the right hand side of the equation: ­("21 ("12 ); "22 ) ¡ ­("11 ; "12 ) + "12 t+1 t+1 t+1 t+1 = ­("21 ("12 ); 0) ¡ ­(0; "12 ) + "12 t+1 t+1 t+1 t+1 16 are a strictly increasing function of "12 . Thus, a change in the pattern of next- t+1 period gross ‡ows with a given L vector will always result in a di¤erent value for the right hand side of (5.1). Now suppose that the tari¤-a¤ected steady state behavior of the model continues until T ¡ 1, so that Lt = L¤ , f"ij g = "¤ , and t fmij g = m¤ for t < T . Then LT = L¤ as well, so the right hand side of (3.9) t will be di¤erent from what its tari¤-a¤ected steady-state value would be, and so "12¡1 must also be di¤erent from ("¤ )12 , a contradiction. Thus there must be a T deviation from tari¤-a¤ected steady-state behavior at some point before period T . Now suppose that the tari¤-a¤ected steady state behavior of the model continues until T ¡ 2. Then LT ¡1 = L¤ , so "12¡1 must di¤er from ("¤ )12 . But then the right- T hand side of (5.1) for t = T ¡2 will be di¤erent from its tari¤-a¤ected steady-state value, and so "12¡2 must also di¤er from ("¤ )12 , a contradiction. Thus, there must T be a deviation from the tari¤-a¤ected steady state behavior at some point before T ¡ 1. Proceeding in this way, we can see that the adjustment process to the new policy must begin immediately at time t = 0. The reasoning behind this has to do once again with idiosyncratic shocks. Even if wages are currently equal in the two sectors, if a worker knows that an event will occur shortly in the future that will depress wages in sector 2 for a long time afterward, and if that worker happens to have low moving costs at the moment, understanding that her moving costs may not be so low later on, she may simply jump at the opportunity to move now. For example, a worker who has been separated from one …rm in the sector that will experience the shock, instead of looking for employment with another …rm in the same sector, may simply move to the other now that it is as easy to …nd a job there as in the worker’s current sector. It should be noted that anticipatory movements of labor are also a feature of Mussa-type models, as studied in detail by Dehejia (1997). However, in those models, the anticipatory behavior is a result of the existence of a retraining sector with rising marginal costs, while in the current model it arises purely from the presence of time-varying idiosyncratic moving costs. Anticipatory reorientation of an economy associated with a forthcoming change in trade policy is an important phenomemon empirically, as documented for the case of accessions to trade blocs by Freund and McLaren (1999). This mechanism provides an additional potential source for it. (iv) Anticipatory changes in wages. This is an immediate corollary to the point just made. In the example discussed above, if workers begin to leave sector 2 immediately as soon as the planned future liberalization is announced, then 17 clearly wages in sector 2 will begin to rise right away and wages in sector 1 will begin to fall right away. Of course, sector 2 wages will then drop abruptly at the date of the actual liberalization, and continue to adjust after that. This is important for a number of reasons. First, in doing empirical work on the relationship between tari¤s and wages, the issue of timing could be ex- tremely important. Simply looking at a pair of snapshots taken before and after a liberalization, for example, could miss a large part of the actual movement in wages; further, in the simple story just told, if the pre-liberalization data were collected very shortly before the liberalization, the empirical results would over- state the downward e¤ect of the liberalization on wages in the a¤ected sector. Second, these anticipatory e¤ects on wages can provide a motive for gradualism in trade policy. If the government wishes to compensate the workers harmed by a liberalization but cannot do so through lump-sum transfers, announcing the policy change in advance and allowing these adjustment mechanisms to do their work can in principle be an e¤ective way of doing so. This is a point made by Dehejia (1997) in the context of a Mussa-type model. (v) Persistent wage di¤erentials in long-run equilibrium. A feature of the model that is not obvious is that it generally predicts wage di¤erentials across cells even in the steady state. Consider, once again, a version with two cells and with s constant. Suppose that C 12 = C 21 , and suppose that there is a steady state in which w2 ¸ w1 . Observe that if in that steady state L1 > L2 , then we must have m21 > m12 . From (3.5), this implies that "21 > "12 . From (3.11), this implies that ­("2 ) > ­("1 ). From (4.3) applied recursively, that means that À 2 > À 1 . But from (4.1), this implies that "21 < "12 , a contradiction. Thus, in order to have L1 > L2 in the steady state, we must also have w 1 > w2 . Thus, in the steady state a sector will have a higher wage than the other if and only if it has more workers than the other. This conclusion contrasts sharply with the behavior of a Mussa-type model, in which factor returns are equalized across sectors in the long run (see Mussa (1978)). The reasoning is as follows. Suppose that both cells had the same wage in the steady state, but cell 1 was ten times the size of cell 2. In that case, workers would be indi¤erent between the two cells apart from idiosyncratic e¤ects. In each period, a certain fraction of the workers in either cell would realize negative moving costs, which could be interpreted as boredom with the current job or location or a desire to move to the other cell to realize some personal opportunity. With the wages identical, an identical fraction of the workers in each cell would 18 wish to change sectors in each period. However, this would imply a much larger number of workers moving from 1 to 2 than vice versa. The result would be net migration toward 2, which would push down the wage in cell 2 and pull up the wage in cell 1. The wage di¤erential thus created would then tend to slow down migration out of 1 and speed up migration out of 2, and this process would continue until the aggregate number of workers moving in each direction would be equal. These e¤ects, which might be called ‘frictional’ wage di¤erentials, thus provide a new reason for persistent intersectoral or geographic wage di¤erences, quite in- dependent of compensating di¤erentials, e¢ciency wages and union e¤ects, which have been emphasized in the labor economics literature. It should also be empha- sized that these e¤ects occur even if the average moving costs C ij are all equal to zero. The persistent wage di¤erentials are induced entirely by the variance in idiosyncratic e¤ects. (vi) Limiting behaviour as idiosyncratic shocks become important. Finally, there is a sense in which the aggregate behaviour of the model when idiosyncratic shocks are very important mimics the aggregate behaviour of a static model with no mobility at all. This underlines how crucial it is to take account of gross ‡ows, as is being done here, and to estimate the structural parameters of the mobility costs, because using a reduced-form econometric approach could produce normative conclusions that would be seriously in error. To make this point, consider a class of distributions for the "i ’s indexed by ± > 0 in the following way. For a particular distribution function G1 and associated density g1 , the distribution function G± and density g± are de…ned by G± (") = G1 ("=±) and g± (") = g1 ("=±)=±. Thus, G± is a radial mean-preserving spread of G1 for ± > 1; the probability that " · y with the distribution G1 is equal to the probability that " · ±y with the distribution G± . With this family of distributions, if ± is very small, then idiosyncratic e¤ects are trivial most of the time, but as ± becomes large, idiosyncratic e¤ects become more important and can eventually dwarf wages in their e¤ect on workers’ decisions. The asymptotic e¤ects of increases in ± are summarized in the following. Proposition 5.1. When the distribution of idiosyncratic shocks is given by the family G± , as ± ! 1 the matrix of gross ‡ows mij converges uniformly in equi- librium over the whole state space to a matrix each of whose components is equal to 1=n. Thus, if ± is very large, regardless of the labor demand shocks, workers would 19 always be approximately evenly distributed across the cells of the economy. In the extreme case, the number of workers in each cell would be completely insensitive to, for example, the elimination of tari¤s, and all of the adjustment would occur in the form of changes in wages. Aggregate data would suggest that each industry has in e¤ect a captive labor force, and the cost of the elimination of a tari¤ on textiles, for example, would be borne entirely by workers in the textile sector, while all other workers would enjoy a net bene…t through lower textile prices. However, this would be quite wrong. In such an economy, far from being captive, workers would be very footloose, and a typical textile worker would face only a 1=n chance of continuing in the textile sector next period. Therefore, particularly if n is large, the cost borne by the textile workers would be very low; for most of such a worker’s future career, she would be in other sectors, enjoying the bene…t of lower prices. It may in fact be a Pareto-improving liberalization, while the reduced- form approach would mistakenly conclude that one sector of workers would be badly hurt and would bitterly oppose the liberalization. Thus, a focus on gross ‡ows in equilibrium, and attention to the variance of mobility costs as well as their means, are, in principle, crucial to getting the normative conclusions right. 6. A special case, and empirical implementation. The model takes a particularly tractable form when a judicious choice of func- tional form is made. Assume that the "i are generated from an extreme-value t distribution with parameters (¡°º; º), which implies:7 E["i ] = 0 8i; t t i ¼2º 2 V ar["t ] = 8i; t 6 Note that while we make the natural assumption that the "’s be mean-zero, we do not impose any restrictions on the variance, leaving º (which is positively related 7 The cumulative distribution, mean, and variance for an extreme-value distribution with parameters (®; º) are given by: n o F (") = exp ¡e¡("¡®)=º E(") = ® + °º ¼2 º 2 V ar(") = 6 For further properties of the extreme-value distribution, see Patel, Kapadia, and Owen (1976). 20 to the variance) as a free parameter to be estimated. It can easily be shown that, with this assumption: "ij ´ ¯Et [Vt+1 ¡ Vt+1 ] ¡ C ij = º[ln mij ¡ ln mii ] t j i t t (6.1) and: ­("i ) = ¡º ln mii t t (6.2) Both these expressions make intuitive sense. The …rst says that the greater the expected net (of moving costs) bene…ts of moving to j, the larger should be the observed ratio of movers (from i to j) to stayers. Moreover, holding constant the (average) expected net bene…ts of moving, the higher the variance of the idiosyncratic cost shocks, the lower the compensating migratory ‡ows. The second expression says that the greater the probability of remaining in cell i, the lower the value of having the option to move from cell i.8 Moreover, as one might expect, when the variance of the idiosyncratic component of moving costs increases, so too does the value of having the option to move. Substituting from (6.1) and (6.2) into (3.14) we get: C ij + º[ln mij ¡ ln mii ] = ¯Et [wt+1 ¡ wt+1 + C ij + º[ln mij ¡ ln mii ] t t j i t+1 t+1 +º[ln mii ¡ ln mjj ]] t+1 t+1 This expression can be simpli…ed and rewritten as the following conditional mo- ment restriction: · ¸ ¯ j i ij jj (1 ¡ ¯) ij ij ii Et (w ¡ wt+1 ) + ¯(ln mt+1 ¡ ln mt+1 ) ¡ C ¡ (ln mt ¡ ln mt ) = 0 º t+1 º (6.3) This has the virtue that it can be estimated with data on gross ‡ows and wages, using standard Generalized Method of Moment techniques. This is an ongoing project. 7. Conclusion. This paper has articulated an equilibrium model of labor adjustment to external shocks, which has been designed to be useful for trade policy analysis and to be 8 Note that 0 < mii < 1, so ­("i ) = ¡º ln mii > 0. t t t 21 empirically estimable. The key features are an in…nite horizon in which all workers have rational expectations; the possibility of shocks to labor demand in a sector (as caused, for example, by a change in trade policy) or in a geographic location; publicly observable costs of moving or of changing sectors; and time-varying, id- iosyncratic private costs as well. We have shown that the equilibrium solves a particular social planner’s dynamic programming problem, which facilitates anal- ysis of the equilibrium. In addition, the equilibrium exhibits gross ‡ows in excess of net ‡ows (and indeed, constant movement of workers even in a steady state), which is an important feature of empirical labor adjustment; gradual adjustment to a shock; anticipatory adjustment to an announced policy change; and persis- tent ‘frictional’ wage di¤erentials across geographic locations or sectors, which will exist even if the average moving costs are zero, and which provide a new and independent theoretical rationale for wage di¤erentials in long-run equilibrium. Finally, it is shown that the key equilibrium condition takes a particularly simple form when the functional forms are chosen in a particular way, making the econometric estimation of the parameters of the model feasible with data on gross ‡ows and wages over time for a particular economy. This is the subject of ongoing work. 8. Appendix. Proof of Proposition (3.1). Clearly Â(") is decreasing and continuous, with Â(") ! 0 as " ! 1 and Â(") ! mij as " ! ¡1. Clearly »(") is increasing and t continuous, with »(") ! mii as " ! 1 and Â(") ! 0 as " ! ¡1. Thus, we can t …nd an "¤ such that Â("¤ ) = »("¤ ). If Â("¤ ) = 0, we are done. If not, then we have a positive mass of i workers who have "i ¡ "j < "¤ and who remain in i, and an equal mass of i workers who have "i ¡ "j > "¤ and who move to j. Clearly, if we simply reversed their roles, making the movers stay and the stayers move, the next-period allocation of labor would be unchanged, and the total surplus would be higher. Therefore, the original allocation rule could not have been optimal. Proof of Proposition (3.2). Suppose that for some set A(1) µ "k +"ik and yet Dik ("; Lt ; st ) > 08" 2 A(1). Without loss of generality, assume that for all " 2 A(1), "j + "ij ¡ ("k + "ik ) ¸ e > 0. For any positive N, consider the ball of radius 1=N around the point " 22 " = (¡"i1 ; ¡"i2 ; : : : ; ¡"in ), and note that within such a ball will be points for 0 0 which the expression "i ¡ "i ¡ "ii is negative for all i0 , points for which it is positive for all i0 , and points with every other possible combination of signs (note 0 0 that at the center of the ball "i ¡ "i ¡ "ii = 08i0 ) . For N = 1; : : : ; 1, de…ne a 0 0 subset of such a ball, B(N ) µ "ii 8i0 6= j; "i ¡ "j < ¯ i0 0¯ "ij ; and maxi0 ¯" + "ii ¯ < 1=Ng. (Note that at the center of the ball, "i + "ii = 0 0 08i0 .) By the previous proposition, Dij = 1 everywhere on B(N ) for all N. De…ne a sequence A(N ) of subsets of A(1), where for each N the probability R Q measure p(N ) ´ A(N) Dik ("; Lt ; st ) n (f("k )d"k ) of workers in A(N) who go to k=1 k is equal to the smaller of p(1) and the measure of B(N ). For large enough N, we will have "j + "ij ¡ ("k + "ik ) < e for all " 2 B(N), and a measure of workers " in A(N) going to k that is equal to the measure of workers in B(N ) who go to j. But then for every worker in A(N), "j ¡ "k ¸ "ik ¡ "ij + e, and the worker moves " j k ik ij to k; while for every worker in B(N ), " ¡ " < " ¡ " +e, and the worker moves " to j. Clearly, if for " 2 A(N), we simply reduced Dik ("; Lt ; st ) to 0 and increased Dij ("; Lt ; st ) by Dik ("; Lt ; st ); and if for " 2 B(N), we reduced Dij ("; Lt ; st ) to 0 and increased Dik ("; Lt ; st ) to 1; then the total number of workers going to each cell would be unchanged. However, a positive mass of workers in A(N ) and in B(N ) will have reversed their roles; B(N ) workers with lower values of "j ¡"k now move to k and the A(N ) workers with higher values of "j ¡"k move to j. Thus, the next-period allocation of labor would be unchanged, and the total surplus would be higher. Therefore, the original allocation rule could not have been optimal. Proof of Proposition (3.3). First, we derive some information about the derivatives of mij . They are as follows: e Z 1 Y @ mij ("i ) e ¡i 0 0 =¡ f ("j )f ("j + "ij ¡ "ii ) F ("j + "ij ¡ "ik )d"j < 0 @"ii ¡1 k6=j;i0 if i0 6= j, and Z 1 X Y 0 0 0 0 0 0 f ("i ) f ("i + "ii ¡ "ik ) F ("i + "ii ¡ "il )d"i > 0 (8.1) ¡1 k6=i0 l6=i0 ;k if i0 = j. 23 Note that if i 6= i0 , X @ mij ("i ) e ¡i @mii ("i ) ¡i ii0 = ¡ ii0 j6=i @" @" Z 1 Y 0 = f ("i )f("i ¡ "ii ) F ("i ¡ "ik )d"i ¡1 k6=i;i0 > 0. Thus, the matrix of derivatives µ ¶ i¢ @ mij ("i ) e ¡i e rm ´ ii0 , @" j;i0 6=i which is the Jacobian of the mi¢ function, is a dominant diagonal matrix with pos- e itive elements on the main diagonal and negative elements o¤ the main diagonal. This implies that it has an inverse (see Theorem 1 in McKenzie (1960)), and that the inverse has only positive elements (see Theorem 4 in McKenzie (1960)). This information is useful in the remainder of the proof. The following notation will be helpful. For any vector x, let x[k] denote the vector made up of its …rst k elements; let x¡[k] denote the vector made up of all of its elements after the k th; and again let x¡k denote the vector made by deleting the k th element of x (if x has fewer than k elements, x¡k = x). Now, …x i. The proof will proceed by induction. De…ne the induction hypoth- esis P (n0 ) for n0 · n as follows. P P (n0 ): For any "i 2 n0 + 1 and that "i;k for k · n0 are adjusted to keep the …rst n0 0 elements of the ‡ow vector equal to (m¤ )[n ] . The ¹ function is di¤erentiable by ¡i construction. The derivative of its …rst n0 + 1 elements is equal to: µ 0 ¶· ¸ " ¡ ! # (@ mi¢ )[n +1] ~ @^ " i;n0 +1 0 @¹ " = d¹n +1 0 . (@"i )[n0 +1] ¡i 1 0 d"i;n +1 The left hand side of this equation is an n0 + 1-square matrix of derivatives multiplied by an n0 + 1-by-1 vector. The right hand side is an n0 + 1-by-1 vector that has n0 zeroes, due to the de…nition of the ^ function. Once again, by the " properties of dominant diagonal matrices, the inverse of the …rst matrix on the left hand side exists and has only positive elements. Therefore, every element 0 0 of the vector on the left-hand side has the same sign as d¹n +1 =d"i;n +1 . Since 0 +1 0 +1 0 +1 1 > 0, this means that d¹n =d"i;n > 0. Further, d^=d"i;n " is positive in each element. 0 0 From (3.5), we can see that ¹n +1 ! 0 as ¹i;n +1 ! ¡1. (For example, " i;n0 +1 n0 +1 i;n0 +1 i;n as ¹ " ! ¡1, F (" +" ¡ " ) ! 0 pointwise, so by the dominated 0 0 convergence theorem mi;n +1 ! 0.) Further, ¹k ! 0 as ¹i;n ´ ! 1 for k > ³ " +1 0 P 0 0 n0 + 1 (by a parallel argument), so ¹n +1 ! 1 ¡ n (m¤ )j as ¹i;n +1 ! 1. j=1 ¡i " 0 Therefore, by continuity, there exists a value of ¹i;n +1 such that " 0 0 0 0 0 0 0 (mi¢ (^(¹i;n +1 ; ("¤ )¡[n +1] ; (m¤ )[n ] ); ¹i;n +1 ; ("¤ )¡[n +1] ))[n +1] = (m¤ )[n +1] . ~ "" ¡i ¡i " ¡i ¡i 0 0 0 Finally, since d¹n +1 =d"i;n +1 > 0, as noted above, this value of ¹i;n +1 is unique. " Thus, P (n0 + 1) holds. 25 Proof of Proposition (3.4). Claim (i) is straightforward, since the planner could always set Dii = 1 for all i, which would ensure a non-negative value for (3.2) since C ii ´ 0. Claim (ii) follows from assumption (2.1). The proof of claim (iii) is as follows. Return to the original form of the problem, (3.2). For any L 2 S L and for any n £ n matrix D of functions Di;j : ®T (W )(La ; s) + (1 ¡ ®)T (W )(Lb ; s). The …rst inequality follows from optimization, and the fact that Dc is feasible. The last inequality follows from the concavity of X i and W , and from the fact that Da is optimal at point a and Db is optimal at point b. Therefore, if W is bounded and concave, so will be T k (W ) for any k, and so must be the limit function, which is the true value function V . This completes the proof. Proof of Proposition (3:5). Note that the derivative of U with respect to the choice variable is given by: @U (L; s; ") 0 (8.2) @"ii XZ 0 Y = Li ("j ¡ C ij )f ("j )f ("j + "ij ¡ "ii ) F ("j + "ij ¡ "ik )d"j (8.3) j6=i0 k6=j;i0 Z 1 X Y i 0 0 0 0 0 0 0 0 ¡L ("i ¡ C ii )f ("i ) f ("i + "ii ¡ ¹ik ) " F ("i + "ii ¡ "il )d"i . (8.4) ¡1 k6=i0 l6=i0 ;k 0 Using the change of variables " = "j ¡ "ii + "ij on the …rst integral and rearranging yields: @U (L; s; ") X 0 ij i ii ij ii0 ij @m 0 = L (" ¡ " + C ¡ C ) ii0 @"ii j6=i0 @" Xn @mij i = L (¡"ij ¡ C ij ) ii0 . j=1 @" 27 (The equality follows, …rst, because the term in parentheses equals zero when j = i0 , so we can lift the restriction that j 6= i0 without a¤ecting the equation; and second, the sum of derivatives of the ‡ows across all cells resulting from a change 0 in "ii must equal zero.) The …rst order condition for the Bellman equation is, then: Xµ n ¶ @V @mij i ij ij L ¡" ¡ C + ¯E ii0 = 0. e @ Lj @" j=1 De…ne the function ~i¢ as the inverse of the function mi¢ discussed in Section " e (3.1). Then the …rst order condition implies, if i 6= 1: à ! X Xµ n ¶ @V @mij @eii " 0 Li ¡"ij ¡ C ij + ¯E ii0 e @ Lj @" @mi1 i0 6=i j=1 Xµ n ¶ @V X @mij @eii " 0 i ij ij = L ¡" ¡ C + ¯E 0 =0 e @ Lj 0 6=i @"ii @mi1 j=1 i Now, note that X @mij @eii0 " ii0 @mi1 0 6=i i @" takes a value of 1 if j equals 1, ¡1 if j equals i, and zero otherwise. Thus, the …rst order condition reduces to: µ ¶ 1 i1 i1 @V ii ii @V L ¡" ¡ C + ¯E + " + C ¡ ¯E = 0, or e @ L1 e @ Li µ ¶ i1 i1 @V @V " + C = ¯E ¡ . e e @ L1 @ Li This equation says that the marginal cost of moving a worker from i to 1 is equal at the optimum to the expected discounted marginal bene…t of doing so. This can be repeated for any pair of cells i and j with i 6= j, to yield the indicated condition. 28 Proof of Proposition (3.6). Using (3.8) and (3.4), we have: @V (L; s) e @ Li à ! X Z n 1 Y n X @V~ i j ij j j ij ik j = X1 + (" ¡ C )f(" ) F (" ¡ " + " )d" +¯ mij , ¡1 e @ Lj j=1 k6=j j=1 e e e where V stands for E[V (L; s)js] from (3.8). Rearranging, this becomes à ! X Z 1 n Y i X1 + "j f ("j ) F ("j ¡ "ij + "ik )d"j j=1 ¡1 k6=j à à !! n X @V~ @V~ @V~ + mij ¡C ij + ¯ ¡ +¯ , e @ Lj e @ Li e @ Li j=1 which from (3.9) becomes à ! X Z 1¡ n ¢ Y @V~ i X1 + "j + "ij f ("j ) F ("j ¡ "ij + "ik )d"j + ¯ . ¡1 e @ Li j=1 k6=j This is the indicated condition. Proof of Proposition (5.1). Fix ± > 0. Rewrite the planner’s objective function (3.4): X Z Y X X(Lt ; st ) + Li "j t G± ("ij ¡ "ik + "j )g± ("j )d"j ¡ t t t Li mij ("t )C ij , t ± ij k6=j i;j where mij denotes the gross ‡ow from i to j as calculated from (3.5) using the ± distribution G± , and, as before "ii = 08i. We can rewrite this function once again as follows. U± (L; s; e) ´ " X Z Y X X(L; s) + Li "j G± (±(eij ¡ eik ) + "j )g± ("j )d"j ¡ " " Li mij (±e)C ij , ± " i;j k6=j i;j where e is an n-square matrix of real numbers with eii = 0. In other words, e is " " " simply ", scaled down by a factor of ±. 29 Since Z Y mij (±e) ± " = G± (±(eij ¡ eik ) + "j )g± ("j )d"j " " k6=j Z Y "j "j 1 = G1 (eij ¡ eik + " " )g1 ( )( )d"j ± ± ± k6=j Z Y = G1 (eij ¡ eik + ")g1 (")d" " " k6=j ij = m1 (e), " the gross ‡ows resulting from any given choice of e are independent of ±. " Further, X Z Y L i "j G± (±(eij ¡ eik ) + "j )g± ("j )d"j " " i;j k6=j X Z i "j Y ij ik "j "j 1 j = ± L G1 (e ¡ e + )g1 ( )( )d" " " i;j ± k6=j ± ± ± X = ± Li Ai (ei ), " i where i XZ Y i A (e ) ´ " " G1 (eij ¡ eik + ")g1 (")d". " " j k6=j Each of these Ai functions takes a unique maximum at ei = 0. To see this, " consider a sample of n independent draws from the distribution G1 , and call the realized values "1 ; : : : "n . The function Ai (ei ) is the expectation of the j ¤ th of " these, where j is the value of j that maximizes feij + "j g. On the other hand, ¤ " A (0) is simply the expectation of the highest of the "j ’s. Thus, Ai (0) must be i higher. We can now rewrite the objective function once again: X X U± (L; s; e)=± = " Li Ai (ei ) + [X(L; s) ¡ " Li mij (e)C ij ]=±. 1 " (8.5) i i;j 30 The maximization of (3.3) is, of course, equivalent to maximizing the expected present discounted value of U± (L; s; e)=±. Further, we can speak in terms of the " optimal choice of e in each state instead of the optimal choice of " without making " any substantive di¤erence. b P " P Fix ¢ > 0. Let ¢ = i Li Ai (0)¡supjej¸¢ i Li Ai (e) > 0, where jej indicates " " the absolute value of the element of e that is farthest from zero. The point will " be to demonstrate that if ± is large enough, we will have jej < ¢, regardless of " the value of L and s. P From (2.1) and the fact that i Li ´ L, the last two terms of (8.5) can be made uniformly arbitrarily small by choosing ± su¢ciently high. Choose ± high b enough that those two terms are always less than (1 ¡ ¯)¢=2 in absolute value. Now, suppose that the optimal rule for choosing e has at some state (L¤ ; s¤ ) a " value of e with jej > ¢. Now, replace that rule with one that is identical except " " that at that state, and at all other states after that state has once been reached, e is set equal to 0. In the …rst period in which the change takes e¤ect, that " b would increase the value of the …rst term of (8.5) by at least ¢. Thereafter, it could not reduce the value of that term, because with e = 0, that term would " be at its maximum. On the other hand, in the …rst period of the change or in any subsequent period, the second two terms together could fall by at most b (1 ¡ ¯)¢=2, so the expected present discounted value of the reduction in those b b terms would be at most [(1 ¡ ¯)¢=2]=(1 ¡ ¯) = ¢=2. Thus, the change in the value of the objective function due to the change in rule evaluated at the state b b b (L¤ ; s¤ ) would be at least equal to ¢ ¡ ¢=2 = ¢=2 > 0. 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