WPS7089
Policy Research Working Paper 7089
Some Simple Analytics of Trade
and Labor Mobility
Erhan Artuc
Shubham Chaudhuri
John McLaren
Development Research Group
Trade and International Integration Team
&
Poverty Global Practice Group
November 2014
Policy Research Working Paper 7089
Abstract
This paper studies a simple, tractable model of labor adjust- Pre-announcement induces anticipatory flight from the
ment in a trade model that allows researchers to analyze liberalizing sector, driving up wages there temporarily and
the economy’s dynamic response to trade liberalization. giving workers remaining there what this paper calls “antici-
Since it is a neoclassical market-clearing model, duality pation rents.” By this process, pre-announcement makes
techniques can be employed to study the equilibrium and, liberalization less attractive to export-sector workers and
despite its simplicity, a rich variety of properties emerge. more attractive to import-sector workers, eventually making
The model generates gross flows of labor across industries, workers unanimous either in favor of or in opposition to
even in the steady state; persistent wage differentials across liberalization. Based on these results, the paper identifies
industries; gradual adjustment to a liberalization; and many pitfalls to conventional methods of empirical study
anticipatory adjustment to a pre-announced liberalization. of trade liberalization that are based on static models.
This paper is a product of the Trade and International Integration Team, Development Research Group; and the Poverty
Global Practice Group. It is part of a larger effort by the World Bank to provide open access to its research and make a
contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the
Web at http://econ.worldbank.org. The author may be contacted at eartuc@worldbank.org.
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development
issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the
names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those
of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and
its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent.
Produced by the Research Support Team
Some Simple Analytics of Trade and Labor Mobility1
c
Erhan Artu¸
World Bank
Shubham Chaudhuri
World Bank
John McLaren
Department of Economics
University of Virginia
JEL codes: E24, F13, F16
Keywords: Trade Policy, Sectoral Mobility, Wages
1
The authors are grateful to seminar participants at Brown University, the University of Calgary, Dart-
mouth, and Syracuse, as well as participants at the Tuck-Dartmouth Summer Workshop in International
Economics and the Conference on Trade and Labour Perspectives on Worker Turnover at the Leverhulme
Centre for Research on Globalisation and Economic Policy (GEP), University of Nottingham. Special thanks
are due to Doug Irwin, Carl Davidson, Steven Matusz, and Marta Aloi. This project was supported by NSF
grant 0080731 and by the Bankard Fund for Political Economy at the University of Virginia. The ﬁndings
do not necessarily represent the view of the World Bank’s Board of Executive Directors or the governments
that they represent. Any errors or omissions are the authors’ responsibility.
1 Introduction
This paper presents a model of labor mobility incorporated into a simple trade model.
The goal is to provide a general-equilibrium framework that is rich enough to capture the
main empirical features of labor mobility in practice and yet simple enough to be tractable
with the tools of analysis familiar to trade economists. We thus hope that this framework
can become a useful part of a trade economist’s toolkit.
All aspects of an economy’s response to trade policy, and particularly the distributional
aspects, depend crucially on how labor adjusts, and the costs workers face in doing so (see
Davidson and Matusz (2009), Harrison and McMillan (2007), Slaughter (1998), and Magee
(1989) for extensive discussions of this point from diﬀerent angles). However, with important
exceptions discussed below, for the most part trade theory has avoided realistic modelling of
labor mobility, assuming either frictionless mobility or complete immobility. Moreover, most
of the few existing theoretical treatments of labor mobility are hard to reconcile with key
empirical features of labor mobility, in particular with the evidence on gross ﬂows. Here, we
set up a model that takes on these questions head on.
One can identify three ways in which trade theorists have handled labor adjustment in
trade models. First is the traditional static approach used as a benchmark by many trade
economists, which uses a model with ‘speciﬁc factors,’ factors that cannot adjust at all, for
analysis of the short run, but a model with frictionless adjustment for analysis of the long
run. Mussa (1974), for example, uses the ‘Ricardo-Viner’ model for the short run, which
features instantaneous adjustment for labor but no adjustment for any other factors, and the
Heckscher-Ohlin model, which assumes no mobility costs for any factor, for the long-run. Of
course, this approach oﬀers no insight into dynamics, and assumes away mobility costs for
workers.
Second are explicitly dynamic models with only net ﬂows, such as Mussa’s (1978) seminal
model of intersectoral capital adjustment, in which adjustment is gradual because of convex
adjustment costs for capital. A similar approach has sometimes been adopted for labor,
by stipulating convex retraining costs, as in Karp and Paul (1994) and Dehejia (1997). A
feature of these models is that marginal adjustment costs are assumed to be zero when
2
net movements of labor are zero, and so the long-run steady state is the same as in a
model with no adjustment costs (as in Heckscher-Ohlin). Dixit and Rob (1994) present
a model with a constant cost to switching sectors, equal for all workers, in a stochastic
environment. Feenstra and Lewis (1994) use a one-period model of worker adjustment to
study compensation policies. Matsuyama (1992) models intergenerational labor adjustment.
All of the above papers have the property that all workers who switch sectors move in the
same direction: Gross ﬂows always equal net ﬂows. This is a problem, since empirically gross
ﬂows of labor tend to be an order of magnitude larger than net ﬂows (see Artuc, Chaudhuri
and McLaren (2010), or Jovanovic and Moﬃtt(1990), Table 1). The third approach meets
this diﬃculty by adopting search models of labor reallocation. Hosios (1990), Davidson,
Martin and Matusz (1999), and Davidson and Matusz (2004) explore diﬀerent aspects of
this type of model, in which reallocation in response to liberalization is gradual because it
takes time for workers to ﬁnd jobs in the expanding sector. The pioneering paper by Lucas
and Prescott (1974) ﬁts into this category as well, but it is not useful for analyzing adjustment
to trade policy because it is a steady-state analysis only, and the model is restricted to have
a large number of industries, each of which is small so that a shock to any one industry
would have a negligible eﬀect on the economy. Helpman and Itskhoki (2007) and Helpman
and Itskhoki and Redding (2010) use search frameworks in trade models in a one-period
setting, and Itskhoki and Helpman (2014) in a dynamic setting. The rich implications of
the search approach as applied to trade are explored exhaustively in Davidson and Matusz
(2009). The search approach is complementary to the approach of this paper, which is much
more amenable to analysis of transition paths but does not allow for unemployment.
What we do in this paper is to develop a perfect-foresight neoclassical model of labor
adjustment within a simple trade model, which does generate gross ﬂows in excess of net
ﬂows, as the search models do, but does so through idiosyncratic shocks to workers’ mobility
costs rather than through search. This has the advantage that it allows us to use the powerful
tools of duality theory, well known in trade theory, to derive a wide variety of analytical
results not only about the steady state but also about the whole transition path. This latter
point is important for the potential usefulness of this approach: In order for a model to be
3
really useful in analyzing a trade shock, it must allow us to analyze the welfare eﬀects along
the whole transition path – including (i) announcement eﬀects, (ii) impact eﬀects, (iii) the
path to the new steady state, and (iv) changes to the steady state. Our approach achieves
this.
We thus combine a desirable feature of the neoclassical models (duality) with a desirable
property of the search models (gross ﬂows).
This is a simpliﬁed and more tractable version of a more general model presented in
Cameron, Chaudhuri and McLaren (2007). The general version is stochastic, can accommo-
date any number of industries and geographic regions within the country under study, and
has the virtue that its main parameters can be econometrically estimated, as described in
that paper. Here, we focus on the implications of the model for trade theory in a simple
two-sector special case with one type of labor, which is simple and intuitive to use, allowing
us to derive analytic results for eﬀects of trade shocks on lifetime welfare taking into account
c, Chaudhuri and
the full transition path to the steady state. A companion paper (Artu¸
c, Chaud-
McLaren (2008)) study the model through numerical simulations. Finally, Artu¸
huri and McLaren (2010) estimate the structural parameters of the model on US data and
c and McLaren (2010, 2012) do the same for
simulate the adjustment process, while Artu¸
Turkish data and for sectoral as well as occupational mobility, respectively.
The main results from this paper are:
(i) Gross ﬂows of labor always exceed net ﬂows, so there are always workers moving across
sectors, even in the steady state.
(ii) Equilibrium is unique, and there is no hysteresis, despite the presence of an unavoid-
able ﬁxed cost to switching sectors.
(iii) Wage diﬀerentials persist across locations or sectors even in the long run, despite
the fact that there are always some workers changing sectors. In particular, cancellation of
a sector’s tariﬀ protection leads not only to a short-run drop in that sector’s wage relative
to the other sector, but also to a (smaller but still positive) drop in its relative wage in the
long run. Thus, a frictionless model is not a good predictor for the steady state behavior of
the model.
4
(iv) This last point remains true even if moving costs for workers are zero on average.
(v) The economy adjusts only gradually to changes in policy. This is consistent with
ﬁndings by Topel (1986), Blanchard & Katz (1992), and others. This complicates empirical
work on trade and wages, however, because changes in wages and labor allocation will
continue long after the change in policy has occurred.
(vi) The economy begins to adjust to a policy change as soon as it is anticipated. This
complicates empirical work even more, because it means that adjustment begins before the
policy change takes eﬀect, and in fact changes in wages can reverse their direction when the
policy change actually is executed.
(vii) The incidence of trade policy needs to be analyzed on the basis of lifetime utility,
taking worker mobility and option value into account, and not simply on the basis of wage
levels. We show how to do this. The correction is important: It is theoretically possible, for
example, for a policy change to lower real wages in a sector both in the short run and in the
long run, and yet for the workers in the sector to be better oﬀ as a result of it, because it
raises those workers’ option value. This is far from being only a theoretical possibility, as
c, Chaudhuri and McLaren (2010) show using US worker data.
Artu¸
(viii) Announcing a trade liberalization in advance tends to reduce the diﬀerence between
its eﬀect on export-sector workers and its eﬀect on import-competing workers. This is
because it causes workers to begin to leave the liberalizing sector in advance, pushing up
wages for those who remain, and pushing down wages for workers in the export sector. We
say that this process confers ‘anticipation rents’ on workers in the liberalizing sector. We
show that this leads to worker unanimity in the limit, meaning that all workers agree on the
desirability of the liberalization if it is announced far enough in advance.2 However, it is
2
Note that there are three separate reasons that delayed liberalization attenuates losses for import-
competing workers. The ﬁrst and least interesting is simply that if losses from liberalization are postponed,
with time discounting the eﬀect on present value of utility will be diminished. This will of course not
change the sign of the eﬀect on those workers’ welfare, but the other two reasons can. The second reason
is that postponement can give a worker a chance to time her switch to another sector optimally before the
change takes place, lowering adjustment costs; this is a beneﬁt that results only in a model such as ours,
where time-varying idiosyncratic shocks create option value for workers’ reallocation decisions. The third,
the anticipation rent, is a much more subtle eﬀect that results from general-equilibrium forces. We oﬀer
empirical evidence for such rents in Section 6.
5
crucial to note that this can have the eﬀect in the limit of making all workers beneﬁciaries of
trade liberalization or of making them all net losers from it. Thus, the common presumption
that a phased-in liberalization helps to cushion the blow to aﬀected workers can be exactly
wrong in some cases. We oﬀer a simple condition that determines which way it will go.
Note that the dynamics of wages described in our model present a problem for empirical
work. Many studies have used reduced-form regressions to study the eﬀect of trade shocks
on wages, either across industries (Revenga, 1992), across locations within a country (Kovak,
2013), or both (Hakobyan and McLaren, 2010). An implication of a dynamic model such
as ours is that such regressions should be interpreted as measuring the eﬀect of a trade
shock on the wage at one point along its evolution to the new steady state, and of course
the steady-state eﬀect can be very diﬀerent from the short-run eﬀect (and as our model
makes clear, can even be of the opposite sign).3 The hazards that these properties present
for empirical work are eliminated if one estimates the structural parameters of a dynamic
c,
model and uses it as the basis for policy simulations. That is the approach taken in Artu¸
c and McLaren (2012) on US data; Dix-Carneiro
Chaudhuri and McLaren (2010) and Artu¸
c and McLaren (2010) on Turkish data; and Artu¸
(2014) on Brazilian data; Artu¸ c, Lederman
and Porto (2014) on multiple countries’ data.
At the same time, our approach can oﬀer an important piece of assistance to empirical
work: In equilibrium, any change that raises the welfare of workers in one location or industry
relative to another will result in a rise in ﬂows of workers toward that location or industry,
and a reduction of workers in the opposite direction. This can provide a simple but powerful
way of looking for welfare eﬀects in a reduced-form study. For example, Kovak (2013) ﬁnds
that Brazilian workers tend to move out of a state that receives tariﬀ reductions on its major
industries, and Hakobyan and McLaren (2010) ﬁnd an analogous result for US blue-collar
workers in response to NAFTA tariﬀ cuts. These can be taken as prima facie evidence that
welfare for workers in those locations was reduced relative to other locations.4
3
Another problem for reduced-form studies is that a change in tariﬀ in one industry will aﬀect the wages
in all industries. This is generally not accounted for in reduced-form regressions.
4
Although this is an inference about welfare in one industry or location relative to another, at times it
can be used to make a case for an absolute welfare change. Hakobyan and McLaren (2010) cite evidence
in the literature that the aggregate welfare eﬀects of NAFTA were negligible; given this, the evident large
6
The anticipatory eﬀects the model generates resemble the dynamic adjustment observed
in a number of real-life episodes of trade shocks. Section 6 presents a range of evidence from
a variety of sources including from data on the end of the Multiﬁbre Arrangement that can
be interpreted quite naturally with the story in the model.
Plan of the paper: The following section lays out the basic model. Then Section 3 presents
results on the steady state of the model, and the following section presents results on dynamic
adjustment. Section 5 discusses the incidence of policy changes in the model, taking lifetime
utility into account, and analyzes the eﬀect of pre-announcement of liberalization on the
distribution of gains and losses. Section 6 discusses empirical support for the model. A ﬁnal
section summarizes.
2 The Basic Model
There are two sectors, X and Y, each with a large number of competitive employers, who
combine a sector-speciﬁc ﬁxed (latent) factor with labor for production. The two sectors
may be located in two separate regions and may require diﬀerent skills, making it costly for
workers to move between them. Without any costs of moving between sectors, and without
any idiosyncratic shocks to workers, the economy would be a Ricardo-Viner model (Jones,
1971). We shall see that its equilibrium is very diﬀerent from that model, both in its dynamic
character and in its steady state.
The economy’s workers form a continuum of measure L.
2.1 Production
The quantity of aggregate output in sector i in period t is given by:
i
qt = Qi (Li
t ),
negative eﬀect on relative welfare of certain classes of worker can be taken as evidence that their welfare fell
in absolute terms.
7
where Li
t denotes labor used in sector i in period t and we suppress the ﬁxed factor in each
industry to simplify notation. Assume that Qi (·) is increasing, continuously diﬀerentiable,
and strictly concave.
The domestic price of good i, pi , is treated as exogenous. A central example, which will
be treated explicitly, is of a small open economy in which the domestic price is equal to
the given world price plus a tariﬀ (export subsidy) for a good that is imported (exported).
For now, we will assume that domestic goods prices are constant over time, but later we
will extend the analysis to the case of an unanticipated or anticipated change due to a
c, Chaudhuri and
trade liberalization. Cameron, Chaudhuri and McLaren (2007) and Artu¸
McLaren (2010) allow for the more general case of a stochastic process for domestic prices.
The wage in sector i at date t is competitively determined:
i
i i ∂Q (Lit)
wt =p ,
∂Li
t
i
where wt denotes the nominal wage paid by sector i at date t and pi denotes the domestic
price of good i. Thus, the competitive employers in each sector each take the wage as given
and maximize proﬁt; the wage adjusts so that this wage just clears the spot labor market in
that sector.
2.2 Labor Mobility
A worker θ who is in industry i at the end of period t receives a non-pecuniary beneﬁt εi
θ,t .
This can be thought of as enjoyment of the work or of living in the region where industry i is
located, for example. We assume that the εi
θ,t are independently and identically distributed
across workers and sectors and over time, with cdf F (·), pdf f (·), and full support:
f (ε) > 0∀ε ∈ ,
and mean zero:
E ( ε) = εf (ε)dε = 0.
8
The cost to a worker θ, who was in i during period t, of moving from i to j = i at the end
of t is, then:
j
εi
θ,t − εθ,t + C ,
where C ≥ 0 is the deterministic component of mobility costs, common to all workers (such
as retraining or relocation costs, or psychic costs of moving to a new occupation). The
j
variable εi
θ,t − εθ,t is the idiosyncratic component of moving costs, which can be negative as
easily as it can be positive. For example, a worker may be bored with her current job and
j
long for a change in career (εi
θ,t − εθ,t < 0), and a worker with a child in the ﬁnal year of high
school may have a non-pecuniary reason to stay in the current location rather than move,
j
as might be necessary to change jobs (εi
θ,t − εθ,t > 0).
Further, we make the boundedness assumption:
E (max{εX , εY }) = 2 εf (ε)F (ε)dε < ∞.
This ensures that the worker’s problem is meaningful; if it was violated, the worker could
ensure inﬁnite utility simply by choosing the sector with the higher value of ε in each period.
Since what is important for workers’ decisions is the diﬀerence between εi and εj , we can
simplify notation by deﬁning:
j
µi i
θ,t = εθ,t − εθ,t ,
for a worker currently in sector i, where µi
θ,t is symmetrically distributed around mean zero,
with cdf G(·) and pdf g (·) derived from F (·) and f (·).
The transition equations governing the allocation of labor are:
ji j
mii i i
t Lt + mt Lt = Lt+1 i = X, Y ; j = i,
where mji
t denotes the fraction of labor force in j at the beginning of t that moves to i by
the end of t, or in other words, the gross ﬂow from j to i.
The timing of events can be summarized thus:
9
εX Y
θ,t , εθ,t
↓
LX
t −→ X
wt −→ mXX
t , mXY
t LX
t+1
t→ t+1→
=⇒ =⇒
LY
t
Y
−→ wt −→ mY Y YX
t , mt LY
t+1
The stock of workers in each sector in each period is determined by events in the previous
period. The current labor allocations together with current product prices determines wages
through spot labor-market clearing. Then each worker learns her ε’s and decides whether
to remain in her current sector or move. In the aggregate, these decisions determine the
following period’s labor allocation.
2.3 Preferences and Expectations
All agents are risk neutral, have rational expectations and have a common discount factor
β < 1. Further, all workers have identical and homothetic prferences, which allows us to
identify a common cost-of-living index. Letting good X be the numeraire, let the cost-of-
living index be denoted φ(pY ), an increasing function that has an elasticity (by Shepherd’s
i
lemma) equal to good Y ’s share in consumption. Thus, the real wage wt received by a worker
in sector i at date t is given by:
X 1 ∂QX (LXt )
wt (LX Y X Y
t , p ) ≡ wt /φ(p ) = Y X
(1)
φ(p ) ∂Lt
Y Y Y Y Y pY ∂QY (LYt )
wt (Lt , p ) ≡ wt /φ(p ) = Y Y
.
φ(p ) ∂Lt
X
Note that an increase in pY will shift wt down as a function of LX Y
t and shift wt up as a
function of LY
t .
Each worker makes a location decision in each period to maximize the expected present
discounted value of real wage income, net of common (C ) and idiosyncratic (µ) moving
10
costs. (Henceforth we will drop the worker-speciﬁc subscript θ, recalling always that the εt
and µt terms are worker-speciﬁc variables.) Let ui (Lt , εt ) denote the (maximized) value to a
worker of being in i given Lt = (LX Y X Y
t , Lt ) and idiosyncratic shocks εt = (εt , εt ) realized by
the worker. Then v i (Lt ) ≡ Eε (ui (Lt , εt )) gives the expected value of ui before idiosyncratic
shocks are realized, but conditional on (Lt ).
Since the worker is optimizing, ui (Lt , εt ) can be written:
j
i
ui (Lt , εt ) = wt + max{εi i j
t + βEt v (Lt+1 ), εt − C + βEt v (Lt+1 )}
i
= wt + βEt v i (Lt+1 ) + εi i i
t + max{0, µt − µt },
where
µi j i
t = β [Et v (Lt+1 ) − Et v (Lt+1 )] − C , (2)
and i = j . The expression µi
t is the common value of the net beneﬁt of moving from i to j .
If this is greater than the idiosyncratic cost µ, the worker will move; otherwise, the worker
will stay.
Taking expectations with respect to the ε’s (and hence the µ’s):
i
v i (Lt ) = wt + βEt v i (Lt+1 ) + Ω(µi
t ), (3)
where:
µ
Ω(µ) = Eµ max{0, µ − µ} = G(µ)µ − µdG(µ). (4)
−∞
i
In other words, the value, v i , of being in i is the sum of: (i) the wage, wt , that is received;
(ii) the base value, βEt v i (Lt+1 ), of staying on in i; and (iii) the additional value, Ω(µi
t ), of
having the option to move. The expression Ω(µi
t ) is thus interpreted as representing option
value.
11
2.4 Key Equilibrium Conditions
An equilibrium is a moving rule characterized by a value of (µX Y
t , µt ) each period (where
a worker in i moves if and only if µ < µi
t ), such that the aggregate movements of workers
induced by that rule generate a time path for wages in each sector that make the proposed
moving rule optimal. Here we derive a key equation that is useful in characterizing equilib-
rium.
From (2), together with (3) applied to period t + 1, we know that
j j
C + µi i i
t = βEt [wt+1 − wt+1 ] + βEt+1 (vt+2 − vt+2 )
+ Ω(µj i
t+1 ) − Ω(µt+1 ) ,
but using (2) applied to period t + 1, this becomes:
j
C + µi i i
t = βEt [wt+1 − wt+1 ] + C + µt+1 (5)
+ Ω(µj i
t+1 ) − Ω(µt+1 ) .
This is an important relationship for characterising the equilibrium behavior of the model.
The interpretation is as follows. The cost of moving (C + µi
t ) for the marginal mover from i
to j equals the expected future beneﬁts of being in j instead of i at time t + 1. This has three
j i
components: (i) the expected wage diﬀerential next period, [wt+1 − wt+1 ]; (ii) the diﬀerence
in expected continuation values, captured by the expected cost borne by the marginal mover
from i to j at time t + 1, C + µi
t+1 ; and (iii) the diﬀerence in option values associated with
being in each sector, Ω(µj i
t+1 ) − Ω(µt+1 ).
Note that from (2), µX and µY are related:
µX Y
t = −µt − 2C .
Given this and the symmetry of the distribution of µ, the equilibrium reallocations of
12
labor are given by the following relationships:
mij i
t = G(µt ) ; mii i
t = G(−µt ) (6)
mji j i
t = G(µt ) = G(−µt − 2C ) ; mjj i
t = G(µt + 2C ).
As a result, the intersectoral allocation of labor follows the following law of motion:
G(−µi i i i i
t )Lt + G(−µt − 2C )(L − Lt ) = Lt+1 . (7)
3 Characteristics of the Steady State
Here we will derive properties of the steady state, deferring discussion of the path to the
steady state until the next section.
In discussing steady states, we will naturally drop time subscripts. The steady-state level
X
of LX will be denoted L . In addition, we will let µ stand for the steady-state value of µX ,
X
and so the steady-state value of µY is given by −µ − 2C . Then L can be derived from (7)
X
as a function of µ, and written L (µ).
The ﬁrst result is a uniqueness property:
Proposition 1. There is a unique steady-state level of µ and LX .
i
Proof. Because the mij and L derive uniquely from µ, it suﬃces to prove uniqueness
of the threshold µ. The value of µ is implicitly deﬁned by the equilibrium condition, which
comes directly from (5):
β X X
µ+C = wY (L − L (µ); pY ) − wX (L (µ); pY ) (8)
1−β
β
+ [Ω(−µ − 2C ) − Ω(µ)] .
1−β
13
Since
∂ Ω(µ)
= G(µ)
∂µ
and
X mY X G(−µ − 2C )
L = YX XY
L= L, (9)
m +m G(−µ − 2C ) + G(µ)
it is easily shown that the right-hand side of (8) is continuous and strictly decreasing in µ.
Since the left-hand side is continuous and increasing on µ on (−∞, ∞), there is a unique
solution for µ.
This result demonstrates the striking diﬀerence that the idiosyncratic eﬀects make for
the behavior of the model. If there were only common moving costs (C > 0) with no
idiosyncratic shocks (µ ≡ 0), then there would be a range of steady states. Any allocation
of labor such that |wX − wY | < C (1 − β ) would then be a steady state, and this would be a
non-degenerate interval of values of LX .
These diﬀerences are illustrated in Figure 1, which is the standard Ricardo-Viner diagram
adapted to our model. The length of the box is L, the downward-sloping curve is the marginal
value product of labor in X , where the quantity LX is measured from the left axis, and the
upward-sloping curve is the marginal value product of labor in Y , where the quantity LY
is measured from the right axis. All marginal products are deﬂated by the consumer price
index φ(pY ). The top panel shows the unique equilibrium for the Ricardo-Viner model, in
X
which there are no moving costs at all. The equilibrium point is marked as LRV , where wages
in the two industries are equalized. The middle panel shows the range of labor allocations for
which it would be unproﬁtable for workers to move for a model with C > 0 but µ ≡ 0, thus
the range of steady states for a model with common, but not idiosyncratic, moving costs.
Thus, in such a model, there would be hysteresis: The value of LX at which the system
comes to rest would be determined by the initial conditions.
The bottom panel shows the outcome for the present model with both common and
idiosyncratic moving costs. In the present model with gross ﬂows, every year a trickle of
people moves from one cell to another, and this constant stirring of the pot eventually
removes the eﬀect of initial conditions, yielding a unique steady state.
14
A second result, concerning wage diﬀerentials, makes it clear that the frictionless model
does not predict behavior in the steady state of the model.
X Y
Proposition 2. In the steady state, the larger sector must have a higher wage: L L ⇒
wX wY .
X Y
Proof. Suppose L < L but wX wY . Then:
X Y
mY X < mXY since mXY L = mY X L in steady state; so
µX > µY since mY X = G(µY ) < G(µX ) = mXY ; so
β [v Y − v X ] > 0 since µX = β [v Y − v X ] − C ; so
vY > vX .
But:
wX 1
vX = + Ω(µX )
1−β 1−β
wY 1
> + Ω(µY )
1−β 1−β
= vY since Ω(µ) is increasing in µ,
which yields a contradiction.
Thus, we can say that the long-run intersectoral elasticity of labor supply is ﬁnite. Note
that this implies persistent wage diﬀerentials, even in long-run equilibrium, and even though
in each period some fraction of the workers in each sector move to the other. The point is
that if a given sector is to be larger than the other in the steady state, it must have a lower
rate of worker exit than the other sector does. In order for that to be the case, it must have
a higher wage. Put diﬀerently, suppose for the sake of argument that X and Y had the same
wage and µX = µY while LX > LY . Then the rate of exit from each sector would be the
same, so a larger group of X workers would arrive in Y each period than the group leaving
Y . This would put downward pressure on the wage in Y , opening up a wage diﬀerential in
15
favor of X , the larger sector.5
Note as well that this remains true even if C = 0, so that average moving costs are
equal to zero. Idiosyncratic moving costs, sometimes positive and sometimes negative, are
suﬃcient to ensure steady-state wage diﬀerentials.
There is also an unambiguous relationship between the steady state of this model and
the Ricardo-Viner equilibrium (which, recall, is the equilibrium of the model with C, µ ≡
0). Speciﬁcally, the steady-state intersectoral allocation of workers always lies somewhere
between the Ricardo-Viner model and equal division of workers between the sectors. To see
this, ﬁrst let the allocation of workers to the X sector in the Ricardo-Viner model, LX
RV , be
deﬁned implicitly by:
wX (LX Y Y X Y
RV , p ) = w (L − LRV , p ).
Proposition 3. The following inequalties must hold in the steady state:
L X L
LX
RV < ⇒ LX
RV < L <
2 2
L X L
LX
RV = ⇒ LXRV = L =
2 2
L X L
LX
RV > ⇒ LXRV > L >
2 2
Proof. The result follows directly from the labor demand curves being downward sloping
L
and Proposition 2. Suppose LX
RV > 2
, which means the demand curves cross to the right
X
of the midway point as shown in Figure 1. If L were, in contradiction to the claim, to lie
L X X
to the right of LX
RV , i.e., 2
< LX
RV < L , then w would have to be less than wY . This
follows from the deﬁnition of LX
RV and the fact that the labor demand curves are downward
sloping. (Look at the diagram.) But that would contradict the earlier result that the larger
sector (in this case X ) has to have the higher wage in steady state. We thus conclude that
X X
L < LX
RV ; but this also implies that w > wY (again, from the deﬁnition of LX
RV and the
fact that labor demand curves slope downward), so from the previous proposition, we must
5
This result should be taken with caution when the model is brought to data. An industry could also have
high wages because of harsh or dangerous conditions; a region could have high wages because of unpleasant
living conditions. These reasons for compensating wage diﬀerentials are absent from this simple model, but
are accounted for in Artu¸ c and McLaren (2012), for example.
16
L X
have 2
LI . Let the domestic price of Y in the two
models be pY Y Y Y
I and pN , with pI > pN , and let the values of µ be µI and µN respectively. We
will show that the direction of the steady-state impact of a labor demand shock is the same
in our model as it is in the model with no mobility costs:
X X
Proposition 4. Under the stated assumptions, LN > LI .
X
X X ∂L (µ)
Proof. Suppose instead that LN ≤ LI . Then µI ≤ µN because ∂µ
< 0 from (9). At
17
the same time:
β X X
µN + C = wY (L − LN , pY X Y
N ) − w (LN , pN ) + Ω(−µN − 2C ) − Ω(µN )
1−β
β X X
< wY (L − LN , pY X Y
I ) − w (LN , pI ) + Ω(−µN − 2C ) − Ω(µN )
1−β
(since wY is increasing in pY while wX is decreasing in pY )
β X X
≤ wY (L − LI , pY X Y
I ) − w (LI , pI ) + Ω(−µN − 2C ) − Ω(µN )
1−β
X X
(since we assumed LN ≤ LI )
β X X
≤ wY (L − LI , pY X Y
I ) − w (LI , pI ) + Ω(−µI − 2C ) − Ω(µI )
1−β
(since Ω(−µ − 2C ) − Ω(µ) is decreasing in µ and µN ≥ µI )
= µI + C
which is a contradiction.
Finally, as a last comparative static result, we note that wage diﬀerentials induced by
policy persist in the steady state. When the tariﬀ on Y is removed, not only does the X -
industry wage rise relative to the Y -industry wage in the short run, but as Proposition 4
shows, it will also do so in the long run.
Proposition 5. Using the notation of the previous problem, given that pY Y
N < pI , it must be
the case that [wX Y X Y
N − w N ] > [w I − w I ].
Proof. Suppose not. Then:
[w Y X Y X
N − w N ] ≥ [w I − w I ], so from (8)
1−β 1−β
[µN + C ] + Ω(µN ) − Ω(−µN − 2C ) ≥ [µI + C ] + Ω(µI ) − Ω(−µI − 2C ),
β β
which implies µN ≥ µI .
X X
But we already know from the previous proposition that LN > LI , which implies that
µI > µN , given (9). This yields a contradiction.
18
In sum, the steady state of the model is unique and qualitatively diﬀerent from the
equilibrium of a frictionless model. It exhibits persistent wage diﬀerentials in favor of larger
industries, produces more evenly-sized industries than a frictionless model, and does not
maximize GDP. Now, we turn attention to the whole time-path of adjustment.
4 Dynamic Adjustment
4.1 Preliminaries
It is in analyzing dynamics and welfare eﬀects along the transition path that the tools of
duality are useful for this model. Here we will see how. First, we will show the optimization
problem that equilibrium solves (as noted above, it is not maximization of GDP). In the
next subsection, we will see how that establishes basic dynamic properties, and in the next
section we will use it to derive results on distributional eﬀects along the transition path.
A general rule for labor allocation in this model could be characterized by two functions.
The function dXY (LX , µX ) gives the probability that a worker in X will move to Y in the
current period, given the current stock LX of workers in X and the worker’s idiosyncratic
cost µX of moving from X to Y . The function dY X (LX , µY ) gives the probability that a
worker in Y will move to X in the current period, given LX and the worker’s idiosyncratic
cost µY of moving from Y to X . These functions deﬁne a feasible allocation rule if and only
if dij ∈ [0, 1] over the whole domain. For any given LX
0 , these functions induce a sequence
LX
t for t = 1, ∞. The following proposition is proven (in a more general form) as Proposition
1 in Cameron, Chaudhuri and McLaren (2007):
Proposition 6. Any equilibrium maximizes:
∞
t=0 β t [QX (LX Y Y X
t ) + p Q (L − Lt )
(10)
−LX
t (µX + C )dXY (LX X X X Y
t , µ )g (µ )dµ − Lt (µY + C )dY X (LX Y Y Y
t , µ )g (µ )dµ ]
within the class of feasible allocation rules dXY and dY X , subject to LX Y
0 and L0 given, and
with Li ij
t determined by the choice of the d functions for i = X, Y , t > 0.
19
Therefore, studying optimization problem (10) can tell us about the equilibrium. Note
that this is a dynamic analogue to the revenue function maximized by equilibrium in a static
neoclassical trade model (see Dixit and Norman 1980, Ch. 2). Let V (LX , LY ) denote the
maximized value of the objective function (10) from an initial condition of LX workers in
X and LY workers in Y . Equivalently, since L is ﬁxed, we can write the value function as
W (LX ) ≡ V (LX , L − LX ).
Since the problem is stationary, the usual dynamic programming logic will apply, and the
function W can be computed by solving a Bellman equation. Conditional on the domestic
relative price pY , for any bounded function W on [0, L], deﬁne the operator T by:
T (W )(LX ; pY ) ≡ max [QX (LX Y Y X
t ) + p Q (L − Lt ) (11)
{dXY ,dY X }
− LX
t (µ + C )dXY (LX
t , µ)g (µ)dµ
− (L − LX
t ) (µ + C )dY X (LX
t , µ)g (µ)dµ (12)
+ β W (LX
t+1 )],
where LX
t+1 = {(1 − d
XY
(LX X
t , µ))Lt + d
YX
(LX X
t , µ)(L − Lt )}g (µ)dµ. The Bellman equation
is then T (W ; pY ) = W . It is easy to show that T satisﬁes the usual Blackwell properties
with respect to W , and that as a result T has a unique ﬁxed point, which can be found as
the unique uniform limit of iterations on the Bellman equation starting from any bounded
candidate function.
The following observation is helpful in characterizing the system’s dynamics, and is also
proven as Proposition 5 in Cameron, Chaudhuri and McLaren (2007):
Proposition 7. The function W is strictly concave.
This result tells us that the equilibrium is unique, since in strictly concave optimization
problems the optimum is unique. It also tells us that the function W (LX ) is strictly decreas-
ing in LX . The following tells us that the value of this function is always equal to v X − v Y ,
which is very useful for analyzing dynamics, as will be seen shortly.
20
Proposition 8. In equilibrium at each date v X − v Y is equal to W .
Proof. First, note that in the optimization it is never optimal to have workers moving
from i to j and at the same time other workers with lower values of µ who are remaining
in i. (In that case, an equal number of workers from the two groups could have their
actions reversed, leaving the future allocation of workers unchanged but reducing aggregate
idiosyncratic moving costs.) Therefore, in an optimal allocation there is for each value of
LX at each date a number µX such that dXY (LX , µ) = 1 if µ < µX and dXY (LX , µ) = 0
if µ > µX . Simlarly, there is for each value of LX at each date a number µY such that
dY X (LX , µ) = 1 if µ < µY and dXY (LX , µ) = 0 if µ > µY . This means that we can rewrite
the Bellman equation as follows.
W (LX
t )
= max [QX (LX Y Y X
t ) + p Q (L − Lt )
{µX ,µY }
µX
t µY
t
− LX
t (µ + C )g (µ)dµ + (L − LX
t ) (µ + C )g (µ)dµ
−∞ ∞
+ βW ((1 − G(µX X Y X
t ))Lt + G(µt )(L − Lt ))].
The ﬁrst-order conditions for this with respect to µX and µY are
µX X
t + C = −βW (Lt+1 ) and (13)
µY X
t + C = βW (Lt+1 ) .
But then from (2), the result follows.
This tells us that in equilibrium, the attractiveness of either sector relative to the other is
a strictly decreasing function of the number of workers who are located in that sector. Now
we can use this to analyze the economy’s dynamics.
21
4.2 Gradual Adjustment to Unanticipated Changes
The preceding analysis can be used to show a number of properties of the model’s dynamic
adjustment. First, labor market adjustments to any change, such as terms of trade shocks or
policy changes, will be sluggish. In particular, suppose that the economy is in a steady state
associated with an initial value of pY , say p . If a one-time shock occurs (say, elimination of
the tariﬀ on good Y ) that results in a new value of pY , say p , the economy will not reach
the steady state associated with p in ﬁnite time.
To see this, consider Figure 2. This illustrates the ﬁrst-order condition (2), or equivalently,
(13), for choice of µY X X
t and hence of Lt+1 given the current value of Lt . The solid upward
Y Y
sloping curve indicates the locus of points (LX X X
t+1 , µt + C ) such that Lt+1 = G(µt )(L − Lt ) +
(1 − G(−µY X X X Y
t − 2C ))Lt , which we can write as Lt+1 (Lt , µt ). One can, thus, interpret it as
the marginal cost curve for the supply of X -workers: the height is the moving cost for the
marginal worker moving to X , given that the total number who wind up in X at the end of
this period is equal to LX X Y X
t+1 . The downward sloping curve gives β (vt+1 − vt+1 ) = βW (Lt+1 ).
This can be interpreted as the marginal beneﬁt of moving a worker from Y to X . Given LX
t ,
Y
the values of LX
t+1 and of µt are determined as the intersection of these two curves.
Now, note that if C > 0, increasing LX
t by ∆ units shifts the marginal cost curve to the
right at each point by an amount strictly between 0 and ∆.6 Since the marginal beneﬁt
curve is strictly decreasing, this implies an increase in LX
t+1 that lies strictly between 0 and
∆. This can be summarized in Figure 3, which shows the transition function that gives LX
t+1
as a function of LX
t . Provided that C > 0, this curve must be strictly increasing, with a slope
strictly less than 1. Thus, there is a unique steady state, and if the system begins at a point
other than the steady state, it will move toward it without ever reaching or overshooting it.7
6 Y Y Y Y
The function LX X X X
t+1 (Lt , µt ) can be written as G(µt )(L − Lt ) + (1 − G(−µt − 2C ))Lt , or G(µt )L +
Y Y X
[1 − G(µt ) − G(−µt − 2C )]Lt . By the symmetry of G, the expression in square brackets is strictly between
zero and 1 if C > 0.
7 Y Y
In the event that C = 0, from the previous footnote we know that LX X
t+1 (Lt , µt ) = G(µt )L. Therefore,
X
the upward-sloping line in Figure 2 will be vertical, and so both µ and Lt+1 will be the same regardless of
LXt . Therefore, the economy jumps right away to its steady state. However, note that the outcome is not
the same as for the frictionless model: As noted in previous sections, even with C = 0, intersectoral wages
will not be equalized, and the equilibrium will not maximize GDP.
22
This provides the result. For example, if the system is initially at a steady state with a
high tariﬀ that is expected to continue permanently, and then the tariﬀ is suddenly removed
never to be restored, then the system will move toward the new steady state each period,
attaining it only in the limit.
Gradual labor-market adjustment to external shocks and policy changes, and the persis-
tent wage diﬀerentials that they imply, have been documented empirically by, among others,
Topel (1986), Blanchard & Katz (1992) and Rappaport (2000). They appear in the model
of Davidson and Matusz (2004), due to re-training and search delays and exogenous rates of
individual job separation. They can also be rationalized by convex training costs for labor
(as in Karp and Paul (1994) or Dehejia (1997), in analogy with convex adjustment costs for
capital as in Mussa (1978)). Here, they result from the presence of time-varying idiosyncratic
shocks to workers (even though those shocks are serially uncorrelated). Even if a worker is
suﬀering low wages as a result of loss of protection to that worker’s sector, it will often be
in that worker’s interest to wait until her personal moving costs are suﬃciently low before
leaving the sector.
4.3 Anticipatory Adjustment to Pre-announced Changes
Another feature of the model’s dynamics is anticipatory adjustment. Suppose the econ-
omy is in an initial steady state with a tariﬀ on imported good Y , and a domestic price of
pY = p . At time t = 0, the government announces a surprise policy change—elimination of
the tariﬀ—starting at some date t∗ > 0. At that date and thenceforth, the domestic price
of Y will be equal to the world price, p < p . We will see that anticipatory net outﬂows of
labor from sector Y will begin immediately from the time of the announcement.
Although the environment is no longer stationary as in the previous sections, the logic
of Proposition 6 still applies, and the perfect-foresight equilibrium still solves the planner’s
problem (10) with pY = p for t = 0, . . . , t∗ and pY = p for t > t∗ . The value function will,
however, depend on the date as well as the current labor allocation, and it is useful to write
it as W t (LX ; p , p , t∗ ) to keep track of the parameters of the problem.
23
Note from (1) that ceteris paribus, a rise in pY will raise the real wage in the Y sector
X
and lower it in the X sector, or w2 Y
(LX , pY ) < 0 and w2 (L − LX , pY ) > 0. This leads to the
following.
Lemma 1. Assume that C > 0. (i) Let W be a bounded, concave, and diﬀerentiable function
on [0, L]. Then ∂T (W ; p )(LX )/∂LX > ∂T (W ; p )(LX )/∂LX . (ii) Let W and W be bounded,
concave, and diﬀerentiable functions on [0, L], with W > W everywhere. Then, for any
value of pY , ∂T (W ; pY )(LX )/∂LX > ∂T (W ; pY )(LX )/∂LX .
Proof. By the envelope theorem, we have:
∂T (W ; pY )(LX X X X Y Y X Y
t )/∂Lt = w (Lt , p ) − w (L − Lt , p )
µX µY
− (µ + C )g (µ)dµ + (µ + C )g (µ)dµ
−∞ ∞
+ β W (LX X Y
t+1 )[1 − G(µt ) − G(µt )].
Using (13) and (4) and rearranging, this becomes the following.
∂T (W ; pY )(LX X X X Y Y X Y Y Y Y
t )/∂Lt = w (Lt , p )−w (L−Lt , p )+Ω(−µt −2C )−Ω(µt )+µt +C . (14)
(i) Comparing p with p , (13) shows that the optimal choice of µY X
t for a given Lt
will be the same for both. This implies that the only diﬀerence in (14) is in the ﬁrst two
terms, which take a higher value for p since the X wage is higher and the Y wage is lower.
This proves the result. (ii) If we replace W with W , then in the ﬁrst-order condition for
the optimization in T (W ; pY )(LX
t ), the marginal beneﬁt curve shifts up (recall Figure 2).
Y
Thus, for any LX
t , we have a rise in the value of µt that is chosen. The ﬁrst two terms of
(14) are unchanged. The derivative of the last three terms with respect to µY
t is equal to
−G(−µY Y
t − 2C ) − G(µt ) + 1, which by the symmetry of the distribution of µ is equal to
G(µY Y
t + 2C ) − G(µt ) > 0. Therefore, the value of (14) has gone up, proving the result.
Now, consider a model in which pY
t ≡ p forever and call it model I (for ‘initial’), with
value function W I and steady-state value of LX equal to LX
I . Consider in the same way a
24
model in which pY
t ≡ p forever and call it model N (for ‘new’), with value function W
N
and steady-state value of LX equal to LX
N . The two comparative statics results just derived,
I N
applied to the recursions on the Bellman operator T , imply that W1 (LX ) < W1 (LX ) for
any LX ∈ [0, L], and that LX X
I < LN .
Now, return to the problem of the model with the announced policy change at time t∗ .
From time t∗ on, the value function and the transition function mapping LX X
t into Lt+1 and
will be exactly as they are in model N . Consider date t∗ − 1. For a given value of LX , the
value function for date t = t∗ − 1 is given by T (W N ; p )(LX ) (the next–period value function
is W N , but the current value of pY is p ). By the lemma, we can conclude:
I ∂T (W N ; p )(LX )
W1 (LX ) < N
< W1 (LX ) (15)
∂LX t
for any LX . The ﬁrst inequality results from part (ii) of the Lemma because W I = T (W I ; p ),
I N
and W1 < W1 . The second inequality results from part (i) of the Lemma because p > p .
However, referring again to the ﬁrst-order condition for the choice of LX X
t+1 given Lt (see
Figure 2), we see that (15) implies that the transition function for period t∗ − 1 lies strictly
in between the transition function for model I and that for model N. This is illustrated in
Figure 5. By the same logic, the transition function for period t∗ − 2 must lie strictly between
that for t∗ − 1 and that for model I, and so on. Then if we were already in a steady state
of model I and it was announced (to everone’s surprise) at date 0 that the tariﬀ would be
removed at date t∗ , the dynamics of the system would follow the path indicated in Figure
4, drawn for the assumption that t∗ = 3. Adjustment toward the new steady state would
begin immediately at date 0, and would continue permanently, always moving toward the
new steady state but never reaching it.
Of course, anticipatory labor adjustments also imply anticipatory wage changes. In
particular, anticipatory outﬂows from Y prior to the actual tariﬀ removal will progressively
raise wY up to the time of the policy change, at which point it will fall discretely, and will
progressively push down wX up to the time of the policy change, at which point it will rise
discretely due to the drop in the consumer’s price index. This has obvious implications for
25
empirical analyses of the impact of trade liberalization on wages.
In particular, suppose that a researcher obtains data on wages and employment levels at
dates t∗ − 1 and t∗ + 1, and compares the values before and after the liberalization. If the
diﬀerences thus observed are interpreted to be the eﬀects of the liberalization, then because
the anticipatory eﬀects are omitted, the study will greatly overestimate the wage eﬀects and
underestimate the sectoral employment eﬀects.
5 Welfare and Incidence
A large part of the reason for studying the workings of a model with labor mobility costs is
to reﬁne our understanding of who gains and who is hurt from a change in trade policy. Here
we will look at two diﬀerent angles of this question. First, a simple envelope result shows
how the eﬀect on a given worker can be expressed in terms of ﬂow probabilities and changes
in wages only. This result shows the importance of gross ﬂows in analyzing incidence, which
is ignored in the vast majority of empirical work. Second, we derive some results on how
delayed trade liberalization can aﬀect who gains and who loses from a reform.
5.1 An Envelope Result
Return again to the case with constant pY . Consider again an unannounced and permanent
change in tariﬀ, which changes the value of pY once and for all. Returning to (3), we can
see that the change in the utility of a worker in sector i would be:
i∗ i∗ i∗ i∗
dvt /dpY = dwt /dpY + βdvt Y i Y
+1 /dp + Ω (µt )dµt /dp ,
i∗ i∗ ∗
where wt , vt and µi
t denote the equilibrium values of the wage, worker’s utility, and moving
threshold in sector i and at date t respectively. Noting that Ω (µ) = G(µ), this means:
i∗ i∗ i∗ j∗ i∗
dvt /dpY = dwt /dpY + βdvt Y i Y Y
+1 /dp + G(µt )β [dvt+1 /dp − dvt+1 /dp ], or
26
i∗ i∗ i∗ j∗
dvt /dpY = dwt /dpY + β (1 − G(µi Y i Y
t ))dvt+1 /dp + βG(µt )dvt+1 /dp .
Recalling (6) and following the recursive logic forward, this becomes:
∞
i∗ k∗
dvt /dpY = βn ik
πt Y
+n dwt+n /dp ,
n=0 k=X,Y
ik
where πt is the probability that a worker who was in i at time 0 will be in k at time t. Thus,
despite the moving costs, a properly constructed discounted sum of wages alone, using gross
ﬂows to average across sectors, is suﬃcient for evaluating incidence.
Note again that this has implications for empirical work. It suggests that looking at
change in an industry’s wages is not enough to tell whether workers in that industry have
beneﬁtted or not. In particular, it is quite possible to construct examples in which a drop
in tariﬀ lowers wages in sector Y in the short run and in the long run, and yet every worker
in the economy beneﬁts, including those in Y .8 The reason is that real wages in X rise
by enough, and the economy is ﬂuid enough, that current Y workers expect to make up in
future employment in X for what they have lost in Y . This also stands in stark contrast to
the convex-adjustment cost approach (Karp and Paul, 1994; Dehejia, 1997); in equilibrium
in those models, a Y -worker must be indiﬀerent between leaving Y and remaining there
permanently, and so would deﬁnitely be worse oﬀ if Y wages were to fall permanently.
Of course, the need, in principle, to examine the eﬀect on all wages now and in the future
is a problem econometrically. However, the model does oﬀer one useful tool for inferring
welfare eﬀects in a reduced-form regression: Examining mobility decisions. Recalling (2)
and (6), it is clear that if a change in policy results in fewer workers choosing to enter an
industry or location and more workers choosing to leave it, then the welfare in that industry
or location must have declined relative to others. This simple revealed-preference test can
8
For example, consider a model with Ricardian technology. If Qi (Li i i i
t ) ≡ A Lt for positive constants A ,
X X Y Y Y Y Y
i = X, Y , then w = A /φ(p ) and w = p A /φ(p ). Suppose that the two goods are perfect substitutes
in consumption so that φ(pY ) ≡ min{1, pY }. Then for pY > 1, wX = AX and wY = pY AY , while for pY ≤ 1,
wX = AX /pY and wY = AY . Then if pY is initially slightly above 1 but then drops to a point well below 1,
wY will fall slightly but by letting pY fall suﬃciently close to 0, wX can be made arbitrarily large. Eventually,
Y -workers will value the option of moving to X enough to compensate them for their small current wage
loss.
27
be implemented in a wide variety of situations.
5.2 The Eﬀects of Policy Delay on Incidence
Trade liberalization measures are usually phased in over time. For example, the elimination
of the Multi-Fibre Arrangement in the Uruguay round was scheduled to be phased in over
ten years, with most of the reduction loaded at the end of the phase-in period. One reason
for this is to soften the eﬀects of the reform on workers likely to be hurt by it, giving them
time to adjust and perhaps removing an incentive to oppose the reform politically. This
motive is studied by Dehejia (2003), in a Heckscher-Ohlin model with convex adjustment
costs for workers (and zero steady-state ﬂows of workers). In that model, it is shown that
gradual phase-in can bring all workers behind a trade liberalization that would otherwise
have been opposed by import-competing workers, making the reform politically feasible.
We will here study the eﬀect of delayed trade liberalization in the present model. To
study a simple version of the problem that makes the mechanisms clear, we focus on a
stark liberalization that brings the economy from autarchy to free trade. Beginning from an
autarchic steady state, the opening of trade may either occur immediately or be announced
(with full commitment) to occur at a later date. In contrast to Dehejia, we ﬁnd that delay
does not, in general, soften the blow of trade liberalization to workers. What it does do, is
to induce movement of workers out of the import-competing sector beginning the movement
the announcement is made. This forces the wage up in the import-competing sector in the
interval of time between the announcement and the actual liberalization, conferring what we
may call anticipation rents on the remaining workers in that industry. At the same time, it
pushes wages down in the other sector. If the period of delay is long enough, the eﬀect is to
unite all workers, so that they all either beneﬁt from trade or lose from it. To anticipate, it
turns out that if an increase in labor supply increases the economy’s long-run relative supply
of export goods (as it does in Dehejia’s model), then delay fosters uniﬁcation of workers
behind free trade, but if an increase in labor supply decreases long-run relative supply of
export goods, delay fosters uniﬁcation of workers in opposition to free trade.
28
Consider an economy that is as of period 0 in an autarchic steady-state with a relative
price of good Y given by p = p . At date 0 it is announced that the economy will be
opened up to free trade as of date t∗ (which could be equal to zero, representing the case
of unanticipated liberalization). Suppose that the world relative price of good Y is given by
p = p . It is useful to rewrite the period-t value function for the planner’s problem that the
∗
equilibrium solves as W t (LX Y
t , Lt ; p , p , t ). (Of course, the value function is not stationary
before t∗ , although it will be afterward, hence the time superscript for the value function.)
Let t∗ = ∞ represent the case in which no trade liberalization is announced.
It is straightforward to check from the envelope theorem applied to the Bellman equation
for the planner’s problem that the payoﬀ to an individual worker in the X sector at the begin-
X 0 ∗
ning of period 0, just after the policy announcement, is equal to v0 = W1 (LX Y
t , Lt ; p , p , t ),
or the marginal value of an X worker from the planner’s point of view. Similarly, the
Y 0
payoﬀ to a Y worker is given by v0 = W2 . Given that, the question of the value of ad-
vance notice is essentially the question of whether or not an increase in t∗ changes the sign of
∗ ∗
Wi0 (LX Y 0 X Y 0 X Y 0 X Y
t , Lt ; p , p , t )−Wi (Lt , Lt ; p , p , ∞) = Wi (Lt , Lt ; p , p , t )−Wi (Lt , Lt ; p , p , ∞)
for i = 1, 2. Call this diﬀerence an i worker’s ‘willingness to consent.’ It is diﬃcult to obtain
∗
general results on this, but if one can ﬁnd results on the cross derivative Wi0 X Y
4 (Lt , Lt ; p , p , t ),
or the derivative of a worker’s payoﬀ with respect to the world price, evaluated at a value
∗
of the world price equal to the domestic autarchic price, then (since Wi0 (LX Y
t , Lt ; p , p , t ) −
Wi0 (LX Y
t , Lt ; p , p , ∞) = 0) one has signed the i worker’s willingness to consent in an interval
for p that includes p . That is the approach taken here.
Denote the equilibrium employment in sector i at time t, as a function of initial labor
stocks, by Li X Y i X Y
t (L0 , L0 ). Denote the steady state employment similarly by L∞ (L0 , L0 ). Deﬁne
i
qt (LX Y i X Y
0 , L0 ) and q∞ (L0 , L0 ) respectively as output in the i sector at time t and in the steady
state. (In general these functions would be conditioned on p , p , and t∗ as well as LX Y
0 , L0 ,
but we will need to evaluate these functions only at the point p = p , so these additional
arguments will be suppressed.) The following will be useful; the proof is in the Appendix.
Lemma 2. Assume that the value function is twice continuously diﬀerentiable in (LX Y
t , Lt ).
Then:
29
(i) The functions LX Y X Y
t and Lt are diﬀerentiable in (L0 , L0 ).
(ii) The derivatives LX X X Y X Y Y X Y X
t1 ≡ ∂Lt (L0 , L0 )/∂L0 and Lt1 ≡ ∂Lt (L0 , L0 )/∂L0 are all non-
negative, and LX Y X
t1 + Lt1 = 1∀t > 0. Further, Lt1 is decreasing in t.
(iii) The derivatives LX Y Y X
t2 and Lt2 are all non-negative, and Lt2 + Lt2 = 1∀t > 0. Further,
LX
t2 is increasing in t.
(iv) Li i
t1 → L∞1 as t → ∞, for i = X, Y .
Parts (i) and (iv) are technical preliminaries. The result in (iv) requires proof, since even
when a series of functions converges uniformly to a limit function, in general the sequence
of derivatives does not converge to the derivative of the limit series. In this case the result
follows because the functions are solutions to an optimisation problem. Parts (ii) and (iii)
follow directly from the model’s dynamics. For example, if one was to add some workers to
X in period 0, that would result in more X workers in each period, but the number of X
workers would fall over time, as workers reallocate toward the Y sector. With a slight abuse
of notation, we will write LX X Y Y
01 = 1, L02 = 0, L01 = 0, and L02 = 1.
Henceforth, we will assume that the planner’s value function is twice continuously diﬀer-
entiable in all arguments.
With this notation, the eﬀect on the payoﬀ of a worker in X is:
X
∂v0 0 ∗ ∗
|p =p = W14 (LX Y 0 X Y
0 , L0 ; p , p , t ) = W41 (L0 , L0 ; p , p , t )
∂p
by Young’s theorem, which by the envelope theorem becomes:
∞
∂ ∂ 1 ∂ p
βt QX
t + QY
t |p =p
∂LX0 t=t∗
∂p φ(p ) ∂p φ(p )
∞
1 α X
= β t (1 − α)QY
t1 − Q
φ(p ) t=t∗ p t1
∞
1
≡ β t Bt ,
φ(p ) t=t∗
30
where α ≡ p φ (p )/φ(p ) is the share of good Y in autarchic consumption expenditure.9 Note
that this provides a simple way of analyzing the eﬀects of delay; the only eﬀect of an increase
in t∗ on this exression is to eliminate some of the terms from the summation. This is the
key to the results that follow.
Since QX Y
t1 is decreasing in t and Qt1 is increasing in t, Bt is increasing in t, as shown
α
in Figure 6. Since p (1−α)
is the autarchic steady-state ratio of Y consumption to X con-
sumption (and hence the ratio of production as well), B∞ ≡ limt→∞ Bt has the same sign as
QY QX
∞1
QY
− ∞1
QX
. Put diﬀerently, B∞ > 0 if and only if an increase in the economy’s total labor
∞ ∞
supply would increase QY X
∞ /Q∞ at a ﬁxed world price of p = p , and B∞ < 0 if an increase
in labor supply would decrease QY X
∞ /Q∞ . Figure 5 illustrates the case in which B∞ < 0. In
addition, QY
01 = 0, since an exogenous increase in the stock of labor in one sector cannot
have a contemporaneous eﬀect on output in the other sector. Thus, B0 < 0.
X
∂v0
As a result, if B∞ < 0, then Bt < 0 for all t, and |
∂p p =p
< 0. However, if B∞ > 0,
then Bt is negative for t below some threshold and positive for t above the threshold. The
X
∂v0
implication is that ∃t such that if t∗ ≥ t, |
∂p p =p
> 0.
Analogously, it can be seen that
Y ∞
∂v0 1
|p =p = β t Dt ,
∂p φ(p ) t=t∗
α X
where Dt ≡ (1 − α)QY
t2 − p
Qt2 . Clearly, since QX X Y
t2 = 0, Qt2 is increasing in t, and Qt2 is
decreasing in t, while QX Y
t2 and Qt2 are non-negative for all t, we can conclude that D0 > 0
and that Dt is decreasing in t. Again, this is illustrated in Figure 5. Further, by part (iv)
of the lemma and the uniqueness of the steady state which implies that LX X
∞1 = L∞2 and
LY Y
∞1 = L∞2 , we conclude that D∞ = B∞ . This all implies that Bt < Dt for all t. Given that
X
∂v0 Y
∂v0
B0 < 0 < D0 , we conclude that |
∂p p =p
< |
∂p p =p
.
∂v0Y
By extension of the logic just used, if D∞ < 0, then ∃t such that if t∗ ≥ t, |
∂p p =p
< 0.
X
∂v0 ∂v Y
Further, if B∞ = D∞ = 0, then Bt < 0 < Dt for all t, and |
∂p p =p
< 0 < ∂p0 |p =p
9
The switch in the order of diﬀerentiation is analogous to the demonstration that in a static trade model
derivatives of factor prices with respect to output derivates is dual to the derivative of output with respect
to factor prices, as in Dixit and Norman (1980, pp.54-55).
31
regardless of t∗ . To help interpret the sign of D∞ , recall that α is the autarkic share of
spending on Y, so the ratio of α/p to (1 − α) is the same as the ratio of autarkic output of
Y to autarkic output of X. Therefore, D∞ < 0 means that the steady state autarkic supply
of X increases when labor is added to the economy.
In addition, since B0 < 0 < D0 , it is clear that increasing t∗ from 0 to 1, by chopping
X
∂v0
oﬀ a term that is negative in the ﬁrst case and positive in the second, increases |
∂p p =p
Y
∂v0
and decreases |
∂p p =p
. Thus, workers in the export sector are hurt by a bit of delay, while
workers in the import competing sector beneﬁt from a bit of delay.
This can all be summarized as follows. The ﬁrst proposition treats the case with B∞ =
D∞ = 0, and the second treats the case with B∞ = D∞ = 0.
Proposition 9. Suppose that, at the autarchy relative price p , steady state relative supply
of the export good is increased [decreased] by an increase in labor supply. Then, there is an
open interval containing p such that if the world price p is in that interval:
(i) Workers in the export [import-competing] sector as of period 0 will beneﬁt from [be
hurt by] immediate unanticipated opening of trade.
(ii) The net beneﬁt to workers in the import-competing sector from the opening up of
trade is strictly less than the beneﬁt to export sector workers, whether the opening is delayed
or not.
(iii) A one-period delay hurts export-sector workers and beneﬁts import-competing work-
ers.
(iv) A suﬃciently long delay before the opening of trade will make workers in both sectors
net beneﬁciaries [net losers] from trade.
Proposition 10. Suppose that, at the autarchy relative price p , steady state relative supply
of the export good is unchanged at the margin by an increase in labor supply. Then there is
an open interval containing p such that if the world price p is in that interval:
(i) Workers in the export sector as of period 0 will beneﬁt from the opening of trade,
whether it is delayed or not.
(ii) Workers in the import-competing sector as of period 0 will be hurt by the opening of
trade, whether it is delayed or not.
32
(iii) A one-period delay hurts export-sector workers and beneﬁts import-competing work-
ers.
Some examples. Four simple special cases illustrate the diﬀerent eﬀects delay can have.
For all examples, assume that p < p , so that Y is the import good, and that the diﬀerence
between the world price and the domestic autarchy price is small enough that the propositions
i
∂v0
apply. (As a result, sector-i workers beneﬁt from trade if |
∂p p =p
< 0.)
(i) Ricardian technology. If Qi (Li i i i
t ) ≡ A Lt for positive constants A , i = X, Y , an increase
in total labor supply will result in an equiproportionate increase in output of both sectors.
(Note that wX and wY are both independent of labor supplies, and so, by (8), µ is unchanged
by a change in labor supply.) Therefore, B∞ = 0, and in this case delay has no role: export
sector workers beneﬁt from trade and import-competing workers are hurt by it, regardless
of t∗ .10
(ii) Inelastic import-competing labor demand. Suppose that both X and Y are produced by
a sector-speciﬁc asset in ﬁxed supply (K X and K Y respectively) together with labor. Suppose
further that the production function for Y is Leontieﬀ: QY (K Y , LY ) = min{K Y , LY }, and
that the production function for X is CES: QX (LX , K X ) = ((LX )ρ + (K X )ρ )1/ρ , with ρ < 1,
ρ = 0 (implying an elasticity of substitution equal to 1/(1−ρ)). In this case, QY Y
t1 = Qt2 = 0∀t.
Therefore, Bt , Dt ≤ 0∀t, with strict inequality for t > 0, so both groups of worker will beneﬁt
from trade with or without delay. The interpretation is that the increased labor demand
in the export sector forces wages up in the import-competing sector because of its inelastic
labor demand, beneﬁtting all workers.11
(iii) Inelastic export-sector labor demand. Now, reverse the production functions for the
two sectors. In this case, QX X
t1 = Qt2 = 0∀t. Therefore, Bt , Dt ≥ 0∀t, with strict inequality
for t > 0, so both groups of worker will be hurt by trade with or without delay. The
interpretation is that the reduced labor demand in the import-competing sector forces wages
10
Recall that these are ‘local’ results, in the sense that they depend on p being close to p . On the other
hand, where p is substantially below p , workers in both industries can beneﬁt from trade (even though the
real wage for sector Y falls both in the short run and the long run), as illustrated in Footnote 8.
11
This is analogous to a parallel result on the role of elasticities of substitution in Mussa’s (1974) static
Ricardo-Viner model.
33
down in the export sector because of its inelastic labor demand, hurting all workers. In this
example, all of the gains from trade are captured by the owners of the ﬁxed factors.
(iv) An example in which delay tips the scales in favor of trade. Now consider a general
i i i
CES speciﬁcation, in which Qi (Li , K i ) = ((Li )ρ + (K i )ρ )1/ρ , i = X, Y , with ρi < 1, ρi = 0.
Example (ii) shows that with any ﬁnite ρX and with ρY suﬃciently large and negative,
import-competing workers strictly beneﬁt from an unannounced liberalization. Example (iii)
shows that with any ﬁnite ρY and with ρX suﬃciently large and negative, import-competing
workers are strictly hurt by an unannounced liberalization. Choose any two such parameter
pairs, and connect them with a curve in (ρX , ρY ) space. There must be a point on the
curve at which import-competing workers are indiﬀerent between a sudden trade opening
and the autarchic steady state. For this parameter pair, by part (ii) of Proposition 9, export
workers are strict beneﬁciaries of sudden trade. In addition, export workers will remain net
beneﬁciaries of trade if the opening is delayed (B∞ must be negative, because in order for
Y workers to be indiﬀerent, D∞ must be negative; but then Bt < 0∀t). Furthermore, since
D0 > 0 and Dt is decreasing in t, any delay will make the Y workers strict beneﬁciaries of
trade. Thus, an immediate liberalization would beneﬁt X workers but not Y workers, but a
delayed liberalization would strictly beneﬁt both classes of worker. (A slight perturbation of
the (ρX , ρY ) pair could then make the Y workers stictly hurt by immediate trade opening,
making them strict beneﬁciaries of delayed opening.)
The interpretation of this result is as follows. An unexpected trade opening pushes down
the real wages of Y workers immediately, while pushing up the real wages of X workers.
The Y workers move only gradually to take advantage of the higher X wages because of
the moving costs. However, if the trade opening is announced in advance, Y workers who
happen to have low moving costs at the moment begin moving to X in anticipation. This
makes Y workers more scarce, pushing up wages for those who do not move (while pushing
X wages down), even though no change in output prices has yet occurred. If the elasticity
of labor demand in the Y sector is suﬃciently low, this anticipatory wage increase is the
dominant eﬀect.
(v) An example in which delay tips the scales against trade. Exactly the same logic as in
34
(iv) can be used to construct a case in which export sector workers are indiﬀerent between
immediate trade and the autarchic steady state. In this case, by part (ii) of Proposition 9,
import-competing workers are strictly hurt by trade whether it is delayed or not (since D∞
must be positive, because in order for X workers to be indiﬀerent, B∞ must be positive; but
then Dt > 0∀t). Furthermore, since B0 < 0 and Bt is increasing in t, any delay will make
the X workers strictly harmed by trade. Thus, in the event of an immediate liberalization,
workers would be divided over trade, while delay would unite all workers in their opposition.
(Again, a slight perturbation of the example can make X workers strict beneﬁciaries of
immediate trade.)
The interpretation of this example is as follows. From the point of view of the X workers,
immediate trade has three eﬀects. It lowers real wages in the Y sector; it provides an
immediate increase in real wages in the X sector; and it pushes a certain number of Y
workers out of the Y sector and into the X sector over time, pushing X wages down. If labor
demand in X is suﬃciently inelastic, the X wage will be lower in the long run than it was
under autarchy. Thus, in this example, for an X worker, the beneﬁt from free trade is short-
lived, and must be weighed against a long-run cost. On the other hand, if the trade opening
is announced in advance, Y workers begin moving into X right away, pushing X wages down
even before output prices change. Thus, X workers lose the beneﬁt of the short-run wage
increase that they would have enjoyed under unanticipated trade, and jump immediately to
the long-run cost of increased competition from former Y workers.
These examples serve to illustrate the range of possible outcomes. Except in the knife-
edge case with B∞ = D∞ = 0, delay tends to make workers unanimous in their stance
toward trade, but whether it is a positive or negative stance depends on the relative respon-
siveness of labor demand in the two sectors. Of course, whether delay in any given real
world liberalization event is likely to turn workers into beneﬁciaries or victims of trade is an
c, Chaudhuri and McLaren (2008) simulate equilibria of this model
empirical matter. Artu¸
for a range of parameters in order to illustrate these points and to explore the magnitudes
of delay required to achieve unanimity of workers. For middling values of the elasticity of
substitution, unanimity can require decades of delay.
35
6 Some Evidence on Anticipatory Eﬀects
Some of the more surprising results of the model concern the economy’s response to an
anticipated future trade shock. The model predicts that after announcement of a future
negative trade shock to a given sector, workers begin to move out of the sector right away,
resulting in both an anticipatory movement in trade ﬂows and anticipatory movements in
wages, the latter favoring workers in the shrinking sector. We note here that these features
are consistent with empirical evidence in quite a number of cases.12 Note that the present
paper is an attempt to clarify the theory, not an empirical study. Our aim in this section is to
provide examples of phenomena that plausibly could be explained by anticipatory movements
such as in the model. We do not claim to have a proof that any one episode is caused by
the underlying mechanism, which is a diﬃcult enquiry for another day. But it is instructive
to observe that a number of cases seem to ﬁt the story of the model quite well.
One place to look for evidence of anticipatory eﬀects is in accession of a country to a
trade block such as the European Union (EU), since the process of accession tends to take
several years even when the outcome is not seriously in doubt. Freund and McLaren (1999)
demonstrated that trade ﬂows in European countries joining the European Union began
to reorient themselves toward the EU on average four years before the date of accession,
suggesting reallocation of resources in anticipation of the new trade regime. This is consistent
with the behavior of our model, even though the data in that study show only trade ﬂows
and do not isolate movements of labor per se.
Another type of example is a change in trade policy uncertainty. Often economic agents
expect a future change in policy with a positive probability, and an agreement between
countries can prevent those future policy changes, thus causing anticipatory reallocation
12
The anticipation rents that go to the workers in the sector that receives the adverse shock result from
our assumption of immobile capital. If capital was perfectly mobile, both intersectorally and internationally,
then the marginal value product of capital would be ﬁxed by the required world rate of return, and so before
the tariﬀ change, the wage in each sector would also be ﬁxed. With the removal of the tariﬀ, the import-
competing sector wage would jump to its new permanent level and stay there. Thus, with perfect capital
mobility, there would be no anticipatory movement of wages, but only of workers. An extension of interest
would be to include quasi-ﬁxed capital, which would allow for the possibility of an anticipatory movement
of capital, and and even anticipatory movement of wages in the opposite direction from our model. That
would be a far more complicated model, of course.
36
ao (2012) examine the 1986 accession of Portugal to the
of resources. Handley and Lim˜
European Community (EC), and show that much entry of ﬁrms into EC export markets
can be explained by the fact that it eliminated uncertainty about future policy changes.13
ao (2013) similarly show that the entry of Chinese ﬁrms into the US export
Handley and Lim˜
market following the accession of China to the WTO is largely explained by the fact that
it eliminated uncertainty about future US trade policy toward China. Handley (2014) ﬁnds
similar eﬀects on exports to Australia, measuring policy uncertainty by the gap between
WTO tariﬀ bindings and applied tariﬀs. These are examples that show a large reallocation
of resources due to a change in the probability distribution of future policy, but these studies
do not look at workers or wages, since the focus is on ﬁrm behavior and in particular the
extensive margin. Pierce and Schott (2012) do look at workers, showing that the reduction
of trade policy uncertainty following China’s entry into the WTO explains a large movement
of workers out of US manufacturing. But the paper does not look at the behavior of workers’
wages, or at workers’ welfare.
One paper that looks at wages and movement of workers in the context of anticipated
policy changes is Hakobyan and McLaren (2010), who study the North American Free Trade
Agreement (NAFTA). NAFTA included schedules for the elimination of all tariﬀs between
signatories, with much variation across industries in the speed of elimination. An industry
with a 10% initial tariﬀ but only a 2% reduction over the data period is an industry with
an anticipated further reduction of 8%; while an industry with a 2% initial tariﬀ and a
2% reduction over the data period is an industry with no further anticipated reduction. It
turns out that industries with a larger anticipated future tariﬀ reduction saw reductions in
blue-collar employment but increases in wages relative to industries with smaller anticipated
future reductions, controlling for a range of personal characteristics including age and human
capital. Although there may be other explanations, this ﬁnding is directly interpretable as
an anticipation rent as described in our model.
The end of the Multiﬁber Arrangement provides a ﬁnal example. The system of quotas
13
For example, Portugal had a preferential agreement with Spain, which was understood to be temporary
but it was not clear where it would lead. Joining the EC together eliminated any uncertainty about tariﬀs
between the two countries.
37
protecting industrial-country textile and apparel industries from low-income country imports
was dismantled under an agreement worked out as part of the Uruguay Round of the GATT
that entered into force in January 1995. Relaxation of the quota was to be phased in
over 10 years, with the last quotas eliminated completely in January 2005, but most of the
liberalization was deferred until the end of the period. Thus, the period 1995 to 2005 was
largely a period of anticipation of a trade liberalization.14
We examine this episode with data on US manufacturing workers who were between 22
and 64 years old from the IPUMS-CPS database between 1981 and 2010. (See King et
al. (2010)). The sample size is 267,281 workers over all years. We observe workers’ wage,
industry, occupation, age, gender, education, and relation to the head of household. Figure
6 shows the evolution of textile and apparel wages relative to other manufacturing wages (in
other words, the average textile-and-apparel sector wage divided by the average non-textile-
and-apparel manufacturing wage). The vertical lines in the ﬁgure mark the anticipation
period. Clearly, wages in the sector increased sharply during the anticipation period and fell
sharply afterward, as predicted by the model. Figure 7 shows the evolution of the share of
textile and apparel employment within manufacturing. Clearly, the anticipation period saw
a rapid movement of workers out of textiles and apparel, from around 8% to below 4%, and
this movement was not simply the continuation of a pre-existing trend.
A natural question is whether or not these trends are driven by compositional eﬀects. For
example, it could be that the least-educated workers leave the sector ﬁrst, driving up average
wages without raising the wage for any one worker. This would be quite diﬀerent from the
anticipation rent we have discussed. To answer this question, we purge the compositional
eﬀects from wages and labor allocations using the following Mincer wage and logit regressions:
n i,n
log wt = βi,t Dt + Xtn Bt + n
1,t , (16)
n
yt = γ0,t + Xtn Γt + n
2,t , (17)
14
Considerable background on the end of the Multiﬁber Arrangement together with some empirical analysis
can be found in Evans and Harrigan (2005) and Harrigan and Barrows (2009).
38
n n
where wt is the wage of individual n at time t and yt is the dependent variable for the logit
n i,n
regression implying yt > 0 if the individual is employed in textiles or apparel; Dt is the
dummy for the sector ﬁxed eﬀect;15 Xtn is the vector of individual characteristics such as
blue collar, female, household-head, age, age squared and education level (secondary school,
n n
high school or college); βi,t , γ0,t , Γt and Bt are regression coeﬃcients, and 1,t and 2,t are
error terms.
We ﬁrst run these regressions separately for each year, and plot βtextile,t to show the
evolution of wages and plot γ0,t to show the evolution of the ratio of textile workers, net
of any eﬀect caused by composition of individual characteristics or general manufacturing
trends. Figures 8 and 9 show that after purging the data of composition eﬀects in this way,
we can still see the negative trend in textile and apparel employment and the positive trend
in the sector’s wages during the period 1995-2004 as predicted by the model.
Finally, we pool the data across all years to test the signiﬁcance of within-period time
trends using the following regressions:
n
log wt = β0 Dt + βj ∆j Dtextile,n + tαj ∆j Dtextile,n + Xtn B + n
3,t , (18)
n
yt = γj ∆j + tθj ∆j + Xtn Γ + n
4,t , (19)
n n
where wt is the wage of individual n at time t; yt is the dependent variable for the logistic
n
regression implying yt > 0 if the individual is employed in textiles or apparel; Dt is the
dummy for time ﬁxed eﬀect; Dtextile,n is the textile dummy which is equal to one if the
individual is employed in the textile/apparel sector; ∆j is the dummy for period ﬁxed eﬀect:
∆1 = 1 if 1981 ≤ t ≤ 1994, ∆2 = 1 if 1995 ≤ t ≤ 2004, and ∆3 = 1 if 2005 ≤ t ≤ 2010; Xtn
is the vector of individual characteristics (such as blue collar, female, household-head, age,
age squared and education level, i.e. secondary school, high school or college); β0 , βj , αj , γj ,
n n
θj , Γ and B are regression coeﬃcients, and 3,t and 3,t are error terms.
15
The ten manufacturing sectors are: 1. Furniture and wood; 2. Metals and minerals; 3. Machinery; 4.
Vehicles, etc.; 5. Miscellaneous Durables; 6. Food; 7. Textiles and apparel; 8. Paper; 9. Chemicals, and 10.
Miscellaneous non-durables.
39
We are seeking a trend rather than a one-time change, so we use intercept and slope
variables and test if the changes in wages and labor allocation are statistically signiﬁcant.
Since we have time dummies for each year of the data, the time trends we estimate are trends
relative to other manufacturing industries. We also cluster the errors by 1-digit manufac-
turing sector to allow for correlated shocks within each sector. In the wage regression, βj
is the intercept and αj is the slope coeﬃcient for the time-trend for period j . In the logit
regression, γj is the intercept and θj is the slope coeﬃcient. In particular, we expect α2 to
be positive and signiﬁcant, indicated a rise in wages between the announcement and shock,
and θ2 to be negative, indicating an exodus of workers between the announcement and the
shock.
Table 1: Mincer and Logit Regression Results
Wage Regression Logit Regression
D 80 94 -0.293 (-39.03) -2.879 (-38.96)
D 95 04 -0.329 (-31.59) -2.806 (-37.98)
D 05 09 -0.280 (-23.86) -3.493 (-47.88)
D 80 94*(t-1980) 0.004 (3.81) 0.006 (2.07)
D 95 04*(t-1995) 0.015 (5.29) -0.077 (-13.12)
D 05 09*(t-2005) 0.004 (0.57) -0.039 (-4.43)
Blue col 0.347 (47.34) -0.449 (-21.49)
Female -0.359 (-58.54) 1.203 (41.74)
Not head -0.113 (-9.54) 0.271 (11.84)
Max high 0.174 (30.17) -0.800 (-34.88)
Max college 0.429 (26.81) -0.637 (-20.33)
Age 0.036 (46.92) -0.008 (-1.39)
Age Squared -0.001 (-46.67) 0.000 (3.39)
(t-statistics are in parentheses)
The results are shown in Table 1. The slope coeﬃcient of the wage regression during the
anticipation period, α2 , is equal to 0.015 with a t-statistic of 5.39. This shows that there was
a strong positive trend in wages between 1995 and 2004. We also ﬁnd some evidence of a mild
increase in wages in the previous years but the slope is much smaller, 0.004, and statistically
weaker. Additionally, we ﬁnd that the slope coeﬃcient of the logit regression during the
40
same period, θ2 , is equal to −0.077 with a t-statistic equal to −13.12, consistent with the
predictions of the model. We ﬁnd that the slope coeﬃcients before the announcement and
after the shock are also statistically signiﬁcant but they have much smaller magnitude and t-
statistics. Therefore the trends in wages and labor allocations in textiles are much stronger
(both statistically and in terms of magnitude) between the announcement and the shock
as predicted by the theoretical model, consistent with the visual evidence we provided via
ﬁgures.
7 Conclusion
We have studied a simple trade model with costly labor mobilty and idiosyncratic moving-
cost shocks. The model is shown to have non-trivial dynamics, including gradual adjustment
and anticipatory eﬀects, resulting solely from ﬁxed moving costs plus time-varying idiosyn-
cratic moving cost shocks to individual workers. A key conclusion is that the steady state
does not resemble a model with frictionless labor mobility. Speciﬁcally, intersectoral wage
diﬀerentials persist permanently, making workers more expensive in larger sectors. In addi-
tion, trade liberalization lowers the long-run wage in the import-competing sector relative
to the export sector, and delaying liberalization reduces the diﬀerence between the eﬀect of
liberalization on export-sector and on import-competing workers.
Thus, the trade economist’s habit of assuming that a frictionless model will be a good
predictor of long-run trade eﬀects is called into question.
41
Appendix
Lemma 2. Assume that the value function is twice continuously diﬀerentiable in
(LX Y
t , Lt ). Then:
(i) The functions LX Y X Y
t and Lt are diﬀerentiable in (L0 , L0 ).
(ii) The derivatives LX X X Y X Y Y X Y X
t1 ≡ ∂Lt (L0 , L0 )/∂L0 and Lt1 ≡ ∂Lt (L0 , L0 )/∂L0 are all non-
negative, and LX Y X
t1 + Lt1 = 1∀t > 0. Further, Lt1 is decreasing in t.
(iii) The derivatives LX Y Y X
t2 and Lt2 are all non-negative, and Lt2 + Lt2 = 1∀t > 0. Further,
LX
t2 is increasing in t.
(iv) Li i
t1 → L∞1 as t → ∞, for i = X, Y .
Proof. First, note that since µX Y X Y
t = −µt − 2C , once µt has been speciﬁed, µt can be
X
computed from it, and then LX Y X Y
t+1 and Lt+1 can be computed from µt , Lt and Lt by (7).
X
We can thus write LX X Y
t+1 as a function of µt , conditional on Lt and Lt . Note that it is
a strictly decreasing function, allowing us to deﬁne its inverse: µX (LX X Y
t+1 ; Lt , Lt ), which is
clearly diﬀerentiable.
Part (i) can be seen as follows. We can write the planner’s ﬁrst-order condition (13) in
this form:.
µX (LX X Y X Y X Y
t+1 ; Lt , Lt ) + C = β [V1 (Lt+1 , Lt+1 ) − V2 (Lt+1 , Lt+1 )], or
µX (LX X X Y X
t+1 ; Lt , L0 + L0 − Lt ) + C
= β [V1 (LX X Y X X X Y X
t+1 , L0 + L0 − Lt+1 ) − V2 (Lt+1 , L0 + L0 − Lt+1 )].
The diﬀerentiability of LX
1 can be inferred by applying the Implicit Function Theorem
to this equation for t = 0. Given the diﬀerentiability of LX X
t , the diﬀerentiability of Lt +1 can
be inferred by applying the same logic to the equation for t = t . Thus, the result follows by
induction.
Parts (ii) and (iii) simply follow from our results on the dynamics of adjustment, plus
the requirement that LX Y X Y
t + Lt = L0 + L0 ∀t.
42
To see (iv), diﬀerentiate the planner’s ﬁrst-order condition with respect to LX
0 to get:
µX X X X X Y X Y X Y
1 Lt+1,1 + µ2 Lt1 + µ3 Lt1 = β [V11 Lt+1,1 + V12 Lt+1,1 − V21 Lt+1,1 − V22 Lt+1,1 ], or
(µX X X X X X
2 − µ3 )Lt1 = (β [(V11 − V12 − V21 + V22 )] − µ1 )Lt+1,1 + β (V12 − V22 ) − µ3 ,
where the derivatives of µX X X Y
1 are evaluated at (Lt+1 ; Lt , Lt ) and the derivatives of the value
function are evaluated at (LX Y X
t+1 , Lt+1 ). Now, since Lt1 is positive but decreasing in t, it must
take a limit, say, η . Taking limits of this equation as t → ∞, we ﬁnd:
β (V12 − V22 ) + µX3
η= ,
µX
2 − µ X
3 − β [(V 11 − V12 − V21 + V22 )
where the derivatives of µX are evaluated at (LX X Y
∞ ; L∞ , L∞ ) and the derivatives of the value
function are evaluated at (LX Y
∞ , L∞ ). Now, noting that the steady-state values must satisfy
µX (LX X Y X Y X Y
∞ ; L∞ , L∞ ) + C = β [V1 (L∞ , L∞ ) − V2 (L∞ , L∞ )],
we can diﬀerentiate this, and, using LX Y X Y X Y
∞ + L∞ = L0 + L0 (and thus L∞1 + L∞1 = 1), solve
for LX X
∞1 . But this then yields L∞1 = η .
43
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48
X
L t+1
o
45
0
L 1 L 2 Steady
X X X X
L0 Lt
State.
Figure 3: The Transition Function.
X Transition function: Model N.
L t+1
Transition function: Time t*-1.
Transition function: Time t*-2.
Transition function: Model I.
o
45
0 X X
L0L1 L2
X
New X
Lt
Old Steady
Steady State.
State.
Figure 4: Anticipated Trade Liberalization..
Figure 5: Derivation of the effect of delay.