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Barriers to Trade and Union
Wage Dynamics
E llen R. R issm a n
W o rk in g P a p e rs S e rie s
M a c ro e c o n o m ic Issu e s
R e s e a rc h D e p a rtm e n t
F e d e ra l R e s e rv e B a n k o f C h ic a g o
D e c e m b e r 19 92 (W P -9 2 -2 2 )
FEDERAL RESERVE B A N K
OF CHICAGO
Barriers to Trade and U n io n W age D y n a m ics
E llen R . R issm an 1
Federal Reserve Bank of Chicago
D ecem ber 3, 1992
1The author would like to thank George Neumann, Dale Mortensen, Bruce Petersen, and
the participants of the Northwestern University Labor Workshop for their helpful comments
and suggestions. The views expressed here are not necessarily those of the Federal Reserve
Bank of Chicago or the Federal Reserve System.
A b stra c t
The U.S. union wage premium expanded over the seventies and collapsed in the
eighties as imports rose. This paper reconciles these facts by examining the effect
of imports on unionized labor markets where feedback from the union wage rate to
imports is introduced. Higher union wages make domestic firms less competitive and
encourage imports. The union optimally balances short term wage gains against long
term employment losses that results from its wage policy. Reductions in barriers to
trade decrease the union wage gap in the long run. In the short term union relative
wages may actually increase.
JEL
Classification Numbers: J51, J58, F13
1
Introd u ction
From the late sixties throughout the seventies and early eighties, imports made sig
nificant advances into U.S. markets that were previously dominated by domestic pro
ducers. For example, imports accounted for 13.7% of domestic U.S. steel sales in
1969 but by 1982 had risen to almost 22% of the market. Even more striking is the
decline in market share of domestic automobile manufacturers from 88.5% in 1969
to 72.1% in 1982. This phenomenon of rising imports was not uniformly distributed
across industries. As noted by Domowitz, Hubbard, and Petersen (1986), markets
that are characterized by high concentration ratios appear to be the ones most af
fected. Because unions are concentrated in industries having high price-cost margins,
it would appear that unionized labor markets have been disproportionately affected
by the increase in foreign competition.
How have unions adjusted to the growth in imports? Union density reached its
peak in 1954 with over 36% of nonagricultural employment and around 27% of the
labor force claiming membership. Unionization has fallen in almost every year since
1954. By 1980 the proportion of the labor force that is unionized declined to less
than 22% and has decreased further since then.1
Union wage patterns seem also to have changed significantly over this time period.
Between 1958 and the late sixties or early seventies the union wage premium drifted
downwards. However, at about the same time that imports started their expansion,
the union/nonunion wage differential began to rise and continued to grow until about
1983. Johnson (1982) estimated the union wage premium increased roughly 14%
between 1969 and 1979 while Lewis (1986) found the union wage differential rose
about 7% over the same period. Additional evidence of an increasing union wage gap
1For more recent data on union affiliation, see the Bureau of Labor Statistics’January issues of
EmploymentandEarnings.
1
in the seventies is found in Ashenfelter (1979), Moore and Raisian (1983), Freeman
(1986), and Linneman and Wachter (1986).
The union wage premium appears to have peaked in 1983 and dropped precipi
tously thereafter according to the Bureau of Labor Statistics’ quarterly Employment
Cost Index. This observation holds true not only for the aggregate economy but also
for manufacturing and nonmanufacturing industries separately. By the third quarter
of 1988, the aggregate union wage gap was 7% lower than it was in 1982 and even
lower than it had been in 1976, the first year for which data are available. The manu
facturing and nonmanufacturing union wage premium fell 5.3% and 7.6% respectively
over this same period.
In models of union wage and employment determination such as analyzed in Dun
lop (1944) and Cartter (1959), the firm unilaterally sets the level of employment so
as to maximize profit each period given the union wage rate. The union in turn is
assumed to know the parameters of the firm’s problem and selects the point on the
derived demand schedule for labor that maximizes some objective function having
employment and wages as its arguments. An exogenous increase in imports causes
an inward shift in the labor demand curve that the union faces. Assuming that both
wages and employment are “normal” goods in the union’s preference ordering, the
effect of rising imports would be to
lower
both the optimal union wage rate and level
of employment. By demanding additional wage increases, as seems to have occurred
in the seventies and early eighties, the decline in union employment is reinforced as
employment decreases along the negatively sloped labor demand schedule.
Why unions should choose to raise wages when confronted with apparently per
manent changes in their employment opportunities is puzzling. One possibility is that
unions are not able to adjust quickly to a changing economic environment because
of inertia introduced via long term contracting. Such an explanation is better suited
to describing the behavior of union wages over the business cycle where disturbances
2
are generally of short duration. Since union wages are typically negotiated in three
year intervals, it is doubtful that a period of expansion in the wage premium lasting
over ten years can be explained by adjustment lags.
An alternative explanation of the rise in the union/nonunion wage differential
relies on an insider-outsider model of union membership such as found in Grossman
(1983) and Lindbeck and Snower (1988). As employment opportunities in unionized
industries are eroded by imports, the composition of union membership shifts towards
those individuals that prefer higher wage premiums over employment. Permanent
shifts in labor demand are accompanied by layoffs of less senior and, therefore, lower
paid workers, thereby causing the optimal wage rate as determined by the median
worker to rise. This changing composition hypothesis resolves the increase in the
union wage gap and concurrent decline in union membership during the seventies and
early eighties. However, it fails to explain why union membership declined steadily
between 1958 and 1968 while the union wage differential exhibited a downward drift.
Furthermore, since the theory relies on adverse shifts in employment opportunities,
it does not address why the union wage premium rose sharply in nonmanufacturing
as well as manufacturing industries nor does it explain the more recent decline in the
union/nonunion wage differential.
Similar to the insider-outsider model in its reliance on compositional shifts, Ed
wards and Swaim (1986) suggested a Darwinian survival-of-the-fittest explanation for
the increase in the union wage premium over the seventies. According to their hypoth
esis, competitive pressures in traditionally unionized product markets caused many
weaker unions to disappear, leaving only those that were most capable of sustaining
large wage increases. The average union wage rate rose as the industrial composition
of unionism changed. The same criticisms apply here as in the previous explanation.
In contrast to other theories, Lawrence and Lawrence (1985) sought to explain
the expansion in the union wage premium by considering how capital stocks adjust to
3
permanent declines in product demand. They argued that imports reduce the wage
elasticity of demand for labor in the short run as substitution possibilities between
labor and existing capital stocks are reduced. In terms of the labor demand curve
equilibrium model, imports do not simply shift the labor demand curve laterally, but
also make it steeper in the short run. Over time, however, capital stocks adjust so
that the labor demand curve eventually becomes more elastic.
The argument articulated by Lawrence and Lawrence relies upon very specific
assumptions about the effect of permanent changes in product demand upon the
elasticity of substitution between labor and capital. The theory explains the rise
and subsequent fall of relative union wages. However, the model fails to explain why
union wage differentials rose in nonmanufacturing industries where product demand
was actually increasing over this time period. It is difficult to believe that the same
putty-clay technology affecting capital movements that drives their model is at work
in the nonmanufacturing sector as well.
Typically, imports are modeled as being exogenously determined. For example,
in Lawrence and Lawrence, imports are simply treated as a shift parameter affecting
the labor demand schedule. The direction of causality runs from imports to wages
and employment. Alternatively, it has been suggested by Abernathy, Harbour, and
Henn (1981) and Gomez-Ibanez and Harrison (1982) among others that higher wage
premiums cause a reduction in the international competitiveness of domestic firms.
In this case the direction of causality runs from wages to imports. According to
this hypothesis, import penetration is most likely to occur in industries that are at
a competitive disadvantage. Because unions raise wages above perfectly competitive
levels, domestic firms are not able to compete with lower cost foreign producers. The
greater is the degree of wage distortion introduced by unions, the larger is the cost
disadvantage of domestic firms in the market. Imports will be attracted precisely to
those industries having high unionization rates and, thus, high wages.
4
The model analyzed in the following sections explicitly introduces this notion of
feedback from union wages to imports. As a result, the choice of the union wage
rate affects the adjustment paths of both imports and employment. The union can
attem pt to stem the flow of imports by reducing its wage demands and boosting the
competitive position of domestic producers. Although the union maintains employ
ment opportunities by following such a policy, it does so at the cost of foregoing
above-competitive wages for its members. Alternatively, the union may ignore the
long run consequences of its wage policy and continue to demand the monopoly wage.
As a result of its high wage policy, the union encourages future imports that continue
to erode the union’s employment opportunities. If this continues indefinitely, the
union essentially prices itself out of the market. The union is likely to follow a strat
egy somewhere in between the two extremes. It is this trade-off between wages in the
short run and market share in the long run that determines the optimal union wage
path and the paths of imports and employment to the new steady state.2
The remainder of this paper consists of three sections. In Section 2 a static model
is presented in which there is no feedback from union wages to imports. Imports
affect only the product price. The model is expanded in Section 3 to analyze the case
of feedback from union wages to imports. Imports are assumed to grow at a rate
proportional to the difference between union wages and the wage rate that would
make foreign producers indifferent to entering the product market.3 An example is
given in which labor demand is assumed to be linear and workers are risk neutral
which shows that the optimal immediate response of a union faced with a reduction
2The model is applied here specifically to the case of imports because the rise in imports and
increase in the union/nonunion wage differential that have been observed are coincident. However,
the theory has much broader applicability. It isrelevant to any occasion in which the union’sproduct
market is threatened by increased nonunion competition, e.g. deregulation.
3There are close parallels between the model as formulated here and the limit pricing model
of Gaskins (1971). I have consciously attempted to use similar notation and terminology so as to
emphasize the similarities.
5
in trade barriers is to
in cr ea s e
the wage rate. Over time the wage rate declines to a
lower steady state level. Conclusions are contained in Section 4.
2
T he S tatic M odel
In this section a simple model of union wage and employment determination is an
alyzed in which imports are introduced via their effect upon product supply. The
model is a variant of the labor demand curve equilibrium models of Dunlop (1944)
and Cartter (1959). The union unilaterally sets the wage rate while the firm chooses
the level of employment so as to maximize profits given the wage. The union knows
the parameters of the firm’s problem and effectively selects the point on the firm’s
labor demand schedule that maximizes the union’s utility.4
The industry faces a downward sloping demand curve for the good, although an
individual firm cannot singly affect the product price. Total supply of the good to
the industry is the sum of domestic supply and foreign supply. Imports are assumed
to be an increasing function of the product price. In equilibrium the product price is
determined by the intersection of the total supply and demand curves. The effect of
an exogenous increase in imports, denoted by the parameter
m
is straightforward. As
the supply of imports rises, the total supply curve shifts outwards and the equilibrium
price falls. Let
p(m )
of imports so that
be the nonnegative product price which is a decreasing function
p'{ni) <
0. Furthermore, assume that
p"(m)
< 0.
The firm acts as both a price- and wage-taker, choosing the level of employment
so as to maximize profits given the level of imports. The profits of the firm are given
4It is well-known that the solution to the labor demand curve equilibrium model is not Pareto
optimal. Alternative models of the bargaining process such as the contract curve equilibrium model
proposed by McDonald and Solow (1981) do not suffer from this criticism. However, without a
specific theory as to how the equilibrium wage and employment combination is selected the model
has little predictive content.
6
by
p(m )f(L) —wL
where
of employment, and
w
f(L
) is the production function of the firm,
L
is the level
is the wage rate. The usual assumptions are made about the
production function, i.e.
f'(L )
> 0 and
f"(L)
< 0. Let
L
=
L (w ,m )
be the labor
demand schedule that solves the firm’s maximization problem. Labor demand is a
decreasing function of the union wage rate. The effect of an increase in imports is
found by solving for the comparative statics:
L m( w , m )
d L ( w , m)
dm
(1)
p{m )f"{L)
which is evaluated at the optimum. Because the product price is decreasing in im
ports, an exogenous increase in imports causes the labor demand curve to shift in
wards.
Following Lazear (1983), the union’s problem is to maximize the expected utility
of the representative union member subject to the constraint that the wage and
employment combination are on the firm’s labor demand curve. There are
members,
L
of whom are employed in unionized jobs and receive a wage of
associated utility tt(tv). The remaining
wa
M —L
union
M
w
with
members receive an alternative wage
which can be thought of as the nonunion wage rate or the monetary equivalent of
the opportunity cost of a union job. It is assumed that
u'(x) >
0 and
u"{x)
< 0. For
the purposes of this model, the existence of feedback from the union wage rate to the
alternative wage is unimportant. For this reason
wa
is assumed to be a constant.
The union’s problem is equivalent to solving:
max [u(u>) —u ( w a) ] L ( w ,
7
m).
(2)
The optim al wage rate, to*, is the solution to the following first order condition:
u'(w*)L(w*, m )
+ [u(io*) —u { w a) \ L w{ w *, m) = 0.
(3)
The effect of an increase in imports on the union wage rate is, in general, ambiguous,
depending upon the relative income and substitution effects. An increase in the
level of imports not only shifts the labor demand curve inwards (income effect) but
also changes the slope of the labor demand curve (substitution effect). The effect of
imports on the optimal union wage rate is given by:
d w* _
dm
where
+ (u ( w ) - u ( w a) ) L wm]
2 u ' { w ) L w + (u(u>) —u ( w a) ) L ww]
—[ u ' { w ) L m
[u"(w)L +
the expression is evaluated at to*. Assuming that
L ww( i v , m )
f"'{L)
< 0, it follows that
< 0, i.e. the labor demand curve is concave in the union wage rate
given the level of imports. The second order condition for the union’s maximization
problem holds so that the denominator of Equation (4) is unambiguously negative.
The numerator of the expression is the sum of the income and substitution effects
respectively. The first term in the expression, i.e. the income effect, is negative. It can
be shown that the same restriction on the third derivative of the production function
that guarantees concavity of the labor demand schedule, namely that
also guarantees that
L wrn( w , m )
f" '(L)
< 0
< 0 so that the substitution effect is unambiguously
negative. An increase in imports not only shifts the labor demand schedule inwards
but also makes it flatter, thereby ensuring that an increase in imports reduces the
equilibrium union wage rate.
To summarize, in the static model an exogenous increase in imports shifts the labor
demand curve constraining the union inwards. The new optimum is characterized by
lower wages provided that the income effect dominates the substitution effect. In
8
order to generate a rise in the union/nonunion wage differential such as observed
in the U.S. during the seventies, the substitution effect not only has to be positive
but also dominate the income effect. Given the structure of the model, this would
entail some assumptions that would produce some nonconcavity in the labor demand
schedule.
3
Feedback from U nion W ages to Im ports
In this section feedback from union wages to imports is introduced. Unions raise
wages above the level that would occur in a competitive equilibrium. The distortion
introduced places domestic unionized firms at a competitive disadvantage since pro
duction costs are increased. Foreign producers, who face lower labor costs, ceteris
paribus, are drawn to the domestic product market as opportunities exist for extra
normal profits.5 The effect of introducing such feedback is to link current union wage
policy with future employment opportunities, thereby adding a dynamic element to
the union’s optimization problem.
In the analysis that follows it is assumed that the union can affect the rate of
growth of imports through its wage policy. By charging a lower wage rate, domestic
producers are better able to compete with lower cost foreign producers. Entry into
the domestic market by foreigners is less attractive since profit opportunities are
reduced. Alternatively, by increasing wage demands the union actually encourages
more rapid growth of imports as domestic production costs rise relative to that of
foreign producers. For the purposes of modeling union wage dynamics, it is important
that international wage differentials be not instantaneously arbitraged. Rather, the
5The automobile industry provides a ready illustration. Abernathy, Harbour, and Henn (1981)
estimated that lower Japanese auto costs were due primarily to differences in compensation between
U.S. and Japanese manufacturers. These lower costs were translated into lower prices and higher
profit margins for the Japanese who rapidly increased their market share.
9
process of setting up deliveries in a foreign country is a slow one. It takes time to
meet foreign regulatory restrictions, increase production, transport goods, set up sales
networks, and market the product. It is plausible that the larger are the perceived
gains of the foreign producer, the more quickly he will respond.
Let
m=
k(w
—w )
(5)
describe the relation between union wages and the rate of growth of imports where,
as before, m is imports. The notation
the union wage rate;
w
x
signifies the time derivative of
x; w
is
is the limit wage, i.e. the domestic wage at which foreign
producers have no incentive to either enter or leave the domestic product market;
and
k
> 0 is a parameter reflecting the speed with which foreigners respond to labor
cost differentials. The limit wage reflects institutionally generated barriers to trade,
such as quotas and tariffs, as well as natural barriers to entry such as high foreign
costs of production, transportation costs, and set-up costs in entering the market.
The larger is the difference between the union wage rate and the limit wage,
w — w,
the greater is the rate of growth of imports.6
The introduction of this feedback from union wages to import growth alters the
nature of the problem facing the union from a static to a dynamic one.
In the
static model developed in Section 2, a permanent decline in imports shifted the labor
demand curve facing the union inwards. The union simply selects a point on the new
labor demand schedule. The union’s problem is more complicated in the expanded
model where feedback is permitted since imports essentially become endogenous. The
union can effectively choose the appropriate level of imports by manipulating the wage
6This simple formulation of the relation between imports and international wage differentials is
attractive for its tractability. However, it is admittedly simplistic. As in the literature on costs of
adjustment, one suspects that there are asymmetries involved. Specifically, ifthe limit wage exceeds
the union wage, the adjustment to equilibrium is likely to be accomplished much more quickly (if
not in fact instantaneously) than ifthe union wage were to exceed the limit wage by a like amount.
10
that it charges domestic manufacturers. For example, the union could keep all imports
out of the market by charging a wage rate equal to the limit wage. By following such
a strategy, however, the union foregoes higher wages in order to maintain market
share. An alternative is for the union to charge a wage rate that is higher than
the limit wage indefinitely. The union gains in terms of wages but continuously loses
employment opportunities until the domestic producer disappears entirely. The union
must somehow balance the short term benefits of higher wages against the long term
losses in employment opportunities that occur as a result of having above-competitive
wages.
The union’s problem is to maximize the discounted utility flow of the representa
tive union member over an infinite time horizon subject to the constraint that import
growth is governed by the expression found in equation (5) and the initial level of
imports at time 0 is mo. The union solves:
max
(6)
f£ ° [ u ( w ( t ) ) — u ( w a) ] L ( w ( t ) , m ( t ) ) e ~ rtd t
s.t.
m=
k(w
—w )
m(0) = m 0,
where
w (t)
is the union wage rate at time
is the exogenous alternative wage rate;
t
and
r
is the discount rate. As before,
L ( w ( t ) , m ( t ) ) is
wa
labor demand at time t which
depends upon the current wage rate and level of imports. It was shown in Section 2
that
Lw(w ,m )
and
L wm( w , m
< 0 and
Lm( w ,m )
< 0. Given that
) < 0. Furthermore, since
p"(m)
f'"(L)
< 0, then
< 0, it follows that
L ww( w , m )
L mm( w , m
< 0
) < 0.7
7The interested reader may find the detailed solution to this maximization problem in the
Appendix.
11
The current value Hamiltonian for this problem is given by:
'H
= [u(iu(f)) —u a] L ( w ( t ) , m(t)) +
where u a = u(w0). The costate variable,
of imports at time
t
n(t),
(7)
fi(t)k(w (t) — w),
is the shadow price of an additional unit
and is necessarily negative. The first term in the Hamiltonian is
the instantaneous utility flow from current wages and employment. The second term
reflects the effect of current wages on future utility.
The optimal union wage rate in the expanded model is lower than in the case
where feedback from union wages to imports is not present. The reason for this is
straightforward. In the static model presented in Section 2 the union’s action had
no effect upon imports so that no trade-off existed between current wages and future
employment opportunities. In the current model where feedback is permitted, the
union considers the effect of its actions on the future and, therefore, tempers its wage
demands to reflect this trade-off.
To see this, solve the first order conditions for the above maximization problem
for
fi*(t)
where the superscript
*d\ -
now denotes the optimal trajectory. Specifically,
[u'(w *)L (w *im *) + (u(w *) ~ ua)Lw(w*, m*)]
M ' “
k
Since the costate variable is negative, the bracketed term in the above expression is
strictly positive. However, from equation (3) in the basic model without feedback,
the optimizing union sets the bracketed expression equal to zero. Because of the
concavity of the union’s utility function in
w,
the result holds.
The solution to the union’s problem with feedback can be expressed as a two
equation system of first order differential equations in
equations of motion about the steady state wage rate,
12
m
w,
and
w.
Linearizing the
and level of imports,
m 3,
gives the following:
w
r
b
w —w
m
k
0
m — ms
(8)
where
(r[u Lm “|" (u
\
u "L
Ha)Liwm] "t* kLmrn(u
+ 2u ' L w
Ua)
>0.
+ (u — u a) L ww
(w = w , m = m s)
The phase diagram shown in Figure 1 is useful in analyzing the movement of this
system about the steady state. From equation (8), the m = 0 locus is horizontal at
w
=
w
and the
= 0 locus is negatively sloped. If the system is initially on the
w
m = 0 locus and the wage rate rises, then imports must also rise. Similarly, if initially
the system is on the
w
= 0 locus and imports rise, then wages must increase. The two
loci effectively divide the
m
x
w
plane into four distinct regions. Regions I and III
are characterized by divergent paths. Any trajectory entering or starting in Region
I (HI) has both the wage rate and level of imports increasing (decreasing) without
bound. These trajectories cannot be optimal since they both imply ever increasing
disutility after some point. The solution is a saddlepoint. The optimal path must lie
along the eigenvector as depicted in regions II and IV.
The optimal strategy for the union to follow depends upon the initial level of
imports. For m0 less than the steady state level of imports, the union charges a wage
rate that exceeds the limit wage, gradually lowering it over time to
w.
As the wage
rate falls, imports increase. Conversely, for m0 greater than the steady state level of
imports, the union sets the wage rate below the limit wage so as to drive out imports.
Over time the wage rate gradually rises to
The parameter
k
w.
is the speed with which foreigners respond to labor cost differ
entials. One might expect that the greater is
13
k,
the larger is imports’ share of the
market in steady state equilibrium. However, the opposite occurs. Increases in
cause the steady state level of imports to
fall
as the
w =
k
0 locus shifts to the left.
The intuition is that the union balances current income against future employment
opportunities. The more quickly foreign producers respond to cost differentials, the
less opportunity there is for future union employment. The union responds optimally
by dropping the union wage rate to
w
more rapidly.
The effect of an increase in the discount rate, r, on the steady state level of im
ports is straightforward. An increase in r causes the union to more heavily discount
the future. As a result, relatively less weight is put on future employment opportuni
ties and relatively more weight is put on current wages in the union’s maximization
problem. By choosing higher wages in the short run, the union’s steady state market
share falls. In terms of the phase diagram, an increase in r has no effect upon the
m = 0 locus but shifts the
w
= 0 locus to the right.
The limit wage is determined by transportation costs and other artificial and
natural barriers to entry into the domestic product market. The greater are these
barriers to entry, the higher is the limit wage. The alternative wage rate,
be thought of as the exogenous nonunion wage and, therefore,
w —wa
w a,
can
is the steady
state union/nonunion wage differential. If the limit wage exceeds the alternative
wage, the long run union/nonunion wage premium is positive and equal to
w
—w a.
Union members would obviously not work for less than the alternative wage so that
for
w < wa
the steady state union wage gap is 0. When the strict inequality holds,
foreign firms capture the entire market.
The extent to which product market conditions limit the ability of the union to
maintain positive wage differentials was examined recently in Stewart (1990). He
found that those unions employed in industries having some degree of market power
received larger union wage premiums. Furthermore, the union/nonunion wage differ
ential was effectively 0 in competitive markets. These facts are consistent with the
14
model’s predictions. In a perfectly competitive market in which either the nonunion
wage rate,
w a,
adjusts freely or in which there are no artificial or natural barriers to
entry, the steady state union wage premium is zero at
w
=
w a.
It is only if there is
some restriction that limits competition can the union maintain positive wage differ
entials.
The limit wage reflects the union’s ability to maintain above-competitive wages.
As the limit wage increases, the growth rate of imports slows, given the union wage
rate. An increase in the limit wage unambiguously causes the steady state level of
imports to fall as the m = 0 locus shifts upwards and the
w
= 0 locus reinforces
this effect by shifting inwards. The effect of changes in the limit wage on the level
of employment is ambiguous. Since imports are reduced, more production occurs
domestically and employment tends to rise. However, offsetting this somewhat, the
steady state union wage rate is now higher and has the opposite effect on employment.
The effect of a decrease in the limit wage on the optimal union wage trajectory
is depicted in Figure 2. Initially, the economy is in steady state equilibrium at point
A with union wages equal to the limit wage,
Wo ,
and imports of m s0. Changes in the
limit wage have two opposing effects upon the optimal wage path. First, as
the m = 0 locus shifts downward to
w\
w
falls,
as shown by the dotted line. The immediate
impact is to decrease the union wage rate. However, offsetting this effect somewhat,
a decrease in the limit wage also shifts the
ib = 0
locus outwards. The effect of this
change is to increase the union wage rate in the short run.
The short run response of union wages to a decrease in the limit wage depends
upon the relative shifts of these two curves. The more responsive is the
w —
0 locus
to changes in the limit wage, the more likely it is that as the limit wage falls, the
union wage rate initially rises. The new steady state equilibrium is shown at point B
with wages equal to
w\
and imports of msl.
In the late sixties, the Kennedy Round of the trade negotiations took effect in
15
which tariffs were reduced by approximately 35% on average and some 50% in indus
tries that were not exempt. Again in the mid-seventies, the Tokyo Round (1973-1979)
further reduced trade restrictions by focusing on the reduction of nontariff barriers to
trade. The limit wage is defined as the wage rate that would induce no foreign entry
into or exit from the domestic product market. By relaxing trade restrictions over
this time period, the limit wage was effectively reduced. The long run effect of this
policy, as noted above, is to reduce the union/nonunion wage differential, assuming
w
>
w a,
and to increase the level of imports. This is roughly consistent with the
decline in the union wage gap observed in the early eighties. However, the short run
effect is unclear.
If unions are aware that their future opportunities are circumscribed by a more
open trade policy, they may optimally choose to increase the wage rate in the short
run. The result depends upon how the
w
= 0 locus shifts relative to the m = 0 locus.
Thus, the immediate effect of a reduction in barriers to entry may be to
in c re a se
the
union/nonunion wage differential while the long term effect is to reduce it. According
to this hypothesis, the increase in the union wage gap between 1969 and 1983 was the
result of an optimal union decision to reap economic rents while the opportunities for
doing so existed. As these opportunities were reduced or eliminated over time, the
union/nonunion wage differential fell, thereby explaining the behavior of the union
wage gap since 1983.
3.1
E xam p le
The following example with linear labor demand curves and risk neutral workers
illustrates how reductions in barriers to trade affect union wage dynamics.
L ( w ym )
=
a
Let
— f3w —7 m be the labor demand curve facing the union. Assuming
union members to be risk neutral, the union’s optimal control problem can be easily
16
solved for the equations of motion for wages and imports:
w
=
r
T7
k
0
—r ( a + j 3 w a ) + k ( w —w a )
2/3
W
2/3
+
rri
—k w
m
The solution to the above system of differential equations is:
w (t)
=
c e Xt
+
(9)
m (t)
=
c(A —r ) — eA<+
w
23
( 10 )
m s,
where
c
+
r(a
m s
=
A =
(m 0 -
-
3 w a — 2 fiw ) — k ( w — w a)
m s)
r7
2/3(A - r)
r - \rz +
2 k r j\
~t
1/2'
)
Analytically, the effect of a decline in the limit wage can be found by differentiating
equation (9) with respect to
w.
Specifically,
d w (t)
dw
= 1+
(2(3r
+ k ) xt
(11)
2 p ( \ - r)
The first term in the above expression reflects the effect of the limit wage on the
m = 0 locus while the second term reflects the shift in the
w
= 0 that occurs. SinceA
is negative, the second term is negative and its magnitude is decreasing over time.
17
As
t
—»• oo, the wage rate falls by an amount equal to the decrease in the limit wage.
The circumstances under which a reduction in the limit wage initially
the optimal union wage rate can be determined from the solution to
in c re a se s
d w (0 )/d w
< 0.
It can easily be shown that a reduction in the limit wage initially increases the union
wage rate so long as
k >
0. In conclusion, in the case of linear labor demand curves
and risk neutral workers a reduction in the limit wage leads initially to increases
in the union wage premium so long as imports respond positively to positive cost
differentials. In the long run, the union wage gap declines to a lower steady state
value.
4
Conclusions
The union/nonunion wage differential rose steadily over the period from 1969 through
1982 in both manufacturing and nonmanufacturing industries. Since then, the union
wage gap appears to have fallen significantly. The model presented here attempts
to explain these recent union wage developments in a dynamic model in which there
exists feedback from union wages to imports. The model suggests that as trade
barriers were lifted in the sixties and again in the mid-seventies, unions responded
optimally to their more limited future opportunities by increasing wages in the short
term while they could do so without suffering large employment losses. The theory
predicts that over time the union/nonunion wage differential falls as imports penetrate
domestic markets and unions lose market share.
The model is broadly consistent with the stylized facts concerning relative union
wage movements and imports. Other explanations of the phenomenon have either
been unsuccessful in explaining both the increase and subsequent decline in the union
wage gap or have resorted to restrictive assumptions about the nature of the produc
tion technology. The novel aspect of the model formulated here is the introduction
18
of dynamic elements that shifts the focus of the union’s problem to consideration of
perm an en t
ru n
versus
te m p o r a r y
changes in the terms of trade and
s h o r t ru n
versus
long
tradeoffs. The theory has the additional benefit of being more broadly applicable
than to the issue of imports alone. The emphasis has been on import penetration
because of the coincidence of the union wage gap explosion and growth in imports.
However, the model is relevant to analyzing the effects of deregulation or, indeed, any
change in the environment that reduces barriers to entry.
The analysis has emphasized the joint determination of union wages and imports.
It has been explicitly assumed that the union cannot affect the responsiveness of
foreigners to cost differentials or non-wage barriers to entry. These simplifications
make the model tractable and provide some interesting insights. However, a richer
theory must consider how the union may influence both
w
and
k.
For example, since
the limit wage depends upon trade restrictions, unions may be able to affect it through
the political process. Discussions in Congress regarding the “unfair” trade practices
of the Japanese suggest that unions may be attempting to influence the limit wage
indirectly through the political arena.
Imports have been assumed to shift the labor demand curve inwards so that the
employment opportunities facing domestic unionized workers are reduced. It has
been explicitly assumed that unions are unable to organize foreign workers because of
geographical and political boundaries. However, one possible option available to the
union that has been ignored in the analysis is for the union to encourage joint ventures
between domestic and foreign producers, such as has occurred in the automobile
industry. In this way unions may be able to maintain employment and membership
without substantial cuts in wages.
19
A p p en d ix A
The necessary conditions to the programming problem found in equation (6) are found by
applying the Pontryagin maximum principle. Thus, there exists a fi*(t) such that:
fi*(t)
=
) - [u(ti>*(f)) - ua]Z/m(u>*(*), m*(<))
fi* (t)e ~ rt
=0
u a]L w(w * (t), m * (t)) + fi* (t)k
=0
(A.2)
k (w * (t) — w )
(A.3)
lim
u '(w * (t))L (w * (t), m * (t))
+ [u(u;*(t)) -
m * (t )
where the superscript
[u(tn) —ua]L (w , m ) in
(A.l)
=
denotes the optimal trajectory. The continuity and concavity of
w
assures the existence of at least one optimal path providing that
the wage rate and imports are contained in compact sets.
Taking the time derivative of equation (A.2 ) and substituting the appropriate expres
sions from equations (A.l) and (A.3) gives a two equation system of first order differential
equations:
. _ -r(u'L + (u - ua)Lw) + kLm [u'(w - w )- (u- ua)] + (u - ua)(w - w)kLwm]
W~
—[u"L + 2u 'L w
+ (t* -
u a)L ww)
m = k {w — w ).
( ‘j
(A.5)
The two equations generate a family of trajectories in the m —w plane. The transversality
condition and initial condition can be used to determine which of the various paths is
optimal.
In steady state equilibrium both
w
and
rh
implies that the steady state wage rate equals
are equal to zero. From equation (A.5) this
w.
Substituting the steady state wage rate
into the expression for the w = 0 locus gives:
0 = r [v !(w )L (w , m s) + (u (w ) - ua)L w( w ym s)] + k L m ( w , m s)(u (w ) - u a)
20
(A.6)
where
ma
is the steady state equilibrium level of imports.
The characteristic equation for the system found in equation (8) is:
A2 —rX
— kb
=0
with solutions:
\
x _
( r 2 + kb)1/ 2
r
(A.7)
M , M = r ± --------r --------
Since both
b
and
k
are positive, it follows that one of the eigenvalues is strictly positive
and greater than the discount rate, say Ai, and the other strictly negative, A2. Thus, the
equilibrium is a saddlepoint. The optimal path, if it exists, must lie within regions II and
IV of the phase plane shown in Figure 2. This path necessarily satisfies the transversality
condition. The transversality condition requires that:
lim
[u '(w * )L {w * ,m * )
t-+oo
+
(u (w *)
k
Since the path converges to the steady state
w
-
ua)L w(w * ,m * ))c _ rt
and
m
_ Q
(A.8)
which are bounded, the condition
holds because the expression is dominated by the negative exponential as t —» 00.
The comparative statics on imports and employment are found by differentiating the
expression in equation (A.6) with respect to the various model parameters. Specifically,
dm s
dk
_____________________ - L
r[u '(w )L m (w , m s) + (u(u>) -
ua]___________________ <
ua)L wm(w , m*)] + k L mm(w , ms)[u(u>) - u a] ~
m ( w ,m s)[u (w ) -
dLs
dk
L m (w , m s)
dm s
~dk
> 0
d m s _________________________ [u'(w)L(w,ms) + L w(w, m s)(u(w) - ua)]____________________
dr
r[u'(w)Lm (w , m s) + (u(w) - ua)Lw m {w, m s)] + kLm m (w, m a)[u(w) - ua]
dLs
dr
21
T t~
Lm
(w,
\d m *
<0
dm a
_ _______________ [rL w(w , m s) +
k L m (w ,
m , ) ] ^ ^ ) _______________
dwa ~ r[u'(th)i,TO(tZ>,m 3) + (u(w) - ua)Lwm(w, m s)] + kLm m (w, m s)[u(u>) - ua]
dL,
dw a
di7ig
^ r[u L ■{*2ti Jjm (u ^o)^wtu]
d w ~ r [ u \ w ) L m{ w , m s) + ( u (w ) - u a)L wm(w ,
L m{ w , m
k L w m tta] kLffiU J
k L mm(w , ms)[w(u>)
ms)] +
dLs
-£=-
s)
=
dm s
dw a
<0
- u0]
\d m *>rx
L w( w ,m s ) + L m ( w , m s) - £ z - <
(I'mg
d Ij g
dm 0
dm o
0
The comparative dynamics are in general more difficult to analyze. However, examina
tion of the slope of the trajectory in the
phase plane offers some insight. The slope
m —w
of any path is given by:
dw
w
dm
m
-r(u'L + (u - ua)Lw ) + kLm [u'{w - w ) - ( u - ua)]+ kLwm(w - w)(u - Ua)
-k(w - w)[u"L + 2u'Lw + (u - ua)Lww]
(A.9)
By differentiating the above expression with respect to the various parameters of the model,
we can find how the slope of the trajectory changes with the different parameters and
attempt to draw conclusions about the comparative dynamics by arguing from the phase
plane.
For example, differentiating equation (A.9) with respect to
d ( dw \ _
dk \ d m )
k
gives:
+ (u — u a)L w\
+ 2u 'L w + (u — u a)L ww]
—r[u'L
k 2(w — w )[u"L
(A.10)
Since the costate variable is negative, the numerator of the above expression is negative
and the sign of the derivative is given by the sign of tv —w . Suppose that
w
is greater than
the limit wage. From equation (??), it follows that the old and new trajectories cannot
22
intersect. If they did intersect, then the new path must move above the old as the slope
decreases. No such path could converge to the new equilibrium point since it would by
necessity imply that the trajectory moved into region I. Therefore, the new trajectory must
initially lie below the old so that an increase in the responsiveness of foreign producers to
the current labor cost differential causes the optimal wage path to be lowered in the short
run. There is no long run effect on wages.
Comparative dynamics results for the other parameters of the model can be found by
differentiating equation (A.9) with respect to r,
w a,
and
w:
_________ [u'L + (u - ua)Lw]_________
k(w - w)[u"L + 2u‘Lw + (u - Ua)Lww)
d
fdw\
_
dwa \ d m )
u 9{ w a){[u"L
-f 2u/Lti,](rLt/,+
k L m ) - r u'LLw w
-k(w
d
f dw
\
dw \ d m j
-
w)[u"L
-
k ( w - w)[(u"L
-f 2 u ,L Xj) + (it-
T*[u L “f*(U
k (w - w )2[u"L
+ 2u'Lw )Lw m —
u fL m L w w ]}
u a)Lw w ]2
VjQ^Lyj^ "I- k ' U ' a )
+ 2u'Lw +
(u - u a)L ww]'
From the above expressions, the effect of an increase in the discount rate is to increase
the wage rate in the short run. Again, there is no long run effect on the union wage. The
effect of an increase in the alternative wage rate is ambiguous. However, if the curvature
of the labor demand schedule is not too great and if
of an increase in
wa
L wm
« 0, then the short term effect
is to raise the optimal wage rate. Without making some additional
simplifying assumptions, the immediate impact of an increase in the limit wage on the
optimal wage trajectory cannot be determined.
23
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