Trade and Labour Demand Elasticity in Imperfect Competition: Theory and Evidence

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Trade and Labour Demand Elasticity in Imperfect
Competition: Theory and Evidence
Daniel Mirza and Mauro Pisu
GEP, University of Nottingham
Preliminary version - Please do not quote
In this article, we extend the Allen (1938)-Hamermesh (1993) the-
oretical relation of labour demand elasticity to imperfect competition
using Dixit’s (1990) framework. In oligopoly, we find that the elasticity
of labour demand depends on a third ’pro-competitive effect’ term that
eases labour demand adjustment to a particular shock like openness.
Besides, this relationship appears to be directly testable in the data.
We use UK firm level data from OneSource database and find that the
long run derived estimators are consistent with our theoretical frame-
work. The short-run estimators however, show only mixed support to
our relation.
In recent years, a growing body of the literature in international trade
tried to investigate whether openness has been increasing labour demand
elasticities. From a labour theory perspective in partial equilibrium, the
Allen-Hamermesh theoretical relation became one of the few general frame-
works to refer to1. This theory states that labour demand elasticity should
be positively affected by its two principal determinants: the elasticity of
substitution between labour and other factors and the elasticity of demand
for goods to prices. Under the assumption that openness is affecting these
factors by increasing the possibility of substitution among factors and goods
respectively, that relation could then predict a consecutive increase in the
elasticity of demand for labour.
Although some empirical studies in the field were inspired by that re-
lation (i.e. Slaughter (2001), Haskel, Slaughter and Fabbri (2002) among
others), two issues remain.
First, Allen (1938) showed that this relation holds in a perfect compe-
tition environment at the industry level. While it is now widely recognized
1One should also note that Leamer (1996), Wood (1995) and also Panagariya (1999)
discussed the effect of trade on labour demand elasticities, but applying HO or specific
factor trade theories in General Equilibrium.

that imperfect competition is one of the most influent basis for the rise in
trade and multinational activities the Allen-Hamermesh relation constrains
the researchers to test the impact of openness on labour demand elasticities
assuming perfectly competitive markets. The first question addressed then
by this article is how the Allen-Hamermesh (henceforth AH) relation could
be extended to an imperfectly competitive setting2. Following Dixit’s (1990)
modelling framework, we show that AH can be generalized to allow for im-
perfect competition. In particular, under the assumption of oligopoly, the
elasticity of labour demand depends on a third term, neglected so far by the
AH relation. This term is reducing the burden on labour demand elasticity:
the elasticity of prices to wages. Actually, an increase in wages has a pure
cost effect, but is reducing at the same time the market share of the firm
and thus its mark-up. As a result of this pro-competitive effect, there might
be incomplete pass through between prices and wages and the adjustment
of labour demand would be then smaller than expected.
The second issue left out by the AH relation is that it does not show
formally the relationship between trade openness mesures and labour de-
mand elasticities3. We try to fill that gap in this article by showing that the
average elasticity of labour demand depends, in a formal manner, on the
import penetration rates. Also, our model provides an explanation for why
the elasticity of demand has not been increasing that much with trade, a
result that was pointed out by previous studies. In fact, it predicts that the
effect of import penetration would be high, if there is complete pass-through
from wages to prices. But in the case of incomplete pass-through, then a
small effect of import penetration should prevail.
We use UK firm level data from OneSource database to test for our
relationship. Our results tend to support our theoretical framework.
The Allen (1938)-Uzawa (1962) relation
Some authors like Slaughter (2001) have been inspired by the following rela-
tion, at the industry level, of the elasticity of labour demand (sensitivity
of labour demand to wages ηl) proved in Allen (1938), further investigated
by Uzawa (1962) and discussed in details by Hamermesh (1993). Consider-
2Note that Krishna et al (2001) have studied the impact of trade on labour demand
elasticities by emphasizing the role of imperfect competition as well. However, the authors
design a framework based on monopolistic competition (i.e. they do not consider strategic
interaction among firms) and assume a Cobb-Douglas production function.
3Jean (2000) could also link trade measures with the elasticity of labour demand but
he uses a different framework than that of Allen-Hamermesh. His work is built on a
perfect competition world in general equilibrium with an Armington type hypothesis on
the demand side and a Leontief production function on the supply side

ing 2 inputs, say labour and capital, the wage elasticity of labour is equal
ηl = −(1 − αl)σ − αlηY
with αl = wl is the share of labour cost to revenue in the industry, σ
is the absolute value of the elasticity of substitution between labour and
capital, and ηY the absolute value of the elasticity of total demand Y to
prices in the sector. Allen (1938) proves this relation by resolving a program
of profit maximisation in perfect competition, considering the case of two
factors and a constant return to scale technology.
What is the intuition behind this relation? If wages increase, and given
a fixed output, employers will want to substitute away labour towards the
other factor of production whose price is now relatively lower (the employers
change the technique of production along the same isoquant). The extent
of this effect depends on α. The higher share of labour cost, the smaller the
pass-through from σ to ηL.
However, industry output is not fixed. In fact, for a given technique of
production, an increase in wages affect positively commodity prices in the
industry which in turn reduce industrial production overall. (The isoquant
moves inward.) This affects downward the demand for the two factors and
a fortiori that for labour. The extent of this decrease in labour demand
following the adjustment of production to the new prices, depends on the
share of labour cost.
Allen (1938) generalizes this relation to m factors of production (m > 2)
by proving that:
ηl = −αlσll − αlηY
Here, σll, as Allen (1938) underlines, represents the elasticity of substi-
tution between l and all all the other factors of production in the industry4.
The positive relation between labour demand elasticity and product demand
elasticity (scale effect) is known to form the third Marshall rule (see Slaugh-
ter (2001) for further discussion).
4The concept of the Allen elasticity of substitution has been called into question not
long ago. Blackorby and Russel (1989) highlight that the AH elasticity of substitution
is completely uninformative about the easy with which inputs can be substituted (when
there are more than two inputs). They stresses as the Morishima (1967) elasticity of
substitution is the true elasticity of substitution. However, this argument does not directly
apply for the purpose of this paper since what we are interested is the own price labour
demand elasticity. Although the σll is not the elasticity of substitution of labour it is an
intermediate figure whereby computing the wage elasticity of labour.

Notice moreover that the factor of pass-through from substitution to
labour demand is now αL. Hence, the effect of substituting away labour
towards all other factors is more harmful on labour demand, the more the
share of labour in total cost is important.
Trade could affect the elasticity of labour demand in two fashions. Slaugh-
ter (2001) and Haskel et al (2002) mention that international commerce
could augment σ by increasing the possibilities of substitution between do-
mestic and foreign factors. Moreover, openness could lead to an increase in
the elasticity of demand to prices ηY by increasing competition. Yet, this
latter prospect makes the inappropriateness, at firm level, of the AH relation
manifest since in perfect competition the product demand is by definition
perfectly elastic. This would imply perfectly elastic input demands as well,
irrespective of the elasticity of substitution or any other factor that might
affect them directly or indirectly.
Generalization of the theory to imperfect com-
The same relation is obtained from a more elegant and simpler formal setting
by Dixit (1990) who minimizes total costs instead of maximizing profits.
We thus follow the same type of formulation than Dixit in what follows but
extend the framework to the case of imperfect competition.
Assume a firm that produces with constant returns to scale. Its total
cost can be written as follows:5
C = yc(W )
Where W = (w1, . . . wm) is the vector of factor prices for m factors
of production. Assuming Cournot competition, the first order conditions
provides an equality between marginal revenue and marginal costs. Prices
could then be represented by:
p =
c(W )
1 − s
where p is the equilibrium price, Y represents total demand (or industry
production), s = y is the market share and η
Y the absolute value of the
elasticity of the product demand to prices faced by the firm. Note in what
5We could have supposed an increasing returns to scale technology by assuming an
alternative expression that includes fixed costs like C = yc(W ) + F , but this does not
affect the relation to estimate hereafter.

follows that
= µ represents the mark up. By Shephard’s Lemma,
1− s
the demand for labor is the derivative of total costs with respect to the price
of labor:
∂c(W )
l =
= y
= ycw
Deriving with respect to wl we have:
∂y ∂Y ∂p
= cwwy +
∂Y ∂p ∂wl
where cww = ∂cw .
We know from Uzawa (1962) and Hammermesh (1993) that when the
cost function is linear and homogenous, the absolute value of the elasticity of
substitution σ
ll equals
c(W )cww . In addition, let η
be the elasticity
p = ∂p
w cw
∂w p
of prices to wages and recall η
Y = ∂Y
the elasticity of total demand to
∂p Y
prices in the industry. Then expressing equation 6 as an elasticity, ηl =
dl wl , gives:
dwl l
ηl = −σllαl − ηY (1/s)ηp
Relation 7 designates the firm-level elasticity of demand in oligopoly
with αl = wl standing as the labour share in total costs. It has some simi-
larities with the traditional relation presented in the prior section. Firstly,
the elasticity of substitution of labour with respect to all other factors (σll)
enters the equation as before and is multiplied by the cost share of labour.
Second, note ηY (1/s) = ηy. This is, for a typical firm, the perceived
elasticity of demand to an increase of its price. Hence, when the firm has a
small market share due for example, to a big number of firms in the industry,
the firm perceives a price-elasticity of demand for its commodity that is high
because it has little market power that enables that firm to set its own price.
Keeping this perceived elasticity in mind, the second term of equation 7
can thus be presented as −ηyηp. How can our framework compare then to
that of AH? To see this, we develop in what follows the expression of the
elasticity of prices to wages ηp = ∂p/p .
Indeed, from 4, we have
∂µc(W )
= cwµ + c(W )
The derivative of costs with respect to wages cw has a positive sign.
Besides, it can be easily shown that the derivative of mark-ups is ∂µ =

µ2(1/ηY ) ∂s . Under the traditional assumption of downward sloping best
response functions, it is well known that market share is decreasing with the
marginal cost of the firm: thus, ∂s is negative and so do the derivative of
mark-ups. To sum up, an increase in wage has 2 opposite effects on prices:
a pure positive cost effect and a negative pro-competitive effect. From that
effect, one can thus observe an incomplete pass-through between a change
in wages and a change in prices. One fraction of the higher cost is now
supported by the firm in the form of lower mark-ups.
Note that the elasticity of mark-ups to wages can be expressed as: ηµ =
− ∂µ wl . Multiplying equation 8 by wl , and recalling the relations 3, 4
∂wl µ
and 5, we obtain the following expression of ηp after simplification:
∂p wl
ηp =
= αl − ηµ
∂wl p
Plugging that expression into equation 7 we obtain:
ηl = −σllαl − ηY (1/s)αl + ηY (1/s)ηµ
Here, the additional term ηY (1/s)ηµ enters the equation. This term is
reducing in absolute values the elasticity of labour demand to wages. It is
doing so because of the pro-competitive effect generated by the variation in
wages on mark-ups. In order not to loose much of their competitiveness,
the firms are constrained to pass a part of the increase in wages on to less
mark-ups, making eventually a relatively small adjustment on prices and
thus demand. Hence, in an oligopoly world, this incomplete pass through
between wages and prices would makes the labour demand less elastic.
Then, from 10 it is possible to see that in a oligopolistic competition,
as it is the case in a perfectly competitive market, labour demand is more
elastic when the cost share of labour αl is high but also when substitutions
between factors (i.e. σll) and goods (ηY ) are high. On the other hand,
relation 7 suggests that labour demand of a firm is less elastic the higher its
market share.
By comparing the two relations 2 and 10, AH appears as a particular
case of the latter equation. Indeed, consider a relatively competitive market
where say, s is small but not too small in order to maintain a finite perceived
elasticity of product demand ηY (1/s) = ∞. In that case, the mark-up tends
to 1 and its elasticity with respect to wages ηµ and ηs would approximate 0.
Hence, the whole increase in wages is passed on prices and the third term
of our equation 10 vanishes, thus retrieving the AH relation.
If the market is perfectly competitive though s ≈ 0, our relationship in-
dicates that the perceived demand elasticity ηY (1/s), would tend to infinity

and so does the elasticity of labour demand. Indeed, in that market the firm
cannot act neither on the wage it offers nor on the price it sets. It takes them
as given from the competitive labour and product markets. Note that this
perfect competition type result is different from that provided by the AH
relation because our unit of study is the firm, not the industry. In AH, factor
and commodity prices should vary evenly across firms and thus affect total
demand as a whole at the industry level. In our firm-level configuration, if
it happens that a firm decides unilaterally to increase its price, consumers
substitute away the product of that firm towards those sold by other firms in
the industry at a lower (market) price. Hence, the composition of suppliers
changes leaving total demand unchanged.
A second particular case of equation η that is worth noting is that of
monopoly with constant elasticity of product demand to prices. In this case,
s = 1, ηµ = ηs = 0: here again a proportionate increase in wages results
in a same proportion of increase in prices. This is also true for the case of
perfect differentiation in, say a monopolistic competition structure: a firm
producing its own variety faces an elasticity of demand compared to that of
a monopolist. In that case, the AH relationship prevails at the firm level.
Elasticity of Labour Demand and Trade
How would openness to imports affect labour demand elasticity? So far, the
literature noted that trade could affect labour demand elasticity by affecting
both the elasticity of substitution and the elasticity of product demand. The
first elasticity, could indeed be affected as openness increase the possibility
of combining domestic and foreign factors in the production process of a
In addition, international trade may augment or reduce the scale effect
due to the perceived elasticity depending on whether it causes an increase
or decrease in the market share and in the cost share of labour. These are
not obvious questions. The answer hinges on how firms react to increased
foreign competition.
The literature concerning international trade and labour demand has am-
ply investigated the effect of trade on the cost share of skilled and unskilled
workers Generally, trade has not been found to affect directly the cost share
of labour while technology has.The relationship between the market share
and trade has not been investigated sufficiently thus far. Consider first that
the number of domestic firms is given in the market. An increase in foreign
firms’ shares due to a larger openness would then reduce mechanically the
domestic firms market shares and hence, exacerbate the elasticity of product
and labour demand. However, the number of firms might be endogenous to

trade. Under this new assumption an increase in foreign competition leads
to least productive firms to exit the market thereby resulting in a more
concentrated market and hence higher market share for surviving firms.
To sum-up, at the firm level the effect of import penetration on market
shares is ambiguous. However, the effect becomes unambiguous at the in-
dustry level, irrespective of whether the number of firms is endogenous or
not. In order to see this, we relate formally hereafter import penetration to
the mean labour demand elasticity prevailing in the industry.
From equation 7, we multiply each term by the market share of the firm
among its peers in the market (sd = yl )6 and sum over all domestic firms
in order to obtain an expression for the weighted mean elasticity of labour
demand in the industry, η =
l sdη
l. After simplification we obtain:
η = −σll
(αlsdl) − ηY ηpN
with S being the total market share of domestic firms on their market.
It is also equal to 1 − M P where M P is the import penetration rate in the
market. Hence, at the industry level, import penetration is directly affecting
market share of domestic firms as a whole. This results in a higher mean
elasticity of labour demand in the industry. In this relation, we can thus
estimate the pure effect of import penetration via its impact on perceived
product demand on average in the industry: η
y = ηY
Data set
UK firm level data were used in this paper. These come from the One-
Source database from 1993 to 1999. It includes information on all public
limited companies, all companies with employees greater than 50, and the
top companies based on turnover, net worth, total assets, or shareholders
funds (whichever is largest) up to a maximum of 110,000 companies, in both
manufacturing and service industries. Companies that are dissolved or in
the process of liquidation are excluded from the OneSource sample. In this
paper we concentrate on manufacturing firms from this data source.
The data set was screened to keep only those firms for which there were
a complete set of information about output and inputs. This left an unbal-
anced panel data of around 36500 observations regarding 11100 firms. This
firm level data set was used to obtain data about number of workers, cost
shares, and mark-ups.
6Y D is total sales of domestic firms

The relationship under scrutiny requires market shares as well. To com-
pute it the firm level data set was merged by industry and years with the
Production Standard Extracts, provided by the British Office of National
Statistics. This procedure was necessary to have reliable measures of total
industry production.
Further detail about the variables used can be found in Appendix A.
Empirical results
Formula 10 can be estimated on firm level data using variables of employ-
ment, wages and market shares. Taking discrete approximations, accounting
for firm i cross-section and time t variations, equation 10 could be estimated
∆ log (lit) = −σ [αit∆ log (wit)] −
− β1 [(αl/sit)∆ log (wit)] + β2 [(1/sit)∆ log (wit)]
+dt + di + εit
where di and dt represent individual and time effects and εit is standard
error term. The ∆ operator expresses here first differences. The parameter
β1 represents the product demand elasticity (i.e. ηY ) whereas β2 estimates
the interaction between the product demand elasticity and the elasticity of
the mark-up to wages (viz. ηY ηµ). Independent variables, ∆ log w, αl and
(1/s) can be easily computed from our data (see appendix) which would
allow us to estimate the above regression.
The relationship in 10 is derived form theory and therefore refers to the
long term. It does not allow for any adjustment and dynamics, which in
actual facts may be important. Here, we are interested in the long run
behaviour of our variables since we want to test the proposition implied by
the theory.
It is widely believed that cross-sectional studies tend to yield results
concerning the long-run while time series studies conduce to short-run es-
timates. This association was investigated by Kuh (1959). By the same
token, the Between estimator in a panel data set is supposed to generate
long term results since it exploits the cross-sectional (i.e. between) variation
of the data whereas the Within estimator is thought to yield short-term coef-
ficients because this methodology uses the time series (viz. within) variation
of the data (Baltagi 2001).
Various authors have tackled the issue of why the Between and the
Within estimates often differ markedly while theoretically they should not.

The culprits have been identified in unobserved individual heterogeneity,
dynamic misspecification and measurement errors. The former leads to a
source of bias which arises if the individual effect is correlated with the
regressors. In this occurrence the Within estimates would be consistent
while the Between would not since the former wipes out the individual ef-
fect whereas the latter does not. Dynamic misspecification could also be
the cause of the discrepancy between the Within and Between estimates
as underlined by Baltagi and Griffin (1984). Indeed, long-lags and short
panel data, as is the norm in firm level data set, may result in dynamic
misidentification. With regards this problem, Pirotte (1999) shows that the
static Between estimator converges towards the long-run coefficient even
though the true data generating processes is dynamic7. Griliches and Haus-
man (1986) have investigated the bias induced by measurement error. They
claim that any transformation that eliminates the individual fixed effect such
as the within and first difference transformation is likely to exacerbate the
measurement error bias. This point is further analysed by Mairesse (1990)
who underlines that, provided the measurement error is not autocorrelated,
the Between estimators minimizes the bias induced by measurement errors,
because of the averaging, whereas Within estimates magnify it.
In the estimation of regression 13 biases due to the unobserved individual
heterogeneity, measurement errors and dynamic misspecification are likely
to be encountered. Between the first two, the measurement error seems to
be the gravest since the discrete approximation of continuous derivative ad-
mittedly cannot be expected to be very precise. Thus, this would favour
the Between estimator since, as discussed before, the Within may aggra-
vate the measurement error bias (Griliches and Hausman 1986). Besides,
the bias related to the unobserved individual effect is probably lessened by
first differencing the data in order to obtain the discrete approximation. In
addition the Between estimator has the advantage to be robust to dynamic
misspecification as Pirotte (1999) has shown.
Therefore, regression 13 has been estimated by means of the following
panel data model
yit = δ0 + xiδ1 + (xit − xi) δ2 + µt + εit
This model permits to distinguish the effect of a temporary variation in
the independent variables from the impact of a permanent one. Indeed, it
assumes that changes in the mean of variables (xi) to wit long run changes,
affect the dependent variable differently than temporary deviations from it
(xit − xi).
7For the converge it is required that T tends to infinity, N is fixed and the parameters are
homogeneous. In addition this result holds irrespectively to the number of autoregressive
and distributed lag terms appearing in the true data generating process