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In this article, we extend the Allen (1938)-Hamermesh (1993) the- oreticalrelation 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-competitiveeect' 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 UKfirmlevel data from OneSource database and find that the long run derived estimators are consistent with our theoretical framework. The short-run estimators however, show only mixed support to our relation.

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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Preliminary version - Please do not quote

Abstract

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.

1

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.

1

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

2

ing 2 inputs, say labour and capital, the wage elasticity of labour is equal

to:

ηl = −(1 − αl)σ − αlηY

(1)

with αl = wl is the share of labour cost to revenue in the industry, σ

pY

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

(2)

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.

3

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.

2

Generalization of the theory to imperfect com-

petition

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 )

(3)

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:

1

p =

c(W )

(4)

1 − s

ηY

where p is the equilibrium price, Y represents total demand (or industry

production), s = y is the market share and η

Y

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.

4

follows that

1

= µ represents the mark up. By Shephard’s Lemma,

1− s

ηY

the demand for labor is the derivative of total costs with respect to the price

of labor:

∂C

∂c(W )

l =

= y

= ycw

(5)

∂wl

∂wl

Deriving with respect to wl we have:

dl

∂y ∂Y ∂p

= cwwy +

cw

(6)

dwl

∂Y ∂p ∂wl

where cww = ∂cw .

∂w

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

w

ll equals

c(W )cww . In addition, let η

be the elasticity

c

p = ∂p

w cw

∂w p

of prices to wages and recall η

p

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

(7)

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-

C

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 .

∂w/w

Indeed, from 4, we have

∂p

∂µc(W )

∂µ

=

= cwµ + c(W )

(8)

∂wl

∂wl

∂wl

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 =

∂wl

5

µ2(1/ηY ) ∂s . Under the traditional assumption of downward sloping best

∂wl

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

∂wl

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

p

and 5, we obtain the following expression of ηp after simpliﬁcation:

∂p wl

ηp =

= αl − ηµ

(9)

∂wl p

Plugging that expression into equation 7 we obtain:

ηl = −σllαl − ηY (1/s)αl + ηY (1/s)ηµ

(10)

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

6

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.

3

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

ﬁrm.

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

7

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

l

Y D

in order to obtain an expression for the weighted mean elasticity of labour

demand in the industry, η =

l sdη

l

l. After simpliﬁcation we obtain:

1

η = −σll

(αlsdl) − ηY ηpN

(11)

S

l

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

1

y = ηY

.

S

4

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

8

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.

5

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

as:

∆ log (lit) = −σ [αit∆ log (wit)] −

(12)

− β1 [(αl/sit)∆ log (wit)] + β2 [(1/sit)∆ log (wit)]

+dt + di + εit

(13)

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.

9

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

(14)

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

10

Competition: Theory and Evidence

Daniel Mirza and Mauro Pisu

GEP, University of Nottingham

Preliminary version - Please do not quote

Abstract

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.

1

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.

1

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

2

ing 2 inputs, say labour and capital, the wage elasticity of labour is equal

to:

ηl = −(1 − αl)σ − αlηY

(1)

with αl = wl is the share of labour cost to revenue in the industry, σ

pY

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

(2)

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.

3

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.

2

Generalization of the theory to imperfect com-

petition

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 )

(3)

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:

1

p =

c(W )

(4)

1 − s

ηY

where p is the equilibrium price, Y represents total demand (or industry

production), s = y is the market share and η

Y

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.

4

follows that

1

= µ represents the mark up. By Shephard’s Lemma,

1− s

ηY

the demand for labor is the derivative of total costs with respect to the price

of labor:

∂C

∂c(W )

l =

= y

= ycw

(5)

∂wl

∂wl

Deriving with respect to wl we have:

dl

∂y ∂Y ∂p

= cwwy +

cw

(6)

dwl

∂Y ∂p ∂wl

where cww = ∂cw .

∂w

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

w

ll equals

c(W )cww . In addition, let η

be the elasticity

c

p = ∂p

w cw

∂w p

of prices to wages and recall η

p

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

(7)

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-

C

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 .

∂w/w

Indeed, from 4, we have

∂p

∂µc(W )

∂µ

=

= cwµ + c(W )

(8)

∂wl

∂wl

∂wl

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 =

∂wl

5

µ2(1/ηY ) ∂s . Under the traditional assumption of downward sloping best

∂wl

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

∂wl

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

p

and 5, we obtain the following expression of ηp after simpliﬁcation:

∂p wl

ηp =

= αl − ηµ

(9)

∂wl p

Plugging that expression into equation 7 we obtain:

ηl = −σllαl − ηY (1/s)αl + ηY (1/s)ηµ

(10)

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

6

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.

3

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

ﬁrm.

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

7

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

l

Y D

in order to obtain an expression for the weighted mean elasticity of labour

demand in the industry, η =

l sdη

l

l. After simpliﬁcation we obtain:

1

η = −σll

(αlsdl) − ηY ηpN

(11)

S

l

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

1

y = ηY

.

S

4

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

8

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.

5

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

as:

∆ log (lit) = −σ [αit∆ log (wit)] −

(12)

− β1 [(αl/sit)∆ log (wit)] + β2 [(1/sit)∆ log (wit)]

+dt + di + εit

(13)

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.

9

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

(14)

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

10

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