Journal of International Economics 79 (2009) 238–247

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Journal of International Economics

j ourna l homepage: www.e lsev ie r.com/ locate / j i e

Unions, competition and international trade in general equilibrium☆

Paulo Bastos a,b, Udo Kreickemeier b,⁎a Research Department, Inter-American Development Bank, United Statesb GEP, University of Nottingham, United Kingdom

☆ We would like to thank two anonymous referees aFalvey, Gabriel Felbermayr, Kjell-Erik Lommerud, Frodparticipants at the Göttingen Workshop of Internacomments. Financial support from the Leverhulme Trusis gratefully acknowledged. Paulo Bastos thanks Funda çfor financial support. The views expressed in this paperdo not necessarily correspond to those of the Inter-Ame⁎ Corresponding author. School of Economics, Universi

NG7 2RD, United Kingdom. Tel.: +44 115 951 4289.E-mail address: udo.kreickemeier@nottingham.ac.uk

1 See e.g. Rodrik (1997, pp. 23ff.) and the references c2 Data on collective bargaining coverage refer to the ye

average is based on information for 20 European countr

0022-1996/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.jinteco.2009.07.003

a b s t r a c t

a r t i c l e i n f o

Article history:Received 17 July 2008Received in revised form 14 July 2009Accepted 15 July 2009

Keywords:Trade unionsProduct market competitionGeneral oligopolistic equilibriumTrade liberalisation

JEL classification:F15F16L13

We develop a two-country, multi-sector model of oligopoly in which unionised and non-unionised sectorsinteract in general equilibrium. The model is used to study the impact of trade liberalisation, deunionisationand firm entry on wages in unionised and non-unionised sectors, and on welfare. We find that a shift fromautarky to free trade increases non-union wages and welfare, whereas the effect on union wages isambiguous. We also show that partial deunionisation leads to higher wages in both unionised and non-unionised sectors, but only increases welfare when the proportion of unionised sectors is sufficiently low.Finally, wages in non-unionised sectors necessarily increase with firm entry, while the response of unionwages and welfare depends on the trade regime.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

In recent years the labour market effects of increased globalisationhave inspired many passionate discussions. Advocates of globalisationoften argue that fiercer product market competition and trade liberal-isation have the potential to induce a general increase in living standards.Less enthusiastic observers, however, often voice the concern that amorecompetitive product market goes hand in hand with the erosion of tradeunion power, thereby implying the end of decent pay for many workers.1

This concern has been particularly noticeable in Europe, where thelabour market in most countries is characterised by a strong unionpresence. According to the OECD (2004), the proportion of theworkforce covered by union agreements was over 67% on average inEuropean nations, versus only 14% in the US.2 In addition to the widercoverage, the typical European collective bargaining system is more

s well as Hartmut Egger, Rode Meland, Peter Wright andtional Economics for helpfult (Programme Grant F114/BF)ão para a Ciência e a Tecnologiabelong to the authors only andrican Development Bank.ty of Nottingham, Nottingham,

(U. Kreickemeier).ited therein.ar 2000, and the (unweighted)ies.

l rights reserved.

centralised than its US counterpart, with wage negotiations takingplace predominantly at the industry-level. In recent years, however, anumber of countries have moved towards a more decentralised wagesetting system, with union bargaining at the level of the individualplant or firm being favoured instead.3

Howmight we expect harsher competition and trade liberalisation toimpact on labour market outcomes when unions are present? This is thequestionwe set out to address in our paper. There is, of course, a sizeablebodyof theoretical research that does just that. The frameworkused in thecentral contributions to this literature is the partial equilibrium oligopolymodel, augmented to allow for union wage setting in the labour market.An important early result in this literature, due to Huizinga (1993) andSørensen (1993), shows that in a symmetric two-country model wherelabourmarkets in both countries are unionised, thewage under free tradeis lower than in autarky. Using the same framework, Naylor (1998, 1999)looks at a complementary question and shows that in a situation ofrestricted trade a reduction in trade barriers increases wages. The publicperception that international trade reduces the power of labour unions istherefore supported by the model if one compares the two extremesituations of autarky and free trade, but not for the intermediate case ofgradual liberalisation.4 In addition to the papers that look at the situation

3 These include the UK and some Central and Eastern European nations. Wagenegotiations at the national level were once prominent in the Northern Europeancountries, but are now rarely observed in practice. For documentation see OECD(2004).

4 Munch and Skaksen (2002) allow for the presence of both fixed and variable tradecosts and show that the results are sensitive as to which of these costs is lowered.

239P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

where unions are present in both countries, there are some high-profilecontributions that look at the asymmetric case where unions are onlypresent in one of the countries.5

A key advantage of the oligopoly framework is to allow for strategicinteractions between firms and unions within an industry, and toinvestigate how these interactions are affected by lower trade barriers.An important shortcoming of that modelling approach, however, isthat it abstracts from general equilibrium linkages between sectors orbetween goods markets and factor markets that have traditionallybeen of interest to trade economists. In fact, the analysis in thisliterature focuses on a single industry, where oligopolistic competitiongenerates rents that organised labour seeks to capture in the form ofhigher wages. Although it is assumed that workers of the unionisedindustry can always find employment in a non-unionised sector, thewage rate in that industry (which constitutes the reservation wage ofunion workers) is exogenously given, and hence unaffected through-out the analysis.

In this paper, we develop a framework that allows for theinteraction between unionised and non-unionised sectors in generalequilibrium. To this end we build on the model by Neary (2009) whoprovides a theoretically consistent but tractable model of generaloligopolistic equilibrium (GOLE).6 There are a small number of firmsoperating in each of a continuum of sectors, yielding a framework inwhich firms are large in their own sector but small in the economy as awhole. Hence they behave strategically against other firms in theirown sector but treat factor prices and national income parametrically.As a distinguishing feature of our setup, we assume that unions arepresent in an exogenous subset of sectors — thereby transformingNeary's GOLE framework into a unionised general oligopolisticequilibrium (UGOLE) model. As each sector represents an infinitesimalpart of the economy, firms and unions behave as in partial equilibriummodels. In particular, as in Naylor (1998, 1999), unions set their wagedemands in partial equilibrium, taking as given the wage rate in non-union sectors. Aggregation across sectors allows for the endogenousdetermination of economy-wide variables, most importantly thecompetitive wage rate and aggregate welfare. The model is used tostudy the impact of trade liberalisation, deunionisation and firm entryon wages in unionised and non-unionised sectors, and on welfare.7

Our main results are as follows. Within a context of intra-industrytrade, further product market integration impacts on union wagesthrough two different channels. Firstly, as shown by Naylor (1998,1999), by reducing labour demand elasticity, integration leadsmonopoly unions to set higher wages. Secondly, by causing anincrease in aggregate labour demand, integration causes an increase inthe competitive wage, inducing a further rise in unionwages. Becauseof this additional positive (general equilibrium) effect, union wagesmay actually be higher under free trade than in autarky, a result thatcontrasts with the previous literature. Another well established resultfrom the unionised oligopoly in partial equilibrium, due to Dowrick

5 See Brander and Spencer (1988), and Mezzetti and Dinopoulos (1991). Theasymmetric oligopoly model has been extended by Lommerud et al. (2003) to allowfor FDI, while Straume (2003) and Lommerud et al. (2006b) look at internationalmergers, and Lommerud et al. (2006a) focus on technological change.

6 See also Neary (2003) for a non-technical overview, and Neary (2007) for anapplication to cross-border mergers.

7 There is an earlier literature introducing unions into the Heckscher–Ohlin trademodel, the classic paper being Johnson and Mieszkowski (1970). They model unions ina minimal way by simply assuming that their existence in one sector of the economyleads to an exogenous inter-sectoral wage differential. Indeed, other contributions tothis literature like Jones (1971) or the textbook treatment of Bhagwati et al. (1998,ch. 25) do not even mention unions as the source of the wage differential. Theliterature gave a lot of attention to the possibility that the capital-intensity ranking ofthe two sectors might depend on whether physical or value intensities were looked at.This case of “diverging factor intensities” was shown to yield the possibility ofparadoxical results, such as downward sloping goods supply functions — see Neary(1978) for a critical discussion.

(1989), states that firm entry does not have a direct impact on unionwages in a closed economy whenwages are set at the industry-level.8

In the UGOLE framework firm entry in all sectors of a closed economyincreases aggregate labour demand, leading to a higher competitivewage which in turn leads to a higher union wage. It is shown that thisresult has to be qualified somewhat in the open economy. Generalequilibrium links are also important when looking at the effects of“deunionisation”, i.e. a reduction in the proportion of sectors that areunionised: Aggregate labour demand increases, putting upwardpressure on both competitive and union wages.

Besides the wage effects, we also consider the aggregate welfareeffects of the different policy scenarios.We find that aggregatewelfareincreases as the economy moves from autarky to free trade, and riseswith harsher product market competition in the open economy, butnot in the closed economy. Our results also indicate that, although aperfectly competitive labour market always leads to the welfaremaximum, partial deunionisation may reduce aggregate welfare. Theremainder of the paper is organised as follows. Section 2 sets out thebasic model. Section 3 shows how the partial equilibrium in eachsector is determined, before Section 4 explores the general oligopo-listic equilibrium. The comparative statics of themodel are analysed inSection 5. In Section 6 we consider two extensions to our model. Firstwe look at the casewhere unionwage setting is at the firm level ratherthan the sector level, and second we introduce technology differencesacross sectors in a parsimonious way by assuming that unionisedsectors are “low-tech” while non-unionised sectors are “high-tech”.Section 7 concludes.

2. Model setup

In this section we present a model of oligopoly in generalequilibrium which allows for labour market unionisation in part ofthe economy. In doing so, we generalise the model developed byNaylor (1998, 1999) in two ways: firstly, we extend Naylor's partialequilibrium analysis to the case where n firms operate in each countryand union wage setting occurs at the industry-level; secondly, weembed the resulting framework into the GOLE model introduced byNeary (2009).

Consider, then, a world consisting of two countries, 1 and 2, whichare assumed to be identical in all respects. We describe the economyof country 1, simply noting that analogous conditions hold in country2.

2.1. Technology

In country 1 there is continuum [0,1] of imperfectly competitiveindustries, each producing a differentiated good. Each industry has nsymmetric firms, where n is small. Hence, firms are relatively large intheir own industry but represent an infinitesimal part of the economyas a whole. As a result, they have market power within their ownsector but treat economy-wide variables parametrically.

Competition in each industry is Cournot. There are unspecifiedbarriers facing new firms, and hence oligopoly rents are not eroded byentry. All income accrues to the aggregate household. Labour is theonly factor of production. The marginal product of labour is constant,and is normalised to unity so that we can discuss output andemployment interchangeably. In line with Brander (1981), nationalmarkets are assumed to be segmented and there is a specific tariff tper unit of commodity traded internationally.

8 See also Dhillon and Petrakis (2002) and Naylor (2002).

240 P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

2.2. Preferences, demand and utility

The representative consumer in country 1 has an additivelyseparable utility function defined over the continuum of goods, witheach sub-utility function quadratic:

U1 x1 zð Þf g½ � =Z 1

0ax1 zð Þ− 1

2bx1 zð Þ2

� �dz ð1Þ

where x1(z) denotes consumption of good z in country 1. Utility ismaximised subject to the budget constraint:

Z 1

0p1 zð Þx1 zð Þdz V I1 ð2Þ

where p1(z) denotes price of the good z, and I1 is aggregate income.The first order conditions give the inverse demand functions for eachgood:

p1 zð Þ = 1λ1

a − bx1 zð Þ½ �; with λ1 p1 zð Þf g; I1½ � = aμ1 − bI1σ2

1

: ð3Þ

Here, λ1 is the marginal utility of income, which is the Lagrangemultiplier attached to the budget constraint, and μ1 and σ1

2 are thefirst and second moment (mean and “uncentred variance”) of prices,respectively:

μ1 =Z 1

0p1 zð Þdz σ2

1 =Z 1

0p21 zð Þdz ð4Þ

Firms treat λ1, which is determined in general equilibrium, parame-trically, and hence the perceived subjective inverse demand functionsare linear.

The indirect utility function can be obtained by substituting thedemand function in Eq. (3) into Eq. (1), which leads to:

U1 =a2 − σ2

1

2bð5Þ

Hence, aggregate consumer welfare is strictly decreasing in thesecond moment of prices.9

2.3. Trade unions and the labour market

There are L workers in country 1. Workers are ex-ante identical inall respects, but their wage depends on the institutional features of theindustry inwhich they are employed. Labourmarket institutions differacross sectors: Trade unions are present in some sectors but not inothers. Sectors are ordered in such a way that those where tradeunions are present have low values of z. There is a threshold sector z ,with z ∈ [0,1], such that trade unions are present in all sectors forwhich z is less than or equal to z, while they are absent in all othersectors, making the labour market in these latter sectors perfectlycompetitive.

In each sector z∈ [0,z], there is a single trade union representing allworkers employed by the firms operating in that sector. We adopt aStone–Geary utility function to represent the union's preferences,assuming that each union aims to maximise rents (weighted by the

9 As stressed by Neary (2009), the quadratic specification of preferences in Eq. (1),though non-homothetic, is a special case of the Gorman (1961) polar form. Thisproperty allows for consistent aggregation over individuals with different incomes(provided the parameter b is the same for all), and enables the use of a singlerepresentative consumer to characterise demands in each country. Furthermore, itfacilitates the normative applications of the model, as it rationalises the use of theindirect utility function of the single representative consumer to evaluate aggregateconsumer welfare in each country.

marginal utility of income). Hence, the utility of each union in countrycan be written as:

X1 zð Þ = λ1 w1 zð Þ− wc1

� �l1 zð Þ ð6Þ

where w1(z) is the nominal union wage in sector z, w1c is the nominal

wage in the non-union sectors, and l1(z) represents total demand forlabour from thefirms that operate in sector z in country 1. Hence, l1(z)=n[y11(z)+y12(z)], where y11(z) and y12(z) represent the output(employment) level of a firm in sector z of country 1 for its home andforeign market, respectively.

As in Naylor (1998, 1999) we adopt a monopoly union frameworkto represent wage determination in each unionised sector: The tradeunion sets the wage and, subsequently, firms choose the level ofemployment. Wage setting occurs simultaneously in all unionisedsectors, and each union treats parametrically the wage set by thecorresponding foreign union. Due to the assumption of a continuum ofsectors, unions are small in their own economy, and therefore takeaggregate income, product prices in the other sectors, and factorprices in the rest of the economy as given when setting their wagedemands.

3. Solving the model in partial equilibrium

In each union sector, wages and employment can be described asthe outcome of a two-stage game. In stage 1, each union chooses itswage, taking as given wc and the wage demand of the correspondingforeign union. In stage 2, each firm chooses its output (and henceemployment), taking as given the wages set in the first stage and theoutput of competitors, both at home and abroad. We solve bybackward induction. In non-union sectors, the model is a simple one-stage Cournot game: Firms choose employment, taking as given thecompetitive wage rate and the output of competitors. Since countriesare identical in all respects, it must be the case that themarginal utilityof income and the competitive wage are the same for both countries:λ1=λ2 and w1

c=w2c , and therefore country indices for λ and wc are

omitted henceforth.

3.1. Production

Profits of a typical firm in sector z of country 1 are given by

π1 zð Þ = p1 zð Þ− c1 zð Þ½ �y11 zð Þ + p2 zð Þ− c1 zð Þ− t½ �y12 zð Þ; ð7Þ

where c1(z) is the marginal cost of labour faced by the typical firm insector z, and we have

c1 zð Þ =w1 if z V z

wc if z N z

(ð8Þ

Maximisation of Eq. (7) leads to the reaction functions of each firm insector z of country 1, which are given by the following standardexpressions:

y11 zð Þ = a − λc1 zð Þb n + 1ð Þ − n

n + 1y21 zð Þ ð9Þ

y12 zð Þ¼ a − λ c1 zð Þþtð Þb nþ 1ð Þ − n

nþ 1y22 zð Þ ð10Þ

By solving these, we may obtain the equilibrium output of each firm,under each possible trade regime.

241P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

For sufficiently high trade costs, there will be no trade, and Eq. (9)simplifies to

y11 zð Þ = a − λc1 zð Þb n + 1ð Þ : ð11Þ

For sufficiently low trade costs, firms in both countries will start toexport. There is two-way trade in equilibrium, and we get

y11 zð Þ = a + λ n c2 zð Þ + tð Þ− n + 1ð Þc1 zð Þ½ �b 2n + 1ð Þ ð12Þ

y12 zð Þ = a + λ nc2 zð Þ− n + 1ð Þ c1 zð Þ + tð Þ½ �b 2n + 1ð Þ ð13Þ

as the outputs (or employment levels) of a firm in sector z for theirrespective domestic and export market. Notably, Eqs. (6), (12) and(13) are homogenous of degree zero in λ−1, w1, wc and t. As in Neary(2003), we choose the marginal utility of income as the numeraire,thereby normalising λ=1. Hence, the union wage w1 is to beinterpreted as a real wage at the margin (Neary, 2007), and similarlyfor the competitive wage wc.

3.2. Union wage setting

We now proceed by analysing union wage setting under autarkyand two-way trade.10 In autarky, trade unions in both countries settheir wage demands in isolation. Union utility is then given by:

X1 = w1 − wc� �n

a − w1

b n + 1ð Þ ð14Þ

By maximising Eq. (14) with respect to w1 we obtain the unionwage demand under autarky:

w1 =a + wc

2ð15Þ

We show in the Appendix that for sufficiently low trade costs eachunion will find it optimal to abandon its previous high-wage strategyand instead lower their wage demand in order to allow thecorresponding firms to compete internationally, benefiting from theemployment gains associated with such strategy. There is, therefore,two-way trade and union utility is given by:

X1 = w1 − wc� �n

a − n + 1ð Þw1 + n w2 + tð Þb 2n + 1ð Þ +

a − n + 1ð Þ w1 + tð Þ + nw2

b 2n + 1ð Þ� �

ð16ÞMaximising Eq. (16) with respect to w1, we obtain the low wage

best reply function of each union in country 1 to the wage demand ofthe corresponding union in country 2, for a given competitivewagewc:

w1 =2a − t − 2 n + 1ð Þwc + 2nw2

4 n + 1ð Þ

Given perfect symmetry, there is an analogous best reply functionfor each union in country 2. The sub-game perfect Nash equilibrium isgiven by:

w1 = w2 = w =2a − t + 2 n + 1ð Þwc

2 n + 2ð Þ ð17Þ

Hence, within the two-way trade regime the wage set by eachunion increases if trade is liberalised, i.e. if t falls. Thus, the key result

10 Since unions treat economy-wide prices parametrically, the choice of the marginalutility of income as the numeraire does not matter for their wage setting strategy.

of Naylor's (1998, 1999) unionised duopoly model remains valid in asetupwithmultiple firms and industry-level wage setting. SubtractingEq. (15) from Eq. (17), we find that the difference between unionwages in two-way trade and autarky is given by

Du −n a − wc� �+ t

2 n + 2ð Þ b 0: ð18Þ

It is therefore clear that, in partial equilibrium, union wages arelower under free trade (t=0) than in autarky, and the earlier findingsof Huizinga (1993) and Sørensen (1993) continue to hold in oursetting. These are partial equilibrium results in the sense that thecompetitive wage wc is treated as a parameter.

4. General oligopolistic equilibrium

Wenow turn to the determinationofwc in general equilibrium. Sinceindustries with trade unions pay a wage premium, workers naturallyprefer to be employed in those industries. However, the numberof high-wage jobs is limited by the labour demand of firms in the unionisedsectors. The allocation of workers to sectors is determined by a lottery.Luckyworkers find employment in a unionised sector, unluckyworkersbecome part of the labour supply available to non-union sectors. Thewage rate in non-union sectors wc is obtained from the equilibriumlabour market condition that exogenous labour supply must equal totallabourdemand in the economy (that is, the sumof labourdemand in theunionised and non-unionised sectors).

4.1. Equilibrium wages in union and non-union sectors

The full-employment condition in country 1 is given by:

L = nZ 1

0y11 zð Þ + y12 zð Þ½ �dz ð19Þ

In autarky, the general equilibrium level of wages in non-unionsectors is obtained by substituting in the full-employment condition fory11(z), using Eqs. (8), (11) and (15), and setting y12(z)=0. This gives

wc = a −2 n + 1ð Þ2− zð Þn bL: ð20Þ

Substituting wc back into the partial equilibrium union wagedemand function Eq. (15), we get the union wage expressed in termsof the model parameters:

w = a − n + 12− zð Þn bL ð21Þ

If there is two-way trade, general equilibrium wages are obtainedin a similar way. In particular, to find the competitive wage wesubstitute in full-employment condition (19) for y11(z) and y12(z),using Eqs. (8), (12), (13) and (17). This gives:

wc = a − t2

− 2n + 1ð Þ n + 2ð Þ2n n + 2− zð Þ bL ð22Þ

Substituting Eq. (22) back into Eq. (17) we obtain the equilibriumlevel of wages in union sectors as a function of the model parameters:

w = a − t2

− 2n + 1ð Þ n + 1ð Þ2n n + 2− zð Þ bL ð23Þ

There is a straightforward link between our equations todetermine the competitive wage in Eqs. (20) and (22) under autarky

Table 1Outputs, prices and profits in union (U) and non-union (NU) sectors.

n(y11+y12) p1 n(π11+π12)

Autarky U 12− z

L a − 12− z

bL b

n 2− zð Þ2L2

NU2

2− zL a − 2

2− zbL 4b

n 2− zð Þ2L2

Trade Un + 1

n + 2− zL a − n + 1

n + 2− zbL ðn + 1Þ2bL

2nðn + 2− zÞ þ nth i

L2n n + 2− zð Þ

NUn + 2

n + 2− zL a − n + 2

n + 2− zbL n + 1ð Þ 2n + 1ð ÞbL

2n n + 2− zð Þ þ nth i

L2n n + 2− zð Þ

242 P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

and trade, respectively, with the corresponding equations in Neary(2009). In particular, the corresponding equations coincide if bothmodels are stripped of the features that lead to asymmetries betweenmarkets and sectors: in our model this would require to eliminateunionisation in all sectors (z =0) and to set t=0, while in Neary'soriginal GOLE model it would mean to eliminate technologydifferences between sectors and countries.

4.2. Output, prices, profits, and welfare

Using the above expressions for general equilibrium wages inunion and non-union industries, it is possible to derive the criticallevel of transport cost t⁎ below which there is two-way trade ingeneral equilibrium as a function of the model parameters. As shownin the Appendix, it is equal to:

tT =2

ffiffiffi2

p− 1

n + 1ð Þ

n 2 + n − zð Þ bL ð24Þ

It is furthermore straightforward to derive the expressions foroutput, prices and industry profits for each sector, under both autarkyand two-way trade.11 Table 1 reports the resulting expressions. As canbe expected, under both autarky and trade output and profits in non-unionised sectors are higher than in unionised sectors, while pricesare lower. Furthermore, sectoral outputs and prices in the tradeequilibrium can be seen to be independent of the tariff level.

Using Eq. (4) and the expressions for prices presented in Table 1 forboth union and non-union sectors it is straightforward to obtain thesecond moment of prices (and hence aggregate welfare) in country 1,for each trade regime. Under autarky in all sectors, the secondmoment of prices can be expressed as:

σ21 = a a − 2bLð Þ + 4− 3 z

2− zð Þ2 bLð Þ2 ð25Þ

Similarly, for the case of two-way trade in all sectors we find:

σ21 = a a − 2bLð Þ + n + 2ð Þ2 − 2n + 3ð Þ z

n + 2− zð Þ2 bLð Þ2 ð26Þ

11 To obtain the output of the representative firm in non-union and union sectorsunder autarky we need to substitute Eqs. (20) and (21), respectively, into Eq. (11).When there is two-way trade in all sectors, we need to substitute Eqs. (22) and (23),respectively, into Eqs. (12) and (13). Once industry output levels are known, it is thenstraightforward to obtain industry prices and profits by using Eqs. (3) and (7),respectively.

5. Comparative statics

Wenowconsider three types of comparative static exercises. First, asin the partial equilibrium model by Naylor (1998, 1999), two tradeliberalisation scenarios are analysed, namely a comparison of autarkywith free trade and a marginal liberalisation of tariffs within the regimeof restricted trade. Second, we assessmore closely the effect that unionshave in the economy by looking at the effects of deunionisation,modelled as a decrease in the number of sectors that are unionised. Thethird shock considered is one of firm entry in all sectors of the economy.

5.1. Trade liberalisation

Consider first the comparison between autarky and free trade. Theeffect of trade on the unionwagew follows directly from a comparisonbetween Eqs. (21) and (23). We find that the union wage is higherunder free trade than under autarky, and hence the well establishedpartial equilibrium result by Huizinga (1993) and Sørensen (1993) isoverturned, if the proportion of sectors that are unionised issufficiently large: zNz ⁎=1−1/(2n−1).

Intuitively, the role of z in determining the general equilibrium effectof trade on the union wage is best understood if one distinguishesbetween two partial effects: The impact effect, i.e. the change in theunion wage for a given level of wc, and the second-round effect, i.e. theinduced change in wc in general equilibrium. We know from Eq. (18)that the impact effect at the sector level is negative, i.e. internationaltrade lowers thewage in every unionised sector, holding the value ofwc

constant. Hence the labour demand in all unionised firms — andtherefore aggregate labour demand — increases. At the pre-trade valueof wc there is therefore an excess demand for labour, putting upwardpressure on the competitivewage.12While the impacteffect at the sectorlevel is independent of proportion of sectors that are unionised, thesecond-round effect is not: With a higher z the impact effect occurs in alarger proportion of sectors, which implies a larger effect on aggregatelabour demand, and therefore a larger (positive) second-round effect onthe competitive wage. If z exceeds the critical level z⁎ the second-roundeffect is sufficiently large toovercompensate the impact effect, leading toan increase in the general equilibrium union wage.

Since both wage rates are expressed in units of the marginal utilityof income, no welfare significance should be attributed to a change inthese wage rates. Rather, from Eq. (5) the welfare effect is determinedby the change in the second moment of prices. Due to the fall in thegeneral equilibrium union wage premium, there is a reallocation ofemployment from non-unionised to unionised sectors and a decline inthe price premium of unionised sectors. As a result, the secondmoment of prices is lower with free trade than in autarky, and hencewelfare is higher. The results are summarised as follows:

12 In addition there is a direct positive effect of free trade on the labour demand of allfirms, as shown by Eqs. (11), (12) and (13).

243P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

Proposition 1. A shift from autarky to free trade leads to (i) an increasein the competitive wage, (ii) an increase in the union wage if z Nz ⁎=1−1/(2n−1), and (iii) an increase in welfare.

We now look at the intermediate case of restricted two-way tradeand consider the effects of a marginal reduction in bilateral tariffs. It isimmediate from Eqs. (22) and (23) that dw/dt=dwc/dt=−1/2, andhence a marginal reduction in tariff rates increases the competitivewage and the union wage by the same amount, leaving the absoluteunion wage premium constant. The effect on the union wage can becompared to the corresponding partial equilibrium effect (Naylor,1998) by combining Eqs. (17) and (22):

dwdt

=AwAt

+AwAwc

Awc

At= − 1

2 n + 2ð Þ +n + 1

2 n + 2ð Þ� �

= − 12

The first term in brackets is the well-known partial equilibriumeffect, the second term is the general equilibrium effect. One can seethat the two effects work in the same direction, but the generalequilibrium effect is larger, with its relative importance increasing inthe number of firms in each sector.

The constancy of the absolute union wage premium w−wc impliesthat a reduction in t also leaves relative employment levels across sectorsand relative goods prices unchanged. Inspection of Table 1 shows theeven stronger result that absolute goods prices in all sectors stay constant.Hence the second moment of prices is constant, and so is aggregatewelfare.13 A marginal reduction in tariffs has therefore a purelyredistributive effect: While wages increase, as just shown, an inspectionof Table 1 shows thatprofits unambiguously decrease,with thedifferencein profits between unionised and non-unionised sectors unchanged.14

The results are summarised as follows:

Proposition 2. Froman initial situation of two-way trade in all industries,a marginal reduction in tariffs increases wages and reduces profits in bothunionised and non-unionised sectors but has no impact on sectoralemployment, product prices and aggregate welfare.

Notably, Propositions 1 and 2 together imply that welfare is higherin all equilibria with two-way trade than in autarky.

5.2. Deunionisation

Consider now the implications of deunionisation, whichwemodel asa marginal decrease in z, the proportion of sectors that are unionised, inboth countries. The effects on the competitive wage and the unionwagecan be directly inferred from Eqs. (20) to (23): Reducing z increases thecompetitivewage and theunionwage,while reducing the absolute unionwagepremium,underbothautarkyand two-way trade. The intuition is asfollows. Since non-unionised sectors pay a lower wage, increasing theirnumber while reducing the number of unionised sectors one-for-oneincreases aggregate labour demand, thereby putting upward pressure onthe competitivewagewc. An increase in the competitivewage induces anincrease of the union wage that is smaller in absolute value — seeEqs. (15) and (17) — and hence the stated results follow.

For the analysis of welfare effects it is useful to look at the polarcases first. For z=0 we have p=a−bL in all sectors, while for z →1we have p→a−bL in the unionised sectors (see Table 1), which in

13 Notably, with identical production technology in all sectors trade liberalisationleaves not only relative outputs across sectors constant, but has also no effect onaggregate output. This is no longer the case with sector specific technologies, seeSection 6.2.14 The unambiguously negative effect of trade liberalisation on profits in unionisedsectors stands in marked contrast to the partial equilibrium result in Naylor (1998),where it is shown that trade liberalisation increases profits for sufficiently low levels oftrade cost.

this case make up “nearly” all sectors of the economy. Hence thesecond moment of prices is equal to p2=(a−bL)2 in the absence ofunionisation, while it approaches the same value if the economyapproaches the limiting case of full unionisation. The second momentof prices is maximised, and hence aggregate welfare is minimised, atsome intermediate level of unionisation zu, which is the thresholdbelow which further deunionisation increases welfare. From partialdifferentiation of Eqs. (25) and (26) we find z u=2/3 in autarky andz u=(2+n) /(3+2n) under two-way trade. Therefore, with two-waytrade in all sectors, the threshold level of unionisation zu is lower thanunder autarky, and this difference increases with the number of firmsoperating in each industry. The results are summarised as follows:

Proposition 3. Deunionisation increases the competitive wage and theunion wage and reduces the union wage premium. Deunionisationincreases welfare once the proportion of sectors that are unionised fallsbelow a threshold level.

5.3. Firm entry

In a closed economy, the wage elasticity of labour demand faced byeach sectoral union is independent of the number of firms. For thisreason, firm entry has no direct impact on union wages, a result that iswell known in the literature (Dowrick,1989;Dhillon and Petrakis, 2002;Naylor, 2002) and can be seen to hold in the partial equilibrium setup inSection 3 of the present paper, Eq. (15). This changes once we move tothe UGOLE model: As in Neary (2003), an increase in n in all sectorsincreases the competitivewage in the closed economy, and therefore theunion wage as well. Specifically, from Eqs. (20) and (21) we find

dwdn

=12dwc

dn=

bLn2 2− zð Þ N 0

As unionwages increase less thanproportionallywithwc, the unionwage premium necessarily falls as the number of firms operating ineach sector increases. With a declining union wage premium therelative output of firms in unionised sectors increases. This does nottranslate into higher output in unionised sectors though, because thenumber of firms in all sectors is increased, and the firms entering non-unionised sectors are larger. In fact, both effects compensate each otherexactly, and relative sectoral output levels as well as goods prices stayconstant (see Table 1). Hence, welfare is unaffected as well. Firm entryin the closed economy therefore has a pure redistribution effect, whereTable 1 shows the decrease in profits that accompanies the wageincrease.

With two-way trade in all sectors, it can easily be verified frominspecting Eqs. (22) and (23) that firm entry increases thecompetitive wage, has an ambiguous effect on the union wage, andreduces the union wage premium w−wc. In the open economy, theresulting increase in the relative size of unionised firms is alsoreflected in an increasing relative output of unionised sectors (seeTable 1). The price of unionised sectors falls, the price of non-unionised sectors increases, and the resulting lower price variancemeans that aggregate welfare goes up. In summary we have:

Proposition 4. In autarky, symmetric firm entry in all sectors leads tohigher competitive and union wages while welfare stays constant. Withtwo-way trade, symmetric firm entry in all sectors increases thecompetitive wage and has an ambiguous impact on the union wage.Welfare increases.

6. Extensions

In this section we extend the analysis in two directions. First, wedeviate from our assumption of sector level unions made in the mainpart of the paper, and instead look at the case where wage setting is

Table 2Outputs, prices and profits with firm level unions.

n(y11+y12) p1 n(π11+π12)

Autarky Un

1 + n − zL a − n

1 + n − zbL n

1 + n− zð Þ2bL2

NU1 + n

1 + n − zL a − 1 + n

1 + n − zbL 1 + nð Þ2

n 1 + n− zð Þ2bL2

Free Trade U2n

1 + 2n − zL a − 2n

1 + 2n − zbL 2n

z−1−2nð Þ2bL2

NU 1 + 2n1 + 2n − z

L a − 1 + 2n1 + 2n − z

bL 1 + 2nð Þ2n z−1−2nð Þ2

bL2

244 P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

less centralised, with unions that are specific to each firm.15 Second,we allow for technological asymmetries between sectors in aparsimonious way, by assuming that there are “low-tech” (traditional)sectors and “high-tech” sectors. In doing so, we focus on what isarguably the empirically plausible case in which only the low-techsectors are unionised.

6.1. Firm level unions

We now look at the case where in each sector z∈ [0,z] wages aredetermined by firm-specific rent-maximising monopoly unions. Wefocus on solving the model for the cases of autarky and free trade.Doing this, and going through the same steps as in Section 3.2, we get

w =a + nwc

n + 1ð27Þ

for the partial equilibrium union wage under autarky, and

w =a + 2nwc

2n + 1ð28Þ

for the partial equilibrium union wage under free trade.16 Comparingthis to Eqs. (15) and (17), respectively, one can see that in partialequilibrium wages are lower with firm-specific unions than with aunion that sets the wage for all firms in the sector, as long as n isgreater than one.17 In autarky it is only with the introduction of firmlevel unions that the wage becomes dependent on the number offirms, as shown in Eq. (27). With centralised wage setting in autarky,the union covers all workers and internalises the change in n, asshown in Eq. (15).

The difference between the free trade wage and the autarky wagein partial equilibrium is given by

D = − n a − wc� �n + 1ð Þ 2n + 1ð Þ b 0; ð29Þ

15 The UGOLE framework is not suited for looking at cases where wage setting ismore centralised than in our original setup, for example the case of one union settingthe wage for workers in all unionised sectors. This is because Neary's GOLE modelrelies on agents that are large in their sector but — due to the assumption of acontinuum of sectors — small in the economy as a whole. A union spanning multiplesectors would by contrast be the only large player in the economy. This fact would sitawkwardly with the assumption that all agents in the economy, including the union(s),treat goods prices and the marginal utility of income parametrically.16 We limit the analysis in this section to the polar cases of autarky and free trade,since with 2n unions in each sector in the open economy the determination of thecritical level of transport cost below which unions switch to a low wage strategy inorder to induce exports is not straightforward.17 Clearly, with only a single firm per sector (n=1) the cases of firm-specific andsector-specific unions are identical.

and comparing Eqs. (18) and (29) we find that the partial equilibriumeffect of trade on the unionwage—while still negative— is weakenedonce unions are firm-specific. Intuitively, the difference between theautarky and trade situations is less pronounced with wage setting atthe firm level since competition between unions in this case existsalready in autarky. Hence the transition to trade is less significant inthis respect than in the case of a sector level union, where competition(in the form of the foreign union) is only introduced by internationaltrade.

The link between partial and general equilibrium is still given byfull-employment condition (19). The general equilibrium competitivewage and union wage, respectively, follow as:

wc = a − n + 1ð Þ2n n + 1− zð Þ bL ð30Þ

w = a − n + 1n + 1− z

bL ð31Þ

The competitive wage and the union wage under free trade are:

wc = a − 2n + 1ð Þ22n 2n + 1− zð Þ bL ð32Þ

w = a − 2n + 12n + 1− z

bL ð33Þ

In analogy to Section 4.2 one can derive sectoral outputs, prices,and profits, and the results are shown in Table 2.

Finally, the second moment of prices can be shown to equal

σ21 = a a − 2bLð Þ + n + 1ð Þ2 − 2n + 1ð Þ z

n + 1− zð Þ2 bLð Þ2 ð34Þ

in autarky, while with free trade in all sectors it equals

σ21 = a a − 2bLð Þ + 2n + 1ð Þ2 − 4n + 1ð Þ z

2n + 1− zð Þ2 bLð Þ2: ð35Þ

Comparing the equations determining the general equilibrium valuesof variables to those in Section 4, one can easily check that with firmlevel unions union wages are strictly lower, whereas the competitivewage is strictly higher, in both trade regimes. Hence, sectoral outputsin union (non-union) sectors are higher (lower) than with industry-level unions, which translates into a lower variance of prices andhigher aggregate welfare. We now highlight how some of the keycomparative static results of our analysis above are affected by thechange in the scope of unions from sector level to firm level.

245P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

In a key departure from our results in Section 5 the comparison ofEqs. (31) and (33) shows that the general equilibrium union wage isnow higher under free trade than under autarky for all values of z . Inanalogy to the case of sector level unions, the intuition behind thisresult is most easily grasped by decomposing the general equilibriumeffect of trade on the unionwage into the impact effect onw for a givenlevel ofwc, and the second-round effect, i.e. the induced effect onwc ingeneral equilibrium. As argued above in our discussion of Eq. (29), inour current setup trade lowers partial equilibrium unionwages by lessthan in a situation of centralised wage setting at the sector level. Theimpact effect on the union wage, while still negative, is thereforesmaller than in our benchmark situation discussed in Section 5. Due tothe smaller impact effect the second-round effect on the competitivewage is smaller as well, but it is still large enough to overturn theimpact effect, even if the proportion of unionised sectors is low.

One situation in which it is well known from the partialequilibrium literature that the scope of unions matters is the marketentry of firms. While, as discussed above, for a closed economy inpartial equilibrium the effect of firm entry on the union wage is zerowith sector level unions, it is known fromDowrick (1989), Dhillon andPetrakis (2002) and Naylor (2002) that this wage effect turns negativeif unions are firm specific. Moving from partial to general equilibrium,the effect on the competitive wage and the union wage of an increasein n in all sectors is found by differentiating Eqs. (30) and (31). We get

dwc

dn=

n + 1ð Þ n + 1 + n − 1ð Þ zð Þn2 n + 1− zð Þ2 bL and

dwdn

=z

n + 1− zð Þ2 bL;

and hence the positive general equilibrium effect of the increase in non the competitive wage is sufficient to make the effect on the unionwage positive as well, thereby overturning the negative partialequilibrium effect. As in Section 4 the increase in n can furthermorebe seen to reduce the unionwage premium, leading to an expansion inthe relative size of unionised firms in the closed economy. Now,however, this is also reflected in an increase in the relative output ofunionised sectors and a relative decrease in union sector prices (seeTable 2). The second moment of prices falls, and hence firm entry inthe closed economy is now welfare increasing.

6.2. Technology differences across sectors

Each sector requires a strictly positive labour input per unit ofoutput, denoted θ1(z). We assume as before that θ1(z)=1 in all union

Table 3Outputs with sector specific technologies.

Autarky U

NU

Total

Trade U

NU

Total

where A≡ z+2θ2(1−z ) and B≡(n+1)z+(n+2)θ2(1−z ).

sectors. In non-union sectors the unit labour requirement is nowstrictly lower than unity: θ1(z)=θb1. The marginal cost of the typicalfirm operating in each sector is therefore given by:

c1 zð Þ =w1 if z V z

θwc if z N z

(ð36Þ

The model is solved exactly as before, and the partial equilibriumwage Eqs. (15) and (17) still hold. The full-employment condition isnow as follows:

L1 = nZ 1

0θ1 zð Þ y11 zð Þ + y12 zð Þ½ �dz; ð37Þ

andwhen substituting for sectoral output levels Eq. (36) now replaces(8). The competitive wage under autarky is now given by:

wc =n 2θ − 2θ − 1ð Þ z½ �a − 2 n + 1ð ÞbL

n 2θ2 − 2θ2 − 1� �

z� � ð38Þ

Substituting back into Eq. (15) one can derive the union wage as:

w =n θ 1 + θð Þ− θ 1 + θð Þ− 1½ � zf ga − n + 1ð ÞbL

n 2θ2 − 2θ2 − 1� �

z� � ð39Þ

Analogously, competitive and union wages under two-way tradeare given by

wc =n n + 2ð Þθ − n + 2ð Þθ − n + 1ð Þ½ � zf g 2a − tð Þ− 2n + 1ð Þ n + 2ð ÞbL

2n n + 2ð Þθ2 − n + 2ð Þθ2 − n + 1ð Þ� �z

�ð40Þ

and

w =n n + 1 + θð Þθ − n + 1 + θð Þθ − n + 1ð Þ½ � zf g 2a − tð Þ− 2n + 1ð Þ n + 1ð ÞbL

2n n + 2ð Þθ2 − n + 2ð Þθ2 − n + 1ð Þ� �z

� ;

ð41Þ

respectively. It is easily checked that the expressions in Eqs. (38) to (41)collapse to the respective expressions in Eqs. (20) to (23) if we set θ=1.

As in the previous section, rather than trying to be exhaustive wehighlight those results for which the model extension consideredchanges the results from our baseline framework in interesting ways. Inparticular, we analyse the case of marginal trade liberalisation undertwo-way trade and the caseoffirmentry in autarky, since inourbaseline

n(y11+y12)

1AL − a 1− zð Þ 1− θð Þθ

bAn

n + 1

2θAL + a z 1− θð Þ

bAn

n + 1

z + 2θ 1− zð ÞA

L + a 1− zð Þ z 1−θð Þ2bA

nn + 1

n + 1B

L − n n + 1ð Þθ 1− θð Þ 1− zð Þ 2a − tð Þb 2n + 1ð ÞB

n + 2ð ÞθB

L + n n + 1ð Þ z 1− θð Þ 2a − tð Þb 2n + 1ð ÞB

n + 1ð Þ z + n + 2ð Þ 1− zð ÞθB

L + n n + 1ð Þ 1− zð Þ z 1−θð Þ2 2a − tð Þb 2n + 1ð ÞB

246 P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

model with identical technology in all sectors these two comparativestatic exercises leave welfare constant. This is in line, of course, with theobservation by Neary (2009, p. 10) that in his model the possibility of apositivewelfare effect of increased competition crucially dependson theexistence of technology differences between sectors: Increased compe-tition in this case leads to labour being reallocated to more efficientsectors, thereby allowing aggregate output to increase.

Both comparative static exercises considered in the context of ourmodel with technology differences across sectors — marginal tradeliberalisation and firm entry— are forms of an increase in competition.Following the steps laid out in Section 4.2 it is straightforward toderive expressions for sectoral and total outputs, and the resultingexpressions are given in Table 3. It is immediate from inspection of theexpressions in the table that an increase in n in autarky and a decreasein t under two-way trade lead to a reallocation of labour to moreefficient sectors, with aggregate output unambiguously increasing.

Despite these unambiguous output effects, simulations show thatthe welfare effect in both cases can have either sign. From Eq. (5),welfare is strictly decreasing in the second moment of prices, which inturn is affected by increased competition via two effects that work inopposite directions: First, the relative goodsprice chargedbyfirms in themore efficient non-unionised sectors decreases. This effect increases thegoods price differential and thereby, ceteris paribus, the secondmomentof prices. Second, increased competition leads to a decrease in averagegoods prices, thereby decreasing the second moment of prices, ceterisparibus. While the same two effects are present in Neary (2009), thepresence of unions in the less efficient sectors means that the pricedifferential between sectors for a given technology difference is larger inour model. As a consequence the net effect of increased competition onthe second moment of prices, and hence the overall welfare effect, isambiguous.

7. Concluding remarks

We have developed a model of oligopoly in general equilibriumfor investigating the effects of trade liberalisation, deunionisationand firm entry onwages in unionised and non-unionised sectors, andon aggregate welfare. In this framework, unions and firms behave asin partial equilibrium models, but aggregation across sectors allowsfor the endogenous determination of economy-wide variables, mostimportantly the competitive wage and aggregate welfare. Generalequilibrium interactions between unionised and non-unionisedsectors play an important role in our analysis of the wage effects inthe different policy scenarios. Indeed, we have shown that tradeliberalisation and firm entry impact not only on union wage settingincentives directly, but also on the outside option of unionisedworkers. For this reason, wages in unionised sectors may increasewith firm entry and be higher with free trade than in autarky.Furthermore, we have shown that partial deunionisation increaseswages in both unionised and non-unionised sectors, but only leads tohigher welfare when the proportion of unionised sectors issufficiently low.

While these results are interesting in their own right, an importantcontribution of this paper is to offer a general equilibrium frameworkfor studying the implications of competition policy and globalisationwhen trade unions are present. One way in which this frameworkmight usefully be extended in the future is by introducing asymmetriesbetween the countries, e.g. in labour market institutions, labourproductivity, or market size. Other promising avenues for futureresearch include allowing for national and international mergers, orforeign sourcing.

Appendix A

Here, we derive the value of t below which there is two-way tradein all sectors. In union sectors, the transition between autarky and

two-way trade occurs when bilateral tariffs are sufficiently low suchthat each union finds it optimal to abandon its previous high-wagestrategy and instead lower their wage demand in order to allow thecorresponding firms to compete internationally. In non-union sectors,firms treat wc parametrically and hence the trade regime dependssolely on their decisions. While the trade regime is determined by theactions of unions and/or firms in partial equilibrium, the boundaryconditions may also be expressed in terms of the model parameters ingeneral equilibrium.

We begin by deriving the boundary condition under which there istwo-way trade in union sectors. To represent a Nash equilibrium, thewage under two-way trade in Eq. (17) needs to exceed a critical level,which has been defined by Naylor (1999) as the switching wage. Fromthe point of view of country 1, the switching wage is the wage ŵ2 ofcountry 2 firms that makes each union in country 1 indifferent betweena high-wage strategy and a low wage strategy, given that the firms ofcountry 2 are exporting. The maximum union utility associated with ahigh-wage strategy is given by:

XH1 = w1 − wc� �

ny11 ðA:1Þ

where y11 is given by Eq. (12) and w1=arg maxΩ1H. After straightfor-

ward computations, we may express Eq. (A.1) as:

XH1 = n

a−wc + n t−wc + w2� �� �2

4b 1 + nð Þ 1 + 2nð Þ ðA:2ÞSimilarly, themaximumutility associatedwith a lowwage strategy

is given by:

XL1 = w1 − wc� �

n y11 + y12ð Þ ðA:3Þ

where from Eqs. (12) and (13) we have y11+y12=(2a− t−2(1+n)w1+2nw2)/(b(2n+1)) and w1=argmaxΩ1

L . Therefore, it is readilyshown that:

XL1 = n

t−2a + 2 1 + nð Þwc−2nw2� �2

8b 1 + nð Þ 1 + 2nð Þ ðA:4Þ

From comparison of Eqs. (A.2) and (A.4), each union in country 1will be indifferent between the two strategies if the wage of country 2firms equals:

w2 = − an

+n + 1

nwc +

2 +ffiffiffi2

p+ 2n + 2

ffiffiffi2

pn

2nt ðA:5Þ

The switching wage ŵ2 is increasing in t, while w2 under two-waytrade as given by Eq. (17) is decreasing in t. Hence, settingŵ2 equal tow2 and solving for t gives us the critical level of trade costs t⁎ belowwhich both unions would choose the low wage strategy. We get:

tT =4 1 + nð Þ

4 + n 7 + 2nð Þ +ffiffiffi2

p2 + nð Þ 1 + 2nð Þ a − wc� � ðA:6Þ

There is, therefore, two-way trade in unionised sectors if tb t⁎, withthe equilibrium wage given by Eq. (17).

We now turn to the analysis of the boundary condition belowwhich there is two-way trade in non-union sectors. From Eqs. (10)and (8) it follows that the critical level of t below which therepresentative firm starts exporting (y12N0) is given by:

tTT =1

n + 1a − wc� � ðA:7Þ

Comparing Eqs. (A.6) and (A.7) it can be easily checked thatt⁎⁎N t⁎. Hence, we conclude that the equilibrium trade regime is two-way trade in all sectors if t∈ [0,t⁎). Substituting Eq. (22) into Eq. (A.6)

247P. Bastos, U. Kreickemeier / Journal of International Economics 79 (2009) 238–247

we obtain the boundary condition t⁎ in terms of the modelparameters:

tT =2

ffiffiffi2

p− 1

n + 1ð Þ

n 2 + n − zð Þ bL ðA:8Þ

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