two-stage bargaining with coverage extension in a dual labour market

European Economic Review 44 (2000) 181}200

Two-stage bargaining with coverage extensionin a dual labour market

Mark A. Roberts!, Karsten Stvhr", Torben Tranvs#,*! University of Nottingham, Nottingham, UK

" Bank of Norway, Oslo, Norway# Institute of Economics, University of Copenhagen, Studiestr~de 6, DK-1455 Copenhagen K, Denmark

Received 1 March 1997; accepted 1 February 1998

Abstract

This paper studies coverage extension in a simple general equilibrium model witha dual labour market. The union sector is characterized by two-stage bargaining whereasthe "rms set wages in the non-union sector. In this model "rms and unions of the unionsector have a commonality of interest in extending coverage of a minimum wage to thenon-union sector. Furthermore, the union sector does not seek to increase the non-unionwage to a level above the market-clearing wage. In fact, it is optimal for the union sectorto impose a market-clearing wage on the non-union sector. Finally, coverage extensionincreases welfare in this model. ( 2000 Elsevier Science B.V. All rights reserved.

JEL classixcation: J42; J31; J50

Keywords: Coverage extension; Two-stage bargaining; Minimum wages; Dual labour market;Centralization

1. Introduction

A characteristic feature of continental European labour markets is that wagesnegotiated in a unionized sector may extend to other sectors.1 Coverage extends

*Corresponding author. Fax: #45 35 323 000, e-mail: [email protected] It is not just wage agreements that may extend beyond the unionized sector. Standard hours of

work, vacation length, and other working conditions may also be copied by or imposed on thenon-unionized sector (see OECD, 1994).

0014-2921/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 4 - 2 9 2 1 ( 9 8 ) 0 0 0 5 6 - 7

Fig. 1. Union density and coverage of collective contracts, 1994.Source: OECD (1997), Chapter 3.Legend: AUS } Australia, AUT } Austria, BEL } Belgium, CAN } Canada, CHE } Switzerland,DEN } Denmark, DEU } Germany, ESP } Spain, FIN } Finland, FRA } France, GBR } GreatBritain, ITL } Italy, JPN } Japan, NLD } The Netherlands, NOR } Norway, NZL } New Zealand,PRT } Portugal, SWE } Sweden, USA } United States.

considerably beyond the unionized sector in these countries as shown by Fig. 1.Moreover, while the share of the work force belonging to trade unions in manyindustrialized countries has fallen since 1980, the coverage of negotiated agree-ments has remained high in continental Europe.

Some interesting questions are raised when the parties bargaining on behalf ofthe unionized sector take coverage extension into account. Do the parties withinthe union sector, "rms and unions, have con#icting interests with respect to thenon-union wage? What would be the outcome if the union sector determines thenon-union wage? And, "nally, can coverage extension improve welfare?

The paper addresses these questions by constructing a two-sector generalequilibrium model with a dual labour market, a primary sector where unions areactive and a secondary sector which is non-unionized. A dual labour market isargued to be a plausible feature of reality by many authors, for instance.2Initially, the case considered is where the union sector bargains only locally andthe secondary sector is monopsonistic. The secondary sector wage will, thus, bebelow its market clearing level. We then consider the case where the primary

2A dual labour market is argued to be a plausible feature of reality by many authors, for instanceMcDonald and Solow (1985), Bulow and Summers (1986), Layard et al. (1991) and, more recently, byManning (1995) and Saint-Paul (1996).

182 M.A. Roberts et al. / European Economic Review 44 (2000) 181}200

sector engages in a "rst stage of central bargaining in order to establish a basewage to cover the whole economy, including the monopsonistic sector.3 Thebase wage can be interpreted as a minimum wage, which may be exceeded bythe primary sector union wage under a second stage of local bargaining.

There are di!erent interpretations of this model. There might be scope for theunion sector to set a minimum wage in absence of a statutory equivalent. Thischaracterizes the Nordic model where minimum wages are negotiated centrallyand uno$cially enforced.4 Alternatively, there might be a statutory minimumwage, but the authorities use wages determined in the union sector as a point ofreference for setting minimum wages. This procedure is followed in manysouthern European countries. Whatever the means of extension, the centralbargaining parties of the primary sector ought to take into consideration the factof extension of the base wage to other parts of the economy.

The assumption of monopsony in the secondary sector in the absence ofcoverage extension is used for two reasons. First, it causes the minimum wageset by the primary sector to be the actual non-union wage as long as theminimum wage is above the monopsony wage. Thus, the primary sector can setthe secondary sector wage at will, below or above the market clearing wage.Second, the assumption of monopsony in the absence of high minimum wagesor coverage extension has some empirical support. Local monopsony power orfrictions in labour mobility between regions or occupational submarkets createsome degree of monopsony at the bottom of wage hierarchies. Recently, someauthors have argued that this is likely to be the case (see Card and Krueger,1995), while others are sceptical (e.g. Neumark and Wascher, 1995). See also thesurvey by Boal and Ransom (1997).

Some interesting results emerge from our model because of the assumptionthat coverage extension is taken into account. First, "rms and unions in theprimary sector enjoy a commonality of interest with respect to the determina-tion of the non-union wage. The primary sector parties will seek to raise thevaluation of their joint surplus.

Second, it is optimal to impose a market clearing wage on the secondarysector so that the price of the secondary sector production is the lowest possible.When the wage in the non-union sector is below the market clearing level, anincrease in the wage increases employment and production and lowers theoutput price. Likewise, it will never be optimal for the primary sector to increasethe base wage above the market clearing level as the price of the secondarysector product would increase. Consequently, the optimal minimum wage is at

3This assumption is made to simplify the analysis; in reality, an agreement is likely to cover thenegotiating sector plus only some part of the rest of the economy (see Fig. 1).

4 In practice, the extension of agreements to non-union segments of the labour markets comeabout through the &persuasion' of "rms who depart from the agreement (see the discussion in OECD,1994).

M.A. Roberts et al. / European Economic Review 44 (2000) 181}200 183

the level clearing the labour market of the secondary market; the primary sectorhas no incentive to attract extra demand by inducing a higher secondary sectorprice via a non-clearing wage for this sector. This result does not depend onwhether the goods are substitutes or complements.

The results capture what appears to occur in many European labour marketswith a primary sector with high non-market clearing wages and job queues, anda secondary sector where the market clears. Market clearing in the secondarylabour market arises endogenously in our model from the actions of the primarysector participants.5

Our results also cast some light on another question, the widespread existenceof two-stage bargaining. This is a puzzle: what can the parties agree upon if theynegotiate twice that cannot be agreed upon negotiating only once? One wouldexpect the initial round of bargaining to be redundant, except, possibly, toestablish a higher #oor level for the ultimate bargain (see Moene et al., 1993;Holden, 1988). We show, nevertheless, that the wage setting in the union sectorcan be una!ected by the initial stage of bargaining, and yet both workers and"rms are still interested in a "rst centralized stage where central agreementsfacilitate coverage extension. So, if the existence of centralized agreements isa precondition for extending the coverage of certain labour market conditionswhich are desired by the union sector, such as a minimum wage, this constitutesat least one reason why a union sector may want to augment local "rm-levelbargaining with an initial stage of central bargaining.6

Finally, minimum wages seem to be higher in relation to average earnings inEurope than in the USA (Dolado et al., 1996). In the USA, however, polls haveshown overwhelming support from the public in general to an increase in theminimum wage (The Economist, 1996). When large well-paid groups in the USAare in favour of raising the minimum wage this could be for their own self-interest.

The rest of the paper is set up as follows. Section 2 presents the economy.Section 3 considers the case without central bargaining in the primary sectorbut monopsony power in the secondary sector. Section 4 addresses the incen-tives of the parties in the primary sector to introduce collective bargaining overa minimum wage at the central level. Section 5 discusses some of our assump-tions and concludes.

5Market clearing in the secondary labour market is not contradicted by empirical evidence fromEurope (see Dolado et al., 1996).

6Agreements do seem to spread to the economy outside the union sector more when centralizedbargaining is of some importance like in continental European countries. Looking at Fig. 1, thecountries close to the 453 line are those with the most decentralized labour markets, wherebargaining is predominately taking place at "rm level (the USA, Japan, Canada, and the UK) } withSwitzerland as an exception. In the Nordic countries the whole labour market tends to be organizedand, thus, there is less to cover outside the union sector. So naturally these countries tend to beplaced close to the 453degree line as well.


2. The economy

The economy is populated by a large number of households divided intocapitalists and workers. A capitalist household owns a "rm and receives all itsincome from the pro"ts of this "rm (i.e. they supply no labour). A workerhousehold does not own "rms but supplies labour for its only source of income.

The economy has two di!erent sectors producing two distinct consumptiongoods sold at perfectly competitive markets. Within each sector production iscarried out at spatially dispersed locations. There is a "xed number of locationsin each sector and at each of these locations one "rm and a mass of workers arepositioned. There is no entry and neither "rms nor workers can change theirlocation. All locations of each of the two sectors are identical and all the "rms inthe same sector produce the same output.

The two sectors have di!erent labour endowments. The primary sectorlocations are unionized whereas the secondary sector locations are withoutunions as the history and characteristics of this sector simply limits the power ofpotential unions, and existence of unions require a minimum level of unionpower (see Section 2.3 below).

Throughout it is assumed that the primary sector is the high-wage sector andthe secondary sector the low-wage sector (i.e. in Walrasian equilibrium, the wagein the former is the higher). What we have in mind is a labour market that ispartitioned according to both skill and occupation. Those supplying labour tothe primary sector can be thought of as skilled and those attached to thesecondary sector as unskilled. But each group is again located in each of theiroccupational or geographical groups.

2.1. The households

Household preferences are assumed to be separable in consumption anddisutility of work, and homothetic with respect to the consumption goods. Withthese assumptions we can represent the preferences of household i by thefunction

ui(x

1, x

2, e

i)"u#

i(x

1, x

2)!e

i. (1)

The variables x1

and x2

denote the consumption of the good produced in theprimary and the secondary sector, respectively. The function u#

iis homogeneous

of degree one measuring utility from consumption. The term ei3R

`can be

interpreted as the &cost' or disutility of working; employers demand the same"xed amount of e!ort from each employee and the disutility from delivering this"xed e!ort varies across households. Alternatively, e

*can be viewed as the value

of household production forsaken if household i takes up paid work.Assume that demand functions are well-de"ned and let x

i(p

1, p

2, X

i)"

(xi1(p

1, p

2, X

i), x

i2(p

1, p

2, X

i)) be the demand for consumption goods of


household i as a function of nominal income, Xi, and the output prices of each

sector, p1

and p2. The assumptions on the preferences imply that

xi(p

1, p

2, X

i)"X

ixi(p

1, p

2, 1). From this we can derive the indirect utility func-

tion, vi, of household i receiving income X

i, having disutility from work e

i, and

given the prices p1

and p2:

vi(p

1, p

2, X

i)"X

iu#i(x

i(p

1, p

2,1))!e

i.

The output price of the primary sector, p1, is normalized to unity and the

relative price is labelled p ("p2). Household i's indirect utility function then

becomes

vi(p, X

i)"h(p)X

i!e

i, (2)

where the function h(p),u#i(x

i(p

1, p

2, 1)) converts one unit of nominal income

into utility from consumption under the assumption that the household maxi-mizes utility. h(p) is assumed twice di!erentiable and h@(p)40 as the utilitycannot increase when the price increases.

2.2. Firms

In each of the two sectors there is a continuum of size unity of identical "rmseach occupying a location and producing a homogeneous product. The outputof the representative "rm of sector k"1, 2 is produced according to theproduction function

yk"f

k(lk), (3)

where yk

is the real output and lk

is labour input. The "rm has positive butdecreasing marginal product, i.e. f @

k(lk)'0 and f A

k(lk)(0.

Each "rm is owned by one capitalist household. Capitalists have utilityfunctions (2) with e

i"0, and each "rm maximizes the utility of its owner, i.e. it

maximizes pro"t measured in utility terms. Speci"cally, a "rm operating insector k"1, 2 maximizes

nk"h(p)[p

kfk(lk)!w

klk]. (4)

Note that since we have set p1equal to unity the wages are real wages in terms

of sector 1's output price and thus w1

is the real product wage of sector 1.

2.3. Workers

We now turn to the workers' labour supply decision. If employed in sector k,worker i receives the wage w

kand has the disutility e

i. If, alternatively, the

worker does not work, the household receives no income and has no disutilityfrom working (e

i"0). Therefore, using Eq. (2) it follows that worker i's labour

supply is 1 if h(p)wk5e

iand 0 otherwise, i.e. the worker works if the utility

derived from the wage exceeds the disutility from having to work.


There is a continuum of worker households indexed by i in each location.7The value of e

iis speci"c to each worker and private information; it is distrib-

uted according to gk(e

i) in each location in sector k; g

k(e

i) has full support and

is publicly known so that there is no aggregate uncertainty. For later usethe function e

k(lk) is de"ned. Workers supplying labour in a "rm in sector k are

ordered by increasing values of ei. The function e

k(lk) then denotes the aggregate

disutility from work of all workers employed by this "rm if the employment islk

and only those with the lowest disutility from work are employed;

ek(lk)"P

lk

0

eigk(e

i) di. (5)

The marginal disutility at employment level lk

is e@k(lk),Le

k(lk)/Ll

k, which is

the disutility of the employed worker with the highest value of ei. Thus, by

construction the function ek(lk) is continuously di!erentiable with e@

k(lk)'0.

Furthermore, it is assumed that eAk(lk)'0 and e@@@

k(lk)50.

The workers supplying their labour in a "rm in the primary sector areorganized in a local union. The members of the representative union are thecontinuum [0, m], ordered according to increasing values of e

i. Membership, m,

is a function of the utility from employment so that for a given wage w1, all

workers of the representative location for whom h(p)w15e

iare members of the

union.8It is assumed that each union has a utilitarian utility function maximizing the

aggregate utility of its members (cf. Oswald, 1985). The union has l1

employedmembers receiving the wage w

1. Furthermore, the union has m!l

1unemployed

members having no disutility from working and receiving no wage. Thus, thelocal union's utility ;

1can be written as

;1"P

l1

0

[h(p)w1!e

i]g

1(e

i) di#P

m

l1

[0!0]g1(e

i) di.

Of the total membership only l14m is employed by the "rm, and we shall

assume that the union will supply any amount of labour as e$ciently aspossible: workers will be employed according to the disutility of work, e

i, in

increasing order, so that worker i"l1

is the marginally employed worker.9

7 It makes no di!erence whether the continuum of workers is unbounded or large and bounded aswill be clear below.

8The assumption of endogenous membership size ensures that only those who would be willing towork at the negotiated wage join the union. The results would not change if alternatively we assumea "xed (large) number of members.

9Thus, either the union simply knows the disutility of work of its members, or this is learnedthrough a revelation mechanism, or it is possible for the union members to trade jobs.


The utility function of the union can using Eq. (5) be rewritten as

;1"l

1h(p)w

1!e

1(l1). (6)

Notice that a union cannot ful"l a contract (l1, w

1) unless employment, l

1, and

the wage, w1, are such that h(p)w

15e@

1(l1) and, as will become clear below, even

that will require some bargaining power.Finally, the labour supply in each "rm in the secondary sector is derived. If the

wage in the secondary sector is w2, all workers for which the condition

h(p)w25e

iholds will supply one unit of labour. Therefore, the labour supply in

the secondary sector, l2, is given implicitly by the condition

e@2(l2)"h(p)w

2. (7)

Using the conditions on e2(l2), it follows that the labour supply l

2is a convex

function of the wage w2.

3. Local bargaining and monopsony

In this section we consider the case where the employers in the secondarysector can exploit their monopsony power while the primary sector is character-ized by decentralized bargaining.

3.1. Local bargaining in the primary labour market

The primary labour market is locally unionized and "rms and workersbargain at a decentralized level.

The objectives of "rms and unions are given above in Eqs. (4) and (6). Thelocal unions and "rms engage in e$cient bargaining in the sense of McDonaldand Solow (1981) over wages and employment. It is assumed that in case of noagreement there is a strike so that there is no production and no wages are paid.In this case both pro"t and union utility are zero. Hence, the union and the "rmmaximize } with respect to w

1and l

1} the Nash bargaining product

;c1n1~c1

, (8)

where c3 [0, 1] de"nes the bargaining power of the union. Product prices aretaken as given, cf. the assumption of perfect competition. The employmentresulting from the bargaining, l*

1, is given implicitly by the su$cient "rst-order

condition

h(p) f @1(l*1)"e@

1(l*1). (9)

The wage solution on the contract curve is then

w*1"c

f1(l*1)

l*1

#(1!c)1

h(p)

e1(l*1)

l*1

. (10)


Condition (9) states that the marginal product of labour measured in utilityunits has to equal the marginal disutility from work. When employment isdetermined, the wage can be found from Eq. (10) which shows the sharing of thesurplus implied by e$cient bargaining. If c"1 the union extracts all surplusand the wage is equal to the average product. Assume for a moment that the"rms had all the bargaining power, c"0, then the wage would be equal to[1/h(p)][e

1(l*1)/l*

1] and the union's utility becomes ;

1"0. However, if

h(p)w1"e

1(l*1)/l*

1, then, as discussed at the end of Section 2.3, the union cannot

deliver the required labour, because e1(l*1)/l*

1(e@

1(l*1) implies that membership is

strictly less than desired employment. Therefore, we have to assume that c islarge enough to ensure a wage w

15e@

1(l*1)/h(p).

Substitution of the employment and wage solutions into the pro"t and unionutility functions yield the Nash bargaining pro"t n*

1and utility ;*

1:

n*1"(1!c)[h(p) f

1(l*1)!e

1(l*1)], (11)

;*1"c[h(p) f

1(l*1)!e

1(l*1)], (12)

where employment, l*1, is given implicitly by (9).

3.2. Monopsony in the secondary labour market

Each "rm in the secondary sector exploits its monopsony power, i.e. the "rmrealizes that a lower employment leads to lower wages, cf. the labour supplyfunction (7). The objective function of the "rm is given in Eq. (4). The "rm doesnot know the disutility of work of the individual workers and pays all employeesthe same wage. The problem of the monopsony "rm can then be written as

maxl2

n2

s.t. e@2(l2)"h(p)w

2.

Product prices are taken as given. The su$cient "rst-order condition to the"rm's problem implies that the monopsony employment in the secondary sectoris the l.0/

2satisfying

pf @2(l.0/2

)!e@2(l.0/2

)

h(p)!

eA2(l.0/2

)

h(p)l.0/2

"0, (13)

and the corresponding wage in the secondary sector w.0/2

is

w.0/2

"

e@2(l.0/2

)

h(p). (14)

The wage}employment combination (w.0/2

, l.0/2

) given by Eqs. (14) and (13) ison the labour supply curve. Employment as well as the wage are below theWalrasian equilibrium level.


4. Two-stage bargaining and extended coverage of minimum wages

In this section we consider the possibility of the unionized sector imposinga uniform minimum wage covering the whole economy. The idea is that byimposing a minimum wage also for the secondary sector, the monopsony powerin this sector is o!-set. That causes the production to increase and the relativeprice to fall. The result is that under suitable conditions the "rms as well as theworkers in the primary sector are better o!.

The bargaining now takes place in two stages. At the "rst stage, the centralbargaining parties of the union sector negotiates a minimum wage extending toboth sectors. Ultimate wage determination is retained at the local level althoughthe two sectors have to subject to the condition that the wage is greater than orequal to the minimum wage. As previously, no agreement at the local bargaininglevel leads to con#ict in which neither workers nor "rms receive any compensa-tion.10 Hence, the minimum wage does not in#uence the threatpoints of thelocal unions and "rms in the local stage bargaining (compare Holden, 1988).

Central bargaining is carried out by representatives of the local unions withthe representative union's utility as its objective and an employers' federationwith the representative "rm's pro"t as its objective. The union group, or centralunion, and the employers' federation of the primary sector take into account thee!ect of the minimum wage on subsequent local wage determination.

4.1. Local wage formation

The model is solved backwards starting with the wage formation at the locallevel. First local bargaining in the primary sector is considered. The Nashproduct (8) is again maximized with respect to w

1and l

1but now under the

restriction that w15w.*/. The wage determined in the primary sector at the

local level given the "rst stage minimum wage w.*/ becomes

w1"max[w.*/, w*

1], (15)

where w*1

is the function (10).Wage and employment determination in the secondary sector are now con-

sidered. The minimum wage is imposed by the central union and the employers'federation, covering the whole economy. Therefore, "rms and workersin the secondary sector take the minimum wage as given, i.e. the wage in sector2 is

w2"max[w.*/, w.0/

2]. (16)

10The existence of threats of con#ict at the local level can be based on national legislation ordecided by the central bargaining parties in separate negotiations.


We will only consider the more interesting case where the (optimal) minimumwage lies between the monopsony wage and the negotiated wage of sector 1.Thus, it is assumed that

w.0/2

4w.*/4w*1. (17)

We consider employment in the secondary sector with reference to theWalrasian wage, which equilibrates labour supply and demand. Labour demandis taken from the "rst-order condition to the "rm's problem: pf @

2(l2)!w

2"0.

Labour supply is given by Eq. (7). Walrasian employment, lK2, can then by found

where pf @2(lK2)"e@

2(lK2)/h(p). Combining these two equations yield the Walrasian

wage

wL2"pL f @

2(lK2), (18)

where pL is the equilibrium price given clearing of the good markets, marketclearing in the secondary labour market, and local e$cient bargaining insector 1.

Employment is supply determined if the minimum wage is below wL2, and

demand determined if the minimum wage is above wL2. In conclusion, the

employment, l2, in the secondary sector is given implicitly as a function of w.*/,

Ge@2(l2)!h(p)w.*/"0 if w.*/(wL

2,

pf @2(l2)!w.*/"0 if w.*/5wL

2.

(19)

4.2. The central stage of bargaining: Setting the optimal minimum wage

At the "rst stage, the central union and the employers' federation bargain overthe minimum wage. It turns out to be convenient to assume that the two partiesbargain over the real consumer minimum wage, i.e. the wage in utility terms.11

The factor h(p) converts a wage into utility and can be interpreted as theinverse of the (ideal) consumer price index. Let w.*/ be the (real) consumer (pricede#ated) minimum wage, i.e.

w.*/"h(p)w.*/. (20)

A given w.*/ could, for example, be implemented through an indexation schemein which the consumer minimum wage is adjusted when the price index changes.

The Walrasian equilibrium price is pL as derived in Eq. (18) and the equilib-rium wage in sector 2 is wL

2. The corresponding consumer wage in Walrasian

equilibrium is denoted w;2

and found as w;2"h(pL )wL

2, cf. Eq. (20).

11The central bargaining parties could alternatively bargain over the real product minimumwage, w.*/, directly. The results found in that case are qualitatively similar to the results reported inthis section although an additional assumption has to be imposed, see Roberts et al. (1997).


The problem of the central union and the employers' federation is to maxi-mise the Nash product of the parties of a representative location with respect tow.*/. Maximization is subject to the wage setting at the local level and to marketclearing at the output markets. At the second (local) stage the "rm and the unionbargain e$ciently obtaining the pro"t n*

1in Eq. (11) and the union utility;*

1in

Eq. (12). The formal problem of the central bargaining parties is

maxw.*/

(;*1)c(n*

1)1~c

s.t. h(p) f @1(l*1)!e@

1(l*1)"0,

Ge@2(l2)!w.*/"0 if w.*/(w;

2,

pf @2(l2)!w.*//h(p)"0 if w.*/5wL

2,

h@(p) f1(l*1)#[h(p)#h@(p)p] f

2(l2)"0. (21)

The constraints have the following interpretations. The "rst constraint takesinto account that the employment in the primary sector is found by e$cientlocal bargaining (from Eq. (19)).12 The constraints in the curled bracket ensurethat the employment in the secondary sector is determined by the short side ofthe market (from Eq. (19) using Eq. (20)). Finally, the last constraint followsfrom the market clearing condition for the sector 2 output market.13

The problem of the central bargaining parties is solved using theKuhn}Tucker method for constrained optimization. The two cases w.*/(w;

2and w.*/5w;

2are considered separately. Solving the constrained optimization

problem yields the "rst order condition w.*/"w;2

(see the appendix).We continue by analysing when the "rst-order solution constitutes a global

maximum. The Nash bargaining product is continuous in w.*/ and, therefore, itis su$cient to demonstrate that the Nash bargaining product is increasing inw.*/ when w.*/4w;

2and decreasing in w.*/ when w.*/5w;

2. This holds under

a weak condition.When the consumer minimum wage changes, the Nash bargaining product

changes according to

L(;*1)c(n*

1)1~c

Lw.*/"cc(1!c)1~cf

1(l*1)h@(p)

Lp

Lw.*/

#cc(1!c)1~c[h (p)f @1(l*1)!e@

1(l*1)]

Ll*1

Lw.*/. (22)

12Notice that an increase of the relative price p leads to a lower, not a higher, employment in theprimary sector.

13Applying Roy's identity to Eq. (2) gives us the demand for sector 2 outputx2"!f

k(lk)h@(p)/h(p), where f

k(lk) is sector k's income. Market clearing requires that total demand

!f1(l1)h@(p)/h(p)!f

2(l2)h@(p)/h(p) equals total supply f

2(l2). The market clearing condition in Eq. (21)

then follows from simple rearranging.


An increase in the consumer minimum wage w.*/ in#uences the objective(;*

1)c(n*

1)1~c via two variables, namely the relative price and the employment in

sector 1. The relative price channel is captured by the "rst term in Eq. (22) andthe employment or income channel by the second term. The expressioncc(1!c)1~cf

1(l*1)h@(p) denotes how an increase in the relative price p changes the

Nash bargaining product given that the employment and production in sector1 is constant. The term is non-positive re#ecting that } when sector 1 productionis constant } an increased relative price p leads to non-increasing welfare in theprimary sector. The term cc(1!c)1~c[h(p)f @

1(l*1)!e@

1(l*1)] captures the e!ect of

changed employment in the primary sector. The employment l*1

determined atthe local stage is given by the "rst constraint in Eq. (21) from which it followsthat the second term in Eq. (22) is zero, cf. the envelope theorem. The intuitionis that on the margin the local bargaining in the primary sector always adjuststhe employment so that the net contribution to the welfare from changes in theemployment is zero. Thus, the primary sector does not bene"t from pricingthe secondary sector "rms out of the market by increasing the minimum wageabove the market clearing level. In conclusion, the common interest of the "rmsand the unions in the primary sector will be to induce a reduction of the relativeprice of sector 2 output, p.

The following two subsections analyse how a change in the consumer min-imum wage, w.*/, is transmitted into a change in the relative price, p, throughthe restrictions to the maximization problem.

4.2.1. Below the market clearing wageIn the case w.*/(w;

2, employment is determined by the labour

supply schedule and the constraints impose the following conditions on p, l*1,

and l2:

e@1(l*1)!h(p) f @

1(l*1)"0, (23)

e@2(l2)!w.*/"0, (24)

h@(p) f1(l*1)#[h(p)#h@(p)p] f

2(l2)"0. (25)

The system of constraints (23)}(25) is block recursive. By total di!erentiationit can be demonstrated that a change in the consumer minimum wage, w.*/,leads to the following change in the relative price, p:

dp

p"

1

l!2k1#k

e2

g2

#kg1

e1!m

1

e2

g2

dw.*/

w.*/, (26)


where the following elasticities are de"ned:

k"h@(p)p

h(p), l"hA(p)

p

h@(p),

gk"f @

k(lk)

lk

fk(lk), m

k"f A

k(lk)

lk

f@k(lk), e

k"eA

k(lk)

lk

e@k(lk).

The elasticity k denotes the percentage change of the indirect utility when theprice p increases one percent; k40 as h@(p)40. Furthermore, k can be shown tobe equal to (the negative of ) the share of expenditures spent on sector 2 productsand, hence, k5!1. The elasticity l denotes the percentage change in (thenegative of ) the sector 2 product's expenditure share when the relative priceincreases one percent. The representative agent has homothetic preference, soGi!en goods are ruled out, i.e. l40. It can be shown that (l!2k)/(1#k) is theelasticity of substitution.14

The elasticity gkis derived from the production function of the representative

"rm in sector k and denotes the percentage change in output of a one percentagechange in the employment. From the assumptions on the production functionwe have that g

k'0. The elasticity m

kis the inverse of the labour demand

elasticity in sector k, cf. the "rst-order condition to the "rm's problem. Hence,1/m

kdenotes the percentage change in the labour demand in sector k by a one

percentage change in the wage wk. The elasticity m

k(0.

Finally, the elasticity ekis the inverse of the labour supply elasticity in sector k.

This can be seen, for sector 2, from Eq. (7). Therefore, 1/ekis the sector k labour

supply response to a change in wk.15 From the assumptions on the disutility

function it follows that ek'0.

Eq. (26) expresses the response of the relative price to a change in theconsumer minimum wage. When w.*/ is increased, the relative price p isin#uenced partly by direct changes in the employment and production in sector2, partly by induced changes of the production in the primary sector. The directe!ect on the relative price p is captured by the "rst term, (l!2k) e

2/(1#k)g

2,

in the denominator in Eq. (26) (holding the production in sector 1 constant). Thesecond term in the denominator, kg

1e2/(e

1!m

1) g

2, includes the indirect e!ect

on p of the change in the production in the primary sector.

14The own price elasticity of demand for sector 2 goods is l!k.15The term eA

k(lk) is the change in the union's marginal disutility from one more employed. A large

eAk(lk) indicates that the last employed worker has much larger disutility from work than the

preceding worker. Therefore, to make the marginal worker willing to work, the wage has to be large.A large eA

k(lk) implies, ceteris paribus, that the elasticity e

kis large and, hence, that the labour supply

elasticity is small: a percentage change in the wage only leads to a small change in the labour supply.


Two &e!ective' sectors are needed in order for relative price changes to bewell-de"ned. The following assumption ensures that the households alwaysconsume both goods:

!1(k(0. (27)

Given homothetic preferences, assumption (27) implies that the elasticity ofsubstitution is strictly smaller than 0, i.e. (l!2k)/(1#k)(0. Thus, condition(27) is su$cient to ensure that dp/dw.*/(0 as both terms in the denominatorare negative: an increase in the consumer minimum wage leads to a decrease inthe relative price of sector 2 goods.

4.2.2. Above the market clearing wageThe consumer minimum wage negotiated at the central stage will never

exceed w;2. This can be seen by considering the constraints to the maximization

problem in the case where w.*/5w2. Conditions (23) and (25) are unchanged

but the employment in the secondary sector is now determined by the labourdemand,

pf @2(l2)!w.*//h(p)"0. (28)

By total di!erentiation of the simultaneous system (23), (25) and (28) thefollowing relationship between the consumer minimum wage and the relativeprice is found:

dp

p"

1

1#k#l!2k1#k

m2

g2

#kg1

e1!m

1

m2

g2

dw.*/

w.*/. (29)

Again, assuming that condition (27) holds we have that dp/dw.*/'0, i.e.a higher consumer minimum wage leads to an increase in the relative price, p,when w.*/5w;

2.

The "rst term in the denominator, 1#k, denotes the direct e!ect on the price,cf. (28); a higher cost of labour increases the marginal cost of production and,ceteris paribus, the price of the sector 2 product. The second term,(l!2k)m

2/(1#k)g

2, accounts for the e!ect of reduced production in the

secondary sector resulting from the increased labour costs. The "nal term,kg

1m2/(e

1!m

1)g

2, incorporates the e!ect of reduced production in the primary

sector moderating the increase in the price p. In total, an increase in theconsumer minimum wage, w.*/, increases the relative price, p, when condition(27) is satis"ed.

4.2.3. Summary of the resultsNow, we can characterize the over-all behaviour of the central bargaining

parties.


Proposition 1. Assume that the central bargaining parties set the consumerminimum wage w.*/. If condition (27) is satis,ed for all w.*/5w.0/

2, then the

central bargaining parties set the consumer minimum wage, w.*/, such that thesecondary labour market clears (i.e. the consumer minimum wage is equal tothe=alrasian wage of Sector 2, w.*/"w;

2).

As an immediate corollary to Proposition 1 we observe that the two-stagebargaining institution dominates single stage local bargaining from the view-point of the primary sector. Moreover, we have the following second best result:

Proposition 2. Assume that the central bargaining parties set the consumerminimum wage w.*/. If Eq. (27 ) is satis,ed for w.*/5w.0/

2, then

(i) when w.*/(w;2

the central bargaining parties gain by increasing w.*/ ;(ii) when w.*/'w;

2the central bargaining parties gain by reducing w.*/.

Assume that } for some reason } the consumer minimum wage is below orabove wL

2. Then, if it is not possible to move the consumer minimum wage all the

way to wL2, it is still making the central bargaining parties better o! to bring the

minimum wage towards the Walrasian wage, cf. Proposition 2.Notice, as long as the products are not perfect complements or perfect

substitutes,16 Propositions 1 and 2 do not depend on the degree of substitutionbetween the two goods in the representative household's utility.17

4.3. Welfare

As an interesting corollary to Proposition 1 above, we have that the imposi-tion of a minimum wage at the central stage leads to a socially e$cient outcome} if condition (27) holds.

The social optimum is straightforward in our model: employment in bothsectors need to equilibrate the marginal product and the marginal disutility oflabour, i.e.

h(p) f @k(lk)"e@

k(lk), k"1, 2.

This is always achieved in the unionized sector even with decentralized bargain-ing only (because of e$cient bargaining), and as we have just seen, it is theaspiration of the parties in the unionized sector to set w.*/ such that this is,indeed, achieved also in the non-union sector. Therefore, in this model an

16 If the two products are perfect complements, the function h(p) is not di!erentiable. Perfectsubstitutes are ruled out by assumption (27).

17The sensitivity of the relative price p with respect to a change in the consumer minimum wagew.*/ does, however, depend on the degree of substitutability between the two products.


inclusion of a central bargaining stage in which a minimum wage is extended tothe whole economy increases not only the welfare of the primary sector but alsoaggregate welfare, i.e. coverage extension increases welfare.

5. Discussion

In Section 4 we studied a dual labour market with a unionized primary sectorthat is able to extend the coverage of a minimum wage agreement to the entireeconomy. It was shown that, under fairly general assumptions, there are incen-tives for the primary sector to extend coverage of the minimum wage to thewhole economy. The setup of the model ensures that the addition of a centralbargaining stage has no direct e!ects on the welfare of the primary sector but hasindirect general equilibrium e!ects.18 The bene"t of coverage extension isattained only through relative price changes caused by shifts in the two sectors'production. The households in the primary sector consume the good producedin the secondary sector and seek the highest possible valuation of it's surplusand are always better o! when the relative price p is lowered.

In order to obtain the lowest possible relative price, the union sector setsa (minimum) wage for the non-union sector equal to the market clearing wage inthis sector. This implies that involuntary unemployment in the non-union sectordoes not exist, even when the sector is covered by a minimum wage. Hence, inour model the union sector will set a socially optimal minimum wage and, in thissense, can be &trusted' to set the wage for the non-unionized sector. Below wediscuss some of the assumptions behind our results.

5.1. Local strikes and lockouts

In Section 4 it was assumed that workers can strike and "rms use lock-out incase of disagreement during local bargaining. This implies that the minimumwage becomes an outside option to the local bargaining and, thus, does notin#uence the settlement of this stage as long as the minimum wage is not set attoo high a level. This is a precondition for the commonality of interests withinthe primary sector. In most countries the local bargaining parties can call strikesand lock-outs if they fail to reach an agreement, although in some countriesstrikes and lock-outs are prohibited at the local level. In this case the locallydetermined wage in the primary sector depends directly on the minimum wageset at the central stage (see also Holden, 1988). Thus, if local strikes and lockoutsare prohibited, an element of con#ict between the employers' federation and thecentral union at the "rst stage of central bargaining would arise. In this case the

18This rationalization of a two-stage bargaining procedure is in contrast to the one in Holden(1988) where the central stage is redundant when strikes are allowed locally.


unionized sector is unlikely to set the minimum wage at a level clearing thesecondary labour market.

5.2. Mobility and entry

It was assumed that there was no mobility of workers between "rms orbetween sectors. Likewise, each "rm was unable to change location and no entryor exit was possible. Thus, the model has no links between the two labourmarkets and, so, changes in the minimum wage has no direct e!ect on the laboursupply schedules in the two sectors. A relaxation of the assumptions on mobilityand entry would be an interesting avenue of research as e.g. worker mobility islikely to generate a more direct relationship between the minimum wage and theprimary sector wage in addition to the general equilibrium price e!ect.

5.3. Bargaining institutions

Throughout the paper only e$cient bargaining has been considered but thisassumption was chosen for convenience. The central bargaining parties havea common interest when setting the minimum wage in seeking to reduce therelative price. This "nding is likely to hold for other bargaining institutions aswell, for example the &right-to-manage' or monopoly union institutions. As longas "nal determination of employment and wages in the primary sector takesplace at the local level, the central bargaining parties have no incentive toin#uence these variables. The only variable which the bargaining parties per-ceive that can be in#uenced at the central level but not at the local level is therelative price and here their interests coincide because of the assumptiondiscussed above.19 Thus, with homothetic preferences in consumption theminimum wage will be set so that the relative price p is minimized } irrespect-ively of the bargaining institution.

6. Final comments

Coverage extension is an important feature of many European labourmarkets. This paper has demonstrated that coverage extension of a minimumwage can be bene"cial for the parties of the union sector. The minimum wagewould be set at a level clearing the secondary labour market. In our modelcoverage extension would leave the economy in the social optimum.

19With other wage setting institutions the primary sector employment could rise when therelative price increases. However, the e!ect on the primary sector surplus would still be of secondorder, cf. the envelope theorem. So, still, the primary sector wants the relative price as low aspossible.


The only interaction between the labour markets was through the relativeprice based on the assumptions of local strikes and lock-outs being allowed andno mobility of "rms and workers. Further research into the causes and conse-quences of coverage extension could focus on the e!ects of various degrees ofsubstitution between organized and non-organized labour.

Acknowledgements

We would like to thank two anonymous referees as well as Claus T. Hansen,Henrik Jensen, and Christian Schultz for valuable suggestions. The usualdisclaimer applies. This research was supported in part through the project&The Welfare State: Threats, problems, and some Solutions' "nanced by theDanish Social Science Research Council.

Appendix A

Below we solve the problem of the central bargaining parties (21) of Sec-tion 4.2. The problem is solved using the Kuhn}Tucker method for constrainedoptimization. The two cases w.*/(w;

2and w.*/5w;

2are considered separately.

First, the case where w.*/(w;2

is considered. The Lagrangian is:

£"cc(1!c)1~c[h(p) f1(l*1)!e

1(l*1)]!j

1Mh(p) f @

1(l*1)!e@

1(l*1)N

!j2Me@

2(l2)!w.*/N!j

3Mh@(p) f

1(l*1)#[h(p)#h@(p)p] f

2(l2)N

!j4Mw.*/!w;

2N,

where j1,2,j

4are the Lagrangian multipliers. The "rst-order conditions are:

j2h(p)!j

4"0,

h(p) f @1(l*1)!e@

1(l*1)"0

e@2(l2)!w.*/"0,

h (p) f @1(l*1)#[h(p)#h@(p)p] f

2(l2)"0,

w.*/!w;240,

j1, j

2, j

3O0,

j450,

j4Mw.*/!w;

2N"0.


The last condition is the complementary slackness condition. If j4"0, then

j2

must equal 0 which is not permissible as we require the employment to be onthe labour supply schedule. It follows that j

4'0 and, from the complementary

slackness condition, w.*/"w;2.

Similar arguments can be used in the case where w.*/5w;2

and the result isagain that the Kuhn}Tucker "rst-order conditions yield w.*/"w;

2.

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