Profit Sharing, Capital Formationand the NAIRU

Jürgen Jerger and Jochen Michaelis

Revised Version: May 1998

Forthcoming in: Scandinavian Journal of Economics

Abstract:

This paper reconsiders the impact of different remuneration systems on aggregate

employment. Contrary to common wisdom on this topic, we show that the switch from a

Fixed Wage Economy to a Share Economy results in lower aggregate unemployment. An

additional innovation of this paper is the endogeneity of the capital stock decision of the firm

which enables us to assess and reject the common ‘disincentive-to-invest’-criti cism of a Share

Economy.

Keywords: Share Economy, Investment, NAIRU

JEL classification: J23, J33, J51

corresponding author:

Dr. Jürgen Jerger PD Dr. Jochen Michaelis

Institut für Allg. Wirtschaftsforschung Institut für Finanzwissenschaft

Universität Freiburg Universität Freiburg

Platz der Alten Synagoge Maximili anstr. 15

D-79085 Freiburg D-79100 Freiburg

Germany Germany

1

PROFIT SHARING, CAPITAL FORMATION AND THE NAIRU

Jürgen Jerger and Jochen Michaelis*

I. Introduction

The surge in European unemployment over the past twenty years is widely attributed to a rise

of equili brium unemployment (Layard et al. 1991, Lindbeck 1993), or to speak in familiar

acronyms, to a rise of the NAIRU (non-accelerating inflation rate of unemployment). Hence,

possible remedies should be directed towards lowering the NAIRU. This paper will conclude

by suggesting that Martin Weitzman’s (1985) celebrated profit-sharing idea, which was

originally intended to avoid employment fluctuations due to variations in aggregate demand,

also has its merits with regard to equili brium unemployment. This idea was first put forward

by Weitzman (1987) himself. He shows that even in a unionised economy the introduction of

share contracts has a positive employment effect at the firm level. His claim, however, that

this property of a Share Economy carries over to the aggregate level, rested on an intuitive

rather than an analytical argument. Layard/Nickell (1990; henceforth LN) provided the

missing piece of analytical work, but concluded with the discouraging assertion that – for a

Cobb-Douglas technology – aggregate employment is independent of the remuneration

system. A similar result was derived by Holmlund (1990).

* We are grateful for comments on earlier drafts of this paper to participants of the SeventhWorld Congress of the Econometric Society (Tokyo, 1995), the Tenth Annual Congress of theEuropean Economic Association (Prague, 1995) and research seminars at the London Schoolof Economics and the universities of Freiburg and Chemnitz-Zwickau. Particular thanks go toH.-H. Francke, R. Jackman, O. Landmann, R. Layard, M. Pflüger, J. Safford, M. Weitzmanand the referees as well as the editor of this journal. Financial support from the DeutscheForschungsgemeinschaft is gratefully acknowledged by J. Michaelis. The usual disclaimerapplies.

2

In this paper we show that this neutrality result stems from an inadequate modelli ng of

the sectoral bargaining game. More precisely, LN implicitely assume that a firm is obliged to

offer a share contract. The legal framework in all market economies, however, is such that a

firm has the opportunity to reject profit sharing, since the property rights for profits are with

the firm. We take this feature into account by treating a fixed wage contract either as an

outside option or as a fallback-position during share bargains. Both of these modelli ng

strategies lead to a substantial modification of the LN result at the sectoral and − more

importantly − at the aggregate level. In particular, we can show that the remuneration system

matters for aggregate (un)employment (as well as other variables) and that the intuition of

Weitzman (1987) is backed by the analytical results.

A second theme which is taken up in this paper is the interaction between the

remuneration system and capital formation. Although the NAIRU is a longer run concept,

Weitzman as well as LN assume an exogenous capital stock. This simpli fication is potentially

misleading, however, since the choice of the remuneration system is not neutral with respect

to the firm’s capital stock decision, which in turn has repercussions on the wage/employment

determination. The endogeneity of the capital stock enables us to assess and reject the

common ‘disincentive-to-invest’-criti cism of a Share Economy (cf. Meade 1986, Wadhwani

1987). This criti cism asserts that the optimal capital stock in Share Economy is lower than in a

Fixed Wage Economy, simply because ‘effective’ capital cost have to take into account that

the marginal product of capital must be divided between the firm and workers. We show that

this effect will be offset by the positive employment effect of a Share Economy.

The rest of the paper is structured as follows. Section 2 briefly introduces the main

features of the model and describes the sectoral equili brium solution. The aggregate level is

modelled in Section 3. Section 4 concludes.

3

II. Sectoral Equilibrium

The economy consists of a large number of small sectors i, and in each sector a single firm

bargains with a single trade union. The determination of employment, wages and the capital

stock at the sectoral level is modelled as a three-stage game which we solve by backwards

induction. In the first stage, the firm decides on the capital stock, thereby taking into account

the perfectly foreseen wage and employment outcomes of the following stages. The second

stage consists of the wage bargain between the firm and the union, which we analyse by

means of the well -known Nash bargaining model. In the third stage, the firm sets employment

to ensure maximum profits for the given capital stock and pay parameters. Stages two and

three are treated in Section 2.1, the choice of the capital stock is discussed in Section 2.2.

Fixed Capital Stock

Set-up and the Fixed Wage Economy

The ith industry produces output Yi by means of capital Ki and labour Ni . We specify the

production function as Cobb-Douglas with constant returns to scale:1

Y K Ni i i= α β , α β, > 0, α β+ =1 (1)

Each firm faces a constant elasticity demand curve, Y P Yi ie

d= − , where Pi is the firm´s real

output price, e> 1 is the (absolute value of) price elasticity, and Yd is an index of aggregate

real demand. Thus, the firm´s real revenue R PYi i i= is given by

R K N Y Yi i i i de( , ) /= κ 1 (2)

with κ ≡ −1 1/ e denoting the degree of product market competitiveness.

4

The total compensation per worker, Wi , can be written as the sum of a fixed part ω i , and

a share λi of net profits, which are defined as real revenue minus fixed wages and capital

costs:

WR N rK

Ni i ii i i i

i

= +⋅ − −

ω λω( )

. (3)

r represents the exogenous user costs of capital. The assumption λ i = 0 generates the Fixed

Wage Economy (FWE). Profits are defined by

[ ]π λ ωi i i i i i i i i i i i i iR K N WN rK R K N N rK= − − = − − −( , ) ( ) ( , )1 . (4)

The base wage ω i and the share parameter λ i are the concern of wage negotiations. In the

third stage of the game, i.e. for predetermined values of the capital stock and the pay

parameters, firms maximise profits by setting the employment such that the marginal revenue

of labour RN equals marginal costs:

R

R WN i i

N i i

= >= =

ω λλ

for

for

0

0(5)

Upper case subscripts denote partial derivatives throughout the paper, the sectoral index i is

dropped for notational convenience.

We assume that the union maximises the sum of its M i members’ utilit y and that

workers are risk neutral:2 ~

( )U N W M N bi i i i i= + − . By assumption, the exogenous

membership M i exceeds the firm’s labour demand. The M Ni i− > 0 union members who

are not employed by the firm in question obtain the fallback income b, which is assumed to be

1 The Cobb-Douglas specification simpli fies the following exposition significantly, but doesnot alter the main results of our analysis. Where appropriate we will comment on theimplications of an elasticity of substitution differing from unity.2 In Jerger/Michaelis (1995) we show that our results carry over to the case of constantrelative risk aversion.

5

exogenous at the sectoral level. Hence, M bi is exogenous to the union and the maximisation

of

U U M b N W bi i i i i≡ − = −~( ) (6)

is equivalent to the maximisation of ~Ui . See Pencavel (1991) and Booth (1995) for further

details on union preferences.

In the wage bargain of the second stage the real wage Wi is chosen to maximise the Nash

product Ω i :

Max U UWi

i i i i iΩ = − −−( ) ( )π π γ γ1 s.t. R WN i= , (7)

where U i and π i are defined by (6) and (4), respectively. The parameter γ ( )0 1≤ ≤γ denotes

the union’s relative bargaining power. When no agreement is reached, employment and

production fall back to zero. By assuming that the strike income of a union member is equal to

the fallback income (cf. Binmore/Rubinstein/Wolinsky 1986), we get U i = 0 as the union s

threat point. Since capital costs must be covered by firms regardless of their bargaining

success, the firm’s threat point is given by π i irK= − . It is straightforward to show that (7)

leads to

W biF

iF= η (8)

and

NY K

biF d

ei=

−βκδ

ακ βκ11

1/

(9)

where η δ γ γβκi

F = ≡ − + ≥1 1.

6

How (not) to model a Share Economy

Now, we turn to the description of the wage-employment combination of the Share Economy

(SE). As pointed out in the introduction, the results crucially depend on the exact specification

of the bargaining framework. We offer two alternative, but equally plausible scenarios

(labelled S2 and S3) and also reproduce the LN result (S1). Fig. 1 depicts the several solution

concepts for a given capital stock. In section 2.3 we will show how the different scenarios

affect the capital stock decision. This in turn shifts the labour demand and contract curves, but

for expository convenience, we did not include these shifts in fig. 1.

Wi

FS1S2

S3

RN

Ni

b U iF

π iF

contract curve

Figure 1

LN assume that employment and production fall back to zero, if no agreement is reached

in the share bargain. Hence, the threat points are the same as in the FWE case. In fig. 1, the

solution to this bargaining problem is indicated by point S1. We will show that S1 lies on the

contract curve, but outside the lens formed by the indifference curve U iF and the iso-profit

7

lineπ iF . Such a shift from a FWE to a SE would not be a Pareto-improvement. The union

would be better off , but the firm would be worse off .

Is the union in a position to enforce S1? The answer crucially depends on the credibilit y

of the firm’s and the union’s threat to halt production when there is no agreement. In a FWE

such a threat is clearly credible (see Moene 1988). But in a SE the implicit assumption that the

union has a right to strike for a (higher) share of the firm’s profits contradicts the legal

framework observed in almost all market economies, which ensures that the property rights

for profits are with the firm. To put it another way, the LN scenario would only make sense if

one of the following two assumptions holds:

1) A share system is imposed by law. With one exception we do not know of any market

economy which regulates the remuneration system of its firms. The exception is France,

where any firm with more than 50 employees has to pay a well -specified fraction of its

profits to a fund (cf. European Commission 1991).

2) The firm and the union voluntarily restrict the contract design ex ante (in stage zero of the

bargaining game). They bargain over the remuneration system and agree that the labour

contract has to be a share contract. This agreement has to be verifiable in a court of law.

Given such an agreement the union can insist on a share contract and has the right to strike

for a higher share parameter.

However, regulations imposing a share system on the economy are rare and there are no

voluntary agreements which restrict the contract design to a share system. Since the firm is

absolutely free in deciding how to spend its profits, it is absolutely free to say ‘no’ to any

profit sharing. It will do so, if profit sharing leads to a profit decline (as in S1). Similarly, the

workers always have the opportunity to reject a share system. For these reasons, we claim that

the results of LN are derived from a wrong specification of the share bargain and go on to

present a model more in line with the institutional features mentioned above.

8

Our second share scenario (S2) captures the ‘no’ by assuming that a FWE serves as an

outside option in a bargain over a share contract. Both parties have the option to cancel the

share bargain and to move to a bargain over a fixed wage contract. As pointed out by

Binmore/Shaked/Sutton (1989), such an outside option constrains the outcome of the Nash

bargaining solution. If the bargaining solution assigns one party less than the expected utilit y

of his outside option, the party will prefer to take up the option. Hence, the share contract has

to be a Pareto-improvement, i.e. the union’s utilit y must not be lower than U iF and the firm’s

profits must not be lower than π iF . Only the latter of these two constraints is binding. The

union never takes up the outside option, since the unconstrained Nash solution (S1) always

corresponds to a higher utilit y than U iF . On the other hand, the firm always prefers the outside

option, since S1 entails lower profits than F. However, the outside option principle does not

imply that a FWE will be adopted. Knowing the firm’s outside option, the union will offer a

share contract which ensures the firm a profit π iF ‘plus epsilon’ (cf. Binmore/Shaked/Sutton

1989). Hence, the firm has an incentive just to accept the offer and not to choose the outside

option. Idealising epsilon to be zero, our second share scenario leads to point S2 (see fig. 1).

The third share scenario (S3) assumes a two-stage bargaining, in which the parties first

agree on a fixed wage contract F and then go on to negotiate a share contract. If no agreement

is reached in this share bargain, F will be continued. The solution to this problem can be

shown to be a point on the contract curve within the lens formed by the indifference curve U iF

and the iso-profit li ne π iF .3 This modelli ng strategy is intended to capture the situation for

example in the Scandinavian countries, where there is a national bargain over a fixed wage

3 The only paper, at least to our knowledge, which considers the outcome of the right-to-manage model as the fallback position of an eff icient wage bargaining is Espinoza/Rhee(1989). They model the wage bargaining process as a repeated game and argue that, since theloss from non-cooperation builds up period by period, there is a strong incentive to reach anagreement off the labour demand schedule.

9

and subsequently a firm level bargain over a wage drift (or profit sharing). Strikes are not

allowed at the firm level. During the local bargain the firms have to pay the wage negotiated at

the national level (see Holden 1988). Further examples are Germany and the Netherlands,

where industry level bargains over a fixed wage contract are supplemented by firm level

bargains. Strikes at this level are ill egal. The last example is Japan, where bonuses account for

up to one-third of the average worker´s pay. The typical Japanese worker’s pay is divided into

a ‘wage that is surely paid’ and ‘special cash payments’ that are − more or less − at the

discretion of the employer. The first component, i.e. the base wage, is the primary concern of

the spring wage offensive. After these base wages are settled, there are negotiations over

bonuses, but no union can insist on − and strike for − such a payment (see Freeman/Weitzman

1987). For further details on the legal framework of wage disputes in different countries see

the PEPPER-Report (European Commission 1991) and Hartog/Theeuwes (1993).

The (modelli ng) decision between S2 and S3 depends on the exact institutional design of

the share bargain as pointed out. Hence, we will analyse both scenarios and do not take any

stand whether the one or the other is ‘more appropriate’ . But we claim that the LN scenario

always conflicts with the legal framework of a share bargain.

Wages and Employment in a Share Economy

In a SE, the firm and the union bargain over the parameters of a share contract, ω i and λ i .

Leaving the ‘r ight to manage’ with the firm as in the FWE, the Nash-bargaining problem has

to be solved subject to RN i= ω , and, if the FWE serves as outside option (second scenario),

subject to π πi iF≥ . Using the Lagrangean method, we have:

Max L U U Ri i

i i i i i N i i iF

ω λγ π π γ µ ω ν π π

,( ) log( ) log( ) ( ) ( )= − − + − + − + −1 (10)

10

Wi is now given by the share equation (3). We get the LN scenario S1 by assuming ν = 0,

π i irK= − and U i = 0. The ‘outside option’ scenario S2 is implied by ν >0, π i irK= − and

U i = 0. And finally, the assumptions ν = 0, π πi iF= and U Ui i

F= produce the ‘ threat point’

scenario S3.

The first-order conditions of (10) read:

( )

∂∂ω

γ λπ π

ω∂∂ω

γ ω∂∂ω

λ ν λ

LR

NN

U Ub

NN N

i

i

i

i iN i

i

ii

i ii

i

ii i i i

=− −

−− −

+

−− + −

− − =

( )( )( )

( ) ( )

1 1

1 1 0

(11)

∂∂λ

γπ π

γν

L

U Ui

i i i i i

=− −

−+

−− =

( )10 (12)

∂∂µ

ωL

RiN i= − = 0 (13)

∂∂ν

λ ω πL

R N rKii i i i i i

F= − − − − =( )( )1 0 (14)4

Observing (13), the inspection of (11) and (12) immediately shows that for these two first-

order conditions to hold simultaneously, ( )ω∂∂ωi

i

i

bN

− must be equal to zero. This requires

ω i b= . (15)

4 We restrict ourselves to the case ∂∂νLi = 0 instead of

∂∂νLi ≥ 0, since we assumed that the

firm will not reject a share contract that offers the same profits as in F.

11

The optimal base wage is equal to the fallback income, if unions are risk neutral as implied by

(6). In the case of risk averse unions the optimal base wage would be lower than b (see

Jerger/Michaelis 1995).

Unlike the optimal base wage the optimal share parameter depends on the share scenario

under consideration. Inserting (15) into the first-order condition (12), we get for the LN

scenario

λγ ω

ωiS i i i

i i i i

R N

R N rK1 =

−− −( )

. (16)

If the FWE solution serves as outside option (S2), we derive from (14)

λπ

ωiS i

F

i i iR N rK2 1= −

− −, (17)

and for S3, we get

λγ ω

ωγ γ

ωiS i i i

i i i i

i i

i i i i

R N

R N rK

C U

R N rK3 1

=−

− −−

− −− −

( ) ( ), (18)

where C R K N W Ni i i iF

iF

iF≡ −( , ) and U U N W bi i

FiF

iF= = −( ) . By expressing π i

F as well as

Ci and U i in terms of the exogenous parameters, it is easy to show that λ λ λiS

iS

iS1 2 3≥ ≥ holds.

The ‘equals’ sign is valid, if and only if γ = 0.

Substituting the base wage (15) and the share parameter (16) into the share equation (3)

and making use of (1), (2), and (5), we find the wage/employment combination for S1:

W biS

iS1 1= η (19)

where η δiS1 =

12

NY K

biS d

ei1

11

1

=

−βκ ακ βκ/

(20)

As first noted by Pohjola (1987) and Anderson/Devereux (1989), the SE solution (19) and

(20) perfectly mimics ‘eff icient contracts’ in the sense of McDonald/Solow (1981). In the

Cobb-Douglas case the total compensation per worker in a SE is the same as in a FWE

( )W WiS

iF1 = .5 From the union’s point of view there is no ‘price’ f or increased employment

( )N NiS

iF1 ≥ . This can be explained by the ‘ increase’ in the union’s bargaining power due to

the implicit assumption that a share contract can be enforced. The union is better off , but due

to declining profits the firm is worse off .

The second [third] share scenario can be derived along similar lines. Inserting (15) and

(17) [(15) and (18)] into the share equation (3) and again using (1), (2) and (5) produces

W biS

iS2 2= η (21)

W biS

iS3 3= η (22)

where ηβκ

βκ δβκβκ

iS2 11

1 1= − −

−−( ) and η δ

γβκ

βκ δ βκiS3

2 1

11= −

−

−−( ) .

and N N NiS

iS

iS2 3 1= = .6 The comparison of (8), (19), (21) and (22) reveals an already

expected result: W W W WiF

iS

iS

iS= ≥ ≥1 2 3 . If the FWE serves as an outside option, the total

compensation per worker will be lower than in a FWE. The negative impact on union utilit y,

however, is overcompensated by the increase in employment ( )N NiS

iF2 ≥ , i.e. U Ui

SiF2 ≥ . By

5 If the elasticity of substitution between labour and capital is greater (less) than unity, thetotal compensation per worker in a SE, Wi

S1 , exceeds (falls short of) the wage in a FWE, WiF .

6 This simply states that eq. (20) is valid for all variants of a SE. Since Ki is yet to bedetermined, this does not imply the equivalence of employment across the different sharescenarios.

13

construction, the firm is indifferent between F and S2. If the FWE-solution serves as threat

point, the bargained real wage declines even more. Nevertheless, both the union and the firm

are better off than in a FWE.

Endogenising the Capital Stock

The choice of the capital stock is assumed to be made in the first stage of the bargaining

game, i.e. before the wage negotiations take place. The firm now maximises its profits

π λ ωi i i i i iR N rK= − ⋅ − −( )[ ( ) ]1 with respect to Ki subject to the perfectly foreseen wage/

employment combinations. In a FWE ( )λ i = 0 we get the textbook result that the firm

expands the capital stock up to the point where the marginal revenue of capital equals the user

costs: R rKF = . The share scenarios are more complex to calculate, since the share parameters

depend on the capital stock. In its most general form, the first-order condition reads:

∂π∂

λ∂ ω

∂ω

∂λ∂

i

ii

i i i i

ii i i i

i

iK

R N rK

KR N rK

K= −

⋅ − −− ⋅ − − =( )

[ ( ) ][ ( ) ]1 0. (23)

By differentiating (16), (17) and (18) with respect to Ki and inserting the results as well as the

share parameters itself into (23) we can extract more information from this conditon.

For S1, the first-order condition for capital (23) simpli fies to

Rr

KS1

1=

− γ. (24)

This is the traditional ‘disincentive-to-invest’ argument: the required marginal revenue of

capital in a SE is greater than in a FWE. Contrary to a FWE where the marginal revenue of

capital accrues solely to the firm, in a SE the firm has to share the marginal revenue with its

workers.

14

Note that in a SE the ‘effective’ capital costs are higher than in a FWE even if all capital

costs are deductible from the share base. This result squarely differs from the literature, which,

in examining different share systems, concentrates on the change in the share base only,

without considering the change in the share parameter. Wadhwani (1987) and Hoel/Moene

(1988), for example, argue that a profit sharing agreement does not affect ‘effective’ capital

costs when these costs are deductible. This argument is invalidated if the trade unions must be

compensated for the lower share base with a higher share parameter. The trade union’s

requirement that variations in the share base must be (at least) utilit y neutral, makes it evident

that some such compensation must be made. For further details see Michaelis (1997).

If the FWE serves as outside option (S2), the first-order condition for capital is given by

R rKS2 = . (25)

As argued above and ill ustrated in fig. 1, for a given capital stock the firm is indifferent

between the outside option (F) and a share contract (S2). Hence, the firm chooses the capital

stock, which maximises the value of the outside option, i.e. its profits in a FWE. For this

reason, S2 and the FWE will generate the same capital stock (the proof will be given below).

Now consider S3, where the FWE serves as a threat point. In this case, (23) reads

( )R

r C U

KKS i i

i

3

1

1

1

1=

−−

−− −

γ γ∂ γ γ

∂( )

. (26)

The firm anticipates that its capital stock decision influences the threat points in the

subsequent wage bargain via the employment level N iF and the real revenue Ri

F :

( )C K R K N K W N Ki i iF

i iF

i iF

iF

i( ) , ( ) ( )= − and ( )U K W b N Ki i iF

iF

i( ) ( )= − . Using (8) and (9),

some manipulations produce γ γ βκ γ βκC U b Ni i iF− − = − >( ) ( )( / )1 1 02 and hence

( )∂ γ γ ∂ βκ γ βκ∂∂

C U K bN

Ki i iiF

i

− − = − >( ) / ( )( / )1 1 02 . Thus, in S3 the ‘effective’ capital costs

15

also increase relative to a FWE, but to a smaller extent than in S1, since λi and thus the

fraction of the marginal revenue of capital which has to be paid to the workers is lower.

However, the capital stock decision is not only influenced by capital costs, but also by the

employment level. Since for a given capital stock employment is higher in a SE than in a

FWE, the marginal revenue of capital increases. Hence, the total effect of a switch from a

FWE to a SE on the optimal capital stock is not a priori clear.

In order to derive the net effect we have to compute the long-run equili brium for the ith

industry, which is defined by the simultaneous validity of the first-order conditions for both

labour and capital. Consider the FWE first. We have six equations − the wage equation (8),

the first-order conditions R WN i= and R rK = , the firm’s product demand Y P Yi ie

d= − , the

definition of real revenue R PYi i i= and the production function (1) − to determine sectoral

wages, prices, output, revenue, capital stock and employment. For the share scenarios, these

variables can be computed with the analoguous set of equations. Here, we focus on the

solutions for the capital stock which may be written as

K Yij

ij

d* = φ , j F S S S= , , ,1 2 3 (27)

where φ φ ακ βκδ

βκκ

βκκ

iF

iS

r b= =

−− −1

1

1 1, φ ακψ βκ

βκκ

βκκ

iS

r b3

1

1 1=

−− −

, ψ γ γβκ

δ βκ≡ − + >−

−1 02 1

1 .

The ‘ * ’ indicates the long-run equili brium nature of the solutions. A well -defined long-run

equili brium needs the assumption of decreasing returns to scale of the revenue function. They

may be due to either the production technology or a downward sloping demand function for

the firm’s output ( )κ < 1 .

Proposition 1 evaluates the results of eq. (27).

16

Proposition 1. For γ > 0 and a given index of real aggregate demand Yd , the relative size of

the optimal capital stock under the different scenarios is K K K KiS

iF

iS

iS* * * *1 2 3< = < .

Proof. Immediate from the comparison of the solutions (27).

Proposition 1 sheds some new light on the ‘disincentive-to-invest’-criti cism. As already

expected, a clear distinction between the share scenarios has to be made. For the LN case we

confirm, but for the outside option and the threat point scenarios we reject the hypothesis that

the implementation of a SE lowers the capital stock. In these (more realistic) cases the

negative capital cost effect is (over)compensated by the positive employment effect. Note that

this result holds at the sectoral level and does not rest on the economy-wide introduction of a

SE.

Proposition 2 now turns to the question whether the firms have an incentive to offer a

share contract.

Proposition 2. For γ > 0 and a given index of real aggregate demand Yd , the ranking of

profits under the alternative scenarios is π π π πiS

iF

iS

iS* * * *3 2 1> = > .

Proof. Insert the expressions for revenue, wages, employment and the capital stock derived as

stated in the text above eq. (27) into (4).

Firms have an incentive to offer share contracts voluntarily if S3 is the correct model. They

are indifferent if the outside option scenario applies. This result contradicts the finding of

Weitzman (1987, p.102) who diagnosed a prisoner’s dilemma in the sense that a SE is

beneficial for the society but no single firm has an incentive to offer share contracts. However,

firms must fear any legal obligation to offer a share contract, since such a law would shift

17

bargaining power to the union. Concerning union utilit y, it is possible to show that UiS* 1 , Ui

S* 2

and UiS* 3 exceed Ui

F* . Hence, the union has an incentive to accept share contracts.

Having derived these results, there remains a puzzle: if f irms and unions are better off

with share contracts, why do we not observe these contracts more often? We have no

straightforward answer to this question. One possibilit y might be that firms are afraid of being

obliged to offer share contracts after such contracts have been introduced for some time.

Although the present legal institutions support the solutions S2 and/or S3, the firms might fear

a legal obligation that drives them into S1.

Irrespective of the assumed share scenario, our results do not depend on the Cobb-

Douglas assumption. A low elasticity of substitution, for instance, implies that the optimal

capital stock hardly responds to the increased capital costs (a steep capital demand function).

But on the other hand, the labour demand curve also becomes steep and the decline in

marginal labour costs from Wi to the base wage ω i generates only a small i ncrease in

employment. In the borderline case of a Leontief production function both effects are zero and

a switch to a SE would not have any impact on Ni or Ki .

III. General Equilibrium

Aggregate wage setting and equilibrium unemployment

So far, we have described the effects of the different remuneration systems in a partial-

equili brium model. As LN pointed out, these results do not necessarily carry over to the

general equili brium context. For our analysis of aggregate unemployment and wages, we

employ the well -known concepts of wage-setting and price-setting schedules (Layard et al.

1991, Lindbeck 1993). We show how these schedules and hence the labour market

equili brium are affected by the choice of the remuneration system. Again, it will t urn out that

18

the exact specification of the sectoral bargaining is decisive for the comparison of a FWE and

a SE.

Consider first the ‘ target’ real wage, i.e. the wage intended by the wage-setters. Equations

(8), (19), (21) and (22) expressed the wage as a mark-up ηij ( )⋅ ( )j = F, S1,S2,S3 on the

fallback income:

W bij

ij= η . (28)

From our calculations above, we observe η η η ηiF

iS

iS

iS= > >1 2 3 . In the general equili brium

context the relationship between b and the unemployment rate u has to be taken into account,

which we do in the standard way,

b u W uB= − +( )1 , (29)

where B denotes (exogenous) non-labour income and W the outside wage. Since, by

assumption, all sectors are identical, the general equili brium can be obtained by setting

W W Wi = ≡ ∀i , which implies η ηij j= ∀i . Combining (28) and (29) with this symmetry

condition delivers the wage-setting schedule (WS):

Wu

uB

j

j=

− −η

η1 1( ). (30)

Since for γ > 0,η j j> ∀1 the WS-schedule is negatively sloped in the W-u-space. A higher

non-labour income B and a higher mark-up shift the WS-schedule to the right.

For the sake of simplicity we restrict the analysis to the case of a constant replacement

ratio q B W≡ / (the more complex case of a fixed non-labour income B is discussed in

Jerger/Michaelis 1995). For a constant replacement ratio the WS schedule implies a unique

u j :

19

uq

jj

j= −

−η

η1

1( ). (31)

From this we can deduce

Proposition 3. For γ > 0 , the ranking of aggregate unemployment rate under the alternative

scenarios is u u u uF S S S= > >1 2 3 .

Proof. The result follows immediately from (31) and the ranking of the wage mark-ups

mentioned in the text below eq. (28).

The equality in proposition 3 is a reproduction of the LN result, where the positive

employment effect at the firm level vanishes at the aggregate level.7 If the SE is modelled

more appropriately, however, we will observe an unambiguously lower unemployment rate.

Aggregate price setting and equilibrium real wages

We now turn to the aggregate price-setting behaviour (the PS schedule) in order to pin down

the general equili brium real wage. We first derive the feasible real wage of f irm i in a FWE,

which is defined by the simultaneous validity of the first-order conditions for capital, R rK =

and labour, R WN i= . The differentiation of the revenue function (2) yields R P FK i K= κ and

R P FN i N= κ . Combining these equations and computing the marginal productivities from the

production function (1), we get the long-run labour demand-schedule for the ith firm:

W r Pi i= βκ ακ α β β( / ) / /1 . Since firms are identical, the general equili brium is characterised by

W Wi = and Pi = 1. Thus, the aggregate PS-schedule in a FWE can be written as

7 This result is restricted to the Cobb-Douglas case. If the elasticity of substitution betweenlabour and capital is smaller than unity the employment effect of the partial equili briumanalysis carries over to the general equili brium. Aggregate employment will be lower if theelasticity of substitution exceeds unity (see LN, p. 786).

20

Wr

F =

βκ ακαβ

. (32)

The feasible real wage WF is independent of (un)employment due to the assumptions of a

constant-returns-to-scale production function and a constant price elasticity of product

demand (constant mark-up on constant marginal costs).

The PS-schedules for the three SE scenarios can be computed along similar lines. We

again use (2) and (1) to compute the marginal revenues of Ni and Ki and insert the results

into (5) and (24), (5) and (25), and (5) and (26) for the first, the second and the third share

scenario, respectively. These manipulations combined with ω ij b= give the base wage

( )ω βκ ακ γ α β βiS

ir P1 11= −( ) // / for S1. For S2 and S3 we get ω βκ ακ α β β

iS

ir P2 1= ( / ) / / and

ω βκ ακψ α β βiS

ir P3 1= ( / ) / / , respectively. Since in a SE the feasible real wage also comprises

the share element, we have to insert these expressions into (19), (21) and (22), again using

ω ij b= . Observing the symmetry conditions W Wi

j j= and Pi = 1, we get:

Wr

S1 1= −

βκδ ακ γαβ( )

(33)

Wr

S2 11 1= − −

−−( )βκ δ ακβκ

βκ

αβ

(34)

Wr

S32 1

11= − −

−−βκδ γ

βκβκ δ ακψβκ

αβ

( ) . (35)

The comparison of the feasible real wages for γ > 0 in the four scenarios is not

straightforward, since there are two counteracting forces whose relative strength depends on

21

the parameter constellation.8 Firstly, a switch from a FWE to a SE entails a decline in

marginal labour costs because of a lower base wage. The cost decline allows a price decline,

which in turn raises real revenue and via the share element the workers’ compensation. This

effect, pushing up the feasible real wage, obviously depends on the degree of competitiveness

in the product market. The more competitive the product market, the lower is the price decline

corresponding to a given decline in marginal labour costs. In the borderline case of perfect

competition this effect vanishes. The second effect is the increase in ‘effective’ capital costs,

which implies a decrease in the feasible real wage.

What is the net effect? The easiest case is the ‘outside option’ scenario S2, where we do

not observe an increase in ‘effective’ capital costs. Hence, the feasible real wage WS2 is

always higher than WF . The comparison of (32) with (33) and (34) reveals that WS1 is higher

than WF if

( )1 1− >γ δβα (36)

and similarly, WS3 is higher than WF if

δ γβκ

βκ δ ψβκαβ−

−

>

−−

2 1

11 1( ) . (37)

As a rule both conditions are fulfill ed and the SE would have a positive impact on the feasible

real wage even in the LN and the ‘ threat point’ scenario. However, these conditions are

violated for (unrealistic) high values of α and/or γ as well as for very competitive product

markets. In the case of perfect competition ( )κ =1 , criteria (36) and (37) never hold.

8 It is obvious that in the borderline case of γ = 0 the wage is identical in all scenarios:

W W W WF S S S= = =1 2 3 .

22

Aggregate capital stock

Proposition 1 rejected the ‘disincentive-to-invest’ criti cism of the SE except for the scenario

S1 at the sectoral level. Now we explore whether this result carries over to the aggregate level.

Employment in a representative sector i is given by ( )N M uij

ij= −1 . Normalising the

aggregate labour force to unity and hence Mi to 1 S where S denotes the exogenous number

of sectors and observing sectoral symmetry, (1) and (27) can be solved for the equili brium

capital stock under the alternative scenarios

( )( )K uj j j= −11

φ β (38)

Proposition 4 is based on these computations.

Proposition 4. The relative size of the aggregate capital stock under the different scenarios is

K K K KS S F S3 2 1> > > .

Proof. See text.

Although we stuck − for computational reasons − to some convenient functional forms

e.g. in our specification of the production function and union utilit y, propositions 3 and 4 are

robust with respect to a number of modifications. Probably the most important one is that the

reduction of the NAIRU and the associated effects on aggregate capital stock hold a fortiori i f

we assume a more realistic value of the elasticity of substitution between labour and capital

below unity. With respect to union utilit y, the only assumption that would destroy the result is

that unions do not place some positive weight on all l evels of employment. In this case the

unions’ indifference curves would be horizontal (or at least kinked) and eff icient contracts

would lie on the labour demand curve. However, despite the arguments in Oswald (1993),

23

there are good theoretical (see Pencavel 1991) and empirical reasons (see Doiron 1995) to

stick to an employment motive of the unions.

IV. Conclusions

This paper challenges the conventional wisdom concerning the aggregate effects of a switch

from a Fixed Wage Economy of a Share Economy. Layard/Nickell (1990) suggested that

aggregate unemployment would be unaffected by the choice of the remuneration system, and

hence rejected the Weitzman (1987) conjecture that the undisputed positive employment

effect at the sectoral level carries over to the macro level. We show that the Layard/Nickell

result is only valid under the (unrealistic) assumption that a firm must offer a share contract to

the union. If we allow for the possibilit y that a fixed wage contract serves as an outside option

or as a threat point in the share bargain, the sectoral and - more importantly - the aggregate

outcomes are substantially altered. In particular, we show that the Weitzman conjecture

indeed holds.

Unlike most other studies in this area, we allow for an endogenous adjustment of the

capital stock. This does not only render the analysis of equili brium unemployment more

appropriate, but also rejects the traditional ‘disincentive-to-invest’ result. The most important

message of this paper, however, is that the scope of the profit-sharing idea of Martin

Weitzman extends beyond the time horizon of a business cycle.

References

Anderson, S. and M. Devereux (1989), Profit Sharing and Optimal Labour Contracts,

Canadian Journal of Economics 22, 425-433.

24

Binmore, K., A. Rubinstein and A. Wolinsky (1986), The Nash Bargaining Solution in

Economic Modelli ng, Rand Journal of Economics 17, 176-188.

Binmore, K.; A. Shaked and J. Sutton (1989), An Outside Option Experiment, Quarterly

Journal of Economics 96, 753-770.

Booth, A.L. (1995), The Economics of the Trade Union, Cambridge University Press.

Doiron, D.J. (1995), A Test of the Insider-Outsider Hypothesis in Union Preferences,

Economica 62, 281-290.

Espinoza, M.P. and C. Rhee, (1989), Eff icient Bargaining as a Repeated Game, Quarterly

Journal of Economics 96, 565-588.

European Commission (1991), PEPPER-Report: Promotion of Employee Participation on

Profits and Enterprise Results, Social Europe, Supplement III /91.

Freeman, R.B. and M.L. Weitzman (1987), Bonuses and Employment in Japan, Journal of the

Japanese and International Economies 1, 168-194.

Hartog, J. and J. Theeuwes (eds.)(1993), Labour Market Contracts and Institutions, North

Holland, Amsterdam.

Hoel, M. and K.O. Moene (1988), Profit Sharing, Unions and Investments, Scandinavian

Journal of Economics 90, 493-505.

Holden, S. (1988), Local and Central Wage Bargaining, Scandinavian Journal of Economics

90, 93-99.

Holmlund, B. (1990), Profit Sharing, Wage Bargaining, and Unemployment, Economic

Inquiry 28, 257-268.

Jerger, J. and J. Michaelis (1995), Remuneration Systems, Capital Formation and the NAIRU,

London School of Economics Centre for Economic Performance, Discussion Paper No.

227.

25

Layard, R. and S. Nickell (1990), Is Unemployment Lower if Unions Bargain over

Employment?, Quarterly Journal of Economics 105, 773-787.

Layard, R., S. Nickell and R. Jackman (1991), Unemployment - Macroeconomic Performance

and the Labour Market, Oxford University Press.

Lindbeck, A. (1993), Unemployment and Macroeconomics, MIT Press, Cambridge, MA.

McDonald, I.M. and R.M. Solow (1981), Wage Bargaining and Employment, American

Economic Review 71, 896-908.

Meade, J. (1986), Alternative Systems of Business Organization and of Workers´

Remuneration, Allen & Unwin, London.

Michaelis, J. (1997), On the Equivalence of Profit and Revenue Sharing, Economics Letters

57, 113-118.

Moene, K.O. (1988), Unions’ Threats and Wage Determination, Economic Journal 98, 471-

483.

Oswald, A. (1993), Eff icient Contracts are on the Labour Demand Curve: Theory and Facts,

Labor Economics 1, 85-114.

Pencavel, J. (1991), Labor Markets under Trade Unionism: Employment, Wages, and Hours,

Blackwell , Oxford.

Pohjola, M. (1987), Profit Sharing, Collective Bargaining and Employment, Journal of

Institutional and Theoretical Economics 143, 334-342.

Wadhwani, S. (1987), Profit Sharing and Meade s Discriminating Labour-Capital Partnership:

A Review Article, Oxford Economic Papers 39, 421-442.

Weitzman, M.L. (1985), The Simple Macroeconomics of Profit Sharing, American Economic

Review 75, 937-953.

Weitzman, M.L. (1987), Steady State Unemployment under Profit Sharing, Economic Journal

97, 86-105.

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