On the equivalence of profit and revenue sharing

Economics Letters 57 (1997) 113–118

On the equivalence of profit and revenue sharing Jochen Michaelis* Institute for Public Finance, University of Freiburg, Maximilianstr. 15, 79100 Freiburg i.Br., Germany Received 20 February 1997; accepted 19 June 1997

Abstract This paper shows that even in a unionised labour market net and gross profit sharing as well as revenue sharing are isomorphic. Neither the capital stock nor the wage / employment combination depends on the deductibility of base wages and / or capital costs from the share base.  1997 Elsevier Science S.A. Keywords: Remuneration systems; Profit sharing; Investment JEL classification: J23; J33; J51

1. Introduction The seminal work of Weitzman (Weitzman, 1985, 1987) provoked a lively debate on the pros and cons of different remuneration systems, in particular on the employment effect of a profit-sharing scheme compared to a fixed wage system. However, little attention has been paid to the question to what extent the outcome of a shared agreement depends on the definition of profits, which have to be shared with the employees. Gross profit sharing, net profit sharing and revenue sharing have been used interchangeably, but always without any proof. This procedure is hard to justify, since a redefinition of the share base is neither neutral with respect to the marginal revenue of labour (capital) a firm accrues, nor is it neutral with respect to the bargained parameters of the share contract. This paper provides a rigorous analysis of the impact of different share schemes on the firm’s employment decision, the wage bargain and the firm’s capital stock decision. It is shown that indeed the definition of the share base does not matter, i.e. despite different base wages and different share parameters, all share schemes generate the same capital stock and the same wage / employment combination. 2. The model Consider the labour compensation function R( ? ) 2 mv N 2 n rK w 5 v 1 l ]]]]]] N *Tel.: (49-761) 2032358; fax: (49-761) 2032303. 0165-1765 / 97 / $17.00  1997 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00200-0

(1)

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where w denotes total wage income per worker, v is the base wage, l is the share parameter, R is the firm’s revenue, N is labour, K is capital stock, and r represents the exogenous user costs of capital. The parameters m (0# m #1) and n (0# n #1) measure the degree of deductibility of base wages and capital costs from the share base. If n equals unity, both the opportunity costs that the entrepreneur charges on his own capital and interest payments on external funding are deductible from revenues before the sharing of profits. If only the latter is deductible, the parameter n corresponds to the firm’s debt ratio. The wage formula Eq. (1) encompasses gross profit sharing ( m 51; n 50), net profit sharing ( m 5 n 51) and revenue sharing ( m 5 n 50). Analysing a fixed wage system ( l 50) is beyond the scope of this paper. The determination of employment, wages and the capital stock 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. In the third and final stage, the firm exercises its right-to-manage and sets employment to ensure maximum profits for the given capital stock and pay parameters. The firm’s profit function is given by

p 5 R(K, N) 2 wN 2 rK.

(2)

The revenue function R(K, N) is assumed to be strictly concave and twice differentiable in K and N. The concavity may be due to the production function and / or non-competitive product markets. The assumptions on the revenue function include as special cases a price-taking firm with a decreasing returns to scale production function and a monopolistically competitive firm with a constant returns to scale production function. The right-to-manage privilege of the firm enters the set-up as the additional condition that marginal labour costs equal marginal revenue of labour. Inserting Eq. (1) into Eq. (2) and maximising with respect to N gives the following first-order condition (subscripts denote partial derivatives throughout the paper): 1 2 lm R N 5 ]]] v. 12l

(3)

In case of profit-sharing ( m 51) employment is uniquely determined by the base wage. A change in the share parameter has only a distribution effect, but no employment effect. This result does not carry over to the revenue sharing scheme ( m 50), where only the fraction (12 l) of the marginal revenue of labour accrues to the firm. Since only this fraction can be used to finance the base wage, employment declines with a higher share parameter. If the base wage were identical, revenue sharing would generate lower employment than profit sharing. In the second stage of the game the firm and the union bargain over the base wage and the share parameter. For sake of simplicity we assume a utilitarian trade union, its objective function is represented by U 5 Nu(w) 1 (M 2 N)u(b).

(4)

The function u(w) is increasing in w. The trade union may be risk averse (u ww ,0) or risk neutral (u ww 50). The exogenous union membership M exceeds – by assumption – the firm’s labour demand.

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Hence, M2N .0 union members are not employed by the firm in question, they obtain the fall-back income b. For a discussion of the pros and cons of Eq. (4) and for further details on union preferences compare Oswald (1985); Booth (1995). The bargaining process is characterised by a generalised Nash solution. The base wage and the share parameter are chosen so as to maximise the Nash product V : g 12g Max V 5 (U 2 U¯ ) (p 2 p¯ ) v, l

1 2 lm s.t. R N 5 ]]] v, 12l

(5)

where U and p are defined by Eqs. (4) and (2), respectively. The parameter g (0#g #1) 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 fall-back income ¯ (cf. Binmore et al., 1986), we get as the union’s threat point U5Mu(b). Since capital costs must be 1 covered by the firm regardless of the bargaining success, its threat point is p¯ 5 2rK. By inserting Eqs. (1), (2), (4) into the Nash maximand Eq. (5), observing the definition of the threat points and taking Eq. (3) into account, we can differentiate with respect to v and l to obtain the first-order conditions:

F G (6) (1 2 g )(R 2 mv N 2 n rK) g ≠N ≠ log V ]]] 5 ]]]]] F A ] 1 u (R 2 mv N 2 n rK)G 2 ]]]]]]]] 5 0 ≠l ≠l R 2 wN N[u(w) 2 u(b)] (1 2 g )(1 2 lm )N g ≠N ≠ log V ]]] 5 ]]]]] A ] 1 u w (1 2 lm )N 2 ]]]]]] 5 0 ≠v ≠ v R 2 wN N[u(w) 2 u(b)] w

(7) with

lu w A ; u(w) 2 u(b) 1 lu w (R N 2 mv ) 2 ]] (R 2 mv N 2 n rK). N For both Eqs. (6) and (7) to hold simultaneously, A must be equal to zero. This implies u(w) 2 u(b) l ]]]] 5 ] (R 2 mv N 2 n rK) 2 l(R N 2 mv ). uw N

(8)

By observing the wage formula Eq. (1) and the first-order condition Eq. (3), Eq. (8) can be rearranged to u(w) 2 u(b) ]]]] 5 w 2 R N . uw

(9)

Eq. (9) describes the contract curve, which is characterised by equality of the slopes of a union 1

The modelling of the threat points is standard, but in Jerger and Michaelis (1995) it is argued that the standard may be wrong. It rests on the assumption that the union has a right to strike for a (higher) share of the firm’s profits. But this contradicts the legal framework observed in almost all market economies, which ensures that the property rights for profits (revenue) are with the firm. Jerger and Michaelis (1995) model the outcome of a bargain on a fixed wage contract as fall-back position of the share bargain. However, the different modelling of the threat points has no impact on the main result of this paper, i.e. all share systems are still equivalent.

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indifference curve and an isoprofit curve of the firm (McDonald and Solow, 1981). In short, the outcome of the share bargain is efficient, i.e. pareto-optimal. The wage / employment combination lies on the contract curve. This result applies to both profit sharing and revenue sharing. The substitution of Eq. (9) into Eq. (7) yields an implicit equation for the optimal share parameter:

g (1 2 l)(R 2 v N) l 5 ]]]]]]]]]]]]]]. (1 2 l)(R 2 mv N 2 n rK) 2 (1 2 g )(1 2 m )v N

(10)

Consider profit sharing (PS) first. For m 51, Eq. (10) reduces to

g (R 2 v N) l PS 5 ]]]]]. R 2 v N 2 n rK

(11)

In a competitive labour market (g 50) we will not observe share contracts, i.e. a noncompetitive labour market is a necessary precondition for share contracts. If the capital costs are not deductible from the share base (gross profit sharing: n 50), the optimal share parameter will be equal to g, the parameter or the union’s relative bargaining power. The deductibility of the capital costs leads to a l PS that exceeds g. This reflects the fact that the union members will not accept a share system with a smaller base (from which the share is computed), if they are not compensated by a larger share of this base. Hence, a switch from gross to net profit sharing (n 51) would be accompanied by an increase in the bargained share parameter. The increase in the share parameter offsets the decline in the share base, i.e. in a profit-sharing scheme total income per worker w PS does not depend on the deductibility of capitals costs. This result can be obtained by inserting Eq. (11) into wage formula Eq. (1) and by observing R N 5 v :

S

D

R w PS 5 R N 1 g ] 2 R N . N

(12)

The parameter n has no impact on the bargained wage. The contract curve Eq. (9) combined with Eq. (12) gives the wage / employment combination w PS and N PS . As first noted by Pohjola (1987); Anderson and Devereux (1989), the profit-sharing solution w PS and N PS perfectly mimics ‘efficient’ contracts in the sense of McDonald and Solow (1981). In the case of a revenue sharing (RS) scheme ( m 5 n 50), Eq. (10) simplifies to

l RS 5 g.

(13)

The optimal share parameter equals union power in the bargain. Inserting Eqs. (13) and (3) into Eq. (1) yields after some manipulations

S

D

R w RS 5 R N 1 g ] 2 R N . N

(14)

The wage / employment combination w RS and N RS can be derived from Eqs. (9) and (14). Since Eq. (14) is identical to Eq. (12), it is clear that w RS and N RS are identical to w PS and N PS , respectively. Despite the different share base, profit sharing and revenue sharing are equivalent with respect to both the bargained wage and the firm’s employment decision. This result extends the analysis of Pohjola (1987); Anderson and Devereux (1989), who have shown that profit sharing serves as a means to

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enforce efficient contracts. Our analysis suggests that the implementation of revenue sharing would also mimic efficient contracts. Let us turn to the firm’s capital stock decision. 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. The firm now maximises the profit function Eq. (2) with respect to K subject to the perfectly foreseen wage / employment combination. In the case of a profit-sharing scheme the profit function may be written as p PS 5R2 v N2 l PS (R2 v N2 n rK)2rK. By observing Eq. (11) we get p PS 5R2 v N2g (R2 v N)2 rK. In the case of revenue sharing we make use of Eqs. (1) and (13) in order to get p RS 5R2 v N2 g R2rK. The maximisation of p PS and p RS with respect to K leads to an identical first-order condition for capital: r R K 5 ]]. 12g

(15)

Since the marginal revenue of capital does not accrue to the investor alone, but has to be shared with the employees, any share system creates a distortion of the firm’s investment decision. Compared to a fixed wage system or efficient contracts the ‘effective’ capital costs increase from r to r /(12g ). In particular the equivalence of profit / revenue sharing and efficient contracts breaks down. The higher the union power in the bargain, the higher the bargained share parameter will be, and the lower the firm’s desired capital stock. Since a declining capital stock goes along with a declining employment level strong union power may be counterproductive even for the union’s utility. This result is similar to Grout (1984). Eq. (15) allows for another—and at least from my point of view—even more interesting conclusion: in a profit-sharing system the increase in ‘effective’ capital costs is independent of the deductibility of interest payments from the share base. This result is a complete reversal of the conventional wisdom found in 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); Hoel and Moene (1988), for example, argue that a profit-sharing agreement does not affect capital costs when all interest payments are deductible.2 This argument is invalid since the trade union must be compensated for the lower share base with a higher share parameter. The endogeneity of the share parameter also exposes the limitations of the frequently made comparison between profit sharing by workers and a profits tax. From the firm’s point of view it does indeed make a difference whether the recipient of the profit component is called ‘tax collector’ or ‘employee’. Because the tax rate is constant every reduction in the tax base ‘profits’ leads to a reduction in the tax burden. However, in case of profit sharing the ‘tax rate’, i.e. the share parameter, must rise. The incentive to pursue a policy of wage avoidance, analogous to tax avoidance policy, will disappear. Since the first-order condition for capital Eq. (15) does not distinguish between profit and revenue sharing, both remuneration systems generate the same K. The desired capital stock does not depend on the specific form of the share scheme. Profit and revenue sharing are equivalent, even if the process of 2

Their line of argument may be seen from p PS 5R2 v N2 l PS (R2 v N2vrK)2rK. If l PS is assumed to be constant and not to be determined by bargain (see Eq. (11)), the maximisation of p PS with respect to K will lead to the first-order condition RK 5(12 ln )r /(12 l), which for n 51 simplifies to R K 5r.

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capital accumulation is taken into account. To put it differently, all share systems are isomorphic. Although this result is derived within a partial equilibrium model only, it is straightforward to show that it carries over to the aggregate level (general equilibrium), where there is an economy-wide adoption of share contracts (see Michaelis, 1997).

Acknowledgements ¨ ¨ I would like to thank Jurgen Jerger, Michael Pfluger, Guido Raddatz and Alexander Spermann for helpful comments and suggestions. Financial support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged.

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