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Profit-shares, bargaining, and unemployment.


Weitzman [1987] has argued that in an economy with long term unemployment due to extra-competitive wages, profit sharing can eliminate this unemployment by lowering the marginal cost of labor. Jackman [1988] provides additional microfoundations for this claim by modeling long term unemployment as arising from an externality in decentralized wage setting. In that model, a mandated profit share lowers wages and raises net profits, thereby inducing the entry of firms into the market, unambiguously lowering unemployment and leaving the real wage unchanged in general equilibrium. The generality of this result is put into question by Holmlund [1990] who considers a similar case, but with the capital stock fixed economy wide, and shows that profit sharing may raise or lower unemployment depending on standard properties of the production function. In that short run model, firms implicitly operate on downward sloping labor demand curves, so profit sharing expands employment only to the degree that it reduces wages in general equilibrium. If the labor demand elasticity is great enough, the general equilibrium effect of profit sharing is to raise wages and lower employment. Holmlund also shows that his result is independent of whether the profit share is bargained over or mandated.

In models with variable capital, the question of the effect of profit sharing on employment does not reduce to the question of its effect on wages. The introduction of profit sharing also affects net profitability and so may lead to an inflow or withdrawal of capital in the market in the long run, further influencing employment. In Jackman's model, employment unambiguously increases. However, Jackman only considers a mandated profit share under "monopoly union" wage setting, leaving the impact of bargaining over profit shares and wages unresolved. Further, substantial structural differences in the models of Jackman and Holmlund make direct comparisons of these models difficult.

In the present paper, I present a model in the spirit of Holmlund's, but with free entry of firms subject to overhead costs, and show that under Nash wage bargains the effect of profit sharing on both employment and compensation depends on whether the profit share is bargained over or mandated as well as on the profit accounting used to determine share payments. Of particular interest is the possibility that profit sharing may reduce both aggregate employment and total compensation per worker due to general equilibrium effects. No such possibility exists in the models of Jackman and Holmlund.

Profit Sharing, Employment, and Bargaining

As is well known, if Nash bargaining is over both wages and employment, then profit sharing leaves microeconomic employment and incomes unchanged. Further, under unilateral employment setting by firms, Nash union wage and profit share bargains emulate Nash bargains over wages and employment (Pohjola [1987], Anderson and Devereux [1989], Johnson [1990]). In partial equilibrium, the ability to bargain over either profit shares or employment in addition to wages raises employment and is Pareto improving. In general equilibrium, however, reservation utilities may rise enough to cause wages to rise and employment to fall. This is shown by Layard and Nickell [1990] for bargaining over employment and by Holmlund [1990] for bargaining over profit shares. Both models are short run models in which the capital stocks are fixed economy wide. Holmlund goes further to show that the same mechanism also operates for mandated profit shares.

In the long run model presented here, Holmlund's result for bargained profit shares continue to hold (as does Layard and Nickell's for bargained employment), though for different reasons. The introduction of profit sharing does not produce upward wage pressure, since free entry eliminates profits and thus any direct effect of profit sharing on reservation utilities. Rather, employment is reduced if it reduces profitability and so yields exit of capital sufficient to offset the positive effect of reduced wages on employment. Given the flexibility of bargaining over both wages and profit shares, labor may be in a position to bargain for a compensation package that reduces profitability and thus aggregate employment.

Holmlund's result for mandated profit shares, on the other hand, is not supported. Mandated profit shares have an unambiguously non-negative effect on aggregate employment as long as the profit share accounting reflects firms' full revenues and costs. A mandated profit share constrains the bargain over compensation so that its effect is to raise profitability and thus expand employment.


I consider a very simple one period model of a labor market with an employment externality. Richer variants of the model appear in Georges [1989; 1991; 1994]. The economy consists of a large number of union-firm pairs. Pair i employs [n.sub.i] workers and pays each worker [z.sub.i] in total compensation. This compensation takes the form

(1) [z.sub.i] = [w.sub.i] + [Lambda][(f([n.sub.i]) - [w.sub.i] [n.sub.i] - C)/[n.sub.i]]

where [w.sub.i] is the wage rate and [Lambda] is the share of profits going to the union. Firms face identical concave production functions f(n), and incur overhead costs C.

Firm i's objective function is net profits

(2) [[Pi].sub.i] = f([n.sub.i]) - [z.sub.i] [n.sub.i] - C

Union i's objective function is

(3) [Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is expected compensation elsewhere in the economy.

I assume that any bargaining is resolved using the cooperative Nash bargaining solution with threat points taken to be the impasse utilities (0,- C) which occur if the union and firm fail to come to an agreement.(1) Specifically, an arbitrator seeks to maximize

(4) [Mathematical Expression Omitted]

subject to whatever constraints apply in a given bargaining environment.

The outside or alternate compensation [Mathematical Expression Omitted] is modeled as the expected income of an individual who is currently unemployed. Calling aggregate employment N, and setting the size of the labor force equal to one, let

(5) [Mathematical Expression Omitted]

where [Delta] is an exogenous rate of turnover of jobs.(2) A worker who is unemployed at the beginning of the period has a chance [Delta] N/[(1- N) + [Delta]N] of being reemployed and earning z, the compensation per worker in the aggregate. Otherwise, the worker obtains zero utility. As N approaches one, the probability of being reemployed approaches one,(3) and so [Mathematical Expression Omitted] approaches z. Notice then that, in equilibrium, the divergence between [Mathematical Expression Omitted] and z is an indicator of the degree of unemployment in this model. Specifically, rewriting Equation (5) we can see that the unemployment rate increases monotonically in the equilibrium ratio [Mathematical Expression Omitted] and is zero at [Mathematical Expression Omitted].

(6) [Mathematical Expression Omitted]

It will be convenient below to show the effect of profit sharing on equilibrium unemployment by deriving its effect on [Mathematical Expression Omitted] in equilibrium.

I consider only zero net profit equilibrium, resulting from the free entry and exit of firms. Aggregate employment N is determined by both employment per firm n and the number x of firms in the market: N = n [multiplied by] x. By Equation (2), zero net profits require that in equilibrium

(7) z = (f (n) - C)/n

Further, by Equation (1), if the profit share rate [Lambda] is less than one, then worker compensation must be purely wages in equilibrium and so

(7[prime]) w = (f(n) - C)/n


If the union and firm bargain over both employment n and total per worker compensation z, then the arbitrator solves the problem [max.sub.[n.sub.i],[z.sub.i]] [V.sub.i]. The resulting employment and compensation levels satisfy

(8) [Mathematical Expression Omitted]

(9) [z.sub.i] = (1/2) [(f([n.sub.i]) / [n.sub.i]) + f[prime]([n.sub.i])]

In equilibrium we have the additional conditions that compensation be equalized across firms and that profits be zero for all firms. Equations (7) and (9) imply that equilibrium employment per firm [n.sup.*] is determined by the firm's technology as follows:

(10) f[prime]([n.sup.*]) = (f ([n.sup.*]) - 2C)/[n.sup.*]

Adding Equation (8) we find that

(11) [Mathematical Expression Omitted]

Thus, by Equation (6) the equilibrium unemployment rate is positive.

The composition of z is irrelevant to the Nash bargain for n and z and thus to aggregate unemployment. This is true whether wages w and [Lambda], are both bargained over or [Lambda] is mandated and wages are bargained over. Any increases in the profit share are traded for reductions in the base wage under the Nash bargain. This point has been made by Anderson and Devereux [1989].


Now consider the case in which firms are free to set employment as they desire after compensation per worker has been determined. It is often argued that this "right to manage" form of bargaining is more representative of real world bargaining than bargaining over compensation and employment, even though the former appears to be inefficient.

Note that a firm i that takes [w.sub.i] and [[Lambda].sub.i] as given and selects employment [n.sub.i] will hire up to the point at which it's marginal product of labor is equal to the wage [w.sub.i], not total compensation [z.sub.i],

(12) f[prime] ([n.sub.i]) = [w.sub.i]

since [z.sub.i] (which is what the firm intrinsically cares about) is in this case not independent of [n.sub.i].

A. Mandated Profit Share

First consider the case in which [Lambda] is mandated and only the wage is bargained over. In this case we find that unemployment depends inversely on the profit share rate. Now [Lambda] is given exogenously, and the arbitrator solves the problem [max.sub.[w.sub.i]] [V.sub.i] subject to Equations (1) and (12).

The resulting employment and compensation levels satisfy

(13) [Mathematical Expression Omitted]

Equations (7[prime]) and (12) imply that, in zero profit equilibrium, for [Lambda] [less than] 1, employment per firm is again determined by the technology, now at a lower level [n.sup.**] which maximizes net output per worker ([n.sup.**] = argmax [(f(n) - C)/n]) and so satisfies

(14) f[prime] ([n.sup.**]) = (f([n.sup.**]) - C)/[n.sup.**]

Notice that total per worker compensation z will be greater in equilibrium than under "efficient bargaining" and will be independent of [Lambda].

Combining Equations (13) and (14) we find that at zero profit equilibrium

(15) [Mathematical Expression Omitted]

where [Eta](n) is the demand elasticity - f[prime] (n) / (n [multiplied by] f[double prime] (n)). Thus, increasing the mandated profit share reduces unemployment, and equilibrium unemployment is eliminated in the limit as the profit share rate approaches 100%.(4) Unions prefer to take lower total per worker compensation in return for increased employment as the profit share rate rises. Firms make positive profits, attracting entry which causes unemployment to fall while leaving compensation unchanged in general equilibrium.

The result that unemployment falls monotonically in the mandated rate [Lambda] of profit sharing is contrary to Holmlund's finding that the sign of du/d[Lambda] depends on the elasticity of labor demand. The unambiguous sign here is due to the dual role of the overhead cost C in the profit share and entry problems. Since z is fixed in general equilibrium, any development that lowers w in partial equilibrium (holding [Mathematical Expression Omitted] constant) will raise employment in general equilibrium. Put another way, at a zero profit equilibrium, a change in [Lambda] does not improve workers' compensation and so has no effect on reservation utilities ceterus paribus.

B. Bargained Profit Share

Now consider the case in which wages w and the profit share rate [Lambda] are bargained over, after which the firm sets employment. The arbitrator solves the problem [max.sub.[w.sub.i][[Lambda].sub.i]] [V.sub.i] subject to Equations (1) and (12).

The resulting employment and compensation levels satisfy Equations (8) and (9). By Equation (12), this requires that the wage be set at the reservation utility [Mathematical Expression Omitted]. Under zero profit equilibrium, the rest of the total compensation z is made up by a 100% profit share [Lambda] = 1. Employment and compensation at both the partial and general equilibrium levels are identical to those under efficient bargaining (Section III).

Since [n.sup.*] [greater than] [n.sup.**], the relative size of [Mathematical Expression Omitted] in Equations (11) and (15) evaluated at [Lambda] = 0 depends on the elasticity of f[prime]. Consequently, starting from no profit sharing and "right to manage," a move to a bargained profit share may be employment enhancing or diminishing. The same is true of a move to bargained employment. It can easily be shown for example that, in this model as with the short run models of Holmlund [1990] and Layard and Nickell [1990] respectively, if the production function is CES in labor and capital, with capital fixed at the firm level, then unemployment will rise (fall) in general equilibrium if the elasticity of substitution is greater than (less than) one.

However, we also have the result in the present model that a move to bargained profit sharing or employment will unambiguously reduce total compensation per worker. Thus the possibility exists that both employment and compensation would be reduced. This result runs counter to those of Holmlund [1990] and Layard and Nickell [1990], for whom compensation and unemployment are positively related. In the short run, if firms are effectively operating on negatively sloped labor demand curves, the effect of profit sharing on employment depends exclusively on its effect on wages. This is not so here, as the real wage unambiguously falls, but net profitability is also reduced, and so the number of firms in the market falls in the long run, pushing up unemployment. Starting from a zero profit equilibrium without profit sharing, the profit share bargain emulates the efficient bargain of Section III by trading a reduced wage for a value of [Lambda] [greater than] 1. The resulting long run exit of firms depresses outside employment opportunities, driving [Lambda] to one.

As a final note, the plausibility of bargained profit shares [Lambda] of one or more must depend on the definition of net profits against which the share is taken. Overhead costs C are likely to include sunk costs. Firms will stay in business even if these costs are not recouped, as long as all other costs are covered. Thus, a bargained compensation package may yield negative economic profit without driving firms out of business. However, when generalized economy wide, such a bargain will deter entry of new firms and thus lower aggregate employment in the long run, as indicated above.

As I show below, it is also the case that the effect of profit sharing on employment and compensation is sensitive to the profit share accounting used to determine profit share payments.


In the present model, overhead costs enter both the entry condition (firms will not enter unless they correctly anticipate covering overhead costs) and the Nash bargaining problem (in the calculation of profit against which profit shares are assessed). However, it is plausible that in practice, the profit accounting underlying the entry decision and the bargaining problem might differ. For example, overhead costs C might include some sunk costs which matter to the entry decision but do not appear as current accounting costs.

Suppose then an extreme case in which entry continues to be determined by profits net of overhead costs C, but that the share rate is set on profits gross of overheads. In this case, calling the new share rate [Mathematical Expression Omitted], Equation (1) is replaced by

(16) [Mathematical Expression Omitted]

The equilibrium of the model under efficient bargaining (Section III) is unchanged. Similarly the equilibrium under unilateral employment setting under a bargained profit share (Section IV.B) is unchanged other than the cosmetic distinction that the bargained profit share is [Mathematical Expression Omitted].

However, the problem under unilateral employment setting with mandated profit sharing (Section IV.A) is transformed. The bargaining solution condition Equation (13) is unchanged, but the zero profit entry condition now requires an equilibrium per-firm employment level of [Mathematical Expression Omitted]. Notice that this implies that compensation per worker z falls in [Mathematical Expression Omitted] in equilibrium.

We now have

(17) [Mathematical Expression Omitted]

At [Mathematical Expression Omitted] the equilibrium is identical to that in Section IV.A. However, as [Mathematical Expression Omitted] rises to 1/2, [n.sup.***] rises, and at [Mathematical Expression Omitted] the equilibrium is identical to that in Section IV.B. That is, under a mandated profit share of 1/2 (which corresponds to a profit share of one in Sections II-IV), the wage bargain now mimics the efficient bargain of Section III and the bargained share of Section IV.B.

Thus, the effect of a large mandated profit share (now defined as [Mathematical Expression Omitted]) on unemployment is now ambiguous, since this question is equivalent to the question, is employment greater or smaller if bargaining is over profit shares in addition to wages. The mandated share still lowers wages as before, but now digs deeper into profits, and so may exert a counterveiling negative pressure on entry. Further, since total compensation z per worker falls in [Mathematical Expression Omitted] in equilibrium, we again have the possibility that a mandated share will lower both employment and compensation in general equilibrium.


This paper has considered the effect of profit sharing on unemployment in a simple model with an employment externality, overhead costs, and free entry, under several forms of decentralized union bargaining. The central finding is that under Nash wage bargains, equilibrium unemployment is unambiguously reduced by mandated profit sharing but may be either mitigated or exacerbated by a bargained profit share depending on the elasticity of labor demand. Further, both wages and total compensation per worker are unambiguously reduced by a bargained profit share, so that economy wide profit sharing may have the perverse effect of lowering both employment and compensation in general equilibrium. Finally, if overhead costs relevant to firms' entry decisions are excluded from the profit share accounting, then a mandated profit share may also have such a perverse effect.

Unemployment is aggravated by profit sharing in this paper if it cuts into net profits enough to induce a substantial exodus of capital from the market. Whether or not this is the case depends on, among other things, the bargaining environment. The Nash bargaining solution is used here as a convenient example with which to illustrate the theoretical ambiguity of the effect of a bargained profit share on unemployment. The result that unemployment unambiguously falls in a mandated profit share generalizes beyond the Nash case. Because profits are zero in equilibrium, any bargain that lowers wages in partial equilibrium in response to an increase in the profit share will lead to lower unemployment in general equilibrium. Similarly, the result that bargaining over profit shares in addition to wages will lower equilibrium compensation generalizes to all "efficient" bargaining solutions.

I would like to thank two anonymous referees for their helpful suggestions.

1. A popular justification for this type of solution from non-cooperative game theory is given by Binmore et al. [1986].

2. [Delta] might be a separation and replacement rate within union-firm pairs as in Layard et al. [1991] or, as in Georges [1991; 1994], a turnover rate of firms.

3. As N rises, more jobs turn over and so there are more job vacancies, while fewer workers are searching for jobs.

4. The equilibrium is indeterminate at [Lambda] = 1 since any number of firms can be supported under this rule.


Anderson, Simon, and Michael Devereux. "Profit Sharing and Optimal Labour Contracts." Canadian Journal of Economics, May 1989, 425-33.

Binmore, Ken, Ariel Rubinstein, and Asher Wolinsky. "The Nash Bargaining Solution in Economic Modeling." Rand Journal of Economics, Summer 1986, 176-88.

Georges, Christophre. "Three Essays in the Macroeconomics of Labor Market Dynamics." Ph.D. dissertation, University of Michigan, 1989.

-----. "A Model of Union Wage and Employment Dynamics." Working Paper 91/3, Department of Economics, Hamilton College, 1991.

-----. "Unemployment Persistence Under Profit Sharing." Economics Letters, July 1994, 329-34.

Holmlund, Bertil. "Profit Sharing, Wage Bargaining, and Employment." Economic Inquiry, April 1990, 257-68.

Jackman, Richard. "Profit Sharing in a Unionized Economy With Imperfect Competition." International Journal of Industrial Organization, 6, 1988, 47-57.

Johnson, George E. "Work Rules, Featherbedding, and Pareto-Optimal Union-Management Bargaining." Journal of Labor Economics, 8(1:2), 1990, S237-59.

Layard, Richard, and Stephen Nickell. "Is Unemployment Lower if Unions Bargain over Employment?" Quarterly Journal of Economics, August 1990, 773-87.

Layard, Richard, Stephen Nickell, and Richard Jackman. Unemployment.' Macroeconomic Performance and the Labor Market, Oxford: Oxford University Press, 1991.

Pohjola, M. "Profit-Sharing, Collective Bargaining, and Employment." Journal of Institutional and Theoretical Economics, 143(2), 1987, 334-42.

Weitzman, Martin L. "Steady State Unemployment Under Profit Sharing." The Economic Journal, March 1987, 86-105.

Georges: Associate Professor of Economics, Hamilton College, Clinton, N.Y., Phone 1-315-859-4472 Fax 1-315-859-4477, E-mail
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Author:Georges, Christophre
Publication:Economic Inquiry
Date:Apr 1, 1998
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