Macroeconomic

A Classical Model of the Class Struggle: A Game-Theoretic Approach

Perry G. Mehrling

The Journal of Political Economy, Vol. 94, No. 6. (Dec., 1986), pp. 1280-1303.

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A Classical Model of the Class Struggle: A Game-Theoretic Approach

Perry G.Mehrling f f n n m r t i C.rrzi'r?sztj

T h e class struggle is formalized as a differential game in a strictly supply-side nlodel, an approach that synthesizes the models of Lan- caster ancl Goodivin. Four d i f t t r e n t stead>-state equilibria are de- rived, each corresponding to difttrent assumptions about the degree to which each class is organized to promote its own interests. 111 particular, the Goodwin growth cycle is s h o n n to emerge f'rorn a ivorld c h a r a c t e r i ~ e d by unorganized capitalists and workers, in ~vhich indivitluals ignore the effects of their olvn actions on econo- rny\vicle variables. hlore rele\ ant t r discussion of modern capitalism are the hierarchical equilibria, especially the codeterrninatiot~ equi- l i b r i ~ ~ min which the existence of a full-employtnent e q u i l i b r i ~ ~ n i turns out to be pr-obleniatic. Finally, comparative-statics results sug- gest that the incentives for technological change differ witlely among the four regimes.

I. Introduction

Econonlic growth provides the potential for mitigating the conflict between classes over the product of society. T h r o u g h econonlic g r o ~ v t h everybody can gain something; that much is apparent and agreed o n by all. Whether in fact that potential becomes a reality, whether the gains f r o m growth d o indeed trickle down, is a n issue that economists have long discussed. Less often discussed is the effect

I \$auld like to ackllowledge t h e Ilelpflrl conlrnents b! h l e g h n a d 1)epai a n d Douglas Gale o n a n earlier version of this p a p e r , ~vllic 11 n a s nl) thesis ,it t h e Lolldorl School of Economics. Richard Nelson. S t e p h e n hiarglin. hlichael k i a l ~ t i e l . Paul \Ventit, 1ol111 Lie. T. .''trrq Barruri, a r ~ d t ~ v o referees each r e a d this manuscript in o n e of several later ver- sions allti provided i l ~ \ a l u a b l e suggestior~s f o r ilnprovernents.

[ / o z t , , i i i / ii/ I ' v l i l i i n l F,i,,iorn\. IIIXb, \ol. R4, no. 61 % 1981, h \ 'l'hc L n l \ r r s l t i of (:hlcngo ,111 right, rcacr\cd. 00?L'-:1808 81, 040li-0110!1$01 ,511

M O D E L OF CLASS STRUGGLE 1281

of class conflict on growth, the ways in which conflict may prevent the economy froni achieving an optimal growth path. Part of the diffi- culty in addressing this issue derives froni the widely varying concep- tions of the nature and scope of class conflict, taken informally to encompass everything from negotiations over labor contracts to the struggle for control of the means of production in society. T h e tlvo most prominent formal approaches to this problem in the literature, those of Lancaster (1973) and Goodwin (1967), limit their discussion to the conflict over income shares, and this paper will follow their lead.

Lancaster (1973) and Hoe1 (1978) model capitalism as a differential game in which workers choose a wage level and capitalists choose a rate of investment out of profits. Dynaniic inefficiency results when capitalists fail to invest because the): fear workers will ultimately reap all the gains by raising their wages. These niodels (see also Pohjola 1983, 1984; Basar, Haurie, and Ricci 1985) have the advantage of a strong choice-theoretic basis, but they have no explicit connection to the growth literature.

By contrast, Goodwin (1967) has proposed a growth model of class conflict in which \vorkers push for maximal wage increases and capi- talists invest all profits. He formalizes this behavior as a Volterra- Lotka predator-prey model, borrowed from mathematical ecology (see Pielou 1973), interpreting the enlploynient ratio (the ratio of employed workers to the labor force) and the workers' share of out- put as in some sense competing for survival. In his model the econ- only cycles endlessly around an unemployment equilibrium, and so it might be said that the Goodwin niodel also depicts a situation of dynamic inefficiency. A small literature elaborating the Goodwin niodel has built u p (Atkinson 1969; Desai 1973, 1981; Pohjola 1981; Shah and Desai 1981; Cugno and hlontrucchio 1982; Di Matteo 1982; Wolfstetter 1982; Goodwin 1983; van der Ploeg 1983; Velupillai 1983), but these papers have all assunied that capitalists and workers behave in a certain way, without showing that that behavior is in any sense optimal. '

This paper combines the strengths of the Lancaster and Goodwin approaches to analyzing the dynaniic inefficiency arising from class conflict. A hybrid model of econonlic growth built on a differential

' Velupillai (1983) has cornmented on the lack of a strong choice-theoretic basis for the Goodbvin rnotiel a n d suggested a differential Kame appt-oach. S a r n ~ ~ e l s o r l (197 1) has pointed out that the Voltrr-ra-Lotha equations ar-e the Euler equations associateti \\ith some d\narnic optimization problern, but that pr-oblem has no ready inter-pretation. 2101-e general]), Elster ( 1982) argues that game t h e o r ~ is the appropriate technique for for-malizi~~gclass conflict, though his rejection of all functionalist explanations in the social sciences seel~rs overstateti.

1 2 8 2 J O U R N A L OF POLITICAL E C O N O M Y

game representation of the class struggle allows a fairly simple char- acterization of four different steady-state equilibria, each corre- sponding to different institutional assunlptions that affect the balance of class power. T h e model makes apparent that class conflict may be a significant source of supply-side suboptin~ality in capitalist econonlies, colnparable in importance with the well-known exchange-type ineffi- ciencv pointed out by Samuelson (1958) and Diamond (1965). Sam- ~ielson and Diamond show that the inefficiency that arises from the inability of generations to trade with each other can be overcome by certain institutions, such as the "social contrivance of rnoney" and national debt. In this paper I suggest that the inefficiencies arising from class conflict may be partially overcome if through institutional enforcement classes can precommit to some social contract. That is, a goi.ernment with the power to enforce the contracted distribution as the econonly grows can make possible a dynamic Pareto improve- ment.

T h e paper is organized as follows. Section I1 outlines the structure of the model. Section 111 derives the Goodwin growth cycle as the Nash solution to the differential game with many players of each sort, in which coordination among players is ruled out. If ~vorkers behave in this unorganized way, then capitalists can improve their position by coordinating investment strategies (Sec. IV). Likewise, if capitalists are unorganized, then workers can improve their position by coor- dinating wage demands (Sec. V). Section VI investigates the two- player game, in which both players are organized, and finds a role for a government as the enforcer of a compromise between classes. Sec- tion VII illustrates the use of this model to discuss the distribution of the gains to technological change in each of the four regimes derived in Sections 111-VI. Section VIII concludes the paper with an inter- pretation of the model and suggestions for extensions.

11. The Model

As in Lancaster (1973), the econorny is characterized by fixed coefficient production of one holnogeneous good that can be either consurned or used as capital. Total output Q = ak, and labor input L = hk. Labor is inelastically supplied at wage w , u p to the amount of the entire population, which grows at rate n , so A' = AToe'lt;A',, is set equal to unity as a normalization.

T o make the links with the Goodwin (1967) model clear,2 the analy-

';' Lootiwin's u n t b t - t u n a t e choice of notation has become stantiard, a n d I follow suit 1~ltt1~1.thiitl create m o r e confusion by choosing o t h e r vat-iables.

M O D E L O F CLASS S T R U G G L E

sis is conducted in the p u r e numbers u and zl defined as

T I = - = enlployrnent ratio. N

Two identities follow from differentiating these definitions,

a n d

where g denotes the growth rate of the economy. I n the game out- lined below, .workers choose wage changes (&) a n d capitalists choose investment ( k ) .Equations ( 1 ) a n d (2) demonstrate that we may equiva- lently say that workers choose iL a n d capitalists choose i l , a n d this latter formulation proves more convenient for o u r purposes. This asym- metry in the arenas of action of the two classes is t h e most important assumption o n which the results of the model d e p e n d .

Workers choose iL to maximize t h e discounted present value of their consurnption stream, which is the same as their income stream since we assurne that workers d o not save at all. Workers as a class receive a total income of wL = uak = (alh)Su-il,so discounting at rate m , they nlaximize

where p = m - n is assumed positibe. Capitalists rnaximize their discounted profit income strearn." T h e y

'' H e r e I d e p a r t f r o m Lancaster (1973). \ \ h o assumeti that capit'ilists max~rriire tiis- counteci consumption. Ari alternative a s s u r r ~ p t i o ~ ~ seems rno1.e in t h e Xlarxian spirit of Goodwin's work, w h e r e capitalists a r e concet-rled with acciunulation: "Accurriulate, i2ccum~1late. 'That is hloses anti t h e Pt-ophets" ( l l a r x l l I i 6 , vol. 1 , c h a p . 24). T h i s d e p a r t u r e is Inore a p p a r e n t t h a n real since rnaximiring consurnption ~ \ o u l t i give arl objective function of

which is not too d i f f e r e n t ti-on1 t h e objective functiorl maxinlizirlg p r o h t income. ( T h e second equality follows f r o m illtegratioli b\ parts a n d application of t h e tt-arlsversalit) c o ~ l d i t i o l i . ) Using this objective function would make little t i i f f r r e ~ i c e to t h e model. ,4lthough t h e dynamics a r e somewhat different (i.e.. t h e atomistic solution is not c!clic

~ 1284 J O L K N A L OF P O L I T I C L \ L ~ ~ o h o ~ l

receive (1 - u)ak = (aIX)X(l - u)7l, so discounting at the same rate, they maximize (alX)JTe-"(1 - u)vdt.

Constraints o n t h e actions of capitalists a n d workers t u r n out to be very important since these constraints bite in t h e various equilibria proposed. I n t h e first place a r e direct constraints o n the choice vari- ables iL a n d i l ,

w - 5 ( - 7 + & I ) , y , S > 0, (3) W

T h e first constraint describes wage bargaining. For simplicity, this equation applies equally to each individual worker's wage dernantls a n d to classwide wage demands. Equation (3) enlbodies the idea that the ability of workers to raise wages depends o n the rate of unemploy- ment. At high levels of employment workers can increase their (real) wages faster because the threat of unemployment is reduced. This notion of labor market behavior is expressed in Lipsey (1960) a n d Phelps (1970). For o u r purposes i t is more useful to express (3) as

where ZJ*= 718, the rate of employnlent at which the maximal wage increase is zero. By assumption, 0 < z l * < 1. When the employment ratio is less than v*, wages must be falling. T h e value of 7l* sum- rnarizes the relative power of workers a n d capitalists in the wage bargain.

T h e second constraint follows from assuming that workers d o not save so that investment cannot exceed capitalist profit share. Also, investment is assumed to be nonnegative; capital does not depreciate a n d , once invested, cannot be taken apart and eaten. For o u r pur- poses it is useful to express (4) as

where u* = (a - %)la,the value of u for which the maximal increase of the employment ratio is zero. Xote that u*, unlike T I * ,is a purely technical parameter. When the workers' share is greater than u*, the employment ratio must be shrinking since available investment re- sources a r e insufficient to provide jobs for new ~vorkers.

but I-ather corl\et-ges to u = 1 - [nllcl].1' = I ) " ) , the clualitati\e I-esults of cornparati\e statics a r e the same. 'The form of this objectibe tunctiori sho~vs that it' u i \ tonsitiercd exogenous, capitalists \\ill \ \ a n t iJto be gt-eater \vherl I( > 1 ( ~ n l c ~ )- a n d less \\hen u < 1 - ( T l l l ( ~ ) .

M O D E L OF CLASS STRUC;C;LE 1285

I n acldition there a r e three econorn?wicle constraints o n the state variables u a n d v :

Constraint ( 7 ) means that workers can take n o more of the o u t p u t than is being produced. I n particular, capital cannot be consunletl. If' this constraint is binding, we assume that workers a r e unable to in- crease wages f ~ ~ r t h e r . Condition (8) expresses the idea that there can- not be overfull employment. I n particular, we assume that there can- not be excess capacity; if this constraint is binding, capitalists adjust their investment spending.

Constraint (9) has more interesting economic content. It reflects a n assunlption that a lower limit o n wages is provided by a given conven- tional wage c, conceived as socially determined. (For a discussio~lof' this assunlption see, e.g., Marglitl [1984, chap. 31.) If each ~vor-ker requires at least c units of' the good, then "survival" requires that w L / S r c, v.hich can be re~vritten as (9) using o u r identities. 'rhis constraint may be understood to imply that wages for the unem- ployed a r e provided by the employed (through some unnlocieled transfer system).

When this survival boundary constraint (9) is binding, we assume that workers adjust their wage claims upward to ensure that it is satisfied. T h a t is, o n the survival boundary

a n d capitalists' choice of i, determines u. This adjustment assumption seems in the spirit of' the state variable constraint.''

Given all these constraints, workers a n d capitalists maximize their respective objective functions. T h e workers' problenl in its entirety is then (with constants suppressed) to choose u to

m a r 1P - ~ ~ U L I ~ ~ ( W ) ' O n e exception is allowed. Assume that L' never gets so lo\\ that t h e r e d o not exist

e n o u g h I-esout-ces tot- t h e ecorlorn! to continue. T h a t is, U " ~ J > t A / a . M'llerl this con- straint binds, assume that u (it-ops to u* a n d irlvestmetit is tilaximal. Tliis assumption e n s u r e s continued viability of t h e e c o t i o m \ .

1286 J O C K N A L OF POLITICAL ECONOMY

i l =subject to (.5), ( 7 ) ,(a), and ( 9 ) , = i ( u ,ZJ, t ) , ~ ( 0 ) uO,and v(O) = rlo. In figure 1 the objective of the worker5 is to more in a northeasterly direction.

'The capitalists' problem is to choose il to

subject to (6), ( 7 ) ,(8),and ( 9 ) ,iL = k ( u , v , t ) , u ( 0 ) = uo, and -11(0) = no. In figure 2 the objective of the capitalists is to move in a northwesterly direction.

This conlpletes the outline of the economy. T h e evolution of this econorny is determined by the choices of iL and il. Workers choose the evolution of the wage share, and so also the evolution of the profit rate .rr, since .rr = [ ( l - u)ak]ik .= ( 1 - u ) u . Capitalists choose the evolution of employrnent and thereby also the evolution of output. Some flavor of the symbiotic character of the relations between classes is thereby captured. Though the two groups conflict over irlconle shares, they share an interest in increasing total output. Whether the shared interests prove strong enough to mitigate the conflict is the question addressed in Sections 111-VI.

MODEL OF CLASS STRUGGLE 1287

111. The Goodwin Cycle

I n this section I derive the optimal behavior of workers and capitalists when each agent is an isolated atom and so neglects his irnpact o n the economywide variables in his optimization problem. Each agent has a dominant strategy, a feature of the model that will be very useful in Sections VI a n d V I I when different solution concepts are considered. Optimal atomistic behavior is shown to give rise to the Goodwin growth cycle.

Each worker i faces an individual equivalent of the workers' prob- lem ( W ) . H e can affect his own wage w,directly by his own action, while his impact o n zl is of o r d e r 11". For large iV, each employed worker will ignore this indirect effect and increase his own wage maximally. T h e c o ~ l s t r a i ~ l t o n wage increases (5) will be satisfied with equality, which behavior yields the aggregate equation

Similarly, the individual capitalist j maximizes individual profits. T h e going wage is an economywide variable that the capit, a1'1st can affect only indirectly by affecting 2 1 . But the individual capitalist's effect on zl is of o r d e r 10, where J is the (presumed large) number of capitalists. T h e atomistic capitalist will ignore these indirect effects

1288 JOCRN.AL OF POLITICAL EC'ONOMY

to investment a n d will invest maximally (k equal to its u p p e r bound), which behavior yields the aggregate equation

Equations (1 1) a n d ( 1 2 ) a r e L'olterra-L,otka equations describing Goodwin growth cycles a r o u n d the point (u*, u*) as depicted in figure 3 . (See Goodwin [1967] o r Desai [1973] for proofs.) At high levels of employment, workers increase wages a n d squeeze profits so that new irlvestment is insufficient to provide jobs for the growing population, a n d the u~lemploynlent rate increases. When employment is low, wages a r e falling a n d profits increasing. These profits a r e reinvested to provide growth at more than t h e rate of gr-owth of' population, so the employment ratio increases.

T h e atomistic nature of' decisions in this context leads agents to ignore the social impact of their wage a n d investment decisions o n the economy. Such behavior results in inefficient use of resources (unem- ployment) since o n average the employment ratio is ZJ*. This follows from integrating (1 l ) ,

Choosing T so u(0) = zr(T), which is possible since the economy cycles, we have (u - v*)dt = 0. A similar procedure establishes that the average value of' u is u*. For the purposes of' comparison with the results of' later sections, I shall refer to these average values (u*, ZJ*) as the "atomistic" solution, denoted by the point A , ignoring the fluctua- tions a r o u n d that value:'

Various solution concepts that enable organized groups to irnprove o n this atomistic solution a r e discussed below. T h e hierarchical games (when o n e g r o u p is organized a n d the other is not) that a r e discussed are more easily solved because u n o r g a n i ~ e d players have a dominant strategy, a n d that strategy is a function only of' the state variables. These tmTo features imply that the organized player faces a simple optirnal control problem a n d that the optimal control will also be of'a feedback nature since all the relevant economic information is sum- marized in the c u r r e n t values of the state variables. Needless to say, these features r e n d e r the problem tractable a n d greatly sirnplify its solution.

' This procedure is perhaps less arbitrarv when we realize that this cyclic movement is not robust to various alterations in the model such as induced technical change and rnonev illusion (Desai 1973; Shah and Desai 1981).

MODEL OF CLASS S T R U G G L E 1289

F I ~ , .3.-The (;oodwin g r o w t h cycle: .An "arorni\ticM solution

IV. Workers' Control

I n this section, workers a r e considered to be a united g r o u p acting as a single agent in a hierarchical dynamic game setting. T h e fact that all workers a r e organized should not be taken to mean that the econorny is heavily unionized (many unions), but rather that there is orlr union. hleanwhile, unorganized individual capitalists react to the workers' strategy as in Section 111; their dominant strategy is to invest all profits. Hence the workers' problem is relatively simple. T h e y can take t h e capitalist's behavior as given by (12) a n d solve the optirnal control p r o b l e ~ n (W).

Since whatever is not consumed is invested by capitalists, the work- ers' control problem takes the form of the classical optimal savings problem (Ramsey 1928). I n my model, if investment is productive at all, then desired investment mill be unbounded because of the as- sumption of fixed coefficients a n d constant marginal utility of con- sumption in the objective function. (Costs of investment a n d dimin- ishing n~argirlal product will change this trajectory in well-known \vays, a n d these d o not concerrl LIS here.) Increasing the per capita capital stock by o n e unit forever requires a present investment of o n e unit a n d investment in f u t u r e periods o f a ( 1 - u * ) = 11 units. T h e net i n c o ~ n e frorn this rnarginal investment is a flow of a - a ( l - u*) = au* units. If t h e present value of the cost of investment is less than

1290 J O U R N A L O F P O L I T I C A L E C O N O M Y

the present value of the irlcome stream, then investment is desirable. This condition can be written algebraically as 1 5 $; eCPtau*dt or

1 5 - au* P

Assume throughout that (13) is satisfied; it rnerely says that the econ- omy is productive from the point of view of the workers. If the econ- omy is pr-oductive in this sense, then the optirnal savings program involves a reduction of consun~ption to mere survival until full em- ploymerlt is reached, followed by a jurnp upward in the workers' wage share to u = u*. This share is the largest that allows sufficient product to be reinvested to ensure this income share thereafter. So the work- ers' control steady state is at (u, 21) = (u*, l ) , shown as W in figure 4.

In the formulation of the workers' problem adopted here. where the choice variable is u (which is bounded above by [j]),upward jurnps in u are ruled out. Instead workers m i l l begin irlcreasirlg wages before full employmerlt is achieved. I n figure 4, the separatrix WW shows the values of the state variables at which it is optimal to begin pressing for higher wages. A typical workers' control trajectory is shown in which workers begin by reducing wages to mere survival levels, allowirlg the capital stock to increase rapidly. As ernployrnent rises, workers reach the W W locus at R and begin to elljoy the benefits of their frugality as they raise wages maximally while ernployrnent continues to increase. ( T h e W W locus may cut the survival boundary above o r below a* depending on the parameters of the model.) In ally event, the optimal control converges to the workers' control steady state at W.

One extension is immediate. If capitalists have opportunities out- side the economy to achieve a profit rate of at least (1 - u**)a, then all poirlts o n the trajectory must satisfy (1 - u)a r ( 1 - u**)a, or 11< u**, since even atomistic capitalists will choose the best of alterrlative profitable opportunities."he notatiorl is chosen to emphasize that foreign illvestnle~lt may place an upper limit on the workers' steady- state share by diminishing the productivity of the economy (from the workers' point of view). Assurne that 11** < u* so the co~lstrai~lt is binding. If the ecorlomy is still to be productive in this special sense, then instead of (13) we shall require the stronger

and the steady state rill instead be at (u**, 1).

" One a1ternatit.e opportunity is consumption. If' capit ilists maximize consumption rather than income, as in n . 3, then u** = n - (mla) < u*. As might be intuitively suspected, it can be shown that alternatibe opportunities for capitalists have the effect of increasing their share in all equilibria.

MODEL OF CLASS STRUGGLE 1 2 9 1

W un U - (111) uFIG. 4.-Workers' control. ( I ) i' = - x ; (11) i' = ~ ( Z J Z J * ) ~ ; = - ( i ~ / v ) u= c ~ ( u

- U * ) I I .

V. Capitalists' Control

I n this section capitalists are considered to be a unified group acting as a single agent in a hierarchical dynamic game setting. Meanwhile, unorganized individual workers react to the capitalists' strategy as in Section 111; as a dominant strategy they press for the highest possible share. Hence the capitalists' problem reduces to a relatively simple optimal control problern ( C ) ,with workers' behavior as given by (1 1). This optimal control problem can be shown to yield a productivity condition analogous to (13), which is

When (14) is satisfied, capitalists ~ r o u l d like to reduce zl in order to bring about a more rapid decline in u (an increase in their own share). But since capitalists can invest no more than retained profits, they optimally begin investing before the optimal level of u is attained. T h e CC locus in figure 5 depicts this behavior. T h e analogy uith the work- ers' control solution should be clear.

T o the right of this locus, it is optimal to decrease 21 so 6 = -vn; to the left, it is optimal to increase u so 3 = -a ( u - u * ) v . As drawn, the

FIG.5.-Capitalists'control. ( I ) i ' l = - v r l ; ( 1 1 ) G = c c ( r r - (111) i, = =I I * ) ~ , ; - ( i l / r ~ ) a - 6 ( u - v * ) u .

equilibrium at C* is infeasible because it lies to the left of the sul-viva1 boundary. This reflects ,in assumption that p is quite small. So the optimal control is to stop investing until the survival boundary is reached and then to slide u p the survival boundary to K. On the survival boundary, i, is maximal as long as id is continuing to press against the boundary; otheruise capitalists choose i~ to mairltain course along the boundary by setting (using [ l o ] and [ l l ] )

In any event, the capitalists' optimal control brings the economy to a steady state at (u, ZJ) = (cAlaz~*, v*), the point called K in figure 5, which we shall call the capitalist control steady state. I n order to maintain a high profit rate, capitalists create a certain amount of unemployment in order to prevent wages from rising. This rate is "natural" in the sense that wages will rise if employment is any higher, but these unemployed workers are involuntarily unemployed, uilling to work at the going wage.

M O D E L O F CLASS S T R U G G L E 1293

VI. Codetermination

I n this section, workers act as a unified g r o u p solving (W) a n d capi- talists act as a unified g r o u p solving (C). A noncooperative equilib- rium of this two-player game is a pair of strategy functions h(u, ZJ, t ) a n d i ~ ( u , ZJ, t ) such that ir solves (MI)subject to i~ a n d i~ solves (C) subject to i d . T h e strategy space u n d e r consideration is very large a n d admits multiple equilibria. T o limit the n u m b e r of equilibria, I impose three conditions. T h o u g h n o n e of these conditions is unusual in game- theoretic analyses, they a r e strong e n o u g h t h a t only two possible equi- libria rernain.

CONDITIOX1. Admissible equilibrium trajectories must reach a steady state.

CONDITION Admissible equilibrium trajectories must be dynarn- 2. ically corlsistellt (see Kydland a n d Prescott 1977).

CONDITION Strategy functions must be explicit functions of the 3. current state variables only, not time. This nlay be corlsidered a n assu~nption about t h e information available o r o n e about the types of strategies that the two players a r e institutionally capable of playing (see Basar a n d Olsder 1982).

'To begin with, t h e capitalist control trajectory is a Nash equilibrium of this two-player game that satisfies all three of these conditions. Workers' best response to the capitalists' corltrol strategy is to increase wages maximally, thereby delaying arrival at the capitalist control unemployment steady state. A n d the capitalists' best response to max- irnal wage pressure is the capitalist control strategy, as shown in Sec- tion V. Neither player has a n incentive to change his strategy along the way, so t h e trajectory is dynan~ically consistent. Furthermore, it converges to t h e steady state at K. T h e capitalist control equilibrium is also a codetermirlatio~l equilibrium because capitalists colltrol the key variable, employment. W h e n capitalists reduce ZJ, they also reduce workers' power to raise wages; when z~ is less than v * , workers cannot defend their share a n d capitalists can effectively control income shares as well as employment (subject to [9]).

It is therefore a bit surprising that the workers' control steady state can also be a codetermination equilibrium. Workers cannot force cap- italists to invest in t h e way that capitalists can force workers to reduce their wages. T h e level of 11 can be increased only with capitalists' help. Capitalists will help only if workers' wage d e m a n d s offer them a good deal. State W can be a codeterrnirlation equilibrium only if it is as good as K f o r the capitalists, in some sense. A sufficient corldition for this to be so is that

1294 J O U R N A L OF P O L I T I C A L E C O N O M Y

be satisfied at W.'This corldition states that capitalists receive as much profit at the workers' control steady state W as they do at the capi- talists' control steady state K. Figure 66 shows (15) satisfied with equality.

Also, all along the entire trajectory that converges to W, workers' wage behavior must offer capitalists a good deal if they are to con- tinue to invest. For example, it is easy to see that the workers' control strategy cannot be an equilibrium strategy for the codetermination game. If workers reduce wages to the right of the WlV locus in figure 4 and increase wages to the left, then capitalists' best response is to control 71 so as to choose the best point o n the WW locus; this point is designated R. T h e workers' control trajectory is not a codetermina- tion equilibrium because in its final stages capitalists are becoming worse off, and they can improve their situation given the strategy workers are playing. Codetermination trajectories must involve work- ers' playing a strategy that does not give capitalists an incentive to disinvest. Such trajectories can be shown to exist.

It turns out that K and W are the only equilibria in the game as described thus far. T o see this, we can derive the set of possible steady states for the game and then use the conditions to eliminate most of them from consideration. T o begin, it should be clear that workers and capitalists both can achieve at least the value of the capitalists' control trajectory. No matter what strategy workers play, capitalists can always play capitalist control. No matter what strategy capital- ists play, workers can always increase wages maximally and ensure at least the value of the capitalist control trajectory. Call the capitalists' control trajectory converging to K the "threat" trajectory. Any candi- date for a codetermination steady state must provide both players with at least the value of that trajectory.

Figure 6a shows two loci. T h e KW' locus contains points ( I L , 7 1 ) at which workers are just indifferent between staying at ( u , Z J ) and mov- ing along the capitalists' control trajectory from ( u , 11) to K . ~All points to the right of this trajectory are strictly preferred to the threat trajec- tory and therefore remain candidates for a codetermination steady state.

'T h e KU'' locus is described nlgebraically as the values ( u O ,z l o ) satisfying

where u ( 0 ) = u O ,v ( 0 ) = ii = 6 ( v - v * ) u , and i, = - v n until the survival boundary is z l 0 , reached, after which i~ = - S ( v - U * ) Z J .This is,just the value of the capitalists' control trajectory of Sec. V. Substituting in for the values of u and 71, we can find an explicit solutiori of the equatiori above that can be differentiated to show that riear K it is riegatively sloped and farther away it becomes positively sloped.

MODEL O F CLASS STRUGGLE

Similarly, the KK' locus contains points ( u ,u) at which capitalists are indifferent between staying at ( 1 1 , u) and moving along the threat trajectory.' Note that KK' lies completely to the right of the isoprofit

"The KK' locus is described algebraically as the values ( u O , uO) satisf-ing

1296 J O U R N A L OF P O L I T I C A L EC:ONOSIY

curve (labeled KCD) passing through K ; capitalists will accept lower steady-state profits than they could achieve at K sirice the threat tra- jectory involves a period of even lower profits before the profits at K can be obtained. All points to the left of the K K ' locus are strictly preferred to the threat trajectory.

So we have a roughly triangular area K W ' K ' , which we can call the "zone of' compromise." Points within the zone of compromise are candidates for a codetermination steady-state equilibrium. Hut we can rule out most of these candidates. Suppose the economy were at a steady-state equilibrium (li, 5)somewhere in the zone of compromise. In o r d e r for this to be an equilibrium, workers must gain nothing by increasing their share. T h a t is, for 21 > li we must have i, < 0. If so, then in that region workers would set zi < 0 , preferring the steady state at (li, 5) to a trajectory involving decreasing enlployrnent. But capitalists would respond to such a strategy by playing it > 0 (if they can) in that region, and so (li, zi) cannot be a Nash equilibrium. This argument rules out all points in the zone of conlprornise except those for which either ( i ) 5 5 vY so that unemployment prevents workers from increasing their share o r (ii) li 2 u g SO that employment is decreasing for higher workers' share even if' capitalists invest all profits. Clearly there is at least one such point, K , and that may be the only one, as illustrated in figure 6 a . If' (13) is satisfied, there may be more candidates, as in figure 6 b . But all these additional candidates can also be ruled out, with the exception of' W .

First, in a steady state, condition ii must be satisfied with equality; strict inequality does not permit sufficient investment to keep u p with the growth of the labor force. Second, consider the incentives fbr workers to reduce their share. I n o r d e r for a candidate (u*,5)to be an equilibrium, we must have i~ > 0 to the left. But if workers play this strategy, while they a r e playing i~ < O to the right (by the argument above) then capitalists will optimally play il > O to the left. And by the argument of Section 111, workers' best response to capitalist invest- ment is to decrease their share a bit and allow employment to rise. This argument rules out all remaining candidates except 1.2'. ,4t full employment there is nothing to be gained by reducing u. Ernploy- ment cannot be increased because of' physical limitations o n the avail- ability of' workers.

It remains to check that both K a n d W are also equilibria for capi- talists. Clearly capitalists cannot gain by increasing employment at either K o r W . ,4nd decreasing employment at K unambiguously lvorsens their position. T h e tricky question is whether they will want

where 71 and 7' evolve as i r ~the previous note. It is intuitive that the curve slopes as shown.

M O D E L O F C L A S S S T R U G G L E "97

to decrease employment at W. I n or-der to prevent that from happen- ing, it must be that u is not too negative below W so that capitalists cannot gain by reducing employment. (As noted above, this rules out the worker control trajectory, for which zi = - x below W.) If so, then capitalists will play il > O in that region and W can be a codetermina- tion equilibrium.

Apparently, the full-employment equilibrium is much more tenu- ous than the unemployment equilibrium. In the first place, there may be n o full-employment equilibrium at all if W falls outside the zone of cornpromise. Equation (15) shows that the likelihood of this occurring depends o n all the variables in the economy, the bargaining strength of workers, a n d the conventional wage as well as the production coefficients and growth of the labor- force.

Second, even if' this equilibrium exists in the formal model, there are reasons to doubt whether such a position would be achieved in any real-world economy in which there is uncertainty about future actions. O n e way to interpret the codeternlination equilibrium is that workers believe that capitalists will behave in a certain way as the state variables change in the future and choose an optimal response to that imagined play. For their part, capitalists believe that workers will change their wage behavior in a certain way as state variables change and they choose a certain optimal response. A codeternlination equi- librium is a set of' prior beliefs that lvill turn out to have been correct if both agents act o n them. T h e problem is how these beliefs come to be held, particularly since each individual capitalist and worker, acting myopically, would behave quite differently. T h a t is, capit- 'I 1'1 s t ~must not only believe in a certain set of future wage demands but must also believe that the workers' coalition will continue to hold together, that the o n e big union will keep workers from raising their wages as much as they would each like to d o as individuals. ,4nd workers must believe that capitalists will continue to invest ever1 as the higher level of em- ployment increases the power of workers. ,411 in all, there seerns a great deal of trust underlying this ostensibly noncooperative equilib- rium.

It may be better, therefore, to consider briefly a different sort of codetermination equilibrium, which more explicitly treats the prob- lem of coordinating the beliefs of the two groups. It is much easier to imagine trust in a context in which there is some sort of' social contract between the players a n d provisions for the enforcement of that con- tract. Suppose that the two groups sit down together and draw u p a contract for future investment and income shares. This contract also establishes a government with the power to tax; it can redistribute the proceeds of that taxation o r it can invest and produce goods o n its own. These powers give the government a certain amount of control

1 2 g 8 J O U R N A L OF P O L I T I ( : A L E C O N O M Y

over both the employment ratio and the income share so the govern- ment can effectively police the agreement. In the model ofthis paper, it is clear that the social contract would seek to increase employment as rapidly as possible since both groups gain from such a move. T h e main problem in writing the contract will be in determining the steady-state distribution of income once full employment is achieved. Possible full-eniployment shares lie in the range from W' to K' in figure 6 u . I n this range, by construction, both players are doirlg better than the); could do by then~selves and better than in the noncoopera- tive codetermination equilibrium that converges to K.

T h e problem with such an equilibrium is clear: in general it is not dynamically consistent. Once full e~nploynlent is achieved, capitalists may be loath to reduce their profit share to the previously agreed level. O r workers may be unsatisfied with their share and use their increased bargaining power at full employment to push for higher wages. Fears that the contract will be broker1 in the future of course undermine the contract in the present. Doubts about how the gains from higher e~nployrnerlt will be shared may prevent full employ- ment frorn being achieved at all. A social contract must convincingly precomrnit both sides to a particular path. In this way, the institutiorl of a government capable of enforcing giver1 income shares makes more likely the achievement of full employ~nent."

For the purpose of conlparison with the equilibria of previous sec- tions, we can treat the point CD as a representative full-employment codetermination equilibrium. By assumption the social contract is ~vritten with reference to the zone of compromise and not in absolute terms so that, as the zone shifts because of changes in the data of the econorn);, so too does the contracted equilibrium income share. T h e poirlt CD shifts with any shift in the zone of compromise and so captures the qualitative changes to be expected in any such codeter- mination equilibrium.

VII. Comparative Statics

In this section I derive the steady-state effects of changes in c, u , A, and n. By seeing whether workers or capitalists or both gain from a particular change in the conventional wage, the technology, o r the growth rate of the labor force. I can provide some rough idea of the differing dynarnic evolution of the economy under each of the four regimes discussed above. My contention, which should hardly be con- troversial, is that, in regimes in which a certain type of technological

" See Prze~vorski and Wallerstein (1982) tbr a model specifically addressing this issue in a different f'rame~vork.

- - - - - -

M O D E L OF CLASS S T K U G G L E "99

(or other) change benefits both workers and capitalists, one would be rnore likely to observe that change than another type that helps one and hurts the other. That is, the incentives for different development paths depend on the specific fbrm of the class bargain.

Figure 6 depicts the four steady states under consideration, and table 1 summarizes the average worker's consumption, az~-il/A, and capitalists' total profits, enta(l - u)zl/A, for each.

Table 2 summarizes the steady-state effect on workers' consump- tion and capitalists' profits of increasing the conventional wage c, neutral technical change (a increasing), labor-augmenting technical change ( A decreasing), and increased population growth ( n increas- ing, e.g., by means of' immigration). Each entry in the table puts the effect on workers' corlsuniptiorl first and the effect on capitalists' profits second.

In the atomistic regime there is a bias toward labor-augmenting technical change that increases output without changing income shares so that both sides benefit. One might expect conflict, however, concerning the growth of the labor force. Workers \$.ill fight against immigration and broadening the labor force to include women and other historically excluded groups.'' Capitalists will seek to promote these changes.

T h e capitalist control regime has a tendency toward technological stagnation since workers never gain from technological change. Cer- tain changes may be instituted that improve profits, but the pace of change will be slower than it would be if capitalists could get workers to cooperate. Conflict over the conventional wage occupies both work- ers and capitalists once unemployment is sufficiently high that work- ers are unable to raise wages in the labor market. That is, political conflict over the standard of living emerges out of the repression of wage conflict in the labor market.

T A B L E 1

1Vorkers' (a - n)u* consumption A

Capitalists' nu* au* 12 au* profit ( x em') A A A A

'' If workers are concerned about the corlsumption of households rather than simply their own consumption, then their objection to child and female labor may be tem- pered, but immigration will still be tbught.

1300 J O U R N A L OF POLITI(:AI. EC.ONOhlY

T A B L E 2

U n d e r workers' control, labor-augmenting change is favored. Worker-capitalist conflict continues under the guise of conflict about growth of the labor force.

I n the codeterniination regime, the conventional wage is the focus of political conflict. O n e might expect a codeterniination regime to be the most technologically dynamic since both workers a n d capitalists benefit from both types of technological change. Also, to the extent that there is a n upward drift in the conventional wage as convention catches u p with the actual level of wages, techriological change may lessen the conflict over the conventional wage by raising the actual income of both classes. Technological change shifts the survival boundary to the left, counteracting the rise in c, maintaining o r even increasing the profit share.

VIII. Conclusion

T h e model of class conflict presented here is quite stylized, and "class" is conceived narrowly. Capitalists maxiniize profits while workers maximize income (consumption). Capitalists invest while workers af- fect wages. In this way the nature of the conflict and the possibilities for cooperation a r e well defined if narrow.

Although the model focuses o n the struggle by the t ~ v o groups to increase their income shares, other parameters might also be consid- ered the outcome of a social process of conflict and struggle. T h e lower bound o n wages c is a conventional, not a biological, minimum, socially determined by some unmodeled process. Similarly, the pa- rameters y a n d 6 in the u.age bargaining equation reflect the relative strength of the two groups in changing wages. These parameters a r e functions of the political system (legality of strikes, collective bargain- ing, a n d strikebreaking). I have treated these variables as constant over four widely differing social organization assumptions, although orie main reasori f o r setting u p a u ~ l i o n (pushing for decertification) is precisely to change these parameters in the workers' favor (in the employers' favor).

M O D E L OF C L A S S S T K U G G L E 1301

T h e values of the technological parameters a and A may also reflect some process of struggle over the adoption of' new technologies and new torms of work organization (see Weisskopf, Bowles, and Gordon 1983). In Section VII this idea was used to argue loosely that techno- logical change is more likely to occur when it benefits both parties.

Even the growth of the labor force ma): be considered a parameter affected by the balance of class power. Child labor and immigration ma): be resisted by a strong working class. Some rnodels have stressed the endogeneity of the labor fbrce (Lewis 1954; Marglin 1984), argu- ing that labor supply adjusts to labor demand by pulling people from other sectors of the economy-the household, agricultural, o r tradi- tional sectors. This model does not capture such phenomena directly. T h e model could, however, be extended somewhat in that direction by interpreting it as representing an econornic sector rather than a national economy. It would then be possible, for example, to analyze a dualistic economy as a high-wage codeternlination sector coupled with a low-wage ato~nistic sector with little labor mobility between thern. '4 similar procedure could extend the model to include interna- tional capital and labor flows between countries each of which may (or may not) have a different regime. Capital flows would then place a lower bound on profits as discussed in Section IV, and labor flows could be represented by changes in n.

If the nlodel leaves out all these important aspects of' the notion of class conflict, it nevertheless captures the basic asymmetry people have in mind when they discuss "class conflict." Capitalists and work- ers have different sources of power and different motivations. These differing aims and abilities confront each other in economic relations, delimiting growth opportunities. Depending on the degree of'organi- zatiorl of the two groups, widely differing macroeconomic perfor- rnarlce may be expected. For example, the atonlistic version of' the model may best fit certain historical experiences, such as Britain in the early 1800s. T h e codetermination equilibrium seems most illustrative of modern developed econonlies such as the United States and Ger- many. For such economies, the nod el suggests that prior agreement on the distribution of gains from econo~nic growth is an i~nportant element in achieving that growth. When that agreement fails, growth will tail as well.

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A Classical Model of the Class Struggle: A Game-Theoretic Approach Perry G. Mehrling The Journal of Political Economy, Vol. 94, No. 6. (Dec., 1986), pp. 1280-1303. Stable URL:

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[Footnotes]

1 The Case for Methodological Individualism Jon Elster Theory and Society, Vol. 11, No. 4. (Jul., 1982), pp. 453-482. Stable URL:

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5 Growth Cycles with Induced Technical Change Anup Shah; Meghnad Desai The Economic Journal, Vol. 91, No. 364. (Dec., 1981), pp. 1006-1010. Stable URL:

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References

The Timescale of Economic Models: How Long is the Long Run? A. B. Atkinson The Review of Economic Studies, Vol. 36, No. 2. (Apr., 1969), pp. 137-152. Stable URL:

http://links.jstor.org/sici?sici=0034-6527%28196904%2936%3A2%3C137%3ATTOEMH%3E2.0.CO%3B2-N

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The Case for Methodological Individualism Jon Elster Theory and Society, Vol. 11, No. 4. (Jul., 1982), pp. 453-482. Stable URL:

http://links.jstor.org/sici?sici=0304-2421%28198207%2911%3A4%3C453%3ATCFMI%3E2.0.CO%3B2-P

The Relation between Unemployment and the Rate of Change of Money Wage Rates in the United Kingdom, 1862-1957: A Further Analysis Richard G. Lipsey Economica, New Series, Vol. 27, No. 105. (Feb., 1960), pp. 1-31. Stable URL:

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A Mathematical Theory of Saving F. P. Ramsey The Economic Journal, Vol. 38, No. 152. (Dec., 1928), pp. 543-559. Stable URL:

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Growth Cycles with Induced Technical Change Anup Shah; Meghnad Desai The Economic Journal, Vol. 91, No. 364. (Dec., 1981), pp. 1006-1010. Stable URL:

http://links.jstor.org/sici?sici=0013-0133%28198112%2991%3A364%3C1006%3AGCWITC%3E2.0.CO%3B2-9

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