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Competitive capitalism and cooperative labor management in a dynamic nutshell
499 Europ~iische Zeitschrift fiir Politische Okonomie/ EuropeanJoumal of Political Economy, 2/4 (1986) 499--519 © VVF, Munich
COMPETITIVE CAPITALISM AND COOPERATIVE LABOR MANAGEMENT IN A DYNAMIC NUTSHELL Carl ChiareUa* and Murat R. Sertel**
Summary: Competitive capitalism and co-operative labor management are both formulated as differential games: The former, as a game between capitalists and workers, and the latter as one between an elected council and workers. The two systems are compared against the background of neoclassical optimal growth theory, at their respective equilibria and in terms of their rates of approach toward these states, all within a dynamic "nutshell" model extended from the static one of Dirickx and Sertel (1979). 1. Introduction
The dynamic behaviour of the labor managed firm has been considered by Furubotn (1971) and Atkinson (1973). Furubotn introduces a multi-period model of the labor managed firm in which the time stream of consumption is optimized. Atkinson sets up a model which allows for growth of the firm and decisions about the rate of investment, then examining a number of issues such as whether the labor managed firm will grow more slowly than its capitalist counterpart, how economies of scale affect the behaviour of the labor managed firm and the finance of capital formation by workers plowing back income. Here we work with a differential game model in which some of these issues can be examined. We formulate competitive capitalism as a differential game between, on the one hand, capitalists who hire labor and invest in capital accumulation so as to maximize discounted utility of capitalists' consumption over an infinite horizon and, on the other hand, workers who offer labor so as to maximize their discounted leisure-dependent utility of con* School of Mathematical Sciences, The New South Wales Institute of Technology, Sydney, Australia. ** Department of Economics, Bo~azi¢~iUniversity, Istanbul, Turkey. Acknowledgements - see page 516.
500
Competitive Capitalism
sumption over an infinite horizon. We formulate cooperative labor management as a differential game between, on the one hand, an elected council, which determines the investment program maximizing the discounted infinite horizon utility integral of a typical individual worker, and, on the other hand, the workers, each of w h o m in reaction to a given investment program will offer labor so as to maximize his discounted infinite horizon utility integral. We allow a certain amount of increasing returns to scale in the production function; as we shall see, this amount is limited by stability considerations. We compare the equilibrium of each of the differential games as well as growth rates in a neighbourhood of the equilibrium. We also draw a comparison between the two aforementioned systems and the social optimum that would be obtained by a benevolent social planner, which is to say the neoclassical economy or firm. Our models of competitive capitalism and cooperative labor management are dynamic versions of the corresponding models developed in the "nutshell" of Dirickx and Sertel (1979). Our economic framework is that of the neoclassical one-sector growth model; a single good is produced and it may be either consumed or combined with labor in the productive process to produce further output. Thus output Y is given by (1.1)
Y = K = La,
where K is the stock of productive capital, L is the a m o u n t of labor input and 0 < cx < 1, 0 < ~ < 1. If we assume that there are A workers each offering a fraction x of his total available labor (0 ~< x ~< 1) then, since L = Ax, (1.1) can be written as (1.2)
y = A - e k = xa ,
where we put y = Y/A for the output per worker, k = K/A for the amount of capital per worker, and (1.S)
e= 1 --ol-- ~.
Unlike the standard neoclassical world, our workers do not unstintingiy offer all their available labor to the productive process, as they derive disutility from work. We shall start our analysis by a discussion of the neoclassical optimal growth model with leisure-liking workers. Such a model, more general than ours, has been considered by Chase (1967).
Competitive Capitalism
5 01
2. Neoclassical Dynamic Growth We consider the neoclassical e c o n o m y equipped with A workers. A benevolent social planner will seek the labor participation ratio, x, and per-capital consumption c (and hence the investment program) so as to (2.1)
max f o e-Pt log(c--x~) dt, C,X
subject to the capital accumulation equation (2.2)
l~ = A-ek~x ~ -- c
showing o u t p u t per capita (A -~ k ~ x ~ ) being divided between consumption per capita (c) and investment per capita (~:). The integrand of (2.1) is the typical worker's utility, time discounted at a rate p > 0, the x ~ term representing disutility of work (reflecting utility of leisure). T h r o u g h o u t we assume "r > 1. Using standard techniques in optimal control theory to maximize (2.1) subject to (2.2) we find that along the optimal path (2.3)
x = A -e¢
C9 k n" = x n (k),
where we set (2.4)
e = l/('y -- ~).
Equation (2.3) shows o p t i m a l l a b o r p a r t i c i p a d o n (x) as a concave function of capital per worker (k). Furthermore we find that the optimal path of the neoclassical e c o n o m y is governed by the differential system (2.5) " 1~= (1 - - ~ ) f(k) - l / k , (2.6)
x Co - g(k)),
where for ease of notation we set
I-> with (2.9)
~ = ~,(I - ~) -- ~-
Competitive
502
Capitalism
The quantity e --pt h(t) is the shadow price of capital and is related to per capita consumption by (2.10)
c = x7 + l/h.
To summarise, all quantities of interest are found by solving the differential system (2.5-6) for k and A and then obtaining x and c from (2.3) and 2.10), respectively. The equilibrium of the differential system (2.5-6) is displayed in Table 1 for 1 < 7 < = together with the limiting cases 7 + 00 andr= 1. The last row of Table I displays En, the imputed wage under the neoclassical regime, obtained from dividing the equilibrium consumption per capita (iYn) by the equilibrium labor participation ratio (IQ). Table I: Neoclassical Equilibrium
1 . . pll-4
2 0P
ov(l-4
4-c(l-a] r a4'-a) 0P
The limiting case 7 + 00 reproduces the more familiar neoclassical equilibrium. (See, for example, Cass (1965)). For ease of comparison with the results of Dirickx and Sertel (1979) we note the following relationships between the equilibrium quantities:
Competitive Capitalism
(2.11)
503
~=~__7£* -~n- ~ R'*
The dynamics of the neoclassical economy are illustrated in Figure 1, the phase plane of the differential system (2.5-6). We find that, provided
(2.12) ~+~< 1+~(1-1/"/), the equilibrium (kn, ~n) has the desired saddle point property. This property is desired because given k(0), the initial capital stock, there is a unique choice of ~(0), and therefore c(0), that sets the economy on the unique path nn' tending to the equilibrium (k n, Xn)" Furthermore nn', the stable arm of the saddle point is the only path along which the transversality condition (2.13)
lime -pt ~(t) k(t) = 0, t-* oo
a necessary condition for the maximisation of (2.1) subject to (2.2), is satisfied. We see from Figure 1 that along the dynamic path of the neoclassical economy the capital stock, labour participation and consumption all rise (decline) monotonically to their respective equilibrium values if the initial capital stock is less (greater) than its equilibrium value, whereas ~(t) moves in the opposite direction. k n \ \ ~.o
F,. ----~---
.... ..... ~
o
k{o}[........... ' .............. " " "~"~ n' ,~Col
" ~'
Figure 1: Dynamic path of the neoclassical economy
504
C o m p e t i t i v e Capitalism
The stability condition (2.12) shows the extent to which increasing returns to scale is possible in the neoclassical economy. In the limiting case 7 = 1, (2.12) reduces to (2.14)
c~ -I- fl <
i ,
so that only diminishing returns to scale are allowed in this extreme case. At the other extreme, as 7 -+ 0% (2.12) becomes (2.15)
~
+ fl < 1 + f l ,
the maximum extent of increasing returns to scale allowed in the neoclassical economy. The curve oe +fl = 1 +fl(1 -- 1/3') is displayed in Figure 2. ,a÷0
1+(3
....
I
>~"
Figure 2 The stability condition (2.12) was to some extent anticipated b y Dirickx and Sertel (1979) w h o point out that their static solutions are valid precisely under condition (2.12).
3. Competitive Capitalism We model competitive capitalism as a differential game 1 in which the workers each decide on their respective labor inputs The game we consider here is not a differential game in the strict sense of the term, since the wage w(t) is determined passively by the players via interaction with the auctioneer. We are closer to the simultaneous variational model discussed by Wan
(197z).
Competitive Capitalism
505
and capitalists take the investment decision, each group seeking to maximise a discounted utility integral over an infinite horizon. Our paradigm of competitive capitalism presupposes a Walrasian auctioneer who announces a time path for the wage, w(t). Given this w(t), each worker chooses x(t) so as to maximise his utility, thus coming up with a labor supply function at every point o f time. Simultaneously, given the same w(t), the capitalists determine the a m o u n t of labor to be employed and the investment pattern so as to maximise their utility, coming up with a labor d e m a n d function at every point of time. The auctioneer ends'Li~" setting w(t) so as to equate labor demand and supply at each point in time. Thus, given a wage time path w(t), a typical worker seeks x(t) SO a s t o
(3.1)
max f o e - P ' tl°g ( w x - - x ~) dt, x
where p I > 0 is the discount factor of a typical worker and we take workers' utility to be log (wx - x~ ). Now the typical worker under competitive capitalism, being subject to no differential equation constraint, maximises the integral in (3.1) by choosing c32)
x
which displays for us the labor supply function. The typical capitalist seeks labor input x, and his own consumption c so as to (3.3)
max f ~o e - p ~ t l o g c dt X,C
0
subject to the accumulation equation. (3.4)
1~= A -~ k ~ x ~ - wx -- c/A,
which shows o u t p u t per worker (A-ek~x ~) being divided between the wage p a y m e n t per worker (wx), a capitalist's consumption per worker (c/A) and investment (1~)2. The integrand in Note that the population in the competitive capitalist world is larger than that of the neoclassical world, n o w being A plus the n u m b e r of capitalists. It is perhaps a m o o t p o i n t as to w h e t h e r the population should remain A and we allow some of the workers to become capitalists. We have settled o n our formulation as we wished to see h o w the o u t p u t of afixed group of workers would be affected by differing managerial arrangements.
506
C o m p e t i t i v e Capitalism
[3.3) is the typical capitalist's utility, time discounted at the rate P2 > 0 . The necessar3" conditions for the maximisation of (3.3) subject to (3.4) yield ,3.5)
x = ( : ' - ' k~ ~ v ) l/('-t~)
and
(3.6)
Xc = A,
with the capitalist's shadow price on capital, e -pat termined by the differential equation (3.7)
;k(t),
being de-
J, = X (P2 - "-k-'ake-I xa)"
Given a wage time path w(t), the relations (8.5), (8.6) and the differential equations (8.4), (8.7) together with the transverality condition (3.8)
l i m e -Pat ~.(t)k(t) = 0 t---~~¢
enable the capitalist to determine k(t) and hence, via (3.5), his labor demand function x(t). The wage time path which equates labor supply and demand at every point in time will be such as to equate x in (3.2) to x in (3.5). Thus eliminating w between these latter two equations we find that along the labor market clearing time path, (3.9)
x = x n (k),
where x n (k) is defined by equation (2.3). Thus the functional relation between x and k is the same under competitive capitalism as in the neoclassical economy of Section 2. Of course, this does not mean that the time paths for x and k are the same under the two different economic systems; indeed, as we shall see shortly, they are different 3 . All that we have shown is that the neoclassical and competitive capitalist economies mix labor and capital in the same proportions, but the actual amounts that are mixed in the two economic systems can and do differ. The time paths differ not only for the reasons given later but also because the timc path for k under competitive capit',dism is infuenced by the unequal discount factors p 1 and P2 whereas in the neoclassical world there is just the common discount factor p. However as we see later the difference in time paths for k is not only due t ° P l ~ P2"
CompetitiveCapitalism
507
Substituting (3.5), (3.6) and (5.9)into (8.4) and (3.7) we find that the differential equations for k and ~ reduce to
(3.10)
k=
(1-,6)
f(k)-1/;k,
(3.11)
~. = ;k(p~ - g(k)),
where f(k)and g(k) are defined by equations (2.7) and (2.8). The differential equations (3.10-11) together with the transversality condition (8.8) completely determine the time paths for k and and thereby those of x, w and c. Comparing the differential system of competitive capitalism (8.10-11) with the corresponding differential system for the neoclassical economy (2.5-6), we see that the two systems differ in the differential equation for k and hence will generate different time paths starting from the same initial value of k. In making this and further comparisons we assume that the capitalists discount at the same rate as the social planner in Section 2. Thus, henceforth we put p 2 = p . The equilibrium (kc, ~c ) of the differential system (3.10-11) is given, in terms of the corresponding neoclassical equilibrium values, by (3.12)
k c=k n,
and
(.:.8.15)
X"c = (1 -P/",')
(i --~)
X-n > x- n
so that the equilibrium values of x and w are (3.14)
~c = ~ " ' % = ~ % < %
with ~- (capitalist's equilibrium consumption per worker = c/A) and c-w (a typical worker's equilibrium consumption bundle = Wc~c ) given by (8.15)
~-=(1-~)gn
n , gw=~n<~-
n.
We observe that, in equilibrium, capital stock per worker and labor input per worker are the same in the competitive capitalist economy as in the neoclassical economy, whereas the equilibrium wage is lower under competitive capitalism than the neoclassical imputed wage. In equilibrium, capitalist consumption per worker and the typical worker's consumption bundle in the competitive
508
Competitive Capitalism
capitalist economy are both less than equilibrium consumption per capita in the neoclassical economy. In fact, capitalists and workers share cn in the proportion (1-~) to ~. If we equate the discount factor of the capitalist to the rental on capital in the static framework of Dirickx and Sertel (1979), we find that the equilibrium values of k, x and w that we have obtained correspond precisely with those of their static version of competitive capitalism. It is a straightforward matter to construct the phase diagram for the differential system (3.10-11) and to show that the equilibrium (k,:, ~ ) has the desired saddle point property provided that condition f2.12), which guaranteed the saddle point property for the neoclassical system, is satisfied. Indeed the phase diagram will have the same general features as Figure 1. In Figure 3 we show just the stable arms of the saddle (cc') which constitute the dynamic path of competitive capitalism. We show on the:same diagram the dynamic path of the neoclassical economy, (nn'). It can be seen by comparing relative values of dk/d~ in each economy that the dynamic paths nn' and cc' never cross.
¢
¢0
N° !
\ Figure .3
X,
Competitive Capitalism
509
We see from Figure 3 that the qualitative features of the dynamic path are the same under competitive capitalism as under the neoclassical regime of optimal growth. Thus, given the same initial capital stock per worker ko, both economies have k declining (rising) to the equilibrium value k n if k o is greater (less) than k n . The competitive capitalist economy, however, will start and finish with a higher value for the shadow price of capital. In Section 5 we shall compare the rates at which each economy approaches equilibrium.
4. Cooperative Labor Management Following Dirickx and Sertel (1979) we assume that the workers have identical preferences (now including time preferences) and enter p r o d u c t i o n symmetrically. Our analysis also parallels the "nutshell" of Dirickx and Sertel (1979) in that we analyse the consequences when each worker assumes (self-justifyingly) that his fellows will contribute as m u c h work on the average as he himself does. The workers elect a council whose task is to determine the capital accumulation program which maximizes the utility of a typical worker, given the time path of x(t), the labor participation of a typical worker. The typical worker thus knows what share in output, i.e. income (or i m p u t e d wage) he can expect from a ~iven x(t), and will determine and then supply the x(t) which maximizes his own utility. Note that if we were to allow the council to determine n o t only the optimal capital accumulation program b u t also the optimal time path of labor participation, then we would be in the neoclassical world of Section 2. Thus, taking x(t) as given, the council seeks y(t), the share in o u t p u t going to a typical worker, so as to (4.1)
max f e- " t l o g ( y - x y 0
"~)dt,
the typical worker's utility integral, subject to the accumulation equation (4.2)
I~ = h -e k = x g -- y ,
showing o u t p u t per worker (A-~ k = x g) bein~ divided between o u t p u t share per worker (y) and investment (k). Here p > 0 is the discount factor c o m m o n to all workers and the utility of a
510
Competitive Capitalism
typical worker is log (y - x v ). Since in this model we do not allow workers to save out of their earnings all of y is consumed. The necessary conditions for maximization of (4.1) subject to (4.2) turn out to be (4.3)
y = xv + 1 / ~ ,
with e- ° t ~(t), the council's shadow price on the capital stock per worker satisfying the differential equation (4.4)
~ = ~ ( 0 - A - ~ otk~-I x~) •
Equation (4.3) together with the differential equations (4.2) and (4.4) and the transversality condition (4.5)
l i m e "-°t ~(t) k(t) = 0 ,
summarise the accumulation path chosen by the council in reaction to given x(t). The typical worker takes into account the council's optimal reaction as summarized by equations ( 4 . 2 - 5 ) when seeking x(t) so as to maximize his own utility integral 4 . Thus the typical worker seeks x(t) so as to (4.6)
max f e- m log~ d t , x 0
subject to the differential equation constraints (4.7)
k = A-~ k ~ x~ - x* - ~/',
and (4.8)
~
= 7"/(--p + A - e
k iV- 1 XO).
For convenience we have set
(4.9)
'q =
1IX.
Since the typical worker knows how the council will choose y he is able to express his utility integral as in (4.6) by use of (4.3). The differential equations ( 4 . 7 - 8 ) summarising the council's accumulation program for a given x time path become dyNote that we are assuming an overall cooperative solution of the council and all workers. Alternatively, we may assume that there is some enforcement mechanism in place so that individual workers do not deviate from the overall optimal plan.
Competitive Capitalism
511
namic constraints for the typical worker. Note that the initial values k(0) and ~/(0) of k and ~? are both taken as given. The necessary conditions for the maximisation of (4.6) subject to (4.7-8) (henceforth referred to as problem PW) turn out to be (4.10)
~=x
(k)(l+a~b) e ,
(4.11)
hi = Ul(P - A-e ak~-1 x0( 1 -- (1-~)¢))
where ~b= #2,2[(#1 k), and (4.12)
/h =Pu2 - l/n +u~ - u s
( - o + A-* ~ k ~ - l x ° ) •
Here #1 e-Ot and bt2 e-Pt are Lagrange multipliers, the former of which can be interpreted as the typical worker's shadow price on the capital stock k, while the latter can be interpreted as the typical worker's shadow price on ~/, the argument of his utility function. The dynamic path of the cooperative labor managed economy is given by the differential equations (4.7-8), (4.11-12) with x given by (4.10). We shall refer to this differential system as the differential system (M). The equilibrium of the differential system (M) is given by (4.13)
L =k-n'
(4.14)
~'e
~'n'
ul
and (4.16)
/~'-'2= 0.
Furthermore this equilibrium is unique. The corresponding equilibrium values of x and y are (4.17)
xe = xn ,
(4.18)
Fo =
•
Finally the equilibrium imputed wage We ( = Ye / Xe ) is (4.19)
we = w
.
Thus equilibrium levels of capital stock (k-e ), labor participation (x e ), consumption per worker (~e) and inputed wage (We )
512
Competitive Capitalism
are the same in the labor managed economy as they are in the neoclassical economy. However the dynamic paths leading to the equilibrium will be quite different. The stability analysis of the equilibrium (4.13-16) is more difficult than for the equilibria of the models of the neoclassical and competitive capitalist economies because the larger number of differential equations rules out the phase plane approach. We shall henceforth refer to the equilibrium (4.13-16) as (E), and to the set of equilibrium values (ke, ~'e) as (E'). The task of showing that the solution to the problem PW is a unique path in the (k, ~.) plane tending to the equilibrium (E') is, technically, a complex one. Here we shall indicate the broad outlines of what is involved. First we linearise the differential system (M) about the equilibrium (E) to obtain the linear differential system (4.20)
~ =BEG,
where the column vector ~ has elements ( k - k e , r/-~e , # x - ~ e , # 2 - ~ 2 ) and the matrix B is the Jacobian matrix of the differential system (M) evaluated at the equilibrium (E). The matrix Be is displayed in the Appendix as is the calculation of its eigenvalues. As a result of these calculations we find that B has two positive and two negative eigenvalues provided that the condition (2.12) is satisfied. Hence, since two initial values k(0), r/(0) are given and the other two #I (0),/a 2 (0) are to be determined, the equilibrium (E) has the local saddle point property. Thus in a neighbourhood of the equilibrium (E) there is a unique path (L) tending to (E). That this path is indeed the solution to problem PW can be shown by invoking Theorem 2 of Rockafellar (1976). It is also possible to show that the aforementioned unique path exists globally. (These points are elaborated upon in an appendix in Chiarella and Sertel (1980), where the model here is analysed under the constant returns to scale assumption.) We also show in the Appendix that the eigenvalues of Be are never complex under the assumptions of our model and so there is no oscillatory motion in a neighbourhood of the equilibrium (E). Hence, projecting the solution of the differential system (M) onto the (k,~) plane, we see locally a picture similar to Figure 1, which is characterised by a smooth approach to the equilibrium (E'). However, it has not been possible to make a definite assertion about the relative slopes of nn' and ee' (the projection of L onto (k,~) plane) and possible figurations are displayed in Figure 4.
Competitive Capitalism
513
e
i'1
ke:k:r
e'
n'
Jm
Ae=~n
ke:~ e
,
em
Xe
Figure 4
--'~n
514
Competitive Capitalism
5. Comparison of Growth Rates The differential systems (2.5-6) for the neoclassical economy and (3.10-11) for competitive capitalism may be linearised around their respective equilibria, as was done in the last section, to obtain a second-order linear system
(5.1)
~ = B i t s,
where ~i is the column vector (k-k- i, ?~-X-i) for i = n, c. The matrix Bi is theJacobian matrix of the respective differential system evaluated at its equilibrium. For the differential system (2.5-6) of the neoclassical economy we find that aT/i whilst for the differential system (3.10-11) for the competitive capitalist economy we find that (5.3)
[(1--fl)^leP
Be =
oriel (1-~)k-c~
1/~" ] 0n
The eigenvalues of Bn are (5.4)
b~n) = P-[1 -T-.~/l+--'g'0"]
b~n) '
2
where (5.~)
o = 4~/c~/.
Under the assumption that (2.12) is satisfied we have (5.6)
b~n) < 0 < b~n) ,
which shows from an algebraic point of view the saddle point property of the equilibrium (kn, Xn). Thus in a neighbourhood of the equilibrium (kn, Xn ) the path nn' is asymptotic to the path (5.7)
~i(t) =Vl exp (b] n) t ) ,
where v 1 is a constant vector depending on the initial conditions. Similarly the eigenvalues of B c are
(5.8)
b? :zez[l
l,
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515
where (5.9)
z = (1 --0) I (1 -- OI'Y) < 1.
Here also (5.1o)
b?l < 0 < b Cl ,
provided that (2._12) is satisfied. So in a neighbourhood of the equilibrium (k-c, ~c ) the path co' is asymptotic to the path (5.11)
~ ( t ) = v2 exp (b~c) t ) ,
where v2 is a constant vector depending on the initial conditions. We can see from (5.7) and (5.11) that Ibtn) I and [b~C)I measure the rate at which each of the economic systems approaches its equilibrium. Using the fact that 0 < z < 1 it is not difficult to show that (5.12)
Ib[C)l < Ib]n) l ,
which indicates that, asymptotically at least, the neoclassical economy approaches its equilibrium more rapidly than does the competitive capitalist economy. An examination of the eigenvalues of the matrix Be given in the Appendix indicates that the path ee' is asymptotic to (5.1s)
= v 3 exp
t)
where b t z) = b~n), which indicates that, asymptotically_, the labor managed economy approaches the equilibrium (kn, Xn) at the same rate as does the neoclassical economy.
6. Concluding Remarks We have modelled competititve capitalism and cooperative labor management as differential games and drawn comparisons with, the corresponding neoclassical (socially optimal) solution. The dynamic paths of both systems exhibit the characteristics of the dynamic path of the neoclassical system, namely the monotonic approach to equilibrium (at least asymptotically in the case of cooperative labor management). The equilibrium of cooperative labor management corresponds to that of the neoclassical system, also both systems (asymptotically) approach equilibrium at the same rate. Competetive capitalism yields the same
516
Competitive Capitalism
equilibrium values of k and x as in the neoclassical system, but the typical worker receives, in equilibrium only a fraction/~ of the wage he would receive in the neoclassical system. Also the equilibrium consumption per capita of the neoclassical system is distributed between workers and capitalists in the proportions /3 to (1-13). Futhermore the competitive capitalistic system (asymptotically) approaches equilibrium at a slower rate than does the neoclassical system. In making the above comparisons we have assumed the same discount factor to prevail in all systems. Increasing returns to scale are permitted in all the models discussed here up to the point allowed by the stability condition (2.12). Beyond this the equilibria are unstable. Equating the discount factor of our models to the rental on capital in the static framework of Dirickx and Sertel ( 1 9 7 9 ) w e find that our equilibrium outcome of competitive capitalism agrees with theirs. However our equilibrium outcome of cooperative labor management differs from theirs in the wage, their equilibrium wage being a factor ( l - a ) times what we have obtained. Our models of competitive capitalism and cooperative labor management contain only one market, namely the labor market. Another market which must be considered is the capital (or savings) market. In further developments of our model of competitive capitalism one could allow individual workers a savings-consumption decision, with unconsumed output offered to capitalists on a capital market where some rate of interest, r(t), prevails. The market clearing r(t) could be determined in a free market solution or we could allow e.g., a capitalists' union to be price dictator in the capital market. Simi!arly, we could introduce a capital market into our model of cooperative labor management, the council acting as price setter in the capital market to choose an interest rate that will induce the workers into a savings pattern that is optimal overall.
Acknowledgements: An earlier version of this paper was presented at the Eight Bosphorus Workshop on Industrial Democracy,at Bogazici University Istanbul, July 1984. It benefited from diskussion there with Ahmet Alkan, David Cass. Martine Quinzii and Jaques Thisse. Editorial comments of Manfred Holler have led to improvements. Whatever may be wrong with the paper is despite all wise council we have received.
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517
Appendix We find after some lengthy calculations that p -
(A1)
-Bq
--1
0
0
0
0 -b~--- e
=
0 e
0
q
2
~,
1
p
¢
-~
1
where we have put
(A2)
q=aS(P)2/'7.
The characteristic polynomial for Bq turns out to be b 4 _ 2 p b 3 +(p2_o)h 2 + a p b + r = 0 ,
(A3) where (A4)
a = q(2 + a N a ) ,
and (A5)
r = q2 ( 1 + o4~/~).
If (2.12) is satisfied we can assert that (A6)
8>0,
r>0.
The quartic (A3) has roots (A7)
r'
b m
½[p+-~/P212+e+A+~/p2/2+a-Al,
where
(A8)
A2 = p 4 / 4 + p 2 a + 4 r > p4/4 "
It is a simple matter, using the inequality in (A8), to ascertain that (A7) contains two positive and two negative roots. Thus the
518
Competitive Capitalism
equilibrium E is a local saddle point. The solution to (4.21) may be expressed through
(A9)
}(t) = ~ Viexp(bl')t), i=I
where (A10) b~1)
-3
'
are eigenvalues of B 1 as given by (A7), V i are the corresponding eigenvectors and A i are constants. Since the initial conditions gl (0),/~2 (0) are so far unspecified, we may choose them to set As, A 4 equal to zero and then use k(0), X(0) to calculate A1, A s . Thus the unique path tending to the equilibrium E is asymptotic to the path (All)
~(t)=
2
Z A i V i e x p ( g i 1)t), i=1
which for large t is dominated by
(A12) ~(t)= A 1 V 1 exp (b(lI) t). The dominant eigcnvalue-I h (I) can be expressed in the form (A13)
b~~I =
x -~/1
+ flr(fl/,(1 + fl)),
where (A14)
1
1
+z +
(½ + z -
'
Now some algebraic manipulations reveal that (A15) F ( z ) = l f o r a l l z ~ 0 . (To see this, set u = 1 + 4z, so that for u > 1 we have 1
1
2 r ( z ) = (1 + 2x/-~ + u ) ~ - - (1 - 2x/~ + u)~-= 2.)
Thus, wc have shown that b~ 1 ) = b~n ). Finally, we see that the condition for (A7) to yield complex eigenvalues (and that the approach to equilibrium be hence oscillatory) is
Competitive Capitalism (A16)
519
o2<4r,
w h i c h reduces t o (A17)
(~/]~)2 < 0
which, clearly, never obtains. H e n c e n o oscillatory m o t i o n is possible u n d e r the a s s u m p t i o n s o f o u r m o d e l .
References Atkinson, A.B. (1973). "Worker Management and the Modern Industrial Enterprise", Quarterly Journal of Economics, 87,375-392. Cass, D. (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation", Review of Economic Studies, 32,233--241. Chase, E.S. (1967). 'Leisure and Consumption', appears in Essays on the Theory of Optimal Economic Growth, ed. by K. Shell, M.I.T. Press. Chiarella, C., and M.R. Sertel (1982). "Capitalism, Communism and Labor Management in a Dynamic Nutshell Under Constant Return to Scale", Ch. 10 of Sertel (1982). Dirickx, Y.M.I., and M.R. Sertel, (1979). "The Comparitive Political Economy of Workers' Management and Capitalism in a Nutshell", European Institute for Advanced Studies in Management, Working Paper 79--2, February 1979. Revised version, Ch. 3 of Sertel (1982). Furubotn, E.G. (1971). "Toward a Dynamic Model of the Yugoslav Firm', Canadian Journal of Economics, 1971, 4, 182--197. Rockafellar, R.T. (1976). "Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Nonzero Discount Rate", appears in The Hamiltonian Approach to Dynamic Economics, ed. by D. Cass and K. Shell, Academic Press. Sertel, M.R. (1982). Workers and Incentives, Contributions to Economic Analysis, No. 140, Amsterdam: North-Holland. Wan, H.Y., Jr. (1971). "A Simultaneous Variational Model for International Capital Movement", in Trade, Balance of Payments and Growth, ed. by J. Bhagwati et al., Amsterdam et al.: North-HoUand.
COMPETITIVE CAPITALISM AND COOPERATIVE LABOR MANAGEMENT IN A DYNAMIC NUTSHELL Carl ChiareUa* and Murat R. Sertel**
Summary: Competitive capitalism and co-operative labor management are both formulated as differential games: The former, as a game between capitalists and workers, and the latter as one between an elected council and workers. The two systems are compared against the background of neoclassical optimal growth theory, at their respective equilibria and in terms of their rates of approach toward these states, all within a dynamic "nutshell" model extended from the static one of Dirickx and Sertel (1979). 1. Introduction
The dynamic behaviour of the labor managed firm has been considered by Furubotn (1971) and Atkinson (1973). Furubotn introduces a multi-period model of the labor managed firm in which the time stream of consumption is optimized. Atkinson sets up a model which allows for growth of the firm and decisions about the rate of investment, then examining a number of issues such as whether the labor managed firm will grow more slowly than its capitalist counterpart, how economies of scale affect the behaviour of the labor managed firm and the finance of capital formation by workers plowing back income. Here we work with a differential game model in which some of these issues can be examined. We formulate competitive capitalism as a differential game between, on the one hand, capitalists who hire labor and invest in capital accumulation so as to maximize discounted utility of capitalists' consumption over an infinite horizon and, on the other hand, workers who offer labor so as to maximize their discounted leisure-dependent utility of con* School of Mathematical Sciences, The New South Wales Institute of Technology, Sydney, Australia. ** Department of Economics, Bo~azi¢~iUniversity, Istanbul, Turkey. Acknowledgements - see page 516.
500
Competitive Capitalism
sumption over an infinite horizon. We formulate cooperative labor management as a differential game between, on the one hand, an elected council, which determines the investment program maximizing the discounted infinite horizon utility integral of a typical individual worker, and, on the other hand, the workers, each of w h o m in reaction to a given investment program will offer labor so as to maximize his discounted infinite horizon utility integral. We allow a certain amount of increasing returns to scale in the production function; as we shall see, this amount is limited by stability considerations. We compare the equilibrium of each of the differential games as well as growth rates in a neighbourhood of the equilibrium. We also draw a comparison between the two aforementioned systems and the social optimum that would be obtained by a benevolent social planner, which is to say the neoclassical economy or firm. Our models of competitive capitalism and cooperative labor management are dynamic versions of the corresponding models developed in the "nutshell" of Dirickx and Sertel (1979). Our economic framework is that of the neoclassical one-sector growth model; a single good is produced and it may be either consumed or combined with labor in the productive process to produce further output. Thus output Y is given by (1.1)
Y = K = La,
where K is the stock of productive capital, L is the a m o u n t of labor input and 0 < cx < 1, 0 < ~ < 1. If we assume that there are A workers each offering a fraction x of his total available labor (0 ~< x ~< 1) then, since L = Ax, (1.1) can be written as (1.2)
y = A - e k = xa ,
where we put y = Y/A for the output per worker, k = K/A for the amount of capital per worker, and (1.S)
e= 1 --ol-- ~.
Unlike the standard neoclassical world, our workers do not unstintingiy offer all their available labor to the productive process, as they derive disutility from work. We shall start our analysis by a discussion of the neoclassical optimal growth model with leisure-liking workers. Such a model, more general than ours, has been considered by Chase (1967).
Competitive Capitalism
5 01
2. Neoclassical Dynamic Growth We consider the neoclassical e c o n o m y equipped with A workers. A benevolent social planner will seek the labor participation ratio, x, and per-capital consumption c (and hence the investment program) so as to (2.1)
max f o e-Pt log(c--x~) dt, C,X
subject to the capital accumulation equation (2.2)
l~ = A-ek~x ~ -- c
showing o u t p u t per capita (A -~ k ~ x ~ ) being divided between consumption per capita (c) and investment per capita (~:). The integrand of (2.1) is the typical worker's utility, time discounted at a rate p > 0, the x ~ term representing disutility of work (reflecting utility of leisure). T h r o u g h o u t we assume "r > 1. Using standard techniques in optimal control theory to maximize (2.1) subject to (2.2) we find that along the optimal path (2.3)
x = A -e¢
C9 k n" = x n (k),
where we set (2.4)
e = l/('y -- ~).
Equation (2.3) shows o p t i m a l l a b o r p a r t i c i p a d o n (x) as a concave function of capital per worker (k). Furthermore we find that the optimal path of the neoclassical e c o n o m y is governed by the differential system (2.5) " 1~= (1 - - ~ ) f(k) - l / k , (2.6)
x Co - g(k)),
where for ease of notation we set
I-> with (2.9)
~ = ~,(I - ~) -- ~-
Competitive
502
Capitalism
The quantity e --pt h(t) is the shadow price of capital and is related to per capita consumption by (2.10)
c = x7 + l/h.
To summarise, all quantities of interest are found by solving the differential system (2.5-6) for k and A and then obtaining x and c from (2.3) and 2.10), respectively. The equilibrium of the differential system (2.5-6) is displayed in Table 1 for 1 < 7 < = together with the limiting cases 7 + 00 andr= 1. The last row of Table I displays En, the imputed wage under the neoclassical regime, obtained from dividing the equilibrium consumption per capita (iYn) by the equilibrium labor participation ratio (IQ). Table I: Neoclassical Equilibrium
1 . . pll-4
2 0P
ov(l-4
4-c(l-a] r a4'-a) 0P
The limiting case 7 + 00 reproduces the more familiar neoclassical equilibrium. (See, for example, Cass (1965)). For ease of comparison with the results of Dirickx and Sertel (1979) we note the following relationships between the equilibrium quantities:
Competitive Capitalism
(2.11)
503
~=~__7£* -~n- ~ R'*
The dynamics of the neoclassical economy are illustrated in Figure 1, the phase plane of the differential system (2.5-6). We find that, provided
(2.12) ~+~< 1+~(1-1/"/), the equilibrium (kn, ~n) has the desired saddle point property. This property is desired because given k(0), the initial capital stock, there is a unique choice of ~(0), and therefore c(0), that sets the economy on the unique path nn' tending to the equilibrium (k n, Xn)" Furthermore nn', the stable arm of the saddle point is the only path along which the transversality condition (2.13)
lime -pt ~(t) k(t) = 0, t-* oo
a necessary condition for the maximisation of (2.1) subject to (2.2), is satisfied. We see from Figure 1 that along the dynamic path of the neoclassical economy the capital stock, labour participation and consumption all rise (decline) monotonically to their respective equilibrium values if the initial capital stock is less (greater) than its equilibrium value, whereas ~(t) moves in the opposite direction. k n \ \ ~.o
F,. ----~---
.... ..... ~
o
k{o}[........... ' .............. " " "~"~ n' ,~Col
" ~'
Figure 1: Dynamic path of the neoclassical economy
504
C o m p e t i t i v e Capitalism
The stability condition (2.12) shows the extent to which increasing returns to scale is possible in the neoclassical economy. In the limiting case 7 = 1, (2.12) reduces to (2.14)
c~ -I- fl <
i ,
so that only diminishing returns to scale are allowed in this extreme case. At the other extreme, as 7 -+ 0% (2.12) becomes (2.15)
~
+ fl < 1 + f l ,
the maximum extent of increasing returns to scale allowed in the neoclassical economy. The curve oe +fl = 1 +fl(1 -- 1/3') is displayed in Figure 2. ,a÷0
1+(3
....
I
>~"
Figure 2 The stability condition (2.12) was to some extent anticipated b y Dirickx and Sertel (1979) w h o point out that their static solutions are valid precisely under condition (2.12).
3. Competitive Capitalism We model competitive capitalism as a differential game 1 in which the workers each decide on their respective labor inputs The game we consider here is not a differential game in the strict sense of the term, since the wage w(t) is determined passively by the players via interaction with the auctioneer. We are closer to the simultaneous variational model discussed by Wan
(197z).
Competitive Capitalism
505
and capitalists take the investment decision, each group seeking to maximise a discounted utility integral over an infinite horizon. Our paradigm of competitive capitalism presupposes a Walrasian auctioneer who announces a time path for the wage, w(t). Given this w(t), each worker chooses x(t) so as to maximise his utility, thus coming up with a labor supply function at every point o f time. Simultaneously, given the same w(t), the capitalists determine the a m o u n t of labor to be employed and the investment pattern so as to maximise their utility, coming up with a labor d e m a n d function at every point of time. The auctioneer ends'Li~" setting w(t) so as to equate labor demand and supply at each point in time. Thus, given a wage time path w(t), a typical worker seeks x(t) SO a s t o
(3.1)
max f o e - P ' tl°g ( w x - - x ~) dt, x
where p I > 0 is the discount factor of a typical worker and we take workers' utility to be log (wx - x~ ). Now the typical worker under competitive capitalism, being subject to no differential equation constraint, maximises the integral in (3.1) by choosing c32)
x
which displays for us the labor supply function. The typical capitalist seeks labor input x, and his own consumption c so as to (3.3)
max f ~o e - p ~ t l o g c dt X,C
0
subject to the accumulation equation. (3.4)
1~= A -~ k ~ x ~ - wx -- c/A,
which shows o u t p u t per worker (A-ek~x ~) being divided between the wage p a y m e n t per worker (wx), a capitalist's consumption per worker (c/A) and investment (1~)2. The integrand in Note that the population in the competitive capitalist world is larger than that of the neoclassical world, n o w being A plus the n u m b e r of capitalists. It is perhaps a m o o t p o i n t as to w h e t h e r the population should remain A and we allow some of the workers to become capitalists. We have settled o n our formulation as we wished to see h o w the o u t p u t of afixed group of workers would be affected by differing managerial arrangements.
506
C o m p e t i t i v e Capitalism
[3.3) is the typical capitalist's utility, time discounted at the rate P2 > 0 . The necessar3" conditions for the maximisation of (3.3) subject to (3.4) yield ,3.5)
x = ( : ' - ' k~ ~ v ) l/('-t~)
and
(3.6)
Xc = A,
with the capitalist's shadow price on capital, e -pat termined by the differential equation (3.7)
;k(t),
being de-
J, = X (P2 - "-k-'ake-I xa)"
Given a wage time path w(t), the relations (8.5), (8.6) and the differential equations (8.4), (8.7) together with the transverality condition (3.8)
l i m e -Pat ~.(t)k(t) = 0 t---~~¢
enable the capitalist to determine k(t) and hence, via (3.5), his labor demand function x(t). The wage time path which equates labor supply and demand at every point in time will be such as to equate x in (3.2) to x in (3.5). Thus eliminating w between these latter two equations we find that along the labor market clearing time path, (3.9)
x = x n (k),
where x n (k) is defined by equation (2.3). Thus the functional relation between x and k is the same under competitive capitalism as in the neoclassical economy of Section 2. Of course, this does not mean that the time paths for x and k are the same under the two different economic systems; indeed, as we shall see shortly, they are different 3 . All that we have shown is that the neoclassical and competitive capitalist economies mix labor and capital in the same proportions, but the actual amounts that are mixed in the two economic systems can and do differ. The time paths differ not only for the reasons given later but also because the timc path for k under competitive capit',dism is infuenced by the unequal discount factors p 1 and P2 whereas in the neoclassical world there is just the common discount factor p. However as we see later the difference in time paths for k is not only due t ° P l ~ P2"
CompetitiveCapitalism
507
Substituting (3.5), (3.6) and (5.9)into (8.4) and (3.7) we find that the differential equations for k and ~ reduce to
(3.10)
k=
(1-,6)
f(k)-1/;k,
(3.11)
~. = ;k(p~ - g(k)),
where f(k)and g(k) are defined by equations (2.7) and (2.8). The differential equations (3.10-11) together with the transversality condition (8.8) completely determine the time paths for k and and thereby those of x, w and c. Comparing the differential system of competitive capitalism (8.10-11) with the corresponding differential system for the neoclassical economy (2.5-6), we see that the two systems differ in the differential equation for k and hence will generate different time paths starting from the same initial value of k. In making this and further comparisons we assume that the capitalists discount at the same rate as the social planner in Section 2. Thus, henceforth we put p 2 = p . The equilibrium (kc, ~c ) of the differential system (3.10-11) is given, in terms of the corresponding neoclassical equilibrium values, by (3.12)
k c=k n,
and
(.:.8.15)
X"c = (1 -P/",')
(i --~)
X-n > x- n
so that the equilibrium values of x and w are (3.14)
~c = ~ " ' % = ~ % < %
with ~- (capitalist's equilibrium consumption per worker = c/A) and c-w (a typical worker's equilibrium consumption bundle = Wc~c ) given by (8.15)
~-=(1-~)gn
n , gw=~n<~-
n.
We observe that, in equilibrium, capital stock per worker and labor input per worker are the same in the competitive capitalist economy as in the neoclassical economy, whereas the equilibrium wage is lower under competitive capitalism than the neoclassical imputed wage. In equilibrium, capitalist consumption per worker and the typical worker's consumption bundle in the competitive
508
Competitive Capitalism
capitalist economy are both less than equilibrium consumption per capita in the neoclassical economy. In fact, capitalists and workers share cn in the proportion (1-~) to ~. If we equate the discount factor of the capitalist to the rental on capital in the static framework of Dirickx and Sertel (1979), we find that the equilibrium values of k, x and w that we have obtained correspond precisely with those of their static version of competitive capitalism. It is a straightforward matter to construct the phase diagram for the differential system (3.10-11) and to show that the equilibrium (k,:, ~ ) has the desired saddle point property provided that condition f2.12), which guaranteed the saddle point property for the neoclassical system, is satisfied. Indeed the phase diagram will have the same general features as Figure 1. In Figure 3 we show just the stable arms of the saddle (cc') which constitute the dynamic path of competitive capitalism. We show on the:same diagram the dynamic path of the neoclassical economy, (nn'). It can be seen by comparing relative values of dk/d~ in each economy that the dynamic paths nn' and cc' never cross.
¢
¢0
N° !
\ Figure .3
X,
Competitive Capitalism
509
We see from Figure 3 that the qualitative features of the dynamic path are the same under competitive capitalism as under the neoclassical regime of optimal growth. Thus, given the same initial capital stock per worker ko, both economies have k declining (rising) to the equilibrium value k n if k o is greater (less) than k n . The competitive capitalist economy, however, will start and finish with a higher value for the shadow price of capital. In Section 5 we shall compare the rates at which each economy approaches equilibrium.
4. Cooperative Labor Management Following Dirickx and Sertel (1979) we assume that the workers have identical preferences (now including time preferences) and enter p r o d u c t i o n symmetrically. Our analysis also parallels the "nutshell" of Dirickx and Sertel (1979) in that we analyse the consequences when each worker assumes (self-justifyingly) that his fellows will contribute as m u c h work on the average as he himself does. The workers elect a council whose task is to determine the capital accumulation program which maximizes the utility of a typical worker, given the time path of x(t), the labor participation of a typical worker. The typical worker thus knows what share in output, i.e. income (or i m p u t e d wage) he can expect from a ~iven x(t), and will determine and then supply the x(t) which maximizes his own utility. Note that if we were to allow the council to determine n o t only the optimal capital accumulation program b u t also the optimal time path of labor participation, then we would be in the neoclassical world of Section 2. Thus, taking x(t) as given, the council seeks y(t), the share in o u t p u t going to a typical worker, so as to (4.1)
max f e- " t l o g ( y - x y 0
"~)dt,
the typical worker's utility integral, subject to the accumulation equation (4.2)
I~ = h -e k = x g -- y ,
showing o u t p u t per worker (A-~ k = x g) bein~ divided between o u t p u t share per worker (y) and investment (k). Here p > 0 is the discount factor c o m m o n to all workers and the utility of a
510
Competitive Capitalism
typical worker is log (y - x v ). Since in this model we do not allow workers to save out of their earnings all of y is consumed. The necessary conditions for maximization of (4.1) subject to (4.2) turn out to be (4.3)
y = xv + 1 / ~ ,
with e- ° t ~(t), the council's shadow price on the capital stock per worker satisfying the differential equation (4.4)
~ = ~ ( 0 - A - ~ otk~-I x~) •
Equation (4.3) together with the differential equations (4.2) and (4.4) and the transversality condition (4.5)
l i m e "-°t ~(t) k(t) = 0 ,
summarise the accumulation path chosen by the council in reaction to given x(t). The typical worker takes into account the council's optimal reaction as summarized by equations ( 4 . 2 - 5 ) when seeking x(t) so as to maximize his own utility integral 4 . Thus the typical worker seeks x(t) so as to (4.6)
max f e- m log~ d t , x 0
subject to the differential equation constraints (4.7)
k = A-~ k ~ x~ - x* - ~/',
and (4.8)
~
= 7"/(--p + A - e
k iV- 1 XO).
For convenience we have set
(4.9)
'q =
1IX.
Since the typical worker knows how the council will choose y he is able to express his utility integral as in (4.6) by use of (4.3). The differential equations ( 4 . 7 - 8 ) summarising the council's accumulation program for a given x time path become dyNote that we are assuming an overall cooperative solution of the council and all workers. Alternatively, we may assume that there is some enforcement mechanism in place so that individual workers do not deviate from the overall optimal plan.
Competitive Capitalism
511
namic constraints for the typical worker. Note that the initial values k(0) and ~/(0) of k and ~? are both taken as given. The necessary conditions for the maximisation of (4.6) subject to (4.7-8) (henceforth referred to as problem PW) turn out to be (4.10)
~=x
(k)(l+a~b) e ,
(4.11)
hi = Ul(P - A-e ak~-1 x0( 1 -- (1-~)¢))
where ~b= #2,2[(#1 k), and (4.12)
/h =Pu2 - l/n +u~ - u s
( - o + A-* ~ k ~ - l x ° ) •
Here #1 e-Ot and bt2 e-Pt are Lagrange multipliers, the former of which can be interpreted as the typical worker's shadow price on the capital stock k, while the latter can be interpreted as the typical worker's shadow price on ~/, the argument of his utility function. The dynamic path of the cooperative labor managed economy is given by the differential equations (4.7-8), (4.11-12) with x given by (4.10). We shall refer to this differential system as the differential system (M). The equilibrium of the differential system (M) is given by (4.13)
L =k-n'
(4.14)
~'e
~'n'
ul
and (4.16)
/~'-'2= 0.
Furthermore this equilibrium is unique. The corresponding equilibrium values of x and y are (4.17)
xe = xn ,
(4.18)
Fo =
•
Finally the equilibrium imputed wage We ( = Ye / Xe ) is (4.19)
we = w
.
Thus equilibrium levels of capital stock (k-e ), labor participation (x e ), consumption per worker (~e) and inputed wage (We )
512
Competitive Capitalism
are the same in the labor managed economy as they are in the neoclassical economy. However the dynamic paths leading to the equilibrium will be quite different. The stability analysis of the equilibrium (4.13-16) is more difficult than for the equilibria of the models of the neoclassical and competitive capitalist economies because the larger number of differential equations rules out the phase plane approach. We shall henceforth refer to the equilibrium (4.13-16) as (E), and to the set of equilibrium values (ke, ~'e) as (E'). The task of showing that the solution to the problem PW is a unique path in the (k, ~.) plane tending to the equilibrium (E') is, technically, a complex one. Here we shall indicate the broad outlines of what is involved. First we linearise the differential system (M) about the equilibrium (E) to obtain the linear differential system (4.20)
~ =BEG,
where the column vector ~ has elements ( k - k e , r/-~e , # x - ~ e , # 2 - ~ 2 ) and the matrix B is the Jacobian matrix of the differential system (M) evaluated at the equilibrium (E). The matrix Be is displayed in the Appendix as is the calculation of its eigenvalues. As a result of these calculations we find that B has two positive and two negative eigenvalues provided that the condition (2.12) is satisfied. Hence, since two initial values k(0), r/(0) are given and the other two #I (0),/a 2 (0) are to be determined, the equilibrium (E) has the local saddle point property. Thus in a neighbourhood of the equilibrium (E) there is a unique path (L) tending to (E). That this path is indeed the solution to problem PW can be shown by invoking Theorem 2 of Rockafellar (1976). It is also possible to show that the aforementioned unique path exists globally. (These points are elaborated upon in an appendix in Chiarella and Sertel (1980), where the model here is analysed under the constant returns to scale assumption.) We also show in the Appendix that the eigenvalues of Be are never complex under the assumptions of our model and so there is no oscillatory motion in a neighbourhood of the equilibrium (E). Hence, projecting the solution of the differential system (M) onto the (k,~) plane, we see locally a picture similar to Figure 1, which is characterised by a smooth approach to the equilibrium (E'). However, it has not been possible to make a definite assertion about the relative slopes of nn' and ee' (the projection of L onto (k,~) plane) and possible figurations are displayed in Figure 4.
Competitive Capitalism
513
e
i'1
ke:k:r
e'
n'
Jm
Ae=~n
ke:~ e
,
em
Xe
Figure 4
--'~n
514
Competitive Capitalism
5. Comparison of Growth Rates The differential systems (2.5-6) for the neoclassical economy and (3.10-11) for competitive capitalism may be linearised around their respective equilibria, as was done in the last section, to obtain a second-order linear system
(5.1)
~ = B i t s,
where ~i is the column vector (k-k- i, ?~-X-i) for i = n, c. The matrix Bi is theJacobian matrix of the respective differential system evaluated at its equilibrium. For the differential system (2.5-6) of the neoclassical economy we find that aT/i whilst for the differential system (3.10-11) for the competitive capitalist economy we find that (5.3)
[(1--fl)^leP
Be =
oriel (1-~)k-c~
1/~" ] 0n
The eigenvalues of Bn are (5.4)
b~n) = P-[1 -T-.~/l+--'g'0"]
b~n) '
2
where (5.~)
o = 4~/c~/.
Under the assumption that (2.12) is satisfied we have (5.6)
b~n) < 0 < b~n) ,
which shows from an algebraic point of view the saddle point property of the equilibrium (kn, Xn). Thus in a neighbourhood of the equilibrium (kn, Xn ) the path nn' is asymptotic to the path (5.7)
~i(t) =Vl exp (b] n) t ) ,
where v 1 is a constant vector depending on the initial conditions. Similarly the eigenvalues of B c are
(5.8)
b? :zez[l
l,
CompetitiveCapitalism
515
where (5.9)
z = (1 --0) I (1 -- OI'Y) < 1.
Here also (5.1o)
b?l < 0 < b Cl ,
provided that (2._12) is satisfied. So in a neighbourhood of the equilibrium (k-c, ~c ) the path co' is asymptotic to the path (5.11)
~ ( t ) = v2 exp (b~c) t ) ,
where v2 is a constant vector depending on the initial conditions. We can see from (5.7) and (5.11) that Ibtn) I and [b~C)I measure the rate at which each of the economic systems approaches its equilibrium. Using the fact that 0 < z < 1 it is not difficult to show that (5.12)
Ib[C)l < Ib]n) l ,
which indicates that, asymptotically at least, the neoclassical economy approaches its equilibrium more rapidly than does the competitive capitalist economy. An examination of the eigenvalues of the matrix Be given in the Appendix indicates that the path ee' is asymptotic to (5.1s)
= v 3 exp
t)
where b t z) = b~n), which indicates that, asymptotically_, the labor managed economy approaches the equilibrium (kn, Xn) at the same rate as does the neoclassical economy.
6. Concluding Remarks We have modelled competititve capitalism and cooperative labor management as differential games and drawn comparisons with, the corresponding neoclassical (socially optimal) solution. The dynamic paths of both systems exhibit the characteristics of the dynamic path of the neoclassical system, namely the monotonic approach to equilibrium (at least asymptotically in the case of cooperative labor management). The equilibrium of cooperative labor management corresponds to that of the neoclassical system, also both systems (asymptotically) approach equilibrium at the same rate. Competetive capitalism yields the same
516
Competitive Capitalism
equilibrium values of k and x as in the neoclassical system, but the typical worker receives, in equilibrium only a fraction/~ of the wage he would receive in the neoclassical system. Also the equilibrium consumption per capita of the neoclassical system is distributed between workers and capitalists in the proportions /3 to (1-13). Futhermore the competitive capitalistic system (asymptotically) approaches equilibrium at a slower rate than does the neoclassical system. In making the above comparisons we have assumed the same discount factor to prevail in all systems. Increasing returns to scale are permitted in all the models discussed here up to the point allowed by the stability condition (2.12). Beyond this the equilibria are unstable. Equating the discount factor of our models to the rental on capital in the static framework of Dirickx and Sertel ( 1 9 7 9 ) w e find that our equilibrium outcome of competitive capitalism agrees with theirs. However our equilibrium outcome of cooperative labor management differs from theirs in the wage, their equilibrium wage being a factor ( l - a ) times what we have obtained. Our models of competitive capitalism and cooperative labor management contain only one market, namely the labor market. Another market which must be considered is the capital (or savings) market. In further developments of our model of competitive capitalism one could allow individual workers a savings-consumption decision, with unconsumed output offered to capitalists on a capital market where some rate of interest, r(t), prevails. The market clearing r(t) could be determined in a free market solution or we could allow e.g., a capitalists' union to be price dictator in the capital market. Simi!arly, we could introduce a capital market into our model of cooperative labor management, the council acting as price setter in the capital market to choose an interest rate that will induce the workers into a savings pattern that is optimal overall.
Acknowledgements: An earlier version of this paper was presented at the Eight Bosphorus Workshop on Industrial Democracy,at Bogazici University Istanbul, July 1984. It benefited from diskussion there with Ahmet Alkan, David Cass. Martine Quinzii and Jaques Thisse. Editorial comments of Manfred Holler have led to improvements. Whatever may be wrong with the paper is despite all wise council we have received.
Competitive Capitalism
517
Appendix We find after some lengthy calculations that p -
(A1)
-Bq
--1
0
0
0
0 -b~--- e
=
0 e
0
q
2
~,
1
p
¢
-~
1
where we have put
(A2)
q=aS(P)2/'7.
The characteristic polynomial for Bq turns out to be b 4 _ 2 p b 3 +(p2_o)h 2 + a p b + r = 0 ,
(A3) where (A4)
a = q(2 + a N a ) ,
and (A5)
r = q2 ( 1 + o4~/~).
If (2.12) is satisfied we can assert that (A6)
8>0,
r>0.
The quartic (A3) has roots (A7)
r'
b m
½[p+-~/P212+e+A+~/p2/2+a-Al,
where
(A8)
A2 = p 4 / 4 + p 2 a + 4 r > p4/4 "
It is a simple matter, using the inequality in (A8), to ascertain that (A7) contains two positive and two negative roots. Thus the
518
Competitive Capitalism
equilibrium E is a local saddle point. The solution to (4.21) may be expressed through
(A9)
}(t) = ~ Viexp(bl')t), i=I
where (A10) b~1)
-3
'
are eigenvalues of B 1 as given by (A7), V i are the corresponding eigenvectors and A i are constants. Since the initial conditions gl (0),/~2 (0) are so far unspecified, we may choose them to set As, A 4 equal to zero and then use k(0), X(0) to calculate A1, A s . Thus the unique path tending to the equilibrium E is asymptotic to the path (All)
~(t)=
2
Z A i V i e x p ( g i 1)t), i=1
which for large t is dominated by
(A12) ~(t)= A 1 V 1 exp (b(lI) t). The dominant eigcnvalue-I h (I) can be expressed in the form (A13)
b~~I =
x -~/1
+ flr(fl/,(1 + fl)),
where (A14)
1
1
+z +
(½ + z -
'
Now some algebraic manipulations reveal that (A15) F ( z ) = l f o r a l l z ~ 0 . (To see this, set u = 1 + 4z, so that for u > 1 we have 1
1
2 r ( z ) = (1 + 2x/-~ + u ) ~ - - (1 - 2x/~ + u)~-= 2.)
Thus, wc have shown that b~ 1 ) = b~n ). Finally, we see that the condition for (A7) to yield complex eigenvalues (and that the approach to equilibrium be hence oscillatory) is
Competitive Capitalism (A16)
519
o2<4r,
w h i c h reduces t o (A17)
(~/]~)2 < 0
which, clearly, never obtains. H e n c e n o oscillatory m o t i o n is possible u n d e r the a s s u m p t i o n s o f o u r m o d e l .
References Atkinson, A.B. (1973). "Worker Management and the Modern Industrial Enterprise", Quarterly Journal of Economics, 87,375-392. Cass, D. (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation", Review of Economic Studies, 32,233--241. Chase, E.S. (1967). 'Leisure and Consumption', appears in Essays on the Theory of Optimal Economic Growth, ed. by K. Shell, M.I.T. Press. Chiarella, C., and M.R. Sertel (1982). "Capitalism, Communism and Labor Management in a Dynamic Nutshell Under Constant Return to Scale", Ch. 10 of Sertel (1982). Dirickx, Y.M.I., and M.R. Sertel, (1979). "The Comparitive Political Economy of Workers' Management and Capitalism in a Nutshell", European Institute for Advanced Studies in Management, Working Paper 79--2, February 1979. Revised version, Ch. 3 of Sertel (1982). Furubotn, E.G. (1971). "Toward a Dynamic Model of the Yugoslav Firm', Canadian Journal of Economics, 1971, 4, 182--197. Rockafellar, R.T. (1976). "Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Nonzero Discount Rate", appears in The Hamiltonian Approach to Dynamic Economics, ed. by D. Cass and K. Shell, Academic Press. Sertel, M.R. (1982). Workers and Incentives, Contributions to Economic Analysis, No. 140, Amsterdam: North-Holland. Wan, H.Y., Jr. (1971). "A Simultaneous Variational Model for International Capital Movement", in Trade, Balance of Payments and Growth, ed. by J. Bhagwati et al., Amsterdam et al.: North-HoUand.