A Special Labor-Managed Oligopoly
Ferenc Szidarovszky∗
University of Arizona

Akio Matsumoto†Chuo UniversityAbstract
A special labor-managed oligopoly without product diﬀerentiation isconsidered. The existence of the equilibrium is first proved and a simple
example is presented to show the possibility of multiple equilibrium. The
local asymptotic stability of the equilibria is next examined, the stability
conditions are derived in both discrete and continuous time scales.

1Introduction

The theory of oligopoly is one of the most frequently discussed subjects ofmathematical economics. Since the pioneering work of Cournot (1838), many
researchers examined the classical Cournot model and its extensions including models without and with product diﬀerentiation, multi-product oligopolies,
rent-seeking games, labor-managed models, oligopolies with intertemporal demand interaction and models with product adjustment costs. A comprehensive
summary of the most significant works up to the mid 70s is given in Okuguchi(1976). Multiproduct models with futher extensions and applications are disccussed in Okuguchi and Szidarovszky (1999), and nonlinear dynamic oligopolies
are examined in detail in Bischi et al. (2009).
In this paper a special labor-managed model is examined. The seminal
work of Ward (1958) is considered to be the first to introduce labor-managed
oligopolies. Hill and Waterson (1983) have formulated profit-maximizing andlabor-managed models without product diﬀerentiation and with symmetric firms
and compared the long-term behavior of these models. Neary (1984) generalized this work to the nonsymmetric case. The existence and uniqueness of theequilibrium was proved by Okuguchi (1991) under rather restrictive conditions,furthermore in Okuguchi (1993) comparative statics are presented for profitmaximizing and labor-managed oligopolies. The existence of equilibrium wasalso shown in Okuguchi and Szidarovszky (1999) under slightly more generalconditions, and the asymptotical properties of the equilibrium were also investigated with discrete and continuous time scales. These results were further∗ Department of Systems and Industrial Engineering, Tucson, Arizona, 85721-0020, USA.
szidar@sie.arizona.edu† Department of Economics, 742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393, JPN.
akiom@tamacc.chuo-u.ac.jp

1generalized in Li and Szidarovszky (1999). Okuguchi and Szidarovszky (2008)
have presented a general existence and uniqueness theorem for the equilibriumof labor-managed oligopolies.
In contrary to earlier works, the payoﬀ functions are not concave and thestrategy sets are not compact in the model to be presented in this paper, so
general existence theorems cannot be used. This paper is organized as follows.
In Section 2 the mathematical model will be presented and the existences ofthe equilibrium will be proved. The uniqueness of the equilibrium cannot be
guaranteed in general as it will be demonstrated by a simple example. The
local asymptotical stability of the equilibrium will be examined with discreteand continuous time scales in Section 3. Section 4 concludes the paper.

2

Mathematical Model and Existence Theorem

Consider a single product oligopoly without product diﬀerentiation. Let N
denote the number of firms, and let LkP
be the capacity limit of firm k. If xkNdenotes the output of firm k and Q = k=1 xk is the output of the industry,then the price function is assumed to be f (Q) = A − BQ with A > 0 and B > 0.
PIn order to guarantee nonnegative price, we assume that A/B ≥ N
k=1 Lk . The
amount of labor necessary for producing output xk by firm k is denoted byhk (xk ) = pk x2k . Let dk > 0 be the fixed cost of firm k, and w the competitive
wage rate. The surplus of firm k per unit labor is given asϕk (x1 , ..., xN ) =xk (A − Bxk − BQk ) − wpk x2k − dk,pk x2k(1)

P
where Qk =`6=k x` is the output of the rest of the industry. With thesenotations, an N -person game can be defined in which the set of strategies ofplayer k is the closed interval [0, Lk ] and its payoﬀ function is ϕk . Notice thatϕk → −∞ as xk → 0, so ϕk is not defined for xk = 0.
The best response of firm k can be obtained by maximizing ϕk with fixedvalue of Qk . Sinceµ¶
A − BQkdk
B−−+w ,ϕk (x1 , ..., xN ) =pk xk
pk x2kpk
we have∂ϕk∂xk=−A − BQk2dk+
pk x2kpk x3k(2)1=(2dk − (A − BQk )xk ) .
pk x3kTherefore ϕk increases as xk < 2dk /(A−BQk ), decreases if xk > 2dk /(A−BQk ),so the best response of firm k is the following:⎧ ∗⎨ xk if x∗k ≤ Lk
Rk (Qk ) =(3)⎩Lk otherwise,2

wherex∗k =2dk.
A − BQkQk ≤ALk − 2dk.
BLk

The first case occurs when

The graph of Rk is shown in Figure 1.

Figure 1. Piecewise continuous best response curve
³ ³P
´³P
´´
Notice that the best response mapping R1`6=1 x` , ..., RN`6=N x`QN
maps the compact, convex set k=1 [0, Lk ] into itself, and since it is continuous,
the Brouwer fixed point theorem guarantees the existence of at least one fixedpoint.
Similarly to classical Cournot models, we can present a constructive existence
proof which can be also used to compute the equilibria.
If Q denotes the total output of the industry, then from (3), we havex∗k =2dk
A − BQ + Bx∗kin the first case, so x∗k solves the quadratic equation∗Bx∗2k + (A − BQ)xk − 2dk = 0

and therefore−(A − BQ) +x∗k =p(A − BQ)2 + 8Bdk.
2B3(4)

Hence the best response of firm k can be rewritten as⎧ ∗⎨ xk if x∗k ≤ Lk ,R̄k (Q) =⎩Lk otherwise.

(5)

From (4) we conclude that R̄k0 (Q) = 0 or!
Ã12(A − BQ)B0R̄k (Q) =B− p
2B
2 (A − BQ)2 + 8Bdk
1=2ÃA − BQ
1− p(A − BQ)2 + 8Bdk

implying that in this case

0 < R̄k0 (Q) ≤!

1.
200

00

In the second case of (5), R̄k (Q) = 0 and in the first case the sign of R̄k (Q)is the same as the sign of the following expression:()p2(A−BQ)(−B)− (−B) (A − BQ)2 + 8Bdk − (A − BQ) p,
2 (A − BQ)2 + 8Bdk
which is the same as the sign of(A − BQ)2 + 8Bdk − (A − BQ)2 = 8Bdk > 0.
Therefore in the first case of (5), R̄k (Q) strictly increases and is strictly convex,
and its graph has a similar shape as Rk (Qk ) shown in Figure 1. The equilibrium
industry output is the solution of equation
NP

k=1R̄k (Q) − Q = 0.

(6)

PThe left hand side is continuous, at Q = 0 it is nonnegative and at Q = N
k=1 Lkit is nonpositive implying the existence of at least one equilibrium.
The uniqueness of the equilibium cannot be guaranteed as it is shown in thefollowing example.
Example 1. Let N ≥ 3 be arbitrary, Lk = 4, dk = 8.5 for all k, A = 12 andB = N 2−1 . Clearly all conditions are satisfied: A > 0, B > 0 andNPk=1

Lk = 4N ≤A= 6(N − 1).
B

We can show that bothx̄Sk = 3 −√√0.5 and x̄Mk = 3 + 0.5
4are symmetric interior equilibria. Notice first that for all k,0 < x̄Sk < Lk and 0 < x̄Mk < Lk ,and they satisfy the best response equations xk = Rk (Qk ), since√√
Q̄Sk = (N − 1)(3 − 0.5) and Q̄M
k = (N − 1)(3 + 0.5),
furthermore in these cases
Rk (Q̄k ) =√2dk17√== 3 ∓ 0.5 = x̄k .
A − B Q̄k
12 − 2(3 ∓ 0.5)

It can be also shown that x̄Lk = Lk is a symmetric boundary equilibrium,
since in this case the second case of (3) occurs:
x∗k =

317
17=> 4 = Lk .
24
12 −4(N − 1)
N −1

Local Stability Analysis

From equation (3) we see that Rk0 (Qk ) = 0 or
Rk0 (Qk ) =2dk B.
(A − BQk )2(7)Consider the second case. At the best response
Rk0 (Qk ) =x∗22dk − (A − BQ)x∗kk B=,2dk2dk

where we used equation (4). This expression implies that0 < Rk0 (Qk ) ≤ 1.
Consider first continuous time scales and assume that the firms use adaptive
adjustment toward best responses. Then their outputs satisfy the diﬀerential
equation systemÃ Ã!
!
NPẋk = Kk Rkx` − xk for k = 1, 2, ..., N,(8)`6=kwhere Kk > 0 is the speed of adjustment of firm k. The equilibria of the labormanaged oligopoly game coincide with the steady states of this system. The
local asymptotic stability of this system can be examined by linearization. TheJacobian has the special form⎞
⎛K1 r̄1 · K1 r̄1−K1⎜ K2 r̄2−K2 · K2 r̄2 ⎟⎟ = D + a1T ,J=⎜⎠⎝····KN r̄N KN r̄N · −KN
5where r̄k is the value of Rk0 at the equilibrium,
D = diag (−K1 (1 + r̄1 ), ..., −KN (1 + r̄N )) ,Ta = (K1 r̄1 , ..., KN r̄N )and1T = (1, ..., 1).

The characteristic polynomial of this matrix can be written in the following
form:
ϕ(λ) = det(D + a1T − λI)= det(D − λI) det(I + (D − λI)−1 a1T )=NQk=1µNP(−Kk (1 + r̄k ) − λ) 1 −Kk r̄kK(1+ r̄k ) + λkk=1(9)¶.

For the sake of simplicity, we assume that the Kk (1 + r̄k ) values are diﬀerent,
the other case can be discussed in the same way. The eigenvalues are λ =−Kk (1 + r̄k ) < 0 and the roots of equationg(λ) =NP

Kk r̄kk=1 Kk (1 + r̄k ) + λIt is easy to see thatlim g(λ) = −1,λ→±∞andg 0 (λ) =lim

− 1 = 0.

(10)g(λ) = ±∞λ→−Kk (1+r̄k )±0NP−Kk r̄kk=1 (Kk (1 + r̄k ) + λ)

2 < 0,so g strictly decreases locally. All poles are negative, and since equation (10) isequivalent to a polynomial equation of degree N, there are N real or complex
roots. There is one root between each adjacent pair of poles and one additionalroot after the largest pole. So we found all roots, they are real, and they are
negative if g(0) < 0 orNPr̄k< 1.
(11)
1+r̄k
k=1
Consider next discrete time scales. Then system (9) is replaced by the following system of diﬀerence equations:!
!
Ã ÃNPxk (t + 1) = xk (t) + Kk Rkx` (t) − xk (t) for k = 1, 2, ..., N, (12)`6=kwhere 0 < Kk ≤ 1. The Jacobian of this system is I+J, the eigenvalues of which
are inside the unit circle if the eigenvalues of J are between −2 and 0 which is6

the case if all poles are larger than −2 and g(0) < 0. These conditions can berewritten as
Kk (1 + r̄k ) < 2(13)andNPr̄k< 1.
1+r̄kk=1(14)Notice first that the stability condition (11) (or (14)) does not depend on thespeeds of the adjustments of the firms. They are satisfied if the r̄k derivativevalues are suﬃciently small. Since 0 < r̄k ≤ 1 and by resonable assumption
0 < Kk ≤ 1, condition (13) is almost always satisfied. The only exception is thecase of best response dynamics with Q = A/B where all firms receive zero price.
This is a very unrealistic equilibrium. Hence the stability conditions for discrete
time and continuous time systems are identical. In the case of duopolies andQ 6= A/B, condition (11) (or (14)) holds, since r̄k /(1 + r̄k ) < 1/2, showing thelocal asymptotic stability of the equilibrium with both constinuous and discrete
time scales.
If N ≥ 3, the model may have multiple equilibria and the local stability ofthese equilibria may not be guaranteed as shown in the following example.
Example 2. Take N = 3. The model specified in Example 1 has three equilibria:√√
Lx̄Sk = 3 − 0.5, x̄M
k = 3 + 0.5 and x̄k = Lk
which we call the smallest, middle and largest equilibria. It is apparent
that the largest equilibrium is locally stable as Rk0 (Qk ) = 0. To examine
the stability of the other two equilibria, we calculate the derivative values
evaluated at the equilibria,r̄kS =1717√√' 0.31 and r̄kM =' 0.81.
2(6 + 2)(−6 + 2)2

According to (11) (or (14)), the smallest equilibrium is locally stable
whereas the middle equilibrium is locally unstable.
Example 2 confirms the local stability of the smallest and the largest equilibrium but does not say anything about the global behavior of the trajectories.
The basin of attraction for the discrete dynamic system (12) with N = 3,
Kk = 0.8 for all k and x3 (0) = 5.1 is illustrated in Figure 2. The shape of thebasin depends on the value of x3 (0). It is the set of points in the output space(x1 , x2 ) such that initial points chosen in the red (darker) region converges tothe smallest equilibrium and those in the blue (lighter) region evolve to thelargest equilibrium. We choose two points, denoted as a in the red region and b
in the blue region in Figure 2A, and perform numerical simulations to confirm
the convergence, which are depicted in Figure 2B. In particular, although theinitial point a is close to b, the trajectory starting at point a eventually converges to the smallest equilibrium and so does the trajectory starting at point7

b to the largest equiliubrium.

(A) Basin of Attraction(B) Globally stable time trajectories

Figure 2. Global stability

4Conclusions

The existence of a special labor-managed oligopoly is proved by using fixed
point theorems and also by introducing a solution algorithm. The uniquenessof the equilibrium is not guaranteed in general. Conditions are derived for thelocal asymptotical stability of the equilibrium, and a simple numerical example
illustrates the theoretical results.

8References
Bischi, G-I, C. Chiarella, M. Kopel and F. Szidarovszky (2009): Nonlinear
Oligopolies: Stability and Bifurcation. Springer-Verlag, Berlin/New York(in press).
Cournot, A. (1838): Recherches sur les Principes Mathématiques de la Théorie
des Richesses. Hachette, Paris (English translation (1960): Researches
into the Mathematical Principles of the Theory of Wealth. Kelly, NewYork).
Hill, M. and M. Waterson (1983): Labor-Managed Cournot Oligopoly andIndustry Output. Journal of Comparative Economics, vol. 7, 43-51.
Li, W. and F. Szidarovszky (1999): The Stability of Nash-Cournot Equilibria
in Labor Managed Oligopolies. Southwest Journal of Pure and Applied
Mathematics. vol. 1999, no. 1, 1-12.
Neary, H. M. (1984): Existence of an Equilibrium for Labor-Managed Cournot
Oligopoly. Journal of Comparative Economics, vol. 8, 322-327.
Okuguchi, K. (1976): Expectations and Stability in Oligopoly Models. SpringerVerlag, Berlin/Heidelberg/New York.
Okuguchi, K. (1991): Existence of an Equilibrium for Labor-Managed Cournot
Oligopoly. Mimeo, Department of Economics, Tokyo Metropolitan University, Tokyo, Japan.
Okuguchi, K. (1993): Cournot Oligopoly with Product-Maximizing and LaborManaged Firms. Keio Economic Studies, vol. 30, 27-38.
Okuguchi, K. and F. Szidarovszky (1999): The Theory of Oligopoly with MultiProduct Firms. Springer-Verlag, Berlin/Heidelberg/New York.
Okuguchi, K. and F. Szidarovszky (2008): Existence and Uniqueness of Equilibrium in Labor-Managed Cournot Oligopoly. Rivista di Politica Economica, vol. XCVII, Series 3, no. V-VI, 9-16.
Ward, B. (1958): The Firm in Illyria: Market Syndicalism. American Economic Review, vol. 48, 566-589.

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