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Live and let live type behavior in a multi-market setting with demand fluctuations
Journal of Economic Behavior and Organization 10 (1988) 235444. North-Holland
WE A
ET
D. KANTARELIS and E.C.H. VEENDORP Assumption College, Worcester, MA 01609-1296, USA and Clark University Received July 1986, final version received July 1987 This paper formalizes a cooperative mode of behavior which captures essential elements of the live and let live philosophy, and shows that adoption of such a policy tends to benefit firms in oligopolistic markets. It is indicated that multiplicity of contacts and exposure to demand guctuations may simplify coordination and enhance the potential and profitability of live and let live type behavior.
Attempts to explain interdependence in oligopolistic markets have been made by many economists, most influential among whom has undoubtedly been E.H. Chamberlin (1946). He argues that mutual recognition of interdependence will drive firms to a joint monopoly profit level without formal collusion but instead by reaching some kind of ‘virtual contract’ which implies that there is no specific agreement, but that the parties act as though there were one. How effective this ‘virtual contract’ is depends crucially on such factors as the QZGXYof sellers, cost conditions, and the stability of demand. More recently ecoilumists have started to analyze interdependence of conglomerate firms, pointicg out that with symmetry of conglomerate power incentives for rivalry tend to disappear. C. Edwards (1955) has hypothesized that when such firms operate in a multi-market environment, ‘the multiplicity of their contacts may blunt the edge of their competition.. . Each conglomerate competitor may adopt a live and let live policy designed to stabilize the whole structure of the competitive relationship’. This paper formalizes a policy which reflects the spirit of the live and let philosophy, and shows that the adoption of such a opohstic firms is facilitated by a large number of contact points and fluctuating demands. *Thanks are due to Dan Levitt, Martin Shubik, Richard H. Day, and a referee for helpful comments on previous drafts of this paper. he usualdisclaimers apply.
D.Kantarelis andE.C.H.
236
Veendorp, Live and iet live type behavior
The ‘live and let live’ philosophy is clearly in the spirit of the Chamberlin model. It would seem, however, that the muitipiicity of contacts between these conglomerate firms would generate difficulties for Chamberiin’s ‘virtual contract’. The multiplicity of contacts would typically imply more exposure to demand changes and a more complex technology. A more complex technology is less likely to preserve symmetry among firms, whereas the more frequent exposure to demand changes would enhance the relevance of the complications created by such asymmetries. Jointly they would make the Chamberlin type conditions favoring coordination less likely to apply. This paper indicates that multiplicity of contacts and exposure to demand fluctuations may actually simplify coordination, and enhance the profitability of live and let live type behavior. 2.
he
Y
To our knowledge no explicit formulation of the live and let live hypothesis has been given. But the phrases that seem to be used most frequently to capture the spirit of such behavior, are suggestions ‘to preserve the status quo’, ‘to play it safe’ and ‘not to rock the boat’. Such prescriptions suggest that firms should refrain from taking actions that would significantly alter their positions, described in terms of profits, market shares or some other variables, relative to those of their competitors. Using profits to measure relative positions, adherence to the live and let live philosophy would then imply that firm A considers only those actions that would not increase the profits of its competitor(s) (B) to a much larger or smaller extent than its own, or
In this paper we choose, somewhat arbitrarily, O,=Oz and refer to that part of firm A’s concern with relative positions
as the let live (LL) constraint. To the other half Ax, 2 aAn*,
a=
1/01>O,
we shall refer as the live (L) constraint. aximixing profits subject to these constraints implies that a fi is willing to (tern orarily) forego some profits if a particular move wou either hurt, or u uly benefit the opponent this
tion of such a live and
D. Kantarefis and E.C.H. Veendorp, Live and let live type behavm
237
let live policy by a firm will affect its profits over time when competing with firms that are similarly inclined (though the o[values they use may differ), not how firms might induce their competitors to adopt such a cooperative policy or how they might respond if other firms defect. We explore, in other words, how some kind of virtual contract can be preserved in spite of asymmetries and demand fluctuations, not how it could be established or broken. ive
a&et
The potential for live and let’live behavior by sellers in a traditional Cournot quantity variation model is clearly limited. Increases in output by firm A in response to a demand increase will affect profits of the other firm (B) negatively, while decreases in output in response to a demand decrease will typically increase the profits by firm B by more than those of firm A. We focus therefore on a two market generalization of the Cournot duopoly model, where firms A and B produce the same two undifferentiated products (1 and 2) in amounts Qij(i= 1,2; j= A, B) at zero cost. They face linear demand curves Pi = Qi- QiA-Qie) and are exposed to demand changes that lead to parallel shifts in these demand curves. The constrained maximization problem introduced in section 2 can best be visualized as being based on three components. The first one is the unconstrained profit maximization by each firm on the assumption that the other firm will not change its output levels. Maximizing A’s profits X.A=
C
i=1.2
(ai-QiA-QiB.r-I)QiAr
with respect to QiAyields the Cournot type solution
Qh=(ai-Qi~.t-~)/Zi= 1,2, where QiB,t _ , are the output levels oi firm B in the preceding period, and u1 are the demand intercepts in the present period, which may or may not differ from the preceding ones. Similar expressions can be written down for firm B, with identical time subscriptions in case of simultaneous adjustments, or different ones in case of alternating adjustments. The second component is the LL constraint which requires that the profits of firm B should not be affected negatively by A’s actions. If in response !o a demand change or in response to adjustments by the other firm, firm A revious output levels ( iA,,- J to new ones ( considers changi then the condition that Ax BhO, or
238
D. Rantarelisand E.C.H. Veendorp, Live aid let live type behavior
will upon substitution yield the inequality
The equality part of this constraint is a straight line in going through the p slope -QzE.~-I/QIB.~-I ~VQLW-I~QZA,~-J The third component is the requirement that the increase in
adjusting firm is at least as large as a certain fraction of the increase in profits of the opponent (the L constrzint). This condition, dlr,~ad~~, can be simplified to
SCQ i~.r-i-(a~-(l-9C!Q;~,r-~)/232
The equality part of this condition represents a circle in QIA,QZAspace with }/2, {a2 -( 1-a)Qllr, L_ ,}/2) going through the center M(h-(l-a)Q~~,t-l initial point P(Q 1A,L_ I, QZAer_ I). Figs, 1 and 2 illustrate the role these three components play in the adjustment mechanism for a= 1, and for a demand change in the first market. If firms A and were initially producing the same output bundle, and A’s initial position is given by point P, then t after the demand change will be point Q (with constraints are represented by the str aximization of profits subject to these firm A move to or towards point p, which will typically center of the circle constraint w
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
239
Fig. 1. Firm A’s response to a demand decreme according to 2 live and let live policy in a two dimensional quantity variation model (P = initial; Q=Cournot; d = optimal solution; A4=center of circle constraint).
FIN. 2. Firm A’s response to a demand increase according to a live and let live policy in a two dimensional quantity variation model (P =Einitial; & =Cournot; B= optimal solution; M = center of circle constraint).
240
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
diagram). The suggestion is supported, however, by the results of computer simulations. 4. Results of computer simu
Finding the solution of the inequality constrained maximization problem for arbitrary initial output levels (Qij,1_1) and arbitrary values of the demand intercepts (Qi) is conceptually easy but tedious since one has to verify which constraints, if any, will be binding (details are available upon request from the authors). Given this target solution (Q&,1), firm A chooses new output levels adaptively O<;zSl, Q 1ll.t =QiA.r-l+IZ(Q~,t-QiA.r-l), while firm B responds in a similar fashion, using coefficients /? and p instead of 01and 1. The responses of firms A and B, following a live and let live policy while adjusting to continuous demand fluctuations, were simulated for a large number of different values of the coefficients involved. The results reported in this section refer to the profit levels of these firms while reacting to each other and to parallel demand shifts in both markets. New demand intercepts (ai) were chosen randomly and independently after every four periods from a uniform discrete distribution with range 15 to 20. Firms respond to these changes with different values for the a and #I coefficients, and with different vaiues for the adjustment speeds (A and cc), starting at the Cournot equilibrium output levels for the midpoint of the above range (Ui= 17.5). The simuiations were done for 800 successive demand changes or 3200 periods, and table 1 reports the accumulated (undiscounted) profit levels over the first 8a and last 80 periods. After 800 changes, these accumulated profits became insensitive to continued demand fluctuations. Table 1 also lists the Cournot and joint monopoly profits for the same adjustment speeds. It appears that firms following a live and let live policy fare rather.well in terms of profit compared to Cournot, and that the difference is more pronounced in the last 80 periods than in the first 80 periods. Comparing the different live and let live policies, it appears that the lower levels of the or(p) coefficients lead to higher profit levels, but that these differences are relatively small and getting smaller with re ated fluctuations. The rationale for z;ing higher values of 4,s) is that it offers protection against asymmetry. exceeds b, then firm A has a consistent advantage over its oppon conversely. Table 2 contains similar results for firms with positive costs of The margina ant, but do differ among firms and/or
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
241
Table 1 Profit levels over first 80 (IQ and last 80 (Q) periods for duopolists following a live and let live policy with zero cost of production in response to random demand fluctuations. 01 A B P xr.4 % GA R.B 1 1 1
1 0.5 0 0.5 :
0. :: Cournob Joint max 1
;5
1 A.5 0.5 0 Coumot Joint max
0.1 ::
0.1 0.1 0.1
o:l ::
o:l ::
::
0.1 0.1 E
d0.5
0.4 0.4 ::
x
E
:t
OI4 0.4
&4 0.4
:
5,423 5,497 5,670 5,481 5,649 5,590 5,378 5,993
5,423 5,419 5,375 5,481 5,437 5,590 5,378 5,993
6,638 6704 6,929 6,655 6,879 6,707 6,145 6,870
6,638 6,588 6,419 6,655 6,486 6,707 6,145 6,870
5,463 5,614 5,583 5,890
5,463 5,429 5,353 5,583 5.514 5,702 5,399 6.055
6,463 6,709 6,950 6,557 6,795 6,608 6,163 6.904
6,463 6,325 6,178 6,557 6,377 6,608 6,163 6.994
5,800 5,702 5,399 6,055
Table 2 Proftt levels over last 80 (Q) periods for duopolists following a live and let live policy with different cost coetlicients in response to random demand fluctuations (A=p =O.l). a
B
i-
c IA 3
c ZA
Gn
C2B
%A
%B
E 2:7
2.7 2.7 2.7
4,505 4,529 4,117
4,997 5,086 4,630
--
0 Cournot
A
3 3
: 3
A
;
3
2.7
:
4,813 4,726
4,807 4,776
:
2.7
3
4,370
4,376
Cournot
accommodate asymmetries among tkms, and that these firms preserve roughly the same relative advantage over the Cournot strategy as in the zero cost case. It should be noted that the firm having a 10% cost advantage over its opponent in both markets will accumulate profits which are about 10% higher than those of its competitor for both a(b) values, while profit levels are almost the same if each firm has a 103/, cost advantage in producing one of the products.
ur computer simulations are ba
tion of al cost
D. Kantarelis and E.C.H. Veendorg, Live and let live type behavior
242
of production. One can easily show that the adoption of a live and let live policy by firms that operate in an environment with fluctuating demands will lead to essentially similar results for different duopoly models as long as firms can adjust in more than one dimension. The use of quadratic cost functions, for example, would have transformed the L-constraint in figs. 1 and 2 from a circle to an ellipse. Using a two-market extension of a Quandttype price variation model (1967) would have changed the relative position of the two constraints in these figures: the center of the circle constraint would have been to the southwest of the straight line constraint, explaining why the interaction of these two constraints in an environment with demand fluctuations would gradually lead to higher prices (instead of smaller quantities) and higher profits. The similarities between our live and let live model and traditional oligopoly models are rather obvious. It might be instructive, however, to contrast our model with some relatively recent developments. (A) Crippled
optimization.
Our model is closely related to the ones proposed by Cyert and DeGroot (1973) and Kuenne (1979,80), and referred to by the latter as crippled optimization. Firm A maximizes a function of the following type
WL
X,) + y1Eg(&,X,),
where the Y, can be output levels (Cyert and DeGroot) or price levels (Kuenne), and where the y’s are referred to as coefficients of cooperation (C&D) or consonance coefficients (K). Firms adjust these coefficients in the hope that their rival will make a similar move, or at least on the assumption that the rival will adjust its price or output level accordingly. Their models express more clearly the spirit of an informal coalition than our model. E&u their formulation allows decision by firms that ~
The decision making procedure followed by firms in this paper is closely related to the multiple goal or satisficing models of firm behavior discussed by Ddy and obinson ( 1973) Encarnacion (1944), Ferguson (19&S),and others. Satisficing firms adhere to an hierarchy of goals, maximizing the next goal only after a satisfactory level of the higher ranked goal(s) has been reached and until that goal itself reaches a satisfactory level. The difference is that in our model tisfactory 1s of profits relative to those of tbe opponent can always ile a obtained.
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
243
firm that maximizes sales subject to a minimum profit constraint may not be able to reach that level of profits, our firm can always assure that the relative position is preserved by not making any change in output level.
6. Concludingcomments We have attempted to show that adoption of a live and let live policy tends to benefit firms in oligopolistic markets, given multiple contacts and fluctuating demands. Simulations and various generalizations suggest that the implications are rather robust with respect to the nature of the duopoly model and the specification of the adjustment process, and that they do not depend on symmetry among firms. With symmetry firms could reasonably be expected to reach the joint profit maximizing solution. But with cost conditions that differ among firms this solution loses its appeal especially under changing __L_ _ r;) APmnnrl -C.rrl---- mmAitirmc ~~..~...I..“. Other geieralizations could be attempted. For example, initial explorations suggest that very similar results can be obtained for two firms competing in a single market but with mtiltiple decision variabies, such as price and advertising. Another extension would be to allow more firms and more markets (or other contact points). Keeping the number of firms constant, and increasing the number of markets would seem to increase the scope for live and let live behavior. Not only does it offer more flexibility in meeting the live and let live constraints, but it would also assure exposure to more demand fluctuations, which constitutes the driving force of the adjustment. An increase in the number of firms might have the opposite effect. It seems plausible, therefore, that firms wouid aim for a high ratio of the number of contact points relative to the number of competitors, an hypothesis which finds support in the works of Heggestad and Rhoades (1978), Scott (1982), Feinberg ( 1984,85), and Alexander ( 1985).
efereaces Alexander, D.L., 1985, An empirical test of the mutual forbearance hypothesis:The case of bank holding companies, Southern Economic Journal 52, 122-140. Chamberlin, E.H., 1946, The theory of monopolistic competition, 5th ed. (Harvard University Press,Cambridge, MA). Cyert, R.M. and M.H. DeGroot, 1970, Multiperiod decision models with alternating choice as a solution to the duopoly problem, Quarterly Journal of Economics 84.410429. Cyert, R.M. and M.H. DeGroot, 1973, An analysis of cooperation and learning in a duopoly context, American Economic Review 63, 24-37. Day, R.N. and SM. Robinson, 1973, Economic decision; with L** utility, in: J.L. Cochrane and M. Zeleny, eds., Multiple criteria decision making (University of South Carolina Press, Columbia, SC). Edwards, CD., 1955, Conglomerate bigness as a source of power, in: Rusinessconcentration and price policy (Princeton University Press, Princeton, NJ).
244
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
Encamacion, J., 1964, Constraints and the firm’s utility function, Review of Economic Studies 31,113-119. Feinberg, R.M., 1984, Mutual forbearance as an extension of oligopoly theory, Journal of Economics and Business 36,243-249. Feinberg, R.M., 1985, ‘Sales at risk? A test of the mutual forbearance theory of conglomerate behavior, Journal of Business 58,225-241. Ferguson, C.E., 1%5, The theory of multidiensional utility analysis in relation to multiple-goal business behavior: A synthesis, Southern Economic Journal 32, W-175. Heggestad, A.A. and S.A. Rhoades, 1978, Multi-market interdependence and local market competition, Review of Economics and Statistics 60,523-532. Kwnne, R.E., 1979, Rivalrous consonance and the power structure of OPEC, Kyklos 32, 695-717. Kuenne, R L, 1980, Duopoly reaction functions under srippfdd optimization regimes, Oxford Eumon: :c Papers 32,224-24Q. Quandt, R.E. 1967, On the stability of price adjusting oligopoly, Southern Economic Journal 33, 332-336. Scott, J.T., 198Z, Multimarket contact and economic performance, Review of Economics and Statistics 64, 368-375.
WE A
ET
D. KANTARELIS and E.C.H. VEENDORP Assumption College, Worcester, MA 01609-1296, USA and Clark University Received July 1986, final version received July 1987 This paper formalizes a cooperative mode of behavior which captures essential elements of the live and let live philosophy, and shows that adoption of such a policy tends to benefit firms in oligopolistic markets. It is indicated that multiplicity of contacts and exposure to demand guctuations may simplify coordination and enhance the potential and profitability of live and let live type behavior.
Attempts to explain interdependence in oligopolistic markets have been made by many economists, most influential among whom has undoubtedly been E.H. Chamberlin (1946). He argues that mutual recognition of interdependence will drive firms to a joint monopoly profit level without formal collusion but instead by reaching some kind of ‘virtual contract’ which implies that there is no specific agreement, but that the parties act as though there were one. How effective this ‘virtual contract’ is depends crucially on such factors as the QZGXYof sellers, cost conditions, and the stability of demand. More recently ecoilumists have started to analyze interdependence of conglomerate firms, pointicg out that with symmetry of conglomerate power incentives for rivalry tend to disappear. C. Edwards (1955) has hypothesized that when such firms operate in a multi-market environment, ‘the multiplicity of their contacts may blunt the edge of their competition.. . Each conglomerate competitor may adopt a live and let live policy designed to stabilize the whole structure of the competitive relationship’. This paper formalizes a policy which reflects the spirit of the live and let philosophy, and shows that the adoption of such a opohstic firms is facilitated by a large number of contact points and fluctuating demands. *Thanks are due to Dan Levitt, Martin Shubik, Richard H. Day, and a referee for helpful comments on previous drafts of this paper. he usualdisclaimers apply.
D.Kantarelis andE.C.H.
236
Veendorp, Live and iet live type behavior
The ‘live and let live’ philosophy is clearly in the spirit of the Chamberlin model. It would seem, however, that the muitipiicity of contacts between these conglomerate firms would generate difficulties for Chamberiin’s ‘virtual contract’. The multiplicity of contacts would typically imply more exposure to demand changes and a more complex technology. A more complex technology is less likely to preserve symmetry among firms, whereas the more frequent exposure to demand changes would enhance the relevance of the complications created by such asymmetries. Jointly they would make the Chamberlin type conditions favoring coordination less likely to apply. This paper indicates that multiplicity of contacts and exposure to demand fluctuations may actually simplify coordination, and enhance the profitability of live and let live type behavior. 2.
he
Y
To our knowledge no explicit formulation of the live and let live hypothesis has been given. But the phrases that seem to be used most frequently to capture the spirit of such behavior, are suggestions ‘to preserve the status quo’, ‘to play it safe’ and ‘not to rock the boat’. Such prescriptions suggest that firms should refrain from taking actions that would significantly alter their positions, described in terms of profits, market shares or some other variables, relative to those of their competitors. Using profits to measure relative positions, adherence to the live and let live philosophy would then imply that firm A considers only those actions that would not increase the profits of its competitor(s) (B) to a much larger or smaller extent than its own, or
In this paper we choose, somewhat arbitrarily, O,=Oz and refer to that part of firm A’s concern with relative positions
as the let live (LL) constraint. To the other half Ax, 2 aAn*,
a=
1/01>O,
we shall refer as the live (L) constraint. aximixing profits subject to these constraints implies that a fi is willing to (tern orarily) forego some profits if a particular move wou either hurt, or u uly benefit the opponent this
tion of such a live and
D. Kantarefis and E.C.H. Veendorp, Live and let live type behavm
237
let live policy by a firm will affect its profits over time when competing with firms that are similarly inclined (though the o[values they use may differ), not how firms might induce their competitors to adopt such a cooperative policy or how they might respond if other firms defect. We explore, in other words, how some kind of virtual contract can be preserved in spite of asymmetries and demand fluctuations, not how it could be established or broken. ive
a&et
The potential for live and let’live behavior by sellers in a traditional Cournot quantity variation model is clearly limited. Increases in output by firm A in response to a demand increase will affect profits of the other firm (B) negatively, while decreases in output in response to a demand decrease will typically increase the profits by firm B by more than those of firm A. We focus therefore on a two market generalization of the Cournot duopoly model, where firms A and B produce the same two undifferentiated products (1 and 2) in amounts Qij(i= 1,2; j= A, B) at zero cost. They face linear demand curves Pi = Qi- QiA-Qie) and are exposed to demand changes that lead to parallel shifts in these demand curves. The constrained maximization problem introduced in section 2 can best be visualized as being based on three components. The first one is the unconstrained profit maximization by each firm on the assumption that the other firm will not change its output levels. Maximizing A’s profits X.A=
C
i=1.2
(ai-QiA-QiB.r-I)QiAr
with respect to QiAyields the Cournot type solution
Qh=(ai-Qi~.t-~)/Zi= 1,2, where QiB,t _ , are the output levels oi firm B in the preceding period, and u1 are the demand intercepts in the present period, which may or may not differ from the preceding ones. Similar expressions can be written down for firm B, with identical time subscriptions in case of simultaneous adjustments, or different ones in case of alternating adjustments. The second component is the LL constraint which requires that the profits of firm B should not be affected negatively by A’s actions. If in response !o a demand change or in response to adjustments by the other firm, firm A revious output levels ( iA,,- J to new ones ( considers changi then the condition that Ax BhO, or
238
D. Rantarelisand E.C.H. Veendorp, Live aid let live type behavior
will upon substitution yield the inequality
The equality part of this constraint is a straight line in going through the p slope -QzE.~-I/QIB.~-I ~VQLW-I~QZA,~-J The third component is the requirement that the increase in
adjusting firm is at least as large as a certain fraction of the increase in profits of the opponent (the L constrzint). This condition, dlr,~ad~~, can be simplified to
SCQ i~.r-i-(a~-(l-9C!Q;~,r-~)/232
The equality part of this condition represents a circle in QIA,QZAspace with }/2, {a2 -( 1-a)Qllr, L_ ,}/2) going through the center M(h-(l-a)Q~~,t-l initial point P(Q 1A,L_ I, QZAer_ I). Figs, 1 and 2 illustrate the role these three components play in the adjustment mechanism for a= 1, and for a demand change in the first market. If firms A and were initially producing the same output bundle, and A’s initial position is given by point P, then t after the demand change will be point Q (with constraints are represented by the str aximization of profits subject to these firm A move to or towards point p, which will typically center of the circle constraint w
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
239
Fig. 1. Firm A’s response to a demand decreme according to 2 live and let live policy in a two dimensional quantity variation model (P = initial; Q=Cournot; d = optimal solution; A4=center of circle constraint).
FIN. 2. Firm A’s response to a demand increase according to a live and let live policy in a two dimensional quantity variation model (P =Einitial; & =Cournot; B= optimal solution; M = center of circle constraint).
240
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
diagram). The suggestion is supported, however, by the results of computer simulations. 4. Results of computer simu
Finding the solution of the inequality constrained maximization problem for arbitrary initial output levels (Qij,1_1) and arbitrary values of the demand intercepts (Qi) is conceptually easy but tedious since one has to verify which constraints, if any, will be binding (details are available upon request from the authors). Given this target solution (Q&,1), firm A chooses new output levels adaptively O<;zSl, Q 1ll.t =QiA.r-l+IZ(Q~,t-QiA.r-l), while firm B responds in a similar fashion, using coefficients /? and p instead of 01and 1. The responses of firms A and B, following a live and let live policy while adjusting to continuous demand fluctuations, were simulated for a large number of different values of the coefficients involved. The results reported in this section refer to the profit levels of these firms while reacting to each other and to parallel demand shifts in both markets. New demand intercepts (ai) were chosen randomly and independently after every four periods from a uniform discrete distribution with range 15 to 20. Firms respond to these changes with different values for the a and #I coefficients, and with different vaiues for the adjustment speeds (A and cc), starting at the Cournot equilibrium output levels for the midpoint of the above range (Ui= 17.5). The simuiations were done for 800 successive demand changes or 3200 periods, and table 1 reports the accumulated (undiscounted) profit levels over the first 8a and last 80 periods. After 800 changes, these accumulated profits became insensitive to continued demand fluctuations. Table 1 also lists the Cournot and joint monopoly profits for the same adjustment speeds. It appears that firms following a live and let live policy fare rather.well in terms of profit compared to Cournot, and that the difference is more pronounced in the last 80 periods than in the first 80 periods. Comparing the different live and let live policies, it appears that the lower levels of the or(p) coefficients lead to higher profit levels, but that these differences are relatively small and getting smaller with re ated fluctuations. The rationale for z;ing higher values of 4,s) is that it offers protection against asymmetry. exceeds b, then firm A has a consistent advantage over its oppon conversely. Table 2 contains similar results for firms with positive costs of The margina ant, but do differ among firms and/or
D. Kantarelis and E.C.H. Veendorp, Live and let live type behavior
241
Table 1 Profit levels over first 80 (IQ and last 80 (Q) periods for duopolists following a live and let live policy with zero cost of production in response to random demand fluctuations. 01 A B P xr.4 % GA R.B 1 1 1
1 0.5 0 0.5 :
0. :: Cournob Joint max 1
;5
1 A.5 0.5 0 Coumot Joint max
0.1 ::
0.1 0.1 0.1
o:l ::
o:l ::
::
0.1 0.1 E
d0.5
0.4 0.4 ::
x
E
:t
OI4 0.4
&4 0.4
:
5,423 5,497 5,670 5,481 5,649 5,590 5,378 5,993
5,423 5,419 5,375 5,481 5,437 5,590 5,378 5,993
6,638 6704 6,929 6,655 6,879 6,707 6,145 6,870
6,638 6,588 6,419 6,655 6,486 6,707 6,145 6,870
5,463 5,614 5,583 5,890
5,463 5,429 5,353 5,583 5.514 5,702 5,399 6.055
6,463 6,709 6,950 6,557 6,795 6,608 6,163 6.904
6,463 6,325 6,178 6,557 6,377 6,608 6,163 6.994
5,800 5,702 5,399 6,055
Table 2 Proftt levels over last 80 (Q) periods for duopolists following a live and let live policy with different cost coetlicients in response to random demand fluctuations (A=p =O.l). a
B
i-
c IA 3
c ZA
Gn
C2B
%A
%B
E 2:7
2.7 2.7 2.7
4,505 4,529 4,117
4,997 5,086 4,630
--
0 Cournot
A
3 3
: 3
A
;
3
2.7
:
4,813 4,726
4,807 4,776
:
2.7
3
4,370
4,376
Cournot
accommodate asymmetries among tkms, and that these firms preserve roughly the same relative advantage over the Cournot strategy as in the zero cost case. It should be noted that the firm having a 10% cost advantage over its opponent in both markets will accumulate profits which are about 10% higher than those of its competitor for both a(b) values, while profit levels are almost the same if each firm has a 103/, cost advantage in producing one of the products.
ur computer simulations are ba
tion of al cost
D. Kantarelis and E.C.H. Veendorg, Live and let live type behavior
242
of production. One can easily show that the adoption of a live and let live policy by firms that operate in an environment with fluctuating demands will lead to essentially similar results for different duopoly models as long as firms can adjust in more than one dimension. The use of quadratic cost functions, for example, would have transformed the L-constraint in figs. 1 and 2 from a circle to an ellipse. Using a two-market extension of a Quandttype price variation model (1967) would have changed the relative position of the two constraints in these figures: the center of the circle constraint would have been to the southwest of the straight line constraint, explaining why the interaction of these two constraints in an environment with demand fluctuations would gradually lead to higher prices (instead of smaller quantities) and higher profits. The similarities between our live and let live model and traditional oligopoly models are rather obvious. It might be instructive, however, to contrast our model with some relatively recent developments. (A) Crippled
optimization.
Our model is closely related to the ones proposed by Cyert and DeGroot (1973) and Kuenne (1979,80), and referred to by the latter as crippled optimization. Firm A maximizes a function of the following type
WL
X,) + y1Eg(&,X,),
where the Y, can be output levels (Cyert and DeGroot) or price levels (Kuenne), and where the y’s are referred to as coefficients of cooperation (C&D) or consonance coefficients (K). Firms adjust these coefficients in the hope that their rival will make a similar move, or at least on the assumption that the rival will adjust its price or output level accordingly. Their models express more clearly the spirit of an informal coalition than our model. E&u their formulation allows decision by firms that ~
The decision making procedure followed by firms in this paper is closely related to the multiple goal or satisficing models of firm behavior discussed by Ddy and obinson ( 1973) Encarnacion (1944), Ferguson (19&S),and others. Satisficing firms adhere to an hierarchy of goals, maximizing the next goal only after a satisfactory level of the higher ranked goal(s) has been reached and until that goal itself reaches a satisfactory level. The difference is that in our model tisfactory 1s of profits relative to those of tbe opponent can always ile a obtained.
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firm that maximizes sales subject to a minimum profit constraint may not be able to reach that level of profits, our firm can always assure that the relative position is preserved by not making any change in output level.
6. Concludingcomments We have attempted to show that adoption of a live and let live policy tends to benefit firms in oligopolistic markets, given multiple contacts and fluctuating demands. Simulations and various generalizations suggest that the implications are rather robust with respect to the nature of the duopoly model and the specification of the adjustment process, and that they do not depend on symmetry among firms. With symmetry firms could reasonably be expected to reach the joint profit maximizing solution. But with cost conditions that differ among firms this solution loses its appeal especially under changing __L_ _ r;) APmnnrl -C.rrl---- mmAitirmc ~~..~...I..“. Other geieralizations could be attempted. For example, initial explorations suggest that very similar results can be obtained for two firms competing in a single market but with mtiltiple decision variabies, such as price and advertising. Another extension would be to allow more firms and more markets (or other contact points). Keeping the number of firms constant, and increasing the number of markets would seem to increase the scope for live and let live behavior. Not only does it offer more flexibility in meeting the live and let live constraints, but it would also assure exposure to more demand fluctuations, which constitutes the driving force of the adjustment. An increase in the number of firms might have the opposite effect. It seems plausible, therefore, that firms wouid aim for a high ratio of the number of contact points relative to the number of competitors, an hypothesis which finds support in the works of Heggestad and Rhoades (1978), Scott (1982), Feinberg ( 1984,85), and Alexander ( 1985).
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Encamacion, J., 1964, Constraints and the firm’s utility function, Review of Economic Studies 31,113-119. Feinberg, R.M., 1984, Mutual forbearance as an extension of oligopoly theory, Journal of Economics and Business 36,243-249. Feinberg, R.M., 1985, ‘Sales at risk? A test of the mutual forbearance theory of conglomerate behavior, Journal of Business 58,225-241. Ferguson, C.E., 1%5, The theory of multidiensional utility analysis in relation to multiple-goal business behavior: A synthesis, Southern Economic Journal 32, W-175. Heggestad, A.A. and S.A. Rhoades, 1978, Multi-market interdependence and local market competition, Review of Economics and Statistics 60,523-532. Kwnne, R.E., 1979, Rivalrous consonance and the power structure of OPEC, Kyklos 32, 695-717. Kuenne, R L, 1980, Duopoly reaction functions under srippfdd optimization regimes, Oxford Eumon: :c Papers 32,224-24Q. Quandt, R.E. 1967, On the stability of price adjusting oligopoly, Southern Economic Journal 33, 332-336. Scott, J.T., 198Z, Multimarket contact and economic performance, Review of Economics and Statistics 64, 368-375.