Cournot-like Interaction with Cournot Outcomes

Abstract

This article works out a Cournot duopoly where firms take price as given, but it is not the same price that is taken for given by each firm; it depends on individual expectations about opponent’s behaviour as are shaped by the business cycle. Yet, the result of this interaction is a Cournot equilibrium as if the calculations were based on the same price. It is not the price that would be assumed in the absence of expectations. Expectations make inverse-U-shaped reaction curves peak at higher output levels relative to peaks in their absence, implying a lower price vis-à-vis that which is taken for granted in the traditional analysis. And, output levels are higher because they reflect optimism for the course of the economy. In the static context of the analysis, a Cournot equilibrium comes about as if the business cycle is at each peak. Policy-wise, the message is that observed price variations in an oligopoly do not necessarily indicate Bertrand competition despite the possibility that dynamic interaction may lead to outcomes other than the Cournot one; but it will be about disequilibrium and unstable outcomes.

JEL Classifications: D43, L13, E32

Keywords

Cournot duopoly business cycle inverse-U-shaped reaction curves expectations-dependent price

Introduction

The Cournot duopoly has been examined exhaustively. It is a topic of microeconomic concern and such is the concern underlying its investigation from a dynamic point of view as well (Matsumoto, 2017). The purpose of this note is to add some macroeconomic flavour by introducing into the basic model one way according to which the state of the economy might reflect upon the expectations about the opponents’ behaviour. In the next section, firms take price as given, but it is not the same price that is taken for given by each firm; it depends on individual expectations about opponent’s behaviour, as are shaped by the business cycle. This is the point where the present model departs from the standard Cournot duopoly. The third section concludes with policy message of the analysis.

Formal Considerations

Assume that the demand for a perfectly homogeneous good, Q, is iso-elastic, meaning that the consumer expenditure for this good is constant, so that:

Q = 1 / P \Rightarrow P = 1 / Q .

(1)

This good is produced only by two identical but competing firms, $Q = Q_{i} + Q_{j}$ , $i, j = 1, 2$ , $i \neq j$ The iso-elastic character of the demand implies that $Q_{i}$ is related to $Q_{j}$ negatively. So, given this and the duopolistic structure of the market, each firm i plans its output by taking into account what it expects to be the output of the opponent j. And both firms are aware that this kind of behaviour on their part will influence total market output and equilibrium price. In their calculations, each firm i contemplates demand curve:

P_{i} = \frac{1}{Q_{i} + E (Q_{j})}

(2)

rather than Equation (1). $E (Q_{j})$ is the expectations about $Q_{j}$ , which may differ from j’s expectations about $Q_{i}$ Expectation is the only element that differentiates the two firms, which explains the presence of index i in Equation (2). The ‘fact’ that each firm contemplates a different demand curve in its calculations is the feature of the present modelling that differentiates it from the standard Cournot duopoly.

Under the same unit costs, k, profits are:

Π_{i} = \frac{Q_{i}}{Q_{i} + E (Q_{j})} - k Q_{i}

(3)

with reaction functions:

Q_{i} = \sqrt{\frac{E (Q_{j})}{k_{i}}} - E (Q_{j}) .

(4)

Hence, in the absence of expectations, that is, when $E (Q_{j}) = Q_{j}$ , the equilibrium output of each firm would be:

Q_{i}^{*} = Q_{j}^{*} = 1 / 4 k

(5)

with equilibrium price $P = 4 k$ . These are the quantities against which those under expectations will have to be compared. Expectations have an extrapolative, $E^{e} (Q_{j})$ , and a regressive, $E^{r} (Q_{j})$ , component:

E (Q_{j}) = w_{i} E^{e} (Q_{j}) + (1 - w_{i}) E^{r} (Q_{j}),

(6)

where $w_{i} \in (0, 1)$ and $1 - w_{i}$ are the weights of each component. Once $E (Q_{j}) \neq Q_{j}$ , the former cannot exist without the latter, and vice versa as follows. The former reflects the phase of the business cycle during which expectations are formed.In good times, the opponent is expected to try market more than $Q_{j}^{*}$ , and the opposite, in bad times:

E^{e} (Q_{j}) = Q_{j} + b_{i} (Q_{j} - Q_{j}^{*}) .

(7)

The extrapolation parameter, b, is in any case positive. $Q_{j}$ is adjusted by the term $b_{i} (Q_{j} - Q_{j}^{*})$ , which captures the influence of the state of the economy on expectations. And since $E^{e} (Q_{j}) \neq Q_{j}$ , the ith firm has to consider in addition to the speed according to which the duopoly will attain $Q_{j}^{*}$ after all:

E^{r} (Q_{j}) = Q_{j} + h_{i} (Q_{j}^{*} - Q_{j}) .

(8)

Parameter $h \in (0, 1)$ captures the speed of the adjustment. The weights too are shaped by the state of the economy. The larger the expected deviation from $Q_{i}^{*}$ , the dimmer the prospect of attaining it becomes, implying a small $w_{i}$ and a large $1 - w_{i}$ :

w_{i} = \frac{1}{1 + {(Q_{j} - Q_{j}^{*})}^{2}} .

(9)

Now, inserting Equations (7), (8) and (9) in Equation (6) yields that:

E (Q_{j}) = Q_{j} + \frac{(Q_{j} - Q_{j}^{*}) [b_{i} - h_{i} {(Q_{j} - Q_{j}^{*})}^{2}]}{1 + {(Q_{j} - Q_{j}^{*})}^{2}} .

(10)

Inserting Equation (10) in Equation (4) gives the reaction functions when $E (Q_{j}) \neq Q_{j}$ . Figure 1—whereas in Figure 2, $x \equiv Q_{j}$ and $y \equiv Q_{i}$ —illustrates the resulting reaction functions with $k = 1.2$ and hence, $Q_{j}^{*} = 1 / 4 (1.2) ≅ 0.20833$ , and ( $b = 1.075$ , $h = 0.5$ ; bottom curve), ( $b = 0.5$ , $h = 1.075$ ; upper grey curve) and ( $b = 0.9$ , $h = 0.7$ ; middle dotted curve). All curves are inverse-U-shaped and peak at the same output level.

Figure 1.

Source: Calculations of the author.

In Figure 2, the curves from the beginning of the axes are those in the absence of expectations, the grey curve is that under ( $b = 0.5$ , $h = 1.075$ ), while the curve it cuts at point B is associated with ( $b = 1.075$ , $h = 0.5$ ): The sellers have exactly the opposite expectations. The figure shows that the initial downward sloping parts of the curves do not intersect and hence, for small levels of output as at the trough of a business cycle, there is no equilibrium. Once out of the trough, the equilibrium occurs at the peak of the curves, and hence, the precise form of expectations does not matter; in the presence of expectations, all curves have the same peak. What matters is only the presence of expectations, which raises equilibrium output from that corresponding to point A to that associated with point B. To try solving for the quantities at B is certainly futile, given the complexity of the expressions involved in the calculations.

Figure 2.

Source: Calculations of the author.

The point is that Cournot-like interaction along the business cycle produces a Cournot outcome. Even though calculations by each firm are made by taking each for granted a different price—the one shaped by the differing expectations of each firm, which in turn are influenced by the state of the economy—the equilibrium configuration comes up as if the calculations were based on the same price. But it is not the price that would be assumed in the absence of expectations. Expectations make reaction functions peak at higher output levels relative to the peak in their absence, implying a lower price vis-à-vis that is taken for granted in the traditional analysis. And output levels are higher because they reflect optimism for the course of the economy; only then it makes sense to compete. After a recession, during which reaction functions have become linear and there is no interaction between the two firms at all, the functions start rising for both firms capturing the optimism in the economy. In fact, one might argue that these linear parts are not components proper of the reactions curves, suggesting that firms do not operate at all before a minimum of sales can be ensured. They peak at the peak of the business cycle and only afterwards start declining. They cross at these peaks, indicating that only then rivalry can stop, alas temporarily. In the static context of the analysis, a Cournot equilibrium comes about as if the business cycle is at each peak, but in a dynamic framework, the peak is only temporary.

Conclusion

Even more complex expressions will arise if the $Q_{j}$ ’s in the reaction functions are let to enter them with a time lag, as realism dictates. Puu (1991) has shown that the non-linearities ensuing even if $E (Q_{j}) = Q_{j, - 1}$ are capable of producing quite chaotic dynamics. Policy-wise, the message is that observed price variations in an oligopoly do not necessarily indicate Bertrand competition despite the possibility that dynamic interaction may lead to outcomes other than the Cournot one; but it will be about disequilibrium and unstable outcomes.

Acknowledgement

This article has benefited from the useful comments and suggestions of an anonymous referee.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author received no financial support for the research, authorship and/or publication of this article.

References

Matsumoto

(Ed.). (2017). Optimization and dynamics with their applications . Singapore: Springer.

Puu

(1991). Chaos in duopoly pricing. Chaos, Solutions & Fractals , 1(6), 573–581.