In this paper we develop a stochastic version of a dynamic Cournot model. The model is dynamic because firms are slow to adjust output in response to changes in their economic environment. The model is stochastic because management may make errors in identifying the best course of action in a dynamic setting. We capture these behavioral errors with Brownian motion. The model demonstrates that the limiting output level of the game is a random variable, rather than a constant that is found in the non-stochastic case. In addition, the limiting variance in firm output is smaller with more firms. Finally, the model predicts that firm failure is more likely in smaller markets and for firms that are smaller and less efficient at managing errors.
The starting point of oligopoly theory is the static Cournot model. It provides an example of Nash equilibrium and embeds a variety of possible market outcomes. One is the Cournot Limit Theorem, which states that the Cournot-Nash equilibrium is the monopoly outcome in a market with one firm, and approaches the competitive equilibrium as the number of competitors approaches infinity1.
More sophisticated oligopoly models consider cases where firms interact over time. For example, in a Cournot-type supergame where the Cournot game is repeated in every period from now to eternity, an appropriately defined trigger strategy can support cooperation in every period (Friedman [
In this paper, our goal is to extend the dynamic Cournot model to include management errors. Brownian motion is used to model these behavioral errors. To our knowledge, we are the first to incorporate Brownian motion into an oligopoly model. By adding this stochastic component, we are able to show that output converges to a random variable rather than a constant, as in the deterministic dynamic Cournot model. In the stochastic model, the limiting variance in firm output is smaller in duopoly than in monopoly. Finally, the stochastic model predicts that firm failure is more likely in smaller markets and for firms that are smaller in size and are less efficient at coping with behavioral errors. These results provide us with a better understanding of stochastic markets, in particular the exit patterns of firms when managerial errors are present.
The remainder of the paper is organized as follows. In Section 2 we summarize the deterministic model on which our stochastic model is built. In Section 3 we develop the stochastic version of the dynamic Cournot model by adding Brownian motion. Here, we derive the limiting distribution of the game and investigate the consequences of allowing one firm to be superior at coping with errors. In Section 4 we analyze the effect of management error on firm failure. The final section provides a brief conclusion.
Our stochastic model builds from a deterministic dynamic model that we develop in this section. Several scholars have investigated a dynamic version of the Cournot model. For example, Maskin and Tirole [
Consider a market with two firms (1 and 2) that produce homogeneous goods. The inverse demand function at
time t is linear:
stants. In this model, firm inertia is motivated by the cost of changing its level of output. This adjustment cost is
that both firms participate in this dynamic version of the game. At t = 0, each firm’s goal is to choose a production plan to maximize its discounted stream of profits:
which is subject to the discount factor
Dockner [
Parameter m is a function of the parameters of the model and represents firm i’s long-run sales target in the absence of firm j (i.e., firm i’s monopoly output). Parameter m will increase with the size of the market. These functions capture the strategic dynamics of the model. The solution to this system is:
where
In this section we add a stochastic component to the dynamic Cournot model of the previous section. The stochastic component derives from a behavioral error that influences the dynamic best-reply functions of firms. One potential cause is the presence of boundedly rational managers who make errors when attempting to identify the best course of action in a dynamic setting3. Such errors may result from the cognitive limitations of managers coupled with the time pressure to make quick decisions. In our setup, behavioral errors occur when managers implement the dynamic best-reply functions and are captured by Brownian motion4. Adding Brownian motion (Bt) to the dynamic best-reply Equations (2) and (3) and converting them to stochastic difference equations produces:
Equations (6) and (7) describe each firm’s updating strategy in output, which is impacted by three forces. The first is m, which represents the firm’s sales target. Firm i’s output will increases when
Solving this system simultaneously produces a complex Gausian process, as described in Theorem 1.
Theorem 1: Given the game described above, the optimal paths of
Proof: See Appendix.
If
In the limit, the solutions to Equations (8) and (9) require Ito integration. This result is described in the next theorem.
Theorem 2: Characteristics of the limiting distribution of Equations (8) and (9) as t → ∞ are:
A comparison of the pair of Equations (4) and (5) with the equations in Theorem 2 reveals that unlike the deterministic Cournot model,
ministic case, the optimal output level of each firm converges to
steady state equilibrium is the same on average as in the non-stochastic Cournot model. However, the stochastic model has a non-zero variance as seen in Equation (11). Another feature of the model is that the mean value in the model with Brownian motion does not depend on the initial positions of the two firms.
The final issue of interest is the effect of assuming that firm 1 is superior at dealing with behavior errors
To further analyze the effect of firm ability to cope with behavioral errors, consider the case where firms are equally capable (i.e.,
comes
duopoly. With a single firm, the limiting variance in the single Orstein-Uhlenbeck mean-reverting equation
equals
5For Orstein-Uhlenbeck mean-reverting equation,
Firm failure is common in a free market economy. For example, Dunn et al. [
Consider a setting with the following characteristics. Assume an emerging market where firms enter by initially producing below their long-run optimal levels of output6. Firms are asymmetric, and firm 1 has the initial
strategic advantage, such that
cannot re-enter the market in a later period7. From Equations (8) and (9), we know that the optimal output path for firm 1 is greater than that of firm 2. This suggests that firm 2 is more likely to fail than firm 1. In order to
simplify the analysis and focus on the importance of starting values, we assume that
With these assumptions and by defining firm 1 and 2’s respective exit periods as
Theorem 3: If firm 1 starts out larger than firm 2 in period 0, then the probability that
Proof: See Appendix.
Although there is still a chance that firm 1 will fail, its probability of failure diminishes with the exit of firm 2. When firm 2 exits the market, Equation (6) shows that firm 1 will grow more rapidly toward its long-run optimum. This increase in size helps insulate firm 1 from failure.
The next question of interest is the precise probability that firm 2 will exit. Is this probability less than 1 [i.e.,
Theorem 4: Let the difference in starting values be sufficiently small, such that
Proof: See Appendix.
In order to gain greater insight into the likelihood of firm 2 failure, we calculate the lower bound on this probability. This model produces the following result.
Theorem 5: The lower bound on the probability of firm 2 exit is
Proof: See Appendix.
The lower bound is less than 1, and it falls as m and
In this paper, we have extended the dynamic Cournot model to include behavioral errors on the part of management. The dynamic portion of the model follows the literature by assuming that firms face a cost of adjusting output. Thus, firms do not immediately adjust output to a new equilibrium after a behavioral error is made. Management error in this dynamic setting is modeled with Brownian motion. To our knowledge, we are the first to include Brownian motion in an oligopoly model.
The model produces several interesting results. First, Theorem 2 shows that the limiting output level is a random variable rather than a constant, as is found in the non-stochastic version of the dynamic Cournot model. It also indicates that the limiting variance in firm output is smaller in duopoly than in monopoly. Finally, Theorems 2 - 5 indicate that the probability of firm failure decreases for markets that are larger in size and for firms that are larger and more efficient at dealing with behavioral errors.
We would like to thank Roland Eisenhuth, Jenny Lin, Todd Pugatch, Carol Tremblay, Edward Waymire, Patrick De Leenheer and two anonymous referees for helpful comments on an earlier version of the paper.
HyunghoYoun,Victor J.Tremblay, (2015) A Dynamic Cournot Model with Brownian Motion. Theoretical Economics Letters,05,56-65. doi: 10.4236/tel.2015.51009
Proof of Theorem 1
We present this system of Equations (6) and (7) in matrix form:
The eigenvalues of
We diagonalize this SDE system using the eigenvectors.
Let
Transforming
The solution of this stochastic equations is:
where
From
where
Because of
QED
Proof of Theorem 2
From (A9) and (A11), the mean is:
The variance of (A9) and (A11) is the same as below:
Since
Accordingly,
QED
Proof of Theorem 3
We multiply both sides of (A9) and (A11) by
Let
We put them in differential form,
Since the drifting term of
Proof of Theorem 4
Let
Since
QED
Proof of Theorem 5
We reset (A25):
We have to find a function
We change the drift of (A27) so that we can find an explicit value of the possibility of
Since the new stochastic process has a larger drift term, the possibility of
Then,
A lower bound for business failure is
QED