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Linear Dynamic Cournot Oligopoly Model with Adaptive Expectations Szomolányi Karol, Surmanová Kvetoslava Faculty of Economic Informatics, University of Economics, Bratislava Abstract Cournot oligopoly equilibrium is in general dynamically not stable, if on the market act more than two oligopolies (duopolies).In the model it is presumed that oligopolies consider the production of the others to be constant in time. If we extend the model by assumption that oligopolies expect quantities of the others future production, the system is more stable. Such model is more realistic as oligopolies in reality have access to information helping them to better estimate future productions of the others. In the paper we provide an approach, how to deal with a linear Cournot oligopoly dynamic model augmented by adaptive expectations. Keywords: Cournot oligopoly, economic dynamics, adaptive expectations
Introduction The theory of oligopolies knows two basic approaches to the static oligopoly market equilibrium: Cournot and Stackelberg. The reader can study these approaches in basic microeconomic textbooks (see for example Varian [5]). For both static and dynamic analysis of both approaches is widely used the game theory (more in Lukáčik [2]). If we take dynamics of both Cournot and Stackelberg models, we ask for the stability of equilibriums. Dynamic Cournot models have been already in world economic dynamic literature solved and reader can find them in basic textbooks of the economic dynamics (for example Gandolfo [1] or Shone [4]). According to them the market is unstable if there are more than two oligopolies (duopolies). Mostly there are linearity assumptions and models lack expectations. An interesting paper dealing with nonlinear duopoly dynamics is [3]. A lack of expectations in the Cournot equilibrium model means that oligopolies consider quantities of the production of the others for constant in the time. By the other words; oligopoly i assumes that in time t oligopoly j producec the quantity that it produced in time t – 1. In our opinion oligopolies in practice dispose with techniques to get better information about the others production. Model is needed to be augmented by expectations. This approach also has been discussed in a literature of the economic dynamics 1 . Here we provide our approach of the linear dynamic Cournot model augmented by adaptive expectations. We will show that by an adaptive expectations assumption the model is more stable. The stability of the model is given by the value of the adaptive expectations parameter. The paper can be divided in sections: A short introduction and a literature survey were in Introduction. In the first section is a short introduction to solving a system of n linear difference equations with constant coefficients. In the second and third sections are known static (2nd section) and dynamic (3rd section) linear models of Cournot oligopoly. In the fourth section we provide our approach of the model extension by adaptive expectations. The paper is finished by Conclusions. 1. First Order Simultaneous System of n Linear Difference Equations with Constant Coefficients In this section we briefly show a technique, how to solve the first order simultaneous system of n linear difference equations with constant coefficients. We have to note that there is not place to show all details of solving the system. We choose only parts that are useful for our analysis in sections 3 1
Gandolfo [1] refers: Okuguchi, K.,1976, Expectations and Stability in Oligopoly Models, Berlin and New York, Springer-Verlag
and 4. We refer the reader interested in more details to basic economic dynamics textbooks Gandolfo [1] or Shone [4]. Let us consider the first order simultaneous system of n linear difference equations with constant coefficients: x t +1 = Ax t + b
(1.1)
where xt is a nx1 vector of endogenous variables (x1,t, x2,t, … xn,t)T, A is nxn matrix of known constant coefficients, b is nx1 vector with constant coefficients (b1, b2, … bn)T and t is time index 2 . Homogenous System, General Solution of the Homogenous System The Homogenous system to the (1.1) is: xt +1 = Ax t
(1.2)
Solution of the system (1.2) will be in the form xt = αλt, where α is nx1 vector with constants not all zero. By substituting it into (1.2) and after modifying we get:
λ t [ A − λI] α = 0
(1.3)
As in general λt ≠ 0 we can (1.3) write as:
[ A − λI ] α = 0
(1.4)
As α ≠ 0, using acknowledgements of linear algebra we can (1.4) write as equation by the determinantal form:
A − λI = 0
(1.5)
Equation (1.4) is called characteristic equation and gives n roots (λ1, λ2, … , λn), the characteristic roots of the matrix A.. Some of these roots may be complex conjugate, some may be real distinct and some may be real multiple. To each characteristic root λj a corresponding charactreristic vector αj can be associated through (1.4). The general solution of the system (1.2) will then have form: x t = VΛ t a
(1.6)
where V is nxn matrix with elements αij in ith row and in jth column (αj vector in jth column), Λt is nxn diagonal matrix with λit in ith diagonal and a is nx1 vector of arbitrary constants (A1, …, An)T. 3 Solution (1.6) is the general solution of the homogenous system (1.2) but is not a solution of the system (1.1). Particular solution To get a solution of the (1.1) we need the particular solution. In economic equilibrium models we interpret a particular solution also as an equilibrium solution. We can get the particular solution by the method of undetermined coefficients. As the vector b is vector of constants, as particular solution we try substitute a vector of coefficients to (1.1):
x = Ax + b 2
(1.7)
As we deal with economic dynamics, we assume that independent variable is time. It can be proved that if xt is the solution of the (1.2) Axt is also the solution, where A is an arbitrary constant. General solution of the system (1.2) must have exactly n arbitrary constants. 3
Particular solution then is: x = (I − A) −1 b
(1.8)
General Solution The general solution of the system (1.1) is the sum of the general solution of the homogenous system (1.2) and of the particular solution (1.8): x t = VΛ t a + (I − A ) −1 b
(1.9)
According to the (1.9) endogenous variables move in time. If all characteristic roots are real we can easy investigate the movement of variables: (a) if all characteristic roots are positive, the movement is monotonic; (b) if there is at least one negative characteristic root, the movement is oscillatory; (c) if all characteristic roots are in absolute value less than one, the movement is convergent to the equilibrium (the first term of (1.9) tends to zero, as time rises); (d) if there is at least one characteristic root in absolute value greater than one, the movement is divergent (the first term of (1.9) and so all endogenous variables tend to +(-)infinity, as time rises); (e) if there is at least one characteristic root that equals -1 and all the others are in absolute value less than one, the system after some time will oscillate around its equilibrium with constant amplitudes. We can then easy derive stability conditions for known characteristic roots:
λi < 1 ∀i
(1.10)
We note, that (1.10) can be used only if roots are real. If there is at least one complex conjugate root one can transform it from Cartesian form to the polar one and so the movement is oscillatory. 2. Static Cournot Oligopoly Model The Cournot oligopoly model is in the microeconiomic literature well known (see for example Varian [5]). The brief introduction to the Cournot oligopoly model is in this section. Let us consider that in the market contributes n oligopolies; ith oligopoly produces the quantity of xi units of the market commodity. The whole production of all oligopolies is given by an aggregate production: n
X = ∑ xi
(2.1)
i =1
Demand for the commodity is given by a price demand function p(X). Assume that ith oligopoly costs are given by cost function:
ci = ci ( xi )
(2.2)
Cournot model gives answer to the question: What is an optimal production of each oligopoly xi? For each oligopoly is production of the others as given and it maximizes its profit function: max π i ( xi , X ) = p ( X ) xi − ci ( xi ) qi
subject to the aggregate production (2.1). First order conditions are:
(2.3)
∂π i ( xi , X ) ∂p ( X ) ∂c ( x ) xi + p ( X ) − i i = 0 ∀i = ∂xi ∂xi ∂xi
(2.4)
System of equtaions (2.4) gives with the aggregate production (2.1) Cournot equilibrium. 3. Dynamic Cournot Oligopoly Model Dynamic analysis generally provides answer to questions: If a system is not in its equilibrium, (a) what is the movement of variables and (b) do they move towards their equilibrium? The question (b) videlicet is: Is the system stable? In our case the system is the market with n oligopolies and equilibrium is the Cournot one. We ask, whether the equilibrium is stable. The answer in this question is by some assumptions in economic dynamics literature well known and reader can study it in basic economic dynamics textbooks (Gandofo [1], Shone [4]). In this section we briefly introduce the analysis. We will assume that oligopoly has not perfect information and do not know the quantity of the other oligopolies production xj; j ≠ i. It will simply assume that the production of the others equals to their production in the previous time. In time t every oligopoly for aggregate production considers: n
X i ,t = ∑ x j ,t −1 + xi ,t
(3.1)
j ≠i
instead of (2.1). In time t every oligopoly considers as price demand pi,t = p(Xi,t). Following the similar logic, as we followed in static Cournot Oligopoly Model in section 2, we can Cournot equilibrium write as a first order system of simultaneous difference equations: ∂p ( X i ,t ) ∂xi ,t
xi ,t + p ( X i ,t ) −
∂ci ( xi ,t ) ∂xi ,t
= 0 ∀i
(3.2)
where ci(xi,t) is ith oligopoly’s cost function. In literature (Shone [4], Gandolfo [1]) is the system solved by linearity assumptions – linear price demand function is p(Xi,t) = a - bXi,t and each oligopoly has linear cost function: ci(xi,t) = fi + cixi,t; with parameters a, b (marginal slope to the price demand), fi (fixed costs of ith oligopoly) and ci (variable costs of ith oligopoly). By substituting functions to (3.2) and after modification we can get the system:
xi ,t +1 = −
a − ci 1 n x j ,t + ∑ 2 j ≠i 2b
(3.3)
Respectively in matrix form: x t +1 = Ax t + b
(3.4)
where xt is a nx1 production vector (x1,t, x2,t, … xn,t)T, A is nxn matrix with elements of 0 on the diagonal (ii) and of -1/2 outside diagonal (ij;i≠j) and b is nx1 vector {(a-ci)/2b)}i=1n. The characteristic equation of the system is:
A − λI = 0
(3.5)
It can be shown that characteristic roots of the (3.5) are real numbers; λ = -(n-1)/2 and λ = 1/2 with multiplicity (n – 1) (see for example Gandolfo for more details). It follows that for n = 2 (duopolies) roots will be 1/2 and -1/2 and so realized quantity oscillates toward to Cournot equilibrium. For n = 3
roots will be given by vector (-1, 1/2, 1/2)T and so realized quantity improperly oscillates with constant amplitudes; for n = 4 root vector is (-3/2, 1/2, 1/2, 1/2)T and so realized quantity oscillates with explosive amplitudes from the Cournot equilibrium, and so on. The system is stable only for n = 2. 4. Cournot Oligopoly Model and Expectations In the section 3 oligopolies assumed that the production of other contributors does not vary in time. This assumption we can characterize as “naïve”, as it means that oligopolies do not behave rational. We will replace such “naivety” by an expectation assumption. Assume that each oligopoly expect that in time t oligopoly j will produce xej,t units of the market commodity. In time t every oligopoly for aggregate production considers: n
X i ,t = ∑ x ej ,t + xi ,t
(4.1)
j ≠i
instead of (3.1). We can the system (3.3) rewrite by:
xi ,t = −
1 n e a − ci ∑ x j ,t + 2b 2 j ≠i
(4.2)
There are many ways, how to form expectations. The most used one is the adaptive expectations. Every (ith) oligopoly expects that the production of the jth oligopoly is given by: x ej ,t − x ej ,t −1 = β i ( x j ,t −1 − x ej ,t −1 ) ; 0 < β i < 1
(4.3)
where βi is expectation parameter of the ith oligopoly. The interpretation of the (4.3) is well known and reader can study it for example in Gandolfo (page 40). For simplicity let us assume that all oligopolies have the same expectation parameter β = β0 = β1 =… βi … = βn. Let us define ith oligopoly expectations of the production of the others by taking the sum of (4.3) like this: n n ⎛ n ⎞ e e x x β x x ej ,t −1 ⎟ ; 0 < β < 1 − = − ⎜ ∑ ∑ ∑ ∑ j ,t j ,t −1 j ,t −1 j ≠i j ≠i j ≠i ⎝ j ≠i ⎠ n
(4.4)
By expressing of ith oligopoly expectations of the production of the others in time t and t-1 from (4.3) and substituting them into (4.4) we will get the first order system of simultaneous difference equations in matrix form: x t +1 = Ax t + b
(4.5)
where xt is a nx1 productioun vector (x1,t, x2,t, … xn,t)T, A is nxn matrix with elements of 1-β on the diagonal (ii) and of - β/2 outside the diagonal (ij;i≠j) and b is nx1 vector {β(a-ci)/2b)}i=1n. It can be shown that characteristic roots of the characteristic equation are [2-(n+1)β]/2 and (2- β)/2 with multiplicity (n – 1). There is monotonically movement, if all roots are positive and the system is stable if all roots in absolute value are less than 1. Solving these inequalities we will get the stability condition: n<
4−β
β
(4.6)
and the condition of the monotonic movement: n≤
2−β
β
(4.7)
In comparison of the stability condition (4.6) with the condition of the model without expectations (n < 3) this condition is less strict, as 0 < β < 1 and so expectations make the model more stable. In the case β = 1 the stability condition is the same as the stability condition from the origin model. This can be well understood from the ith oligopoly expectations of the production of the others (4.4). If β = 1, the production of the others will equal to their previous period production and so the model is the same as the one from 3rd section. On the other hand if β = 0 characteristic root is 1 and 0 with multiplicity n. In this case, as it follows from (4.4), expectations do not vary in time and so the production is constant for all t. To let the system be stable it is required to let the parameter β be positive and sufficiently small. The smaller β is (but positive), the better expectations oligopolies have and the more stable system is. In the dynamic Cournot model without expectations (section 3) there were in all cases oscillations. From (4.7) it follows that by the adaptive expectation assumption, there is a possibility of the monotonic movement. Again as β is smaller, the more probability is that movement will be monotonic. Conclusions In our paper we showed that under adaptive expectations Cournot oligopoly dynamics can be stable even if number of oligopolies is more than two. This result can be reached by adaptive expectations assumption, which is by our view, as we already have written, more realistic. The stability of the model is given by adaptive expectation parameter. We can interpret the parameter as ability of oligopolies to estimate the future production of the others. The less the parameter is (but positive), the better this ability is and so the better information makes the model more stable. This we can demonstrate by the intuition under perfect expectations. If all oligopolies have perfect expectations, variables xej,t become xj,t for all j, oligopolies can perfectly determine the production of the jth oligopoly for all j. By substituting for xej,t to (4.1) and solving the model we get the static Cournot equilibrium given by (2.4). The prices and quantities are, under perfect expectations presumption, perfectly elasticit and the system is stable. Acknowledgement This work was supported by the Grant VEGA No. 1/4652/07. References: [1] Gandolfo, G (2005) Economic Dynamics, Springer Verlag, Berlin, study edition. [2] Lukáčik, M. (2007) Game Theory and Economics, AIESA 2007, Bratislava. [3] Mikušová, N. (2007) Dynamic Duopoly Model, AIESA 2007, Bratislava. [4] Shone, R. (2003) Economic Dynamics, Cambridge University Press, 2nd edition. [5] Varian, Hal, R. (1999) Intermediate Microeconomics, W. W. Norton & Company; 5th edition