Een Dynamisch Duopoliemodel

Summary

The basic problem of duopoly theory is the hypothesis about conjectural price variations. Such a conjectural hypothesis is not required if we draw a distinction between induced and autonomous price changes. Induced price changes (for the pricesp ₂ p2 and ∂p2 indicated as ∂p ₁ and ape, respectively) occur as a result of, and after, autonomous price changes, whereas autonomous price (indicated asdp ₁ anddp ₂) are not results of other (autonomous) price changes.

The duopoly model comprises two linear demand functions. For the sake of simplicity it has in the first instance been assumed that induced price changes depend on autonomous price changes according to the following linear price reaction functions:

$$\partial p_2 = q_{12} \cdot dp_1 and dp_1 = q_{12} \cdot dp_2 $$

where the price reaction coefficientsq ₂₁ andq ₁₂ are unknown constants.

The duopoly problem has been investigated in three directions. First, a dynamic process is presented in which both duopolists in each period reconsider their expectation about the price reaction coefficientsq ₂₁ andq ₁₂ respectively. These reconsiderations are based on recent factual price changes. The dynamic process converges to a steady state in which the expected price reaction coefficients equal the factual ones. The steady state refers to the equilibrium state of the duopoly situation because both duopolists cannot then increase their profit by an autonomous price change.

The second subject of investigation concerns the straightforward solution of the equilibrium state. By using the so-called continued partial derivatives (the first-order ones represent the sensitivity of both profits to autonomous price,changes), it is possible to deduct two first-order equilibrium conditions and two first-order reaction conditions that determine endogeneously the values ofp ₁,p ₂,q ₂₁ andq ₁₂ in the equilibrium state.

Finally, the case in which the duopoly model comprises two 5-th degree price reaction functions (instead of the abovementioned linear ones) has been investigated. For convenience in solving the model the 5-th degree has been preferred over the general n-th degree. These 5-th degree functions are

$$\left\{ \begin{gathered} \partial p_2 = q_{21} \cdot dp_1 + q_{21;2} \cdot (dp_1 )^2 + ... + q_{21;5} \cdot (dp_1 )^5 \hfill \\ \partial p_2 = q_{21} \cdot dp_2 + q_{12;2} \cdot (dp_2 )^2 + ... + q_{21;5} \cdot (dp_2 )^5 . \hfill \\ \end{gathered} \right.$$

It-is pointed out that all unknown price reaction coefficients are endogeneously determined by the reaction conditions of first-through fifth-order. Further, it is shown that linearity of the demand functions implies linear price reaction functions.

Generalization of these concepts towards competition amongst many multiproduct enterprises is feasible without any restriction; a publication on this problem is in preparation.