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Dynamic Games and Applications

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21-04-2017

Intertemporal Non-separability and Dynamic Oligopoly

Authors: Curtis Eberwein, Ted To

Published in: Dynamic Games and Applications | Issue 2/2018

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Abstract

We construct a framework for modeling dynamic Cournot oligopoly. We consider models where utility maximizing consumers give rise to demand functions that depend on current and prior period prices. Future demand depends on the current price and consumers, and firms must take this into account when making their decisions. Focusing on problems that yield dynamic demand functions that are linear in current and prior period price, we characterize the unique Markov perfect equilibrium in linear strategies. We then demonstrate the applicability of our framework through a series of practical examples.

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It is straightforward to allow for heterogeneity of consumers (we provide two such examples) provided that conditions ensuring nonnegativity of consumption are satisfied.

Given quadratic utility functions, there will generally exist an equilibrium with “policy functions” that are linear in the state. We focus on such equilibria.

It is straightforward to allow for i.i.d. idiosyncratic shocks, i.e., $c_{it}={\bar{c}}+\varepsilon _{it}$. In this case, rather than taking rival outputs as given, firms must form beliefs over the distribution over rival outputs.

Note that we have specified the problem as one where in equilibrium, rival outputs are functions of the prior period price and an individual firm maximizes discounted expected profits through choice of price. Equivalently, we could invert (1) to get $p_t(x_t^i+X_t^{-i},p_{t-1})$ and specify an individual firm’s problem more traditionally as:

$$\begin{aligned} V^i(p_{t-1},c_t)=\max _{x_t^i}[(p_t(x_t^i+X_t^{-i}(p_{t-1},c_t ),p_{t-1})-c_t)x_t^i+\beta E_tV^i(p_t,c_{t+1})]. \end{aligned}$$

In equilibrium, for every $i, x_t^i$ is a best response to $X_t^{-i}$ (i.e., $x_t^i$ solves the Bellman equation) and $X_t^{-i}=\sum _{j\ne i}x_t^j$.

With more complicated shocks and additional information and behavioral assumptions to ensure solvability, prices can be shown to follow more complicated stochastic processes. We examine one such extension in “Appendix 2.”

As shown in the prior section, given linear consumer demand, expected prices will indeed have this form. Linearity of demand will be verified shortly.

This system is a subset of the system of equations used in Appendix 1 to prove Theorem 1. It is important to bear in mind that this is an artificial system of equations. For example, application of stability conditions to establish properties of the solution (i.e., the correspondence principle in macroeconomics) would be a mistake since stability in the artificial system is meaningless.

It is straightforward to introduce a cost to holding inventory. Suppose for example that inventory decays by the factor $\delta $ so that the following budget constraint becomes: $p_t(i_{t+1}-\delta i_t+y_t)+w_t\le {\bar{w}}$.

This can easily be derived from the fact that $(N-1)^2\ge 0$.

The case where $d<0$ is analogous but requires $\lambda =-\sqrt{(2q-1)/\beta }$.

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Title: Intertemporal Non-separability and Dynamic Oligopoly
Authors: Curtis Eberwein
Ted To
Publication date: 21-04-2017
Publisher: Springer US
Published in: Dynamic Games and Applications / Issue 2/2018
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-017-0220-z

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Other articles of this Issue 2/2018

Evolutionary Games with Sequential Decisions and Dollar Auctions

Tauberian Theorem for Value Functions

Stochastic Differential Games for Which the Open-Loop Equilibrium is Subgame Perfect

One-Dimensional Stationary Mean-Field Games with Local Coupling

Potential Differential Games

Algebraic Formulation and Nash Equilibrium of Competitive Diffusion Games

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