# A differential game analysis of R&D in oligopoly with differentiated goods under general demand and cost functions: Bertrand vs. Cournot

## TL;DR

This paper models R&D competition among differentiated goods firms in oligopolies using differential games, analyzing how firm and industry R&D investments vary with the number of firms, demand, costs, and strategic interactions.

## Contribution

It provides a comprehensive differential game framework for R&D in oligopolies with general demand and cost functions, comparing Bertrand and Cournot models and solution concepts.

## Key findings

- Steady state R&D per firm decreases as the number of firms increases.
- Total industry R&D investment increases with the number of firms.
- Memoryless closed-loop and feedback solutions are shown to be equivalent.

## Abstract

We study a dynamic oligopoly with differentiated goods by differential game approach under general demand and cost functions. We show that the steady state value of the R&D investment by each firm is decreasing with respect to the number of firms, and the steady state value of the industry R&D investment is increasing with respect to the number of firms. Also we show that if there is no spillover, whether the R&D investment of each firm given the cost level in the memoryless closed-loop case is larger or smaller than that in the open-loop case depends on whether the strategic variables are strategic substitutes or strategic complements. Further we show that the memoryless closed-loop solution and the feedback solution (by the Hamilton-Jacobi-Bellman equation) are equivalent.