Nash Equilibrium Analysis in a Two-Firm Quality Differentiation Model

This analysis examines a scenario with two firms, Firm 1 and Firm 2, each offering products with distinct quality levels: Firm 1's product has quality 'H', and Firm 2's product has quality 'L', where 0 < 'L' < 'H'. Assuming zero marginal cost for both firms, each consumer has unit demand and a weight 'w' on quality. A consumer gains a payoff of 'w * q - p' when purchasing a product of quality 'q' at price 'p', and zero payoff if they choose not to buy. Consumers' weights 'w' are uniformly distributed between 0 and 1, meaning the fraction of consumers with 'w' between any 'w1' and 'w2' is 'w2 - w1'. Both firms simultaneously announce their prices, 'p1' by Firm 1 and 'p2' by Firm 2. Consumers then decide which product to purchase.

The goal is to identify all Nash equilibria satisfying the condition 'p1 / p2 ≥ H / L'. For each equilibrium, the fraction of consumers buying from each firm at the equilibrium prices will also be determined.

To find the Nash equilibrium, we need to analyze each firm's best response to the price chosen by the other firm. Let's begin with Firm 1's best response.

If Firm 2 sets a price 'p2', Firm 1 aims to maximize its profit, equivalent to the consumer surplus it generates. The consumer surplus for Firm 1's product of quality 'H' is calculated as '(w - p1)', where 'w' is the consumer's weight on quality. Similarly, the consumer surplus for Firm 2's product of quality 'L' is '(w - p2)'.

Given each consumer's unit demand, Firm 1 strives to maximize the fraction of consumers buying its product, which equates to maximizing the total consumer surplus it generates. Therefore, Firm 1 seeks to set the highest possible price 'p1' that still attracts consumers.

Since consumers' weights 'w' are uniformly distributed between 0 and 1, the highest possible value for '(w - p1)' is '(1 - p1)'. Consequently, Firm 1 wants to set 'p1' as low as possible, ideally 'p1 = 0', to attract all consumers.

Now consider Firm 2's best response. If Firm 1 sets a price 'p1 = 0', Firm 2 aims to maximize its profit by attracting the largest possible number of consumers. Firm 2's profit is equal to the consumer surplus it generates, calculated as '(w - p2)'.

Similar to Firm 1, Firm 2 seeks to set the highest possible price 'p2' that still attracts consumers. The highest possible value for '(w - p2)' is '(1 - p2)'. Therefore, Firm 2 aims to set 'p2' as low as possible, ideally 'p2 = 0', to attract all consumers.

Let's examine the condition 'p1 / p2 ≥ H / L'. With 'p1 = 0' and 'p2 = 0', the condition 'p1 / p2 ≥ H / L' is always satisfied.

Therefore, the Nash equilibrium in this case is for both Firm 1 and Firm 2 to set a price of 0. In this equilibrium, all consumers will buy from both firms, resulting in a 50% fraction of consumers buying from each firm.