Consumer Demand Analysis in a Duopoly with Quality Differentiation

There are two firms, Firm 1 and Firm 2. Firm 1's product has quality 'H' and Firm 2's product has quality 'L', with 0 < 'L' < 'H'. For simplicity, assume that both firms have zero marginal cost. Each consumer has unit demand and a weight 'w' on the quality: she receives payoff 'w'q - 'p' if she buys a product of quality 'q' at price 'p' and zero payoff if she buys nothing. The consumers' 'w' are uniformly distributed between 0 and 1, meaning that the fraction of consumers with 'w' between any 'w1' and 'w2' is 'w2' - 'w1'. The two firms simultaneously announce their prices, 'p1' by Firm 1 and 'p2' by Firm 2. Each consumer then chooses which product to purchase.

To find the fraction of consumers buying from each firm given 'p1' and 'p2', we need to compare the payoffs for consumers buying from each firm.

Let's assume a consumer has 'w' as their weight on quality. If they buy from Firm 1, their payoff would be 'w'H - 'p1'. If they buy from Firm 2, their payoff would be 'w'L - 'p2'. If they choose not to buy anything, their payoff would be 0.

Now, let's consider the different cases:

Case 1: 'p1' ≤ 'p2' In this case, consumers with 'w' > 'p2'/(H - L) will buy from Firm 2, consumers with 'w' < 'p2'/(H - L) will choose not to buy anything, and consumers with 'p2'/(H - L) ≤ 'w' ≤ 'p1'/H will buy from Firm 1.

The fraction of consumers buying from Firm 1 is given by the fraction of consumers with 'p2'/(H - L) ≤ 'w' ≤ 'p1'/H, which is ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L)).

The fraction of consumers buying from Firm 2 is given by the fraction of consumers with 'w' > 'p2'/(H - L), which is 1 - ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L)).

Case 2: 'p2' < 'p1' In this case, consumers with 'w' > 'p1'/H will buy from Firm 1, consumers with 'p2'/(H - L) ≤ 'w' ≤ 'p1'/H will buy from Firm 2, and consumers with 'w' < 'p2'/(H - L) will choose not to buy anything.

The fraction of consumers buying from Firm 1 is given by the fraction of consumers with 'w' > 'p1'/H, which is 1 - 'p1'/H.

The fraction of consumers buying from Firm 2 is given by the fraction of consumers with 'p2'/(H - L) ≤ 'w' ≤ 'p1'/H, which is ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L)).

In summary:

If 'p1' ≤ 'p2':

• Fraction buying from Firm 1: ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L))
• Fraction buying from Firm 2: 1 - ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L))

If 'p2' < 'p1':

• Fraction buying from Firm 1: 1 - 'p1'/H
• Fraction buying from Firm 2: ('p1'/H - 'p2'/(H - L)) / (1 - 'p2'/(H - L))