Consumer Choice with Quality Differentiation: Finding Equilibrium Market Shares

To find the fraction of consumers buying from each firm given p1 and p2, we need to determine the threshold prices at which consumers switch from buying one firm's product to the other.

Let's start by considering the consumers with w ≥ H. These consumers have a preference for higher quality products, so they will only buy from Firm 1 if p1 < H. Otherwise, they will choose not to buy anything. Therefore, the fraction of consumers buying from Firm 1 among this group is 1 if p1 < H and 0 otherwise.

Now let's consider the consumers with L ≤ w < H. These consumers have a preference for quality, but they may still choose Firm 2's product if the price difference is significant. The threshold price at which they switch from buying Firm 2's product to Firm 1's product is p1 = p2 + (H - L). If p1 < p2 + (H - L), these consumers will buy from Firm 1, otherwise, they will buy from Firm 2. The fraction of consumers buying from Firm 1 among this group is (p2 + (H - L) - p1) / (p2 + (H - L) - p2) = (H - L - p1) / (H - L).

Finally, let's consider the consumers with 0 ≤ w < L. These consumers have a preference for lower prices and are unlikely to buy anything if the prices are too high. The threshold price at which they switch from buying nothing to buying from Firm 2 is p2 = L. If p2 < L, these consumers will buy from Firm 2, otherwise, they will choose not to buy anything. The fraction of consumers buying from Firm 2 among this group is 1 if p2 < L and 0 otherwise.

To summarize, the fraction of consumers buying from Firm 1 is:

• 1 if p1 < H
• (H - L - p1) / (H - L) if H ≤ p1 < p2 + (H - L)
• 0 otherwise

The fraction of consumers buying from Firm 2 is:

• 1 if p2 < L
• 0 otherwise