Price Competition with Quality Differentiation: Finding the Consumer Market Share

Let's denote the fraction of consumers buying from Firm 1 as 'x' and the fraction buying from Firm 2 as '1-x'.

If a consumer buys from Firm 1, their payoff is 'wH - p1'. If a consumer buys from Firm 2, their payoff is 'wL - p2'.

Consumers will buy from Firm 1 as long as 'wH - p1 ≥ wL - p2'. Rearranging this inequality, we get 'w(H-L) ≥ p1 - p2'. Since 'w' is uniformly distributed between 0 and 1, the fraction of consumers with 'w' such that 'w(H-L) ≥ p1 - p2' is 'H-L'. Therefore, the fraction of consumers buying from Firm 1 is '(H-L)x'.

Similarly, the fraction of consumers buying from Firm 2 is '(H-L)(1-x)'.

To find the Nash equilibrium, we need to find the values of 'x', 'p1', and 'p2' that satisfy the following conditions:

'x ≥ 0' and 'x ≤ 1' (fraction of consumers buying from Firm 1 must be between 0 and 1)
'1-x ≥ 0' and '1-x ≤ 1' (fraction of consumers buying from Firm 2 must be between 0 and 1)
'p1/p2 ≥ H/L' (price ratio must be greater than or equal to quality ratio)

We can start by assuming that 'p1/p2 = H/L' and solve for 'x':

'(H-L)x / (H-L)(1-x) = H/L'

Simplifying, we get 'x / (1-x) = H/L'

Cross multiplying, we get 'Lx = H(1-x)'

Expanding, we get 'Lx = H - Hx'

Rearranging, we get '(L+H)x = H'

Dividing by 'L+H', we get 'x = H / (L+H)'

Now we need to check if this value of 'x' satisfies the conditions 1 and 2.

Since 'H > L', 'H / (L+H) > 0' and 'H / (L+H) < 1'. Therefore, the condition 'x ≥ 0' and 'x ≤ 1' is satisfied.

Similarly, '1 - H / (L+H) > 0' and '1 - H / (L+H) < 1'. Therefore, the condition '1-x ≥ 0' and '1-x ≤ 1' is satisfied.

Now we can find the fraction of consumers buying from each firm:

Fraction buying from Firm 1 = '(H-L)x = (H-L) * H / (L+H)' Fraction buying from Firm 2 = '(H-L)(1-x) = (H-L) * (1 - H / (L+H))'

Note that consumers whose 'w' are close to 0 may choose not to buy anything, so the actual fraction of consumers buying from each firm may be lower than the calculated fractions.

To summarize, the fraction of consumers buying from Firm 1 is '(H-L) * H / (L+H)' and the fraction of consumers buying from Firm 2 is '(H-L) * (1 - H / (L+H))'.