Nash Equilibrium in a Duopoly with Quality Differentiation

To find the fraction of consumers buying from each firm, we need to consider the payoffs and choices of the consumers.

Let's assume that a fraction 'x' of consumers buy from Firm 1 and a fraction (1-'x') buy from Firm 2.

For consumers buying from Firm 1:

• The payoff for consumers with quality 'H' is 'wH' - 'p1'.
• The payoff for consumers with quality 'L' is 'wL' - 'p1'.

For consumers buying from Firm 2:

• The payoff for consumers with quality 'H' is 'wH' - 'p2'.
• The payoff for consumers with quality 'L' is 'wL' - 'p2'.

Consumers will choose to buy from Firm 1 if the payoff from buying from Firm 1 is greater than the payoff from buying from Firm 2. This can be expressed as:

'wH' - 'p1' > 'wH' - 'p2' 'wL' - 'p1' > 'wL' - 'p2'

Simplifying these inequalities, we get:

'p2' - 'p1' > 0 'p2' - 'p1' > 0

Since 'p1'/'p2' ≥ 'H'/'L', we know that 'p2' - 'p1' > 0.

This means that Firm 1 will attract consumers with quality 'H' and Firm 2 will attract consumers with quality 'L'.

Now let's consider the fraction of consumers buying from each firm based on their quality. Since the consumers' 'w' are uniformly distributed between 0 and 1, the fraction of consumers with quality 'H' is 'H' and the fraction of consumers with quality 'L' is 'L'.

Therefore, the fraction of consumers buying from Firm 1 is 'Hx' and the fraction of consumers buying from Firm 2 is 'L(1-'x')'.

To find the Nash equilibrium, we need to find the value of 'x' that satisfies the condition 'p1'/'p2' ≥ 'H'/'L'.

Let's assume that 'p1'/'p2' = 'z', where 'z' ≥ 'H'/'L'.

From the inequalities 'p2' - 'p1' > 0, we can rewrite 'z' as 'p1'/'p2'.

'p1'/'p2' = 'p1'/('p1' + 'p2')

Simplifying this equation, we get:

'p1' = 'z'p1' + 'z'p2' (1 - 'z')'p1' = 'z'p2' 'p1'/'p2' = 'z'/(1 - 'z')

Since 'p1'/'p2' ≥ 'H'/'L', we know that 'z'/(1 - 'z') ≥ 'H'/'L'.

To find the value of 'x', we need to find the value of 'z' that satisfies this inequality. This can be done by rearranging the inequality:

'z'/(1 - 'z') ≥ 'H'/'L' 'Lz' ≥ 'H'(1 - 'z') 'Lz' ≥ 'H' - 'Hz' ('L' + 'H')'z' ≥ 'H' 'z' ≥ 'H'/('L' + 'H')

Therefore, the fraction of consumers buying from Firm 1 is 'Hx' = 'H'('H'/('L' + 'H')) = 'H'^2/('L' + 'H') and the fraction of consumers buying from Firm 2 is 'L'(1-'x') = 'L'(1 - 'H'/('L' + 'H')) = 'L'('L'/('L' + 'H')) = 'L'^2/('L' + 'H').

So, the fraction of consumers buying from each firm is 'H'^2/('L' + 'H') and 'L'^2/('L' + 'H'), respectively.