A consumer with income m who consumes a product of quality s_i and pays p_i obtains the utility s_im = 6 - p_i. If instead the consumer decides not buy the good, the resulting utility is zero. Consumer income m is uniformly distributed on the interval [2, 8] with the density 1/6. The total mass of consumers is equal to 1. There are two firms in the market. Firms 1 and 2 offer the qualities s₁ and s₂, respectively. We assume that s₁ ≤ s₂ and s₁, s₂∈ [1, 2]. Suppose that firm i has constant marginal cost equal to c · s_i. It is, thus, more expensive to produce higher quality.

1. Derive the demand of firms 1 and 2, and calculate the reaction functions of the two firms.

2. Calculate the Nash Equilibrium in prices and find the equilibrium profits as a function of s₁ and s₂. What are the equilibrium quality choices of the two firms?

3. How does an increase in c affect the profits of the two firms? Provide the economic intuition behind this result. Show that the high quality firm, firm 2, continues to earn higher profits than firm 1 as long as c < 5/6.