Consider that consumers have utility
U={ θs-P if they purchase

0 otherwise}
where s represents the quality, p the price, and θ the taste parameter. θ is distributed over the population with cumulative distribution F. If we normalize N = 1, the demand function is q = 1- F( p/s)
where F^-1, the inverse demand function is an increasing function. To simplify, let θ be uniformly distributed on [0; 1]. The cost function is C(q, s) = 0.5s^2.q

1. Monopoly.
a. Write the monopolist's program and derive the optimal levels of quantity and quality.
b. Write the social planner's program, and derive the optimal levels of quantity and quality.
c. Show that, when the difference in output is taking into account, the monopoly and the social planner choose the same quality.

2. Duopoly.

Consider now that there are two qualities s1 and s2 with s2 > s1 provided by two different  firms 1 and 2. The timing is the following:  first  firms choose their qualities, second they compete in price.
a. Derive the demand for each  firm.
b . For given qualities, write the optimization program for each  firm, and derive their best response functions (price competition).
c. Derive the optimal prices and the payoffs for each  firm.
d. What would be the optimal qualities ( first stage of the game)?