**1.** Consider a linear city model where two firms, 1 and 2, are located at two endpoints of a [0,1] interval. Consumers are located uniformly over the interval and buy at most one unit of a good served by either firm. Firms choose two variables, quality si and price pi , i = 1; 2. Given (si ; pi) chosen by firm i = 1; 2, if a consumer located at xE[0,1] purchases a unit from firm i = 1; 2, her utility is:

v+si-pi-t(i-1-x)2

where v is some large number (large enough to ensure that consumers will purchase from either firm in the equilibria considered below), and t > 0. Suppose that firm i needs to incur a constant marginal cost of production, c(si), per each unit sold for a good with quality si , where c'(.) > 0, c”(.) > 0, and c'(0) = 0, and c'(infinity) = infinity. (e.g., c(si) = si2). Suppose a game in which qualities are given and firms choose prices simultaneously. Characterize the equilibrium in price of this game when the qualities are identical and when they are different.

**2.** 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) = 0.5s2q

**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)?

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