Write the monopolist’s program and derive the optimal levels of quantity and quality

Derive the optimal prices and the payoffs for each firm

 

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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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