Download Problem Set 3

Problem Set 3
Problem 1
Consumers are characterized by a willingness to pay function U(m, x) for a certain
durable commodity, where m is the quality level of the commodity and x is the taste parameter of
the consumer. Assume that
U(m, x) = [a+bx]m
where a,b>0
Consumers’ taste parameter, x, is uniformly distributed along the unit interval.
The technology is characterized by constant returns to scale, where the unit cost of
production depends upon the quality level of the product, which can be any nonnegative number.
Assume unit cost C(m)=m2.
1.
Assume that a single firm exists in the market. It considers producing two types of goods
of qualities m1 and m2 where m1>m2. Find what are the quality levels selected. How
much is produced of each type of product? Are the profits of the monopolist higher when
he offers two types of commodities instead of a single type of commodity?
2.
Assume that two firms exist in the market and each can produce a single type of good.
Characterize the equilibrium assuming that the strategy choice of each firm consists of
the quality level of its product and the quantity to produce of it. Both quality and
quantity are simultaneously selected. Compare this equilibrium with the predictions
obtained when a monopolist produces two quality levels.
Problem 2
Assume consumers are uniformly distributed according to their “ideal points” on a line of length
l. Two firms are located on this line at distances “a” and “b” from the endpoints, respectively.
a
b
Firm 1
Firm 2
Each firm can enhance the quality of its product by m units at a cost of m2 per unit. The utility
derived by a representative consumer of i who is located at distance x from this firm is given as
follows:
U i x   v  tx  mi  pi  ,
where v is a positive parameter.
1
Consider the following two stage game. In stage 1 each firm chooses its location on the
line (Firm 1 chooses “a” and Firm 2 chooses “b”) as well as the quality of its product. In stage 2
firms compete as Bertrand oligopolists in setting prices. Discuss the properties of the equilibria
and demonstrate the possible non-existence of pure-strategy Nash equilibria. How does your
solution change if the transportation cost (disutility) is expressed as a quadratic function tx2.
Problem 3
Two firms produce a differentiated product and face the following linear demand:
pi = a – bqi – dqj
a,b,d>0
0<d<b
The firms expect to compete against each other for an infinite period.
a.
Assume that firms compete as Cournot oligopolists. What is the minimum discount
factor necessary in order to sustain the collusive outcome.
b.
Repeat (a) under the assumption that firms compete as Bertrand oligopolists.
Hint:
(i)
Assume that each firm can, immediately, detect deviations from the collusive
behavior. In response, the firm retaliates by reverting to the non-cooperative
Cournot (Bertrand) production level (pricing), in all subsequent periods.
(ii)
Notice that the “best deviation” the firm can consider in a certain period, is the
quantity (price) which corresponds to the “static best response” on the firm’s
reaction function (under the assumption of cooperative behavior of the rival in the
period of the deviation).
2