Competition, Innovation, and Antitrust. A Theory of Market ... - Intertic

1.1.1 Monopoly and Antitrust Issues
1.1 A Simple Model of Competition in Quantities 5
Our first investigation of the market described above focuses on a monopoly.
Consider a single firm producing q. Itsprofit mustbegivenbyπ =(a −
q)q − cq − F . Its maximization requires an output satisfying the optimality
condition ∂π/∂q = a − 2q − c =0, 3 which can be solved for the monopolistic
output:
q M = a − c
2
The monopolistic price can be derived from the inverse demand function as
p M = a − q M =(a + c)/2, and the associated profits are: 4
(a − c)2
π M = − F
4
Imagine now that another firm enters in the market. When the two firms
compete at the same level, it is natural to imagine that their strategic choices
are taken simultaneously. In the equilibrium of this duopoly, both firms must
choose their output levels independently, and these output levels must be
consistent with each other. The result is a Cournot equilibrium.
Consider firms i and j. If they compete choosing independently their
outputs, firm i has the following profit functionπ i =(a − q i − q j )q i − cq i − F ,
and total production is now Q = q i + q j ; of course the profit offirm j is
the same after changing all indexes. Profit maximization by firm i requires
∂π i /∂q i =0or a − 2q i − q j = c, from which we obtain the so called reaction
function:
q i (q j )= a − c − q j
2
This is a rule of behavior for firm i whichcanbeinterpretedintermsof
expectations: the larger is the expected production of firm j, the smaller
should be the optimal production of firm i. Firmj will follow a similar rule:
q j (q i )= a − c − q i
2
The geniality of Cournot’s idea is that in equilibrium the two rules must be
consistent with each other. In terms of expectations, the equilibrium production
of each firm must be the optimal one given the expectation that the other
firm adopts its equilibrium production. Mathematically, we can solve the system
of the two reaction functions to find out the production of each firm in
3 The second order condition ∂ 2 π/∂q∂q = −2 < 0 guarantees that the profit
function is concave, so that the solution corresponds to a maximum.
4 We will assume that F is small enough to allow profitable entry by one firm in
the market.