Download Contracts as a barrier to entry : A contract between an incumbent

Contracts as a barrier to entry : A contract between an incumbent firm and a customer can create
a barrier to entry for the entrant.
Suppose that there is one buyer who is willing to pay a maximum price of 1 for one unit of an
input. The incumbent seller has cost 5 . The potential entrant’s cost is unknown to the buyer and
the incumbent. Both expect it to be a value uniformly distributed in the [0, 10] interval.
Timeline :
The entrant decides whether to enter. If he stays out the incumbent remains a monopoly and
charges the maximum price which is 10.
If the entrant enters, the two firms (the incumbent and the entrant) play a price-setting game.
(That means : Bertrand competition). In the Nash equilibrium of this game the low cost firm sets a
price just below the high-cost firm’s cost.
As an example, suppose the entrant’s cost is 3. Then in the Nash equilibrium, it will set p = 4.999… and the incumbent
will set p = 5. The buyer will buy from the entrant.
CASE 1 No contract between the incumbent and the buyer.
If the entrant’s cost is greater than 5, then no entry takes place and the incumbent sets the
monopoly price which is 10. If the entrant’s cost is less than 5 , then the entrant enters and sets
price equal to the incumbent’s cost, 5 – ε where ε is a positive and very small number. Without a
contract the buyer’s expected payoff is:
1/2 times zero (no entry) plus 1/2 times (10 – 5 ) (entry takes place)
The buyer’s expected payoff is 2.5.
The incumbent’s expected payoff is:
1/2 (no entry) times (10 – 5 )
This is equal to 2.5.
Note that the probability of entry is 1/2.
CASE 2 Contract between the Buyer and the Incumbent
Before the entrant makes the entry decision, the incumbent offers the following contract to the
buyer. The buyer promises to purchase the input from the incumbent at a price 7.5. If the buyer
later wishes to switch to the entrant, then the buyer must pay the incumbent a penalty of 5.
We will now show that the buyer is equally well off, and the incumbent is strictly better off, with
this contract than without it. First notice that, despite the contract, entry still takes place when the
entrant is very efficient. In fact, if the entrant’s cost is less than 2.5, then the entrant will enter and
set its price equal to 2.5. Given this price, the buyer switches: it now pays 2.5 (price) plus 5
(penalty from breach of contract), a total of 7.5 (the price required by the incumbent). Following
this reasoning, we conclude that entry occurs with probability 1/4, which is the probability that
the entrant’s cost is lower than 2.5. The buyer’s expected payoff is 1/4 times (10 – 2.5 – 5 ) (entry
takes place) plus 3/4 times (10 – 7.5 ) (entry does not take place). This is equal to 2.5 (the same
value as before). As for the incumbent, he now gets 1/4 times 5 (entry takes place and the entrant
receives the damage payment) plus 3/4 times 7.5 – 5 (entry does not take place), or 50/16 = 3.125.
The loser in this process is the entrant. Without contracts, the entrant sells at a price 5, whereas
with the above contract the maximum the entrant can charge is 2.5. Notice that, if the entrant’s
cost lies in the interval [ 2.5 ; 5 ], then entry does not take place even though the entrant is more
efficient than the incumbent.