A short formal proof to show that when the rights to a product are shared between two firms the outcome is worse for the firms and consumers.

1. Consider the market for all 138 Premier League games broadcast over a season.1
92 of
these games are shown by Sky Sports and 46 are shown by Setanta Sports. To watch the
game a consumer must pay a fee to both companies.
Let's suppose that consumers' consent to pay is distributed uniformly along the [0 1]
continuum between. Let P1 and P2 be the prices charged by the rights holders. All
consumers whose consent to pay is higher than P1 + P2 will request a subscription from each
owner and pay the total fee to watch the 138 games.
Owing to the distribution of preferences, demand for subscriptions is equal to 1 – (P1 + P2).
The short run producer surplus of each company (assuming zero marginal cost and ignoring
sunk costs) can be given by P.Q.2
The profit of right holder 1 (Sky Sports) can be written as P1[(1 – (P1 + P2)] and that of right
holder 2 (Setanta Sports) as P2[( 1 – (P1 + P2)]. To find the prices that maximise producer
surplus we get the first derivative with respect to price and set to zero for each rights
holder.
License Holder 1. License Holder 1.
( )[ ]
21
2
11
2111 1
PPPP
PPP
+−=
+−=Π ( )[ ]
21
2
22
2122 1
PPPP
PPP
+−=
+−=Π
12
21
21
21
1
1
=−⇒
+−=
Π
PP
PP
dP
d
12
21
12
12
2
2
=−⇒
+−=
Π
PP
PP
dP
d
As a result of this we have a simple set of simultaneous equations
2P1 – P2 = 1
2P2 – P1 = 1
1
Obviously in cases such as this the market definition chosen is key. In this instance we could consider the
market to be the broadcast of an individual Premier League game, the broadcast of a subset of Premier League
games (a package), the broadcast of all Premier League games, all football broadcasting, all team sport
broadcasting or even all sport broadcasting. We will take it to be the 138 games shown over a season.
2
Producer surplus in the long run (profit) will depend on the sunk costs of obtaining the rights and the fixed
costs of the broadcasts.

2. The solution to this leads to P1= P2 = 1/3. The price of the full subscription to the consumer
will therefore be P1 + P2 = 2/3.
Assume now that there is only one owner of the broadcast rights who sets a single price of
P. His short rub producer surplus is written as P(1 - P), and the derivative is equal to 1 - 2P.
Setting this to zero gives the value P = 1/2.
In other words, when the exclusive rights are in the hands of one firm, the price of the good
will be set at a lower level and more consumers will have access to it.
Note that a concentration of rights is also favourable to owners. When there are two
owners, each makes a profit of 1/9, i.e., a total profit of 2/9, whereas with one owner, the
profit is 1/4, which is more than 2/9.
Two rights holders is a second best outcome for both firms and consumers.