Economies of Scope, Entry Deterrence and Welfare

Appendix

Proof of Lemma 1

If I does not expand to market B, then when E enters there is a duopoly with symmetric cost, and post-entry profits are given by . As a consequence, entry in market A is optimal as long as ■

Proof of Lemma 2

Not entering when I does not expand can only be optimal for . Since , for all (equality holds for ), it follows that , for all . Thus, regardless of the entrant’s beliefs about , it is optimal not to enter when I expands to market B. ■

Proof of Lemma 3

When the entrant’s profit in case of entry is nil, thus entry cannot be profitable. When entry is profitable if the expected profit, given that is uniformly distributed on , is higher than the entry costs. Finally, when expands to market , if the entrant’s profits are , hence E should not enter; if the entrant’s profits are and entry is profitable if and only if ■

Proof of Lemma 4

A monopolist incumbent with no threat of entry would expand to market B if and only if condition [1] holds. Substituting the values of the profits, the condition is equivalent to:

[6]

For it is easy to verify that satisfies the previous condition, thus . On the other hand, for the previous condition is not satisfied even for , implying that no type of incumbent wants to expand to market B. Finally, for condition [6] holds in equality for

Thus the incumbent enters if and only if ■

Proof of Lemma 5

Substituting the equilibrium profits in condition [2], we conclude that expansion to market B is optimal as long as:

[7]

For it is easy to verify that satisfies the previous condition; therefore, . On the other hand, for the previous condition is not satisfied even for , implying that no type of incumbent wants to expand to market B. Finally, for condition [6] holds in equality for

Thus the incumbent enters if and only if ■

Proof of Lemma 6

Substituting the equilibrium profits in condition [3], we conclude that expansion to market B is optimal as long as:

[8]

For it is easy to verify that satisfies the previous condition, thus . On the other hand, for the previous condition is not satisfied for , implying that no type of incumbent wants to expand to market B. Finally, for condition [8] holds in equality for

Thus the incumbent enters if and only if ■

Proof of Proposition 1

We need to check that the incumbent’s strategy is optimal given the entrant’s strategy, that the entrant’s strategy is optimal given beliefs and that beliefs are consistent with Bayes rule and the incumbent’s equilibrium strategy. When Lemmas 5 and 6 imply that and thus the optimal strategy of the incumbent is to expand for all , regardless of the entrant’s strategy. Since all incumbent’s types expand to market B, expansion to market B is not informative about , thus posterior beliefs should be equal to the prior beliefs that is uniformly distributed on . Given these beliefs, the optimality of the entrant’s strategy follows from Lemmas 1–3. Note that since ■

Proof of Proposition 2

When Lemmas 5 and 6 imply that the incumbent follows a cut-off strategy for all the possible strategies of the entrant and that ( is equal to 1 for ). Considering this, the proofs of cases 1 and 2 are immediate consequences of Lemmas 1–3 and Lemmas 5 and 6.

In case 3 one can show that there cannot exist a PBE where the entrant follows a pure strategy when I expands. If E never enters when he observes I expanding to market B, then types would expand to market B. However, considering the posterior beliefs, the entrant would be better off by entering as , a contradiction. Similarly, if E always enters when I expands to market B, only types want to expand to market B, but then it would be optimal for E not to enter as , a contradiction. Thus, if there does not exist a PBE where E follows a pure strategy when I expands to market B.

Let us now check the mixed strategy PBE. In order for it to be optimal for to follow a mixed strategy when I expands to market B, firm E has to be indifferent between entering and not entering. That is, has to be such that condition [4] holds.

Considering the optimal strategy of firm E (entering when I does not expand to market B, entering with probability when I expands to market B), firm I should expand to market B if and only if:

[9]

Thus if is the solution to eq. [5], then type will be indifferent between expanding or not to market B while types strictly prefer to expand to market . Thus it is optimal for I to expand to market B for . Finally, the belief that is uniformly distributed on is consistent with the cut-off strategy of the incumbent. ■

Proof of Proposition 3

When Lemma 5 implies that and Lemma 6 implies that if I expects E to always enter then I does not expand for all .

To prove 1. (a) we just need to note that, given E’s strategy, not expanding to market B is indeed optimal for all . Moreover, when the incumbent does not expand to market , it is optimal for E to enter by Lemma 1 and, given beliefs, it is also optimal to enter as .

The proofs of 1.b and 2 are similar to the proofs of cases 3 and 2 in the previous proposition, respectively.■

Proof of Lemma 7

Consider the equilibrium 1.(a) of Proposition 3 and let be type ’s equilibrium payoff. Let e denotes the off-the-equilibrium path action of expanding and be the set of mixed strategy best responses (MBR) of the entrant when the incumbent expands if the entrant’s beliefs are concentrated on the set of types T that make type strictly prefer expanding to his equilibrium strategy. That is:

and let be the set of mixed strategy best responses of the entrant that make type exactly indifferent between expanding or not. A type is deleted according to criterion when expansion is observed (action e is observed) if there exists a type such that:

[10]

In other words, type is eliminated if the set of the entrant’s responses that make type willing to deviate is strictly smaller than the set of responses that make type willing to deviate. Since the set of mixed strategies for which wants to deviate (i.e. expand) is strictly smaller than the set of mixed strategies for which type wants to deviate, all types except are eliminated according to criterion . But if the entrant believes when he observes expansion, we does not want to enter. Thus 1.(a) does not survive criterion .

The proof that equilibrium 1.(a) of Proposition 3 does not satisfy the universal divinity criterion is similar as the only difference is that in order to eliminate type when e is observed the RHS of eq. [10] is substituted by ■

Proof of Proposition 4

Let us consider first the case of a pure strategy PBE. If expansion by type was prohibited, the entrant would enter but the incumbent of type would not expand, thus the consumer surplus would be . On the other hand, if expansion by type was allowed, E would not enter and consumer surplus would be . However, since the RHS is decreasing with and the inequality holds for . Thus consumers would be worse off if entry deterrence was prohibited.

In the case of a mixed strategy equilibrium, the proof is similar. If entry deterrence expansion was prohibited, E would enter and consumer surplus would be . On the other hand, if expansion by type was allowed, the entrant would enter with probability and not enter with probability . The expected consumer surplus would be

[11]

Note that eq. [11] is decreasing with and for is equal to . Thus, the expected consumer surplus when entry deterrent expansion is allowed is higher than when entry deterrence is prohibited. ■

Proof of Proposition 5

Under the assumptions we know that the PBE is a pure strategy entry deterrence equilibrium (see case 2 of Proposition 3). We first prove that preventing entry deterrence decreases welfare for and . The rest of the result follows from continuity. For , preventing entry deterrence decreases welfare if:

Since and we know that

Since for

For , the RHS of the previous is

which holds for all c.

When , , thus

For , the RHS of the previous is positive if and only if

which holds for all .

Thus, for all , we have shown that for and

Since the LHS is continuous with and , by the sign preserving property of continuous functions there exists an such that if , then the function in the LHS is still positive. ■

Proof of Proposition 6

Under the assumptions we know that the PBE is a pure strategy entry deterrence equilibrium (see case 2 of Proposition 3). We first prove that preventing entry deterrence decreases welfare for and . The rest of the result follows from continuity. For , we know that preventing entry deterrence decreases welfare if

The LHS of this condition is decreasing with . Thus if the condition holds for , then it holds for all . Substituting in the expression and simplifying we obtain

We know that and that, for , which implies

Thus a sufficient condition for prohibition against entry deterrence to decrease welfare is

which holds for .

For , we know that . Thus, a sufficient condition for prohibition against entry deterrence to decrease welfare is

which holds for all .

We showed already that for , and we have

Since the LHS is continuous with , by the sign preserving property of continuous functions, there exists an such that if then the function in the LHS is still positive. Consequently, for close enough to 1 and , a prohibition against entry deterrent expansion decreases welfare for all ■

Proof of Proposition 7

Under the assumptions we know that the PBE is a pure strategy entry deterrence equilibrium (see case 2 of Proposition 2). We first prove that preventing entry deterrence increases welfare for and . The rest of the result follows from continuity. For , preventing entry deterrence increases welfare if:

The RHS is decreasing with , thus this condition is easier to be satisfied for . Let us consider . By Lemma 5, this implies that . For the previous condition is equivalent to

For and , is equal to

Thus when and the condition for entry deterrence prohibition to be optimal for is

which holds for .

When , is equal to

Thus when and the condition for entry deterrence prohibition to be optimal for is

which holds for all . Thus when , and it is optimal to deter entry for type as long as . By continuity, preventing entry increases welfare for and close enough to 1. ■