A Exclusive Licensing Subgame

Suppose that the innovator offers only one license, so that only one D firm adopts the new input. Firm 1 does not get the new input and produces a final good of quality s1=s at price p1; firm 2 adopts the new input and produces a final good of quality s2=ψs>s, at price p2 with s2−s1=sψ−1. Assuming that both firms stay active in the market, the demands for the goods are defined in eqs. (5) and (6). D firms’ profits are:

In order to find the Cournot equilibrium, we assume that the demands for both goods are positive (formally, this means that 0>θ_=p1s>θˆ=p2−p1sψ−1>1), we derive the corresponding candidate equilibrium and we finally check whether this is the effective equilibrium.^[32]Cournot competition leads to the following third stage quantity and price equilibrium:

with q2r2,0;sψ,s>0⟺s2ψ−12>r2. Firm1 profit is then:

As for firm 2:

The U firm chooses the two-part tariff contract for firm 2 r2,F2 such that:

with ΠU=2sψ−s−2r2s4ψ−1r2+F2−f and

if we reasonably assume that r1=r2 (D firms are symmetric so that the equilibrium royalty set by the U innovator would be equal for either D firm). The first constraint comes from the non-negativity of q2 and the second constraint (binding at equilibrium) ensures that firm 2 has the incentive to get the license rather than the outside option, that is not buying the innovation given that the rival firm 1 would get it.^[33] The maximization problem thus becomes

This objective is concave in r2 and the maximum is r2=−s2. Under non-negative royalties, the solution is a contract such that (superscript EL stands for exclusive licensing):

Equilibrium quantities and prices are:

Equilibrium profits are:

Producer surplus, consumer surplus and social welfare are:

ΠUEL defined in eq. (10) is the U patent holder equilibrium profit under exclusive licensing, when selling via a two-part tariff, which reduces to a fixed fee, the new input to only one D firm. We check whether at equilibrium, the conditions for both firms to stay in the market are satisfied, and we find that they are:

Consider now the case of negative royalties, the optimal contract under exclusive licensing is then such that:

Equilibrium quantities, prices, firms’ profits, the external patentee’s equilibrium profit, producer surplus, consumer surplus and social welfare are (superscript ELneg stands for exclusive licensing and negative royalties):

Note that under both cases of non-negative and negative royalties, the D firms are worse off with respect to the status quo.^[34] However, if either firm thinks that the rival is not making an offer, then this firm will have the incentive to make a slightly positive offer and get the innovation and this reasoning holds up to the outside option. Note also that, as one could expect ΠUEL−ΠUELneg=−s44ψ−1>0, the patent holder prefers not being constrained to non-negative royalties. Also, from the industry point of view, ΠEL−ΠELneg=s44ψ−1>0, given that under negative royalties production is subsidized. However, computing the joint profits of the patent holder and the licensee under vertical separation we find that ΠUEL+π2ELsψ,s=ΠUELneg+π2ELnegsψ,s, that is the two separated subjects, jointly, are indifferent between negative and non-negative royalties. As for consumers and social welfare we find that:

Finally, we verify that, also at this equilibrium, the conditions for both firms to stay in the market are satisfied:

B Complete Technology Diffusion Subgame

Suppose the U firm decides to offer two licenses, m=2. In this case, the patent holder will also set a minimum bid b, otherwise no firm would make a positive offer because each firm is guaranteed to have a license irrespective of its bid.\ When the policy m=2, r and b is offered, in equilibrium each firm will offer this minimum bid, b.^[35] The U firm maximization problem is:

where π10,r2;s,sψ is defined in eq. (8) for firm 2. Here the outside option for each firm is not buying the new input given that the rival firm does. The optimal fixed fee corresponding to b is πiri,rj;sψ,sψ−πi0,rj;s,sψ. πiri,rj;sψ,sψ and qiri,rj;sψ,sψ denote the third stage equilibrium D firm i profit and quantity when both firms produce the high-quality good, namely:

As the two constraints are binding at equilibrium, we have

with πi0,rj;s,sψ=sψ+rj24ψ−12s. The maximization problem, thus becomes:

Solving this problem we find that under non-negative royalties, the optimal contract is:

where rT≥0⟺ψ≥ψT=1.5856. This means that when the innovation is small the inventor’s incentive is to set a per-unit price as low as possible, that is the optimal contract is a fixed fee. In other words, given the positive outside option of producing the low-quality good at zero marginal cost, for small innovations the inventor cannot set a positive per unit royalty. In contrast for large innovations we have a positive per-unit royalty.

If we allow for negative royalties, the optimal contract is the first line of eq. (7) for any innovation size. Note that for a small innovation size (ψ>ψT), the U monopolist is willing to subsidize the D production, however in this case the explanation is not the incentive to make the D affiliates more aggressive but the presence of the outside option: each D firm can always produce the low-quality good at zero marginal cost and make positive profit. To induce them to buy the small innovation the inventor cannot set a positive per unit royalty.

Equilibrium magnitudes are for ψ≥ψT, and for any innovation size under negative royalties:

with QT>1 as 16ψ2+4ψ+1−32ψ2−7ψ+2=11ψ−16ψ2−1>0, which also implies pT<sψ;

Whereas for ψ>ψT, in case of non-negative royalties, equilibrium magnitudes are:

Notice that at equilibrium it always holds that pdTsψ=sψ3<sψ.

C Proof of Proposition 1

Given the equilibrium analysis of Appendix Sections 1 and 2, direct comparisons of eqs. (12), (10), (16) and (17) show that:

D Proof of Corollary 3

Straightforward comparisons with respect to the status quo show that the consumer surplus is higher: CST−CS>0, CSdT−CS>0, CSEL−CS>0, CSELneg−CS>0; as for social welfare, as long as f is not too large SWT−SW>0, SWdT−SW>0, SWEL−SW>0, SWELneg−SW>0; as for the licensees, they are worse off, πiELsψ,s−πi>0, πiELnegsψ,s−πi>0, πTsψ−FTrT,rT−πi>0, and πdTsψ−FT0,0−πi>0.

E Proof of Corollary 4

As already pointed out, SWEL<SWELneg, this comparison is relevant for ψ>ψ‾. Also:

F Vertical Integration Subgame

Consider the quantity competition between the VI firm and firm 1 producing the high-quality final good. The VI firm has zero variable production costs as the new input is transferred at the marginal cost c2=0, whereas firm 1 incurs marginal cost r1. The third stage equilibrium quantities and profits are:

where qVI0,r1;sψ,sψ and q1r1,0;sψ,sψ are obtained from expression (15) substituting properly ri and rj; πVI0,r1;sψ,sψ and π1r1,0;sψ,sψ are obtained from expression (14) substituting properly ri and rj. The VI firm offers firm 1 the two-part tariff contract r1,F1 such that:^[36]

where π10,0;s,sψ=ψ2s4ψ−12, obtained from eq. (8) is firm 1 outside option. As the first constraint is binding at equilibrium, we have:

The optimal contract is then:

If we let the VI firm to set negative fees, the vertical merger implements the monopoly outcome by inducing the nonintegrated firm to produce a nil quantity (foreclosure) and compensating it for the outside option. Equilibrium magnitudes are:

However negative fees would be clearly held to be illegal by antitrust authorities. It is clear from the analysis above that the VI firm wants to restrict as much as possible the quantity produced by the non-affiliate firm so as to (at least) partially internalize the vertical externality.

If the VI firm is constrained to nonnegative fees, it will optimally let the nonaffiliate firm to produce a positive quantity as low as possible (up to its outside option) q1r1:π1r1=ψ2s4ψ−12. Given the Cournot equilibrium quantities, we have

The optimal contract is then:

with rVI<sψ2 for any ψ and ∂∂ψrVI>0.^[37] Equilibrium quantities and price are:

with QVI>1 as 4ψ2−ψ+ψ32−24ψ−1ψ=−ψ4ψ−ψ−1>0. Equilibrium profits, CS, PS and SW are:

Note that the optimal contract is the same if we allow for negative royalties, as the VI firm has always incentive to set a positive royalty.

G Proof of Proposition 5

The proof comes from the VI firm maximization problem, given that it could always decide not to sell the license and get profit equal to πVI0,0;sψ,s=2ψ−12sψ4ψ−12>ΠVI defined in expression (19).

H Proof of Proposition 6

As far as the private profitability is concerned, we wonder whether the patent holder prefers to stay out of the market or to vertically integrate with either firm given the outcome of the possible subgames previously analyzed. We find that vertical integration is always privately profitable. Namely, comparing the profit under VI (ΠVI) with the joint profit under VS (the profit of the external patent holder plus the profit of either potential licensee), we obtain under non-negative royalties:

and under negative royalties:

As for the social profitability, we find that vertical integration is socially profitable for large innovations, whereas it is welfare detrimental for small innovations. Namely, we make the following comparisons, under non-negative royalties: