Risk sharing in licensing

International Journal of Industrial Organization 16 (1998) 535–554

Risk sharing in licensing Alain Bousquet, Helmuth Cremer*, Marc Ivaldi, Michel Wolkowicz University of Toulouse, Institut Universitaire de France, Place Anatole, F-31042 Toulouse, France Received 7 February 1995; accepted 11 November 1996

Abstract This paper studies the design of linear license contracts under demand or cost uncertainty. The optimal contract consists, in general, of a mix of a fixed fee and royalties. The source of uncertainty has a crucial impact on the type of royalties that must be used. In particular, under demand uncertainty at most two of the instruments are used. The contract generally combines a fixed fee with an ad valorem royalty. When cost is uncertain, a wider variety of cases can arise. The contract may involve a combination of either type of royalties, coupled with a fixed fee. Alternatively, it may be optimal to use all three available instruments.  1998 Elsevier Science B.V. Keywords: License; Royalties; Risk sharing; Product innovation; Process innovation JEL classification: D81; L14; D32

1. Introduction Risk sharing has for long been recognized as a major rationale for the financial arrangements between a patentee and a licensee. Because the demand for a new product is uncertain and / or the potential cost reduction of a new technology is not perfectly known, both seller and buyer may be better off if the payment for the right to use an innovation includes a state-contingent royalty (rather than consisting of just a fixed fee). The inventor wants to benefit from a growing

* Corresponding author. 0167-7187 / 98 / $19.00  1998 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 97 )00005-2

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demand for a new product, and the licensee wishes to avoid high payments in case of disappointing sales. A number of authors have acknowledged the risk sharing dimension of license contracts. However, to our knowledge, nobody has yet formalized these aspects. This reflects the apparent belief that the underlying idea is too obvious to deserve any formal modelling. The current paper takes a closer look at the risk sharing properties of licensing contracts. It shows that the relationship between risk and license contracts is too complex to be accounted for by a simple informal conjecture. Patent licensing is a commonly used practice. The license contract grants one or several firms the right to exploit an innovation, in exchange for some payment to the patentee. The specified payment scheme can have various degrees of complexity. In some cases, it simply consists of a fixed license fee. Quite frequently, however, it also includes royalties on a per unit and / or ad valorem basis. Accordingly, the licensee pays a certain amount per unit of output produced with the patented technology. Alternatively, if an ad valorem scheme is used, the compensation is proportional to the sales revenues collected from these units. The licensing practices of CNET (Centre National d’Etudes des Telecommunications, the research center of France Telecom) provide a good illustration of the different payment modes. Within its portfolio in 1990 consisting of 286 contracts, 225 contracts (78 percent) included royalties. Amongst these, only nine contracts specified a per unit royalty while the remaining ones used an ad valorem scheme. The most frequently observed type of contract is based on a combination of fixed fee and ad valorem royalties (179 contracts, representing 63 percent of the total portfolio). Risk sharing is also likely to be an important consideration in a broader range of vertical contracts, including franchising and retail contracts. In many respects, these arrangements resemble licensing contracts. Franchising contracts often include ad valorem royalties, while retail contracts specify a wholesale price which essentially amounts to the use of a per unit royalty. In both cases, a fixed fee may also be included. Our presentation focuses on licensing; however, the implications of our results for retail and franchising contracts will also be discussed, in particular, as far as empirical issues are concerned. The existing formal literature on licensing has focused on non risk-related considerations to justify the use of royalties.1 Two main lines of arguments have emerged. The first one is based on strategic considerations and applies to the case where the patent is licensed to two or more firms. Since royalties increase the licensees’ marginal costs, they can be used by the licensor to induce the

1

A similar comment applies to the franchising and retailing literature; see e.g., Rey (1994) or Rey and Caballero (1996) for recent surveys. Rey and Tirole (1986) do consider a setting where demand is ex-ante uncertain but focus on somewhat different issues (specifically, the usefulness of imposed retail prices).

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downstream firms to restrict their levels of output (see e.g., Katz and Shapiro (1985) and Shapiro (1985)). Starting from an initial non-cooperative equilibrium, the use of royalties increases overall profits in the industry and hence the total amount the licensor can extract. The second justification for royalties is based on a signalling argument in a context of asymmetric information.2 For instance, if the patentee has private information about the value of the innovation, he can signal a high value by offering a contract which relies heavily on royalties and thus requires low payments if the innovation is not very effective.3 To pinpoint the significance of the risk sharing issue, we will ignore both strategic considerations in the downstream market and asymmetric information problems. Though no doubt significant, both aspects would complicate the license fee versus royalties issue and make our arguments less sharp. The strategic aspects are avoided by assuming that the licensee has a monopoly in the downstream market. Moreover, while we assume that the characteristics of the innovation are not (perfectly) known ex ante, there is no informational asymmetry—neither the patentee nor the (potential) licensee have any private information. We shall assume that the uncertainty pertains to either demand or cost. For the sake of interpretation, one can think of demand uncertainty as being particularly relevant in the case of a product innovation. At the time the contract is negotiated, neither party may be able to predict the exact level of demand for the new product. In the case of a process innovation, on the other hand, one can expect cost (rather than demand) to be the major source of uncertainty. Specifically, the potential cost reduction of the innovation may not be known ex ante. The paper considers a particular, but empirically significant, class of license contracts consisting of a fixed fee, a per unit royalty and an ad valorem royalty. It characterizes the optimal license contract and determines the combination of the three instruments that must be used. The problem involves, in particular, the choice of the type of royalties (if any). Not surprisingly, the optimal contract is shown to rely, in general, on a combination of fixed fee and royalties. This property holds whatever the source of uncertainty. A more surprising result is that the source of uncertainty has a crucial impact on the type of royalties that must be used. In particular, if demand is uncertain (while cost is not) at most two of the instruments ought to be used. The optimal contract generally consists of a fixed fee along with an ad valorem royalty. On the other hand, if cost is uncertain (while demand is not) a wider variety of

2 See e.g., Gallini and Wright (1990); Macho-Stadler and Perez-Castillo (1990) and Kamien (1992). Beggs (1992) obtains a similar result for the case where the contract is offered by a potential licensee who has private information. 3 A common feature of all the papers we mentioned so far is that the license contract is determined through some (possibly very simple) bargaining process. Some authors have taken a different approach by considering the possibility of auctioning a license contract; see e.g., McAfee and McMillan (1986) and Katz and Shapiro (1986).

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cases can arise. Depending on the parameter values, and in particular on the distribution of costs, the optimal contract can involve a combination of either type of royalties along with a fixed fee. For a particular range of parameter values, the solution is interior and all three instruments are used.

2. The model Consider an industry with an upstream monopoly who owns a patent for an innovation. The upstream firm is specialized in R&D and the innovation can only be exploited if the patent is licensed to a downstream firm. There is only one potential licensee who is a monopolist in the output market. Production takes place according to a linear technology with a (constant though possibly random) marginal cost of c. There is no fixed cost other than the license fee (if any). Information is symmetric (ex ante) but there is uncertainty; some variables are not known (by either party) at the time the license contract is concluded. We shall assume that this uncertainty pertains to either demand or cost, but not simultaneously to both of these variables. Specifically, the following two cases are considered. (1) Demand uncertainty. The inverse demand is a random variable denoted by p 5 p(q, v ), where q is output and v is the state of nature. The distribution of v is common knowledge. Marginal cost, on the other hand, is not random. In some expressions it is nevertheless convenient to use a function c(v ) with c9(v ) ; 0 for all v. We assume that ≠p / ≠v . 0: a higher value of v corresponds to a more favorable state of nature. For technical reasons it is further convenient to assume that ≠p / ≠v 1 q≠ 2 p / ≠q≠v > 0; that is, marginal revenue is higher in good states of nature.4 As explained in the introduction one can think of this case as dealing with a product innovation, where the demand for the new product is uncertain. (2) Cost uncertainty. The marginal cost is a random variable denoted by c(v ) with c9(v ) , 0 so that a higher value of v continues to refer to a more favorable state of nature. The reduction in cost that may be brought about by such an innovation is not known ex-ante. Demand, on the other hand, is now deterministic so that ≠p / ≠v ; 0. This type of uncertainty is likely to be relevant in the case of a process innovation where demand uncertainty is less likely to be a problem, as the innovation concerns an already existing product. The potential licensee is risk averse. His utility v(pn ) is an increasing and

4 This condition is satisfied for instance if v implies an additive shift in a linear demand or a multiplicative shift in a constant-elasticity demand function. It ensures that the licensee tends to choose a higher output level in more favorable states of the nature; see below.

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strictly concave function of net profits, pn : v9 . 0 and v0 , 0. The licensor, on the other hand, is risk neutral. His objective is to maximize expected revenue. A license contract specifies (i) a (fixed) license fee, A, (ii) a per unit royalty, r, and (iii) an ad valorem royalty at rate t, which is applied to total sales. Fees and royalties are restricted to be non-negative (A > 0, r > 0 and t > 0). If the licensor decides not to use a particular instrument, the value of the corresponding variable is set at zero. The timing is as follows: 1. The (potential) licensor offers (and commits himself to) a contract (t, r, A). The offer is made on a take-it-or-leave-it basis. 2. The (potential) licensee either accepts or refuses the offer. If he refuses, his utility equals its (exogenous) reservation level ]v while the upstream firm’s payoff is equal to zero. If he accepts, the process continues as follows. 3. The state of nature v is realized and observed by the licensee (but not by the licensor).5 4. The licensee chooses his level of output, profits are realized and payments to the licensor are made. Note that while the contract is signed ex-ante (before the resolution of uncertainty), output is chosen ex-post so that the licensee can adjust his output to actual demand. Since the licensor can commit himself to a contract, the licensee will accept the offer if and only if his expected utility is at least equal to the reservation utility level ]v. The following notation is used. • q(t, r; v ) is the level of output that maximizes the licensee’s net profits for a given state of nature. Because output is chosen ex-post, maximization of net profits corresponds to the maximization of utility. Formally, q(t, r; v ) is the solution to the following problem max pn 5 (1 2 t)p(q, v )q 2 [c(v ) 1 r]q 2 A. q

(1)

5 Whether or not the licensor also observes v is of no direct relevance to our analysis (except as far as the issue of ex post renegotiation is concerned; see Section 6). What is crucial is that the state of nature is not verifiable (by a third party) so that state-contingent contracts are not available. In the case of demand uncertainty, this difficulty could, strictly speaking, be sidestepped by contracting on both q and p (thus by indirectly inferring the state of nature from price and output observations). However, for such contracts to be effective, it has to be assumed that the licensee can be prevented from restricting its output below the level of demand (for a given price).It should be pointed out that while the unavailability of state-contingent contracts rules out some ‘‘trivial’’ solutions, the linearity of the contract continues, of course, to be a restriction.

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For simplicity we assume an interior solution so that q.0 for all v.6 While the optimal level of output depends on t and r, as well as on v, it is independent of the license fee A—a constant in the maximization problem. Note in passing that our assumptions on the inverse demand and cost functions imply ≠q / ≠v .0 under both demand and cost uncertainty: output is higher in more favorable states of nature.7 The maximum level of net profits is denoted by pn (t, r, A; v ). Similarly R(t, r; v );p[q(t, r; v ), v ] q(t, r; v ) denotes the licensee’s gross revenue at the optimum level of output. • T(t, r, A; v ) is the revenue of the licensor (the total payment of the licensee) in state v : T(t, r, A; v ) ; tR(t, r; v ) 1 rq(t, r; v ) 1 A.

3. The licensor’s problem We start by setting up the licensor’s optimization problem. Then we derive some properties which are valid whatever the source of the uncertainty (demand or cost). The licensor chooses the (non-negative) parameters of the license contract, (t, r, A), to maximize his expected revenues subject to the acceptance constraint of the licensee. Formally, this problem can be stated as follows: Problem 1. max E[T(t, r, A; v )] 5 E[tR(t, r; v ) 1 rq(t, r; v ) 1 A],

(2a)

] subject to E(v[pn (t, r, A; v )]) >v,

(2b)

t, r, A > 0.

(2c)

t, r, A

where the expectations are taken over v. Condition (2b) ensures that the licensee accepts the contract offer. It requires that the licensee’s expected utility (of net profits) be at least equal to the reservation utility level. It is readily verified that

6 If for some v one has pn , 2 A for all q.0 (for instance because demand is very low in an unfavorable state of nature) the licensee’s optimal policy is to set q50 (so that pn 5 2 A). Allowing for such a corner solution would not change any of our results but it would complicate expressions considerably. In particular E(pn )> 2 A remains true whether or not corner solutions arise. Also note that it would never be in the interest of the licensor to offer a contract which implies q50 in all states of nature. 7 This follows from a standard comparative statics argument based on the first-order condition corresponding to (1). Recall that a higher value of v means a higher demand in the demand uncertainty case and a lower cost in the cost uncertainty case.

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this constraint is necessarily binding. Otherwise, the licensor could increase the license fee A and increase its revenues.8 Let L be the Lagrangian expression associated with Problem 1. The multiplier associated with (2b) is denoted by l .0. An interior solution (if it exists) must satisfy the following first-order conditions

F G ≠L ≠q ] 5 E(q) 1 EF(tR9 1 r) ]G 2 lE(v9q) 5 0, ≠r ≠r

≠L ≠q ] 5 E(R) 1 E (tR9 1 r) ] 2 lE(v9R) 5 0, ≠t ≠t

≠L ] 5 1 2 lE(v9) 5 0. ≠A

(3a) (3b) (3c)

To reduce the length of the expressions, we have dropped the arguments of the various functions. Hence, the symbols R, q and v refer to the functions defined in the previous section. In addition, we have introduced the notation R95p1q(≠p / ≠q) for the licensee’s marginal revenue. To calculate the derivatives of the expected utility terms with respect to the royalty rates we have used the envelope theorem.9 Substituting (3c) into (3a) and (3b) and rearranging, one obtains the following two conditions

F G ≠q E(v9)EF(tR9 1 r) ]G 2 cov(v9, q) 5 0. ≠r

≠q E(v9)E (tR9 1 r) ] 2 cov(v9, R) 5 0, ≠t

(4a) (4b)

where cov(v9, R)5E(v9R)2E(v9)E(R) is the covariance (over v ) between the licensee’s marginal utility of net profits and its gross revenues. Similarly cov(v9, q)5E(v9q)2E(q)E(v9) is the covariance between marginal utility and output. The interpretation of these expressions is as follows. The LHS of (4a) measures the impact on the licensor’s expected revenue of a marginal increase in t if the license fee A is adjusted simultaneously to satisfy the acceptance constraint.10 Similarly, the LHS of (4b) gives the impact of a marginal increase in r combined with the required change in A. Each of these expressions includes two terms. The first one is negative and has a straightforward interpretation. It measures the

8

Recall that a change in A has no impact on output so that it leaves the expected revenues from royalties unaffected. 9 Since the licensee chooses output (ex-post) to maximize net profits, one has ≠pn / ≠q50 for all t, r and v. Hence to derive net profits with respect to say t, we only have to take into account its ‘‘direct’’ impact on profits (and not its ‘‘indirect impact’’ via the induced adjustment in q). 10 This follows from the fact that we have substituted (3c) into the other conditions.

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impact of the output distortion created by the increase in t (expression (4a)) or r (expression (4b)).11 The covariance terms, on the other hand, require some closer scrutiny. First, it can be readily verified that cov(v9, R),0 and cov(v9, q),0.12 They measure the insurance benefits provided by royalties. If there were no uncertainty, they would vanish. To understand the insurance mechanism note that an increase in, say, t reduces the licensee’s expected utility by E(v9R) and generates additional revenues of E(R) to the licensor.13 If these proceeds were used entirely to reduce A, they would increase the licensee’s expected utility by E(R)E(v9). The net impact on the licensee’s utility would then be E(R)E(v9)2E(v9R)5 2cov(v9, R).0. To satisfy the acceptance constraint, the reduction in A can thus be smaller than the proceeds of the increase in royalties. Hence, the licensor’s net revenues do increase. In essence, an increase in the royalty rate is like an increase in a tax on a risky variable, the proceeds of which are redistributed through a lump-sum payment (the reduction in A). Conditions (4a) and (4b) have another interesting implication: A contract which relies on a license fee alone can never be optimal. To see this, note that with t5r50, the LHS of both expressions reduce to the (positive) covariance term. Hence, the licensor’s expected revenues increase both in t and in r; a contract with t5r50 is sub-optimal. We have thus established the following proposition. Proposition 1. A license contract with A.0 and t5r50 is sub-optimal. This result does not come as a surprise. It confirms the intuitively appealing conjecture that the presence of uncertainty provides a rationale for the use of royalties and, thus, has a significant impact on the design of the optimal license contract. If there were no uncertainty, it would obviously be optimal to rely on the licensee fee alone.14 Royalties (of any kind) can only be harmful since they tend to distort (generally reduce) output. This reduces the opportunity of the licensor to

11 It is readily verified that ≠q / ≠t,0 and ≠q / ≠r,0, while R9.0 (recall that R9 is evaluated at the licensee’s profit maximizing output level). 12 As argued in Section 2, q is an increasing function of v. The same is true for R since

≠R ≠p ≠q ] 5q ] 1R9 ] ≠v ≠w ≠v with R9.0 at the optimum level of output. On the other hand, net profits are increasing in v so that v9 is decreasing in v. Hence, v’ is negatively correlated with both R and q. 13 This argument ignores the impact on the licensor’s revenue of the induced change in output. This effect is captured by the other terms of expressions (4a) and (4b). 14 In that case the covariance terms in (4a) and (4b) are equal to zero.

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raise revenue and makes the acceptance constraint harder to satisfy.15 Under uncertainty, royalties continue to have an output distortion effect. However, they also provide a measure of insurance: As the payment depends on the state of nature, part of the risk is borne by the (risk-neutral) licensor. The optimal contract strikes the ‘‘right’’ balance between the positive and negative effects of royalties. Note that at t5r50 the output-distorting effect of an infinitely small increase in either of the royalty instruments is zero.16 Thus starting from zero, a small increase in royalties necessarily has a positive impact on the licensor’s expected revenues, the result stated in Proposition 1.17 This leads us quite naturally to the next question: Should the licensor use both types of royalties or just one of them? Formally, this amounts to analyzing whether Problem 1 has an interior solution and, if not, which of the non-negativity constraints is binding. The question is first addressed for the case of demand uncertainty (Section 4) and then re-examined in the context of cost uncertainty (Section 5).

4. Demand uncertainty Assume for the time being that demand is random while cost is not. We shall establish that under this assumption, there never exists an interior solution. At least one of the non-negativity constraints imposed on the contract parameters t, r and A must be binding. This result arises because ad valorem royalties turn out to dominate per unit royalties in the context of demand uncertainty. To establish this result, we proceed as follows. We consider an arbitrary contract in which all instruments are strictly positive. We will then show that the parameter values may be varied in such a way as to increase the licensor’s expected revenue without violating the participation constraint. Roughly speaking, this will amount

15

Specifically, it is in the licensor’s interest that the licensee maximizes gross profits (i.e., net profits plus payments to the licensor) as this ensures that the surplus available to the two firms is maximized. In the absence of uncertainty, the licensor can indeed extract all the surplus through A. If a royalty is imposed, the maximization of net profits results in a level of output which is different from the one that maximizes gross profits. Hence, the total surplus (of the two firms) is smaller and the revenue that can be extracted by the licensor is lowered. 16 The output distortion that is created has no first-order effect on gross profits. This follows from the envelope theorem and the fact that with t5r50, the licensor maximizes gross profits (see footnote 15). 17 Because of the insurance benefits they provide, distortionary taxes may be desirable under uncertainty even if lump-sum taxes are available (see Eaton and Rosen (1980) and Cremer and Gahvari (1994)). Royalties play a similar role in our model, even though the formal settings of the two problems are different.

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to increasing t at the expense of r. The license fee A is adjusted in the process to ensure that the participation constraint is not violated. Formally, this result is stated in the following lemma. Lemma 1. For any contract satisfying A.0, 0,t,1 and r.0, there exists a variation in the contract parameters (dt, dr, dA) with dt.0 and dr,0 which is feasible and results in a higher expected revenue for the licensor. Proof. Consider a small variation in t, dt.0. First we construct dr and dA to ensure feasibility, then we show that the vector of variations increases the licensor’s expected revenue. The variation in r, dr, is determined in such a way that dq50 for all v. Using the first-order condition for the licensee’s choice of output, it is readily verified that this requires ≠q / ≠t c1r dr 5 2 ]] dt 5 2 ]] dt , 0. ≠q / ≠r 12t

(5)

Marginal cost is denoted by c (instead of c(v )) to emphasize the fact that it is the same in all states of nature. The RHS of (5) is finite and a sufficiently small dt ensures r1dr>0 for the constraint (2c) not to be violated. Next, we determine dA to ensure that the expected utility of the licensee is not affected so that variation in the contract does not violate the acceptance constraint (2b). Totally differentiating (2b), using (5) to substitute for dr and rearranging yields c1r E v9(pn ) 2 R 1 ]] q 12t dA 5 ]]]]]]]] dt. (6) E[v9(pn )]

F

S

DG

Once again, it is easily seen that A1dA>0 provided that dt is sufficiently small. Hence, the vector of variations does not violate any of the constraints in Problem 1. Turning to the expected revenue, totally differentiating E(T ), substituting dr from (5) and dA from (6) yields c1r E v9(pn ) 2 R 1 ]] q c1r 12t d[E(T )] 5 E(R) 2 ]] E(q) 1 ]]]]]]]] dt. (7) 12t E[v9(pn )]

3

F

S

DG

4

Factoring out 1 /(12t) and 1 /E[v9(pn )], and using the relation (12t)R2(c1r)q5 pn 1 A results in 1 d[E(T )] 5 ]]]]] (E[v9(pn )]E[pn 1 A] 2 E[v9(pn )(pn 1 A)]) dt (1 2 t)E[v9(pn )] 1 5 2 ]]]]] cov[v9(pn ), pn 1 A] dt . 0. (1 2 t)E[v9(pn )]

(8)

To sign (8), we use the strict concavity of v which implies that v’ and pn are

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negatively correlated. Expression (8) establishes that the variations in the contract parameters increase expected revenue of the licensor. Hence the considered (initial) contract cannot be optimal. j To get an intuitive understanding of the result in the lemma, one has to recall some of the arguments in the previous section. On the one hand, collecting revenues through royalties has the benefit of providing insurance (as it shifts part of the risk to the risk-neutral licensor). On the other hand, royalties also have a negative effect in that they distort the licensee’s output choice. Hence, a variation in royalty rates that allows the licensor to collect more royalty revenues without creating any (further) output distortions is expected to be desirable. This is precisely what the variation constructed in the proof achieves. To see this, consider the following two observations. • We increase t and decrease r so that output in all states of nature remains unaffected. Hence, no additional output distortion is created. • The so-constructed variation in t and r does increase expected royalty revenues [E(T2 A);E(tR1rq)]. To see this, note that the impact on expected royalty revenues of the considered variation is given by 18

F

G

S

D

pn 1 A c1r E R 2 ]] q dt 5 E ]] dt . 0. 12t 12t Part of this increased revenue has to be returned (through a decrease in A) to satisfy the acceptance constraint. However, since a better risk sharing is achieved, the amount that must be returned is less than the amount collected. To sum up, risk sharing is improved while no additional output distortions are created. The following proposition follows immediately from Lemma 1. Proposition 2. The license contract which solves Problem 1 satisfies either (i) t.0, r50 and A>0, or (ii) t.0, r.0 and A50. The proposition shows that at most two of the instruments are used. In addition, it implies that it is always optimal to impose an ad valorem royalty. Lemma 1 has shown that the licensor can gain by increasing in t while decreasing r whenever

18

To sign the expression we have used the fact that pn 1 A>0 (see footnote 6). This property corresponds to a well-known result in taxation theory. For a given after tax output in a monopoly, more revenue is raised through an ad valorem tax than through a per unit tax (see Suits and Musgrave (1953) and Delipalla and Keen (1992)).

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this does not violate the non-negativity constraints. Depending on which of the non-negativity constraints in (2c) (the one on r or the one on A) becomes binding first, this leads him to choose a contract in category (i) or (ii). Intuitively, (i) can be expected to arise if the reservation utility, ]v, is sufficiently small and / or if profit opportunities are sufficiently large (low cost and high expected demand). On the other hand, if ]v is large relative to profit opportunities, case (ii) appears to be a likely outcome. Roughly speaking, these results show that under demand uncertainty ad valorem royalties dominate per unit royalties, as long as the non-negativity constraint on the fixed fee A is not binding. The discussion above provides some insight into the underlying intuition. However, it is admittedly more useful to understand the intuition behind the proof than that behind the result itself. The fact of the matter is that the relationship between risk sharing and royalties is far more complex than one would expect at first; we have found no simple yet rigorous way to summarize this relationship in a few sentences. Nevertheless, the following argument, albeit somewhat loose, may be of some help. Consider the (net) profits of the licensee in a given state of nature. If there is no per unit royalty and no fixed fee, it can be written as follows:

F

G

c pn 5 (1 2 t) R 2 ]] q . 12t In the light of this expression, an ad valorem royalty essentially appears to be a combination of two instruments: a (proportional) ‘‘tax’’ on the licensee’s profits (at rate t) and a per unit royalty (at a rate r such that c1r5c /(12t)). Amongst these two, it is the per unit component which is responsible for output distortions; the profit ‘‘tax’’ only creates an income effect which can be compensated by the fixed fee (but because of the risk-sharing it provides, the required compensation is less than the proceeds of the profit ‘‘tax’’). Consequently, it is possible to switch from a per unit royalty to the ‘‘equivalent’’ ad valorem rate (yielding the same output distortion), while achieving a better risk sharing through the profit-tax component (a profit-tax yields higher revenues in good states of nature and part of its expected proceeds can be retroceded through a reduction in the fixed fee). Put differently, a per unit royalty introduces an element of profit-sharing which is otherwise impossible with the set of available instruments. Clearly, this argument is valid only so long as the non-negativity constraint on A is not binding; hence the possibility of a corner solution where a positive level of the per unit royalty subsists. Finally notice that the per unit component of an ad valorem royalty (namely the value of r which satisfies c1r5c /(12t)) is the same in all states of nature under demand uncertainty (because c is constant). Consequently, there exists a unique ‘‘equivalent’’ ad valorem rate which can be used to replace any per unit royalty. This will no longer be true in the case of cost uncertainty, to which we now turn.

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5. Cost uncertainty We now turn to the case of cost uncertainty where cost is random, while demand is the same in all states of nature. Interestingly, this turns out to change the nature of our results. In particular, we shall show that it may not be possible for the licensor to increase its expected revenues by increasing t at the expense of r. It may now be optimal to use all three instruments (interior solution) or to use per unit royalties (along with fees) instead of ad valorem royalties. To make these points in the simplest possible way, we concentrate on a special case where demand is linear and given by p5 a 2q, where a is a positive constant.19 This setting is sufficiently general to show that various types of solutions can occur. From (1), the output chosen by the licensee is now given by

S

D

1 c(v ) 1 r q(t, r; v ) 5 ] a 2 ]]] . 2 12t

(9)

Once again, we proceed by studying the impact of simultaneous variations in t and r. Lemma (1) has studied a particular variation: t was increased and r decreased so that output in all states of nature was unaffected. Expression (9) shows that such a variation cannot be constructed under cost uncertainty. The required adjustment in r will depend on the state of nature. However, it remains possible to consider a joint variation in t and r (and A) which leaves expected output constant. This is done in the following proposition. Proposition 3. Assume that the inverse demand function is given by p5 a 2q. Consider a contract satisfying A.0, 0,t,1 and r.0. Let (dt, dr, dA) be a variation in the contract parameters with E(c) 1 r dr 5 2 ]]] dt, 12t

F

S

(10a)

DG

E(c) 1 r E v9(pn ) 2 R 1 ]]] q 12t dA 5 ]]]]]]]]] dt; [E[v9(pn )]

(10b)

that is, given dt, dr is chosen so that dE(q)50 (expected output is unaffected) while dA is adjusted for the acceptance constraint not to be violated. The resulting variation in the licensor’ s expected revenue is then given by

19

We assume a to be sufficiently large (compared to the range of marginal costs) for the expected (gross) profits to be positive.

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H

1 d[E(T )] 5 2 ]]]]] cov[v9(pn ), (1 2 t)R 2 (E(c) 1 r)q] (1 2 t)E[v9(pn )]

J

t 2 ]]]3 Var(c) dt, 2(1 2 t)

(11)

where Var(c) is the variance of marginal cost, c. Proof. The proof is similar to that of Lemma 1. We totally differentiate E(T ) and use (10a)–(10b) to substitute for dr and dA. To simplify the second term on the RHS of (11) (which did vanish in the proof of Lemma 1) we have also calculated ≠q / ≠r and ≠q / ≠t from (9) and made use of the fact that cov[(c /(12t); c] simplifies to Var(c) /(12t). j The first term on the RHS of (11) is similar to the RHS of expression (8); it measures the insurance (or risk sharing) impact of the considered variation in the contract parameters. However, contrary to the demand uncertainty case, its sign is not unambiguous here; see below. Note also that the magnitude of this term depends on the degree of risk aversion of the licensee (the concavity of v). If v is close to a linear function, it will be negligible (there are no insurance benefits), while it will tend to be important if v has a significant degree of concavity. The second term on the RHS captures the impact of output changes; see the proof. It was not present in the case of demand uncertainty, where the considered variation in the contract parameters did leave output in all states of nature unaffected.20 With the second term necessarily negative, and the magnitude of the first term depending on the licensee’s degree of risk aversion, it is clear that the considered variation is not necessarily beneficial for the licensor. Hence, per unit royalties are no longer dominated by ad valorem royalties. To understand the source of this ambiguity it may be useful to recall the intuitive argument presented at the end of Section 4. Under demand uncertainty, it was possible to switch from a per unit royalty to the ‘‘equivalent’’ ad valorem rate (yielding the same output distortion), while achieving a better risk sharing through the ‘‘profit tax’’ component. Formally, this ad valorem rate was determined by solving c /(12t)5c1r so that it was given by t5r /(c1r). Under cost uncertainty, the same switch would only be possible if t were allowed to depend on the state of nature; a constant value of t can, at best, be constructed to hold expected output constant.21 Consequently, the simple generic dominance result of ad valorem royalties no longer holds. Output distortions now have to be accounted for and, as

20 Here expected output is unaffected but actual output in a particular state of nature will (in general) change. 21 With non-linear demand functions even this may not be possible.

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illustrated by the argument below, it is no longer clear that the ad valorem royalty (constructed on the basis of expected cost) has better risk sharing properties than a per unit scheme. An exact prediction of the type of solution that emerges is difficult at this level of generality. However, some understanding can be gained by considering the special case where c takes only two values c 1 ,c 2 with probabilities p1 and p2 512p1 .22 First assume that p1 is close to one so that E(c) is close to c 2 . In that case it is easily seen that (12t)R2(E(c)1r)q, which appears in the RHS of (11) is smaller if the cost is c 1 than if it is c 2 .23 But pn is of course larger and thus v9(pn ) is smaller if cost is low. Consequently, the covariance is positive so that the first term on the RHS of (11) is negative and so is the entire expression (the second term being also negative; see above). To sum up, one obtains the opposite result as in the demand uncertainty case. It is now in the licensor’s interest to increase r while decreasing t. The solution then tends to involve a per unit royalty (along with a license fee) but no ad valorem royalty.24 Put differently, per unit royalties appear to be the most effective instrument and ad valorem royalties are used only if the non-negativity constraint on the fee is binding. Now, assume that p1 is close to 1 so that E(c) is close to c 1 . Then, (12t)R2 (E(c)1r)q is larger if cost is c 1 than if it is c 2 .25 But of course, pn continues to be larger (and v9(pn ) to be smaller) if cost is low. Consequently, the covariance is negative so that the first term in the RHS of (11) is positive. The overall sign of the RHS then depends on the relative magnitude of the two terms. However, it can be shown that if the licensee is sufficiently risk averse (v9 sufficiently differs between the good and the bad state of nature) then the first term dominates and one obtains the same type of solution as in the case of demand uncertainty: the contract uses ad valorem rather than per unit royalties (along with the license fee).26 To illustrate these results consider the following example: c 1 5 0.3, c 2 5 0.9,

22

A possible interpretation of this setting is that c 2 is the pre-innovation cost and that the innovation either reduces the cost to c 1 or is completely ineffective. 23 Recall that output is chosen ex-post to maximize pn 5(12t)R2(c1r)q2 A (which depends on actual cost rather than on expected cost). If E(c) is close to c 2 , the output chosen under c 2 is close to the level that would maximize (12t)R2(E(c)1r)q. 24 Assuming that ]v is small enough so that the constraint A>0 is not binding. If it were binding, the solution would involve r.0, t.0 and A50. 25 See footnote 23: if E(c) is close to c 1 , the output chosen under c 1 is close to the level that would maximize (12t)R2(E(c)1r)q. 26 With a binomial distribution, one has Var(c)5p1 (12p1 )(c 2 2c 1 )2 which has its maximum at p1 51 / 2 and tends to zero as p1 tends to one (or to zero). Now, the covariance term does, of course, also tends to zero if p1 tends to one. However, its magnitude depends on the ratio between v9(pn ) in the bad state of nature and E[v9(pn )]. Consequently, a sufficient degree of concavity of v ensures that the first term dominates the second (which is independent of v).

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Fig. 1. Optimal values of t and r as functions of p1 .

]v 5 0.1, a 5 3.05,

p 0.5 n 21 v(pn ) 5 ]]]. 0.5 The optimal values of t and r are represented in Fig. 1 as functions of p1 . The optimal license fee (which is not represented) is strictly positive in all cases. The emerging solution profile confirms our theoretical findings. If p1 is small, a per unit royalty, along with a license fee, is optimal. No ad valorem royalty is imposed. On the other hand, if p1 is sufficiently large, it is the ad valorem royalty which is optimal, and the solution resembles that obtained under demand uncertainty. As the theoretical arguments have shown, this second result, however, crucially depends on the choice of the utility function. Finally, for intermediate values, the solution is interior and all three instruments are used.

6. Concluding remarks This paper has confirmed the intuitively appealing conjecture that uncertainty is a rationale for using royalties in license contracts. However, it has also shown that under uncertainty the determination of the optimal license contract is a complex problem, even within a setting as simple as ours. Under demand uncertainty ad valorem royalties tend to outperform per unit royalties. Intuitively one would

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expect a licensee who had to choose between the two instruments to opt for the ad valorem royalties. They appear to provide a better targeted insurance as they make the payment dependent on the licensor’s revenues, rather than just on output. However, in our setting the two instruments are not mutually exclusive. The licensee could use both types of royalties. Nevertheless, it is generally not optimal for him to do so. Under cost uncertainty, on the other hand, a number of regimes can arise. For some range of parameter values, the results are similar to the demand uncertainty case and per unit royalties are not used. For some other range, the results are completely reversed. The optimal contract includes a per unit royalty (along with a license fee), but no ad valorem royalties. Finally, the case where all three instruments are set at positive levels can also arise. The crucial factors appear to be the distribution of costs and, particularly, its variance, and the probability that the innovation results in a ‘‘substantial’’ cost reduction. Our theoretical results point to a relationship between the type of uncertainty and the form of the license contract. Specifically, they lead one to expect that ad valorem royalties would be used relatively more often than per unit royalties when uncertainty is on demand rather than on costs. They also suggest that license contracts that solely rely on a fixed fee would be observed only when there is no (significant) demand or cost uncertainty or when the licensee is not risk averse. The license practices of CNET referred to in the introduction provide some support for these hypotheses. Specifically, the nine contracts involving per unit royalties all appear to fall into the category of process innovations.27 On the other hand, a large majority (82%) of the sixty-one fixed fee contracts are established with government agencies (or other public administrations) who do not appear to fit into the small risk averse licensee framework of our model. It seems reasonable to conjecture that considerations of risk sharing did not play a significant role in the design of these contracts. These figures are striking and they bear a crucial responsibility in the genesis of our research project. However, their significance should not be overstated; the evidence they provide is of a too limited scope to provide solid empirical foundation for our model. Some additional illustrative support can be drawn from the bearing that our setting may have within the context of contractual relations between producers and distributors. It is plausible to conjecture that cost uncertainty is not a major factor in franchising relations. Consequently, the predominance of ad valorem royalties in franchising contracts may reflect the uncertainty pertaining to (local) demand conditions. On the other hand, retail contracts stipulating a wholesale price (per unit royalty), may compensate for the uncertainty in the cost of retailing.28 However suggestive, these arguments cannot be built into a full fledged

27 28

Altering the production technologies of already existing products and / or service. We thank the referee for bringing this argument to our attention.

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empirical support for our model. A more rigorous way to test our setting would involve the specification of an econometric model of choice of license contracts in order to test the relationship between the chosen set of instruments variables such as the type of innovation, the degree of demand uncertainty and the level of cost uncertainty. Alternatively, a more structural approach, consisting of estimating the first order conditions associated with the program of instrument choice could also be considered. Quite clearly, though, such a thorough econometric analysis would go beyond of the scope of the current paper and is therefore left for future research. Our analysis relies on a number of restrictive assumptions. First, we have not dealt with the case of simultaneous demand and cost uncertainty.29 In reality, however, this appears to be a relevant situation, especially as far as product innovations are concerned. For a new product, both demand and cost may well be unknown ex ante. What’s more, the two types of risk may in some way be correlated.30 Analytically, this case appears intractable, even under more stringent assumption on demand and / or utility. Some insight can be gained from numerical simulations but a thorough numerical analysis would go well beyond the scope of this paper.31 We present such a numerical approach in a companion paper: Bousquet et al. (1995). It shows that the pattern of results for the double uncertainty case are similar to the results obtained in the cost uncertainty case. In many situations (especially for ‘‘extreme’’ parameter values), one type of royalties is not used. In some ‘‘intermediate’’ region, the solution is interior and all three instruments are used. Second, we have considered a highly simplified bargaining process between licensor and (potential) licensee. The licensor can commit to a take-it-or-leave-it offer and hence extract all the surplus from the licensee. However, the results would not change significantly if, say, a Nash Bargaining solution would prevail. Third, we have assumed that there is a single, monopolistic licensee. In some cases, however, especially (but not exclusively) if there is product differentiation in the downstream market, the licensee might find it profitable to grant a license to more than one firm. Comparing the single licensee case to the outcome under alternative arrangements would be a natural extension of our analysis, which would then become a building block of a more general theory of license contracts. The assumption that the single licensee faces no competition in the downstream market, on the other hand, is quite natural in the case of a product innovation or a

29

The results in Section 3 do, however, remain valid, provided that the expectation and covariance operators are properly redefined. 30 Even though there do not appear to be compelling intuitive arguments pointing towards a systematic positive or negative correlation. 31 The presence of inequality constraints combined with the systematic occurrence of various types of corner solutions call for a careful numerical investigation.

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drastic process innovation. However, it appears to be somewhat restrictive in the case of a non-drastic process innovation. Note that most of our results could quite easily be extended to the case where the downstream markets is a Cournot duopoly. Fourth, ex post renegotiation of the contract has been ruled out. If the contract could be renegotiated after the state of the world is realized, licensor and licensee may agree to switch to a fixed fee contract to avoid output distortions. In essence, this amounts to introducing an additional instrument, namely that of statecontingent fixed fees. Though attractive, this instrument appears to be of limited practical relevance because it relies on rather stringent informational requirements. Specifically, state contingency, whether specified in the initial contract or achieved indirectly through renegotiation, requires that the state of nature is observable to both parties. Finally, and perhaps most importantly, we have limited ourselves to the class of linear contracts. This restriction has to be considered along with the fact that we have not explicitly introduced (ex ante) informational asymmetries. In particular, moral hazard variables like the licensee’s cost reducing or demand enhancing efforts are not explicitly accounted for. A natural extension of our study would be to introduce such phenomena, and to solve for the optimal contract given the informational structure. This issue is included in our research agenda and will be considered in a subsequent paper.

Acknowledgements ´´ Bousquet, CEA-IDEI and CNET-SET, France Telecom; Cremer: GREMAQ and IDEI, University of Toulouse and Institut Universitaire de France, Place Anatole France, F-31042 Toulouse Cedex; Ivaldi: GREMAQ-EHESS and IDEI; Wol´´ kowicz: CNET-SET France Telecom. We would like to thank Emmanuelle Auriol and Etienne Turpin for their valuable remarks. We are especially grateful to Firouz Gahvari and to a referee, whose detailed comments were of tremendous help to us.

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