Technical Appendix to

“Step by step. The benefits of stage-based R&D licensing contracts.”

Proof of Lemma 1.

(a) In the absence of adverse selection and moral hazard, consider a feasible contract (m, r, x) that respects the licensee’s IR constraint and the non-negativity constraints. Since the contract sets both the royalty rate and the effort level, the licensor’s royalties revenue conditional on market launch is rS(x). This contract is then equivalent to the following contract: (m, 0, x) , with all milestone payments identical except the last milestone payment m₂= m₂+ rS(x). It is immediately possible to confirm that this new contract is similarly feasible and yields the same value to the licensor. Thus the licensor is indifferent to the use of royalties.

(b) Under (a), we have shown that the optimal contract can always be formulated as a contract with milestone payments only. Thus the individual rationality constraint defines the maximum amount that the licensor can extract from the licensee, for any combination of upfront and milestone payments:

The right hand side of this equation is the licensee’s expected value from the project and is maximized when the effort level x is set to the licensee’s optimal development effort level, i.e., x^∗ such that p^e_nds(x)/dx = 1. This maximizes the expected value of the milestone payments, i.e., the left hand side of the equation, which the licensor can claim in payment from the licensee.

Proof of Lemma 2

In the presence of moral hazard, a non-zero royalty rate reduces the licensee’s incentive to invest in the project, thus reducing the project value. Given Lemma 1, we know that the licensor maximizes his value when the licensee’s development effort is optimal given her type (Lemma 1 (b)) and that a royalty contract can be transformed into an equivalent contract with milestone payments only (Lemma 1 (a)). Therefore, the optimal contract and its value to the licensor without adverse selection and moral hazard can be achieved if a contract with milestone payments only is offered in the presence of moral hazard.

Proof of Theorem 1

First, we invoke Lemma 2 to restrict our search to contracts with milestone payments only.

This transforms the optimization problem into a linear problem. The dual of the licensor’s problem is:

miny V_k^e(0, 0, x_k)y

subject to y ≥ 1 p^o₁y ≥ p^e₁ p^o₁p^o₂y ≥ p^e₁p^e₂

The optimal solution puts y to its lower bound jointly allowed by the three constraints of the dual. Note that each constraint corresponds to a different timing of the milestone payment. This directly yields the conditions listed in Theorem 1. The shadow price of the binding constraint is the optimal milestone payment for the licensor’s original optimization problem.

Furthermore, for any licensee estimates vector p^e, there is always a i such that the above equations hold. To prove this, we start by assuming a vector p^e such that there is a i satisfying the conditions above. We show that by changing the probability estimates one stage at a time, there will always be a new κ satisfying the set of conditions. From our starting point, we generate all possible licensee estimates, with at least one stage satisfying the conditions above.

First, assume that the probability estimate of a stage l > i is changed. An increase in p^e_l will never invalidate the inequalities holding for i. Thus, let us decrease p^e_l until a first condition is violated, i.e. for one κ ≤ l we have Qκ

j=1p^e_i+j ≤ Qκ

j=1p^o_i−j. If κ < n, we know by definition of i that Qs−i

j=1p^e_i+j ≥ Qs−i

A similar reasoning can be applied for a change in the estimate of stage l ≤ k.

Proof of Lemma 3

Suppose that the risk-neutral licensor’s optimal milestone payment is m^∗_i, scheduled at the end of research stage i. Set any milestone payment m_i+k > 0, k > 0 and correspondingly set mi = m^∗_i −Qk

j=1p^e_j+imi+k such that the contract remains feasible.

We need to show that this new contract m_i, m_i+k will never outperform m^∗_i for a risk-averse

This follows from the concavity of the objective function:

 The last inequality follows from the optimality of m^∗_i for the risk-neutral licensor, i.e.,

Qk

j=1p^o_i+j ≤Qk

j=1p^e_i+j.

Conversely, set any milestone payment m_i−k > 0, 0 < k ≤ i and correspondingly set mi = m^∗_i − m_i−k/Qk

j=1p^e_i−k+j such that the contract remains feasible.

We need to show that there exists conditions under which this new contract m_i−k, m_i out-performs m^∗_i for a risk-averse licensor:



Using standard economic theory approach, we relax the licensor’s problem by dropping the individual rationality constraint for the high-type licensee and the incentive compatibility constraint for the low-type licensee. This yields the licensor’s relaxed optimization problem:

mk≥0, rmax_k≥0,q_L(m_0L+ p^o₁m_1L+ p^o₁p^o₂(m_2L+ r_Ls(x^∗_LL)) + q_H(m_0H+ p^o₁m_1H+ p^o₁p^o₂(m_2H+ r_Hs(x^∗_HH))

s.t. − c₁− m_0L− p^e_1L(c₂+ m_1L+ x^∗_LL) + p^e_1Lp^e_2L((1 − r_L)s(x^∗_LL) − m_2L) ≥ 0

−m_0H − p^e_1H(m_1H+ x^∗_HH) + p^e_1Hp^e_2H((1 − r_H)s(x^∗_HH) − m_2H) ≥

−m_0L− p^e_1H(m1L+ x^∗_HL) + p^e_1Hp^e_2H((1 − rL)s(x^∗_HL) − m2L) m^j_i ≥ 0,r^j ≥ 0

Using the low-type licensee’s individual rationality constraint and the high-type licensee’s incentive compatibility constraint, we substitute for m0L and m0H in the objective function, and then write the KKT constraints of the Lagrangian W :

∂W/∂m1L : qL(p^o₁− p^e_1H) + (p^e_1H− p^e_1L) − λ1p^e_1L+ λ2(p^e_1H− p^e_1L) + λ3 = 0

∂W/∂m_1H : (1 − q_L)(p^o₁− p^e_1H) − λ₂p^e_1H+ λ₄ = 0

∂W/∂m_2L : q_L(p^o₁p^o₂− p^e_1Hp^e_2H) + (p^e_1Hp^e_2H− p^e_1Lp^e_2L) − λ₁p^e_1Lp^e_2L+ λ₂(p^e_1Hp^e_2H− p^e_1Lp^e_2L) + λ₅ = 0

∂W/∂m_2H : (1 − q_L)(p^o₁p^o₂− p^e_1Hp^e_2H) − λ₂p^e_1Hp^e_2H + λ₆ = 0

∂W/∂r_L: p^e_1Lp^e_2Ls(x^∗_LL) + (1 − q_L)p^e_1Hp^e_2Hs(x^∗_HL) + q_Lp^o₁p^o₂(s(x^∗_LL) + r_Ls⁰_rL)

− λ₁p^e_1Lp^e_2Ls(x^∗_LL) + λ2(p^e_1Hp^e_2Hs(x^∗_HL) − p^e_1Lp^e_2Ls(x^∗_LL)) + λ7 = 0

∂W/∂rH : (1 − qL)p^o₁p^o₂(s(x^∗_HH) + rHs⁰_rH) − (1 − qL)p^e_1Hp^e_2Hs(x^∗_HH) − λ2p^e_1Hp^e_2Hs(x^∗_HH) + λ8 = 0 The characteristics of the optimal contract menu for the high-type licensee immediately follow from the solution to the second, fourth and last KKT condition of this relaxed problem.

This creates three different scenarios for the high-type licensee, i.e., the optimal contract contains either an upfront payment, a milestone payment after the first phase or a milestone payment at product launch. Solving the first, third and fifth KKT condition under each of these three different scenarios separately, we obtain boundaries for p^o₁ and p^o₂ which determine areas within which different contracts for the low-type licensee will be written.

1. p^o₁ ≤ min{p^e_1H, p^e_1H−^pe1H_q^−pe1L

L } and p^o₁p^o₂ ≤ max{0,^{pe1L(pe2L)2−(1−q}_q ^{L)pe1H(pe2H)2}

Lp^e_2L }

Pooling contract with m_0L= m_0H; m_0L as per the low-type licensee’s project value.

2. p^o₁ ≤ min{p^e_1H, p^e_1H−^pe1H_q^−pe1L

L } and^{pe1L(pe2L)2−(1−q}_q ^{L)pe1H(pe2H)2}

Lp^e_2L ≤ p^o₁p^o₂≤ min{p^e_1Hp^e_2H,^{pe1Lpe2L−(1−q}_q ^L)pe1Hpe2H

L }

Low-type licensee contract (m0L, rL) and high-type licensee contract m0H. 3. p^e_1H−^pe1H_q^−pe1L

L ≤ p^o₁≤ p^e_1H and 0 ≤ p^o₂≤ max{0, p^e_2L−^(1−qL)pe1H_q^{((pe2H)2−(pe2L)2)}

Lp^o₁p^e_2L } Low-type licensee contract m1L and high-type licensee contract m0H.

4. p^e_1H − ^pe1H_q^−pe1L

L ≤ p^o₁ ≤ p^e_1H and p^e_2L − ^(1−qL)pe1H_q^{((pe2H)2−(pe2L)2)}

Lp^o₁p^e_2L ≤ p^o₂ ≤ max{0, p^e_2L −

(1−qL)p^e_1H(p^e_2H−p^e_2L) qLp^o₁ }

Low-type licensee contract (m_1L, r_L) and high-type licensee contract m_0H. 5. p^e_1H ≤ p⁰₁≤ _pe ^qLpe1Lpe1H

1H−(1−q_L)p^e_1L and 0 ≤ p^o₂ ≤ max{0, min{p^e_2H,^{qLpe1L(pe2L)2pe1H−(1−q}_q ^{L)po1(pe1H(pe2H)2−pe1L(pe2L)2)}

Lp^o₁p^e_1Hp^e_2L }}

Low-type licensee contract m_0L and high-type licensee contract m_1H. 6. p^e_1H ≤ p⁰₁ ≤ _pe ^qLpe1Lpe1H

1H−(1−q_L)p^e_1L and min{p^e_2H,^{qLpe1L(pe2L)2pe1H−(1−q}_q ^{L)po1(pe1H(pe2H)2−pe1L(pe2L)2)}

Lp^o₁p^e_1Hp^e_2L } ≤ p^o₂ ≤ max{0, min{p^e_2H,^{qLpe1Lpe2Lpe1H−(1−q}_q ^{L)po1(pe1Hpe2H−pe1Lpe2L)}

Lp^o₁p^e_1H }}

Low-type licensee contract (m_0L, r_L) and high-type licensee contract m_1H. 7. p⁰₁ ≥ max{p^e_1H,_pe ^qLpe1Lpe1H

1H−(1−q_L)p^e_1L} and p^o₂ ≤ max{0,^{(pe2L)2−(1−q}_q ^L)(pe2H)2

Lp^e_2L }}

Pooling contract m_1L= m_1H; m_1L as per the low-type licensee’s project value.

8. p⁰₁ ≥ max{p^e_1H,_pe ^qLpe1Lpe1H

1H−(1−q_L)p^e_1L} and ^{(pe2L)2−(1−q}_q ^L)(pe2H)2

Lp^e_2L ≤ p^o₂ ≤ max{0,^{pe2L−(1−q}_q ^L)pe2H

L }}

Low-type licensee contract (m_1L, r_L) and high-type licensee contract m_1H. 9. p⁰₁ ≤ p^e_1H and p^o₂ ≥ max{0,^{pe1Lpe2L−(1−q}_q ^L)pe1Hpe2H

Lp^o₁ , p^e_2L−^(1−qL)pe1H_q ^{(pe2H−pe2L)}

Lp^o₁ } Low-type licensee contract (m_2L, r_L) and high-type licensee m_0H. 10. p⁰₁ ≥ p^e_1H and max{^{qLpe1Lpe2Lpe1H−(1−q}_q ^{L)po1(pe1Hpe2H−pe1Lpe2L)}

Lp^o₁p^e_1H ,^{pe2L−(1−q}_q ^L)pe2H

L } ≤ p^o₂≤ p^e_2H Low-type licensee contract (m_2L, r_L) and high-type licensee contract m_1H. 11. p^o₂ ≥ p^e_2H and p^e_1Hp^e_2H ≤ p^o₁p^o₂≤ max{p^e_1Hp^e_2H,_q ^{qLpe1Lpe2Lpe1Hpe2H}

Lp^e_1Hp^e_2H−(1−q_L)(p^e_1Lp^e_2L−p^e_1Hp^e_2H)} Low-type licensee contract (m_0L, r_L) and high-type licensee contract m_2H. 12. p^o₂ ≥ p^e_2H and p^o₁p^o₂≥ max{p^e_1Hp^e_2H,_q ^{qLpe1Lpe2Lpe1Hpe2H}

Lp^e_1Hp^e_2H−(1−q_L)(p^e_1Lp^e_2L−p^e_1Hp^e_2H)}

Low-type licensee contract (m_2L, r_L) and high-type licensee contract m_2H.

Finally, using the structure of the optimal contract menu, it is easy to verify that the optimal contracts satisfy the constraints that were previously dropped.

Proof of Lemma 4

Using the relaxation approach outlined in Theorem 2, we drop the high-type licensee’s individual rationality constraint and the low-type licensee’s incentive compatibility constraint.

We can observe from the resulting KKT conditions that the contract for the high-type licensee is not distorted, i.e., follows the optimal structure defined under Lemma 3.

Proof of Theorem 3

In the absence of adverse selection, suppose it is optimal for the risk-neutral licensor to delay licensing for k ≤ n−1 phases and offer a contract with a milestone payment m_l, l ∈ {1, ..., n−k}.

To determine the unique optimal contract, we use Theorem 1.

For notational convenience, let us introduce W^e(k), the value of the project to the licensee at the start of phase k + 1. Then the milestone payment will be m_l= Q^Wk+l^e(k)

i=k+1p^e_i. The licensor’s value from the contract at the time of licensing is: Qk+l

i=k+1p^o_iml.

Should the licensor delay the contract one more phase, to k + 1, Theorem 1 specifies that the milestone payment would still be located at the end of the same phase, and would thus be m0 = W^e(k + 1) if l = 1, and m_l−1 = Q^Wk+l^e(k+1)

i=k+2p^e_i otherwise.

The licensor’s new value from the contract if l > 1 is: −ck+Qk+l

i=k+1p^o_iml, which we can prove is larger thanQk+l

i=k+1p^o_im_l:

Prove similarly for l = 1.

Thus it is not optimal to license the project at a phase such that the optimal contract would contain a milestone payment. Rather, it can be successively further delayed until the optimal contract contains an upfront payment, thus increasing its value, or until no further delay is possible because the licensee’s development effort is needed.

Proof of Lemma 5

Suppose that for a risk-neutral licensor it is optimal to delay the licensing of the project until phase k < n − 1, at which phase he receives an upfront payment m^∗₀. This generates a series of cash flows for the licensor (−c1, ..., −c_k−1, m^∗₀) with their associated probabilities, the product of his PTS estimates.

Should a risk-averse licensor delay the project beyond phase k, he will face the same cash outflows up until phase k. After that, depending on the delay chosen and the corresponding optimal contract terms, the cash flows will change to (z1, ..., zn−k). This generates the following

series of cash flows: (−c₁, ..., −c_k−1, −c_k, z₁, ..., z_n−k) with their associated probabilities, the product of his PTS estimates.

Since the first part is identical, it can be ignored. We know from the fact that delaying the licensing of the project until phase k < n − 1 for an upfront payment m^∗₀ is optimal for the risk-neutral licensor that by Theorem 1:

m^∗₀ ≥ −c_k+ Since u^o(.) is a monotonically increasing concave function:

u^o(m^∗₀) ≥ u^o

The concavity is applied in successive steps until the coefficients are fully extracted from the utility function to obtain the last inequality.

Proof of Lemma 6

(a) Given Theorem 2, the optimal two-stage contract will not include an upfront payment for either licensee type in the following four cases, for which we need to show that delaying is optimal:

upfront payment only, i.e., m_1L = m_1H = −(c₂+ x^∗_LL) + p^e_2Ls(x^∗_LL). Then, p^o₁

p^e_1L[−c₁− p^e_1L(c₂+ x^∗_LL) + p^e_1Lp^e_2Ls(x^∗_LL)] ≤ −c₁+ p^o₁(−c₂− x^∗_LL+ p^e_2Ls(x^∗_LL))

⇐⇒ p^e_1L≤ p^o₁, which is true by assumption.

2. p⁰₁ ≥ max{p^e_1H,_pe ^qLpe1Lpe1H

1H−(1−q_L)p^e_1L} and ^{(pe2L)2−(1−q}_q ^L)(pe2H)2

Lp^e_2L ≤ p^o₂ ≤ max{0,^{pe2L−(1−q}_q ^L)pe2H

L }}

Under immediate licensing, the low-type licensee contract is (m_1L, r_L) and the high-type licensee contract m1H, with m1L = [−c1 − p^e_1L(c2 + x^∗_LL) + p^e_1Lp^e_2L(1 − rL)s(x^∗_LL)]/p^e_1L and m_1H = m_1L− (x^∗_HH− x^∗_HL) + p^e_2H(s(x^∗_HH) − (1 − r_L)s(x^∗_HL)) = m_1L+ ∆. Assume that we delay licensing and ask for a feasible contract menu (m_0L, r_L) and m_0H, with m0L = −c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL) and m0H = m0L+ ∆. Then,

p^o₁

p^e_1L[−c₁− p^e_1L(c₂+ x^∗_LL) + p^e_1Lp^e_2L(1 − r_L)s(x^∗_LL)] + q_Lp^o₁p^o₂r_Ls(x^∗_LL) + (1 − q_L)p^o₁∆

≤ −c₁+ p^o₁(−c2− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)) + q_Lp^o₁p^o₂r_Ls(x^∗_LL) + (1 − q_L)p^o₁∆

⇐⇒ p^e_1L ≤ p^o₁,

which is true by assumption.

3. p⁰₁ ≥ p^e_1H and max{^{qLpe1Lpe2Lpe1H−(1−q}_q ^{L)po1(pe1Hpe2H−pe1Lpe2L)}

Lp^o₁p^e_1H ,^{pe2L−(1−q}_q ^L)pe2H

L } ≤ p^o₂≤ p^e_2H

Under immediate licensing, the low-type licensee contract is (m2L, rL) and the high-type licensee contract is m_1H, with m_2L= [−c₁−p^e_1L(c₂+x^∗_LL)+p^e_1Lp^e_2L(1−r_L)s(x^∗_LL)]/(p^e_1Lp^e_2L) and m1H = p^e_2Hm2L− (x^∗_HH − x^∗_HL) + p^e_2H(s(x^∗_HH) − (1 − rL)s(x^∗_HL)) = p^e_2Hm2L+ ∆.

Assume that we delay licensing and ask for a feasible contract menu (m_1L, r_L) and m_0H, with m_1L= [−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)]/p^e_2L and m_0H = p^e_2Hm_1L+ ∆. Then,

p^o₁

p^e_1Lp^e_2L(qLp^o₂+(1 − qL)p^e_2H)[−c1− p^e_1L(c2+ x^∗_LL) + p^e_1Lp^e_2L(1 − rL)s(x^∗_LL)]

+ qLp^o₁p^o₂rLs(x^∗_LL) + (1 − qL)p^o₁∆

≤ −c₁+ p^o₁

p^e_2L(q_Lp^o₂+ (1 − q_L)p^e_2H)(−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)) + q_Lp^o₁p^o₂r_Ls(x^∗_LL) + (1 − q_L)p^o₁∆

⇐⇒ p^e_1Lp^e_2L≤ p^o₁(q_Lp^o₂+ (1 − q_L)p^e_2H), which is true by assumption.

4. p^o₂ ≥ p^e_2H and p^o₁p^o₂≥ max{p^e_1Hp^e_2H,_q ^{qLpe1Lpe2Lpe1Hpe2H}

Lp^e_1Hp^e_2H−(1−q_L)(p^e_1Lp^e_2L−p^e_1Hp^e_2H)}

Under immediate licensing, the low-type licensee contract is (m_2L, r_L) and the high-type licensee contract is m_2H, with m_2L= [−c₁−p^e_1L(c₂+x^∗_LL)+p^e_1Lp^e_2L(1−r_L)s(x^∗_LL)]/(p^e_1Lp^e_2L) and m2H = m2L+ [−(x^∗_HH− x^∗_HL) + p^e_2H(s(x^∗_HH) − (1 − rL)s(x^∗_HL))]/p^e_2H = m2L+ ∆/p^e_2H. Assume that we delay licensing and ask for a feasible contract menu (m_1L, r_L) and m_1H, with m1L= [−c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL)]/p^e_2L and m1H = m1L+ ∆/p^e_2H. Then,

p^o₁p^o₂

p^e_1Lp^e_2L[−c1− p^e_1L(c2+ x^∗_LL) + p^e_1Lp^e_2L(1 − r_L)s(x^∗_LL)] + q_Lp^o₁p^o₂r_Ls(x^∗_LL) + (1 − q_L)p^o₁∆

≤ −c₁+p^o₁p^o₂

p^e_2L(−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)) + q_Lp^o₁p^o₂r_Ls(x^∗_LL) + (1 − q_L)p^o₁∆

⇐⇒ p^e_1Lp^e_2L≤ p^o₁p^o₂,

which is true by assumption.

(b) Given Theorem 2, the optimal two-stage contract will only include an upfront payment for both licensee types in the following two cases, for which we need to show that delaying will never be optimal:

1. p^o₁ ≤ min{p^e_1H, p^e_1H−^pe1H_q^−pe1L

L } and p^o₁p^o₂ ≤ max{0,^{pe1L(pe2L)2−(1−q}_q ^{L)pe1H(pe2H)2}

Lp^e_2L }

Under immediate licensing, the licensor obtains m_0L = m_0H = −c1 − p^e_1L(c2+ x^∗_LL) + p^e_1Lp^e_2Ls(x^∗_LL).

After delaying, the optimal contract could be either of the following menus:

(a) m_0L = m_0H = −c₂− x^∗_LL+ p^e_2Ls(x^∗_LL). It is immediately obvious that this yields a lower value for the licensor than immediate licensing.

(b) (m_0L, r_L) and m_0H, with m_0L = −c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL) and m_0H = m_0L− (x^∗_HH− x^∗_HL) + p^e_2H(s(x^∗_HH) − (1 − rL)s(x^∗_HL)) = m0L+ ∆. Assuming we license im-mediately with the same (suboptimal) royalty rate r_Land the corresponding feasible milestone payments m0L and m0H, we get:

p^e_1L(−c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL)) + p^e_1H(1 − qL)∆

≥ p^o₁(−c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL) + (1 − qL)∆) This always holds since p^o₁≤ min{p^e_1H, p^e_1H− ^pe1H_q^−pe1L

L }.

(c) (m1L, rL) and m0H, with m1L = [−c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL)]/p^e_2L and m0H = p^e_2Hm_1L− (x^∗_HH− x^∗_HL) + p^e_2H(s(x^∗_HH) − (1 − r_L)s(x^∗_HL)). Assuming we license im-mediately with the same (suboptimal) royalty rate rLand the corresponding feasible

milestone payments m_0L and m_0H, we get:

p^e_1L(−c2− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL)) + p^e_1H(1 − qL)∆

≥ (p^e_1L− C)(−c₂− x^∗_LL+ p^e_2L(1 − rL)s(x^∗_LL)) + p^o₁(1 − qL)∆

where C ≤ (1 − qL)p^e_1H(p^e_1Hp^e_2H − p^e_1Lp^e_2L)/p^o₁p^eL2. This always holds since p^o₁ ≤ min{p^e_1H, p^e_1H−^pe1H_q^−pe1L

L }.

(d) (m_1L, r_L) and m_1H, with m_1L = [−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)]/p^e_2L and m_1H = m1L+[−(x^∗_HH−x^∗_HL)+p^e_2H(s(x^∗_HH)−(1−rL)s(x^∗_HL))]/p^e_2H. Assuming we license im-mediately with the same (suboptimal) royalty rate r_Land the corresponding feasible milestone payments m_0L and m_0H, we get:

p^e_1L(−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)) + p^e_1H(1 − q_L)∆

≥ p^o₁p^o₂

p^e_2L (−c₂− x^∗_LL+ p^e_2L(1 − r_L)s(x^∗_LL)) + p^o₁p^o₂

p^e_2H(1 − q_L)∆

where ^p_p^o1e^po2 2L

≤ p^e_1L and ^p_p^o1e^po2 2H

≤ p^e_1H. 2. p^o₁ ≤ min{p^e_1H, p^e_1H−^pe1H_q^−pe1L

L } and^{pe1L(pe2L)2−(1−q}_q ^{L)pe1H(pe2H)2}

Lp^e_2L ≤ p^o₁p^o₂≤ min{p^e_1Hp^e_2H,^{pe1Lpe2L−(1−q}_q ^L)pe1Hpe2H

L }

Low-type licensee contract (m_0L, r_L) and high-type licensee contract m_0H.

The analysis is similar to the 4 cases elaborated above for the pooling contract at imme-diate licensing.

In document Step by step. The benefits of stage-based R&D licensing contracts (Page 30-40)