A comparison of milestone contract and royalty contract under critical value criterion in R&D alliance

Abstract

This paper attempts to better understand contract type and risk attitude in R&D alliance under information asymmetry, that is, how the marketer designs and chooses optimal contract between royalty contract and milestone contract and how the innovator’s risk attitude and asymmetric information affect the marketer’s optimal contract strategies and profits. We use principal–agent models to formulate the marketer’s contracting problem under asymmetric information about the innovator’s innovation expertise and unobservable efforts. We find that, compared to the case under full information in both contracting structures, the marketer should distort the commission rate upwards under dual asymmetric information when the innovator is risk averse or downwards to lower innovation–expertise and risk-loving innovator; nevertheless, the marketer should offer the first-best contract. Furthermore, investigating the impacts of information asymmetry on the marketer’s profits under two information structures, we find that dual asymmetric information harms the marketer’s profit, especially when the innovator’s effort marginal efficiency is higher. Finally, by comparing the marketer’s profits in two types of contracts, we give the specific regions that milestone contract or royalty contract benefits the marketer better under different information structures.

Appendix

Proof of Proposition 1

Based on the expected value criterion, the marketer’s expected profit

$$\begin{aligned} \Pi _m=(k_1\theta +k_2e_1)(1-w_1(\theta ))+l_1e-\frac{1}{2}e^2-w_0(\theta ). \end{aligned}$$

Based on the critical value criterion, if the innovator accepts milestone contract, his $\alpha$-utility

$$\begin{aligned} \pi _m(\theta ,\theta )= & {} \left( k_1\theta +k_2e_1+\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \\&\quad \times w_1(\theta )+w_0(\theta )-\frac{1}{2}e_1^2. \end{aligned}$$

Because the marketer’s expected profit is decreasing in the fixed payment $w_0$, at optimality, the individual rationality constraint should be binding, i.e., $\pi _m(\theta ,\theta )=0$. Thus, substituting the fixed payment into the marketer’s objective function, we obtain

$$\begin{aligned} \Pi _m= & {} w_1(\theta )\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }+k_1\theta +k_2e_1+l_1e \\&-\frac{1}{2}e^2-\frac{1}{2}e_1^2, \end{aligned}$$

which is concave in $e_1$ and e, respectively. Using the first-order condition, the optimal efforts can be solved as: $e_1=k_2$ and $e=l_1$. By substituting these optimal effort levels into the marketer’s simplified problem above, we simplify the firm’s problem to

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \max _{w_1(\theta )} w_1(\theta )\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }+k_1\theta +\frac{1}{2}\left( k_2^2+l_1^2\right) \\ \text{ subject } \text{ to: }\\ \displaystyle \quad 0\leqslant w_1(\theta )\leqslant 1,\quad \forall \theta \in [\overline{\theta },\underline{\theta }].\\ \end{array} \right. \end{aligned}$$

Because the objective function is piecewise linear in $w_1(\theta )$, we focus on the corner point solutions $w_1(\theta )\in \{0,1\}$. To ensure the marketer can get a positive and maximum profit, if $\alpha >1/2$, the optimal incentive compensation rate is $w_1^F(\theta )=0$; if $\alpha \leqslant 1/2$, the optimal incentive compensation rate is $w_1^F(\theta )=1$. The corresponding optimal fixed payment $w_0^F(\theta )$ for the innovator can be derived based on the binding individual rationality constraint immediately. The proof of the proposition is complete. $\square$

Proof of Proposition2

Similar to the Proof of Proposition 1. $\square$

Proof of Lemma 1

For part (i) and part (ii), if the innovator’s true innovation expertise is $\theta$ but he selects the milestone contract $(w_0(\hat{\theta }),w_1(\hat{\theta }))$, his $\alpha$-utility

$$\begin{aligned} \pi _m(\theta ,\hat{\theta },e_1)= & {} \left( k_1\theta +k_2e_1+\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \\&\times w_1(\hat{\theta })+w_0(\hat{\theta })-\frac{1}{2}e_1^2. \end{aligned}$$

By the first-order condition, the innovator maximizes his profit by exerting an effort $e_1=k_2w_1(\hat{\theta })$. The innovator’s profit under this optimal effort is

$$\begin{aligned} \pi _m(\theta ,\hat{\theta })= & {} \frac{1}{2}k_2^2w_1^2(\hat{\theta })+\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \\&\times w_1(\hat{\theta })+w_0(\hat{\theta }). \end{aligned}$$

Similarly, if the innovator selects the contract $(w_0(\theta ),w_1(\theta ))$ based on his true innovation expertise $\theta$, his profit

$$\begin{aligned} \pi _m(\theta ,\theta )= & {} \frac{1}{2}k_2^2w_1^2(\theta ) +\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \\&\times w_1(\theta )+w_0(\theta ). \end{aligned}$$

The incentive compatibility constraint for adverse selection means that the innovator can obtain his maximal profit $\pi _m(\theta ,\hat{\theta })$ if and only if $\hat{\theta }=\theta$. In other words, the innovator with innovation expertise $\theta$ has no incentive to pretend to be other idea value $\hat{\theta }$. Thus, $\pi _m(\theta ,\hat{\theta })$ satisfies the first-order condition (i.e., local incentive compatibility constraint) $\frac{\partial \pi _m(\theta ,\hat{\theta })}{\partial \hat{\theta }}\bigm |_{\hat{\theta }=\theta }=0$ and the second-order condition $\frac{\partial ^{2} \pi _m(\theta ,\hat{\theta })}{\partial \hat{\theta }^{2}}\bigm |_{\hat{\theta }=\theta }\leqslant 0$. The local incentive compatibility constraint:

$$\begin{aligned}&\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \frac{\mathrm {d}w_1(\theta )}{\mathrm {d}\theta } +k_2^2w_1(\theta ) \frac{\mathrm {d} w_1(\theta )}{\mathrm {d}\theta }\nonumber \\&\quad +\frac{\mathrm {d} w_0(\theta )}{\mathrm {d}\theta }=0. \end{aligned}$$

(1)

And the second-order condition:

$$\begin{aligned}&\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }+k_2^2w_1(\theta )\right) \frac{\mathrm {d}^2w_1(\theta )}{\mathrm {d}\theta ^2} \nonumber \\&\quad +\,k_2^2\left( \frac{\mathrm {d} w_1(\theta )}{\mathrm {d}\theta }\right) ^2 +\frac{\mathrm {d}^2w_0(\theta )}{\mathrm {d}\theta ^2}\leqslant 0. \end{aligned}$$

(2)

If we further differentiate the local incentive compatibility constraint with the respect to $\theta$, we can obtain

$$\begin{aligned}&\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1\!-\!\alpha }{\alpha }\!+\!k_2^2w_1(\theta )\right) \frac{\mathrm {d}^2w_1(\theta )}{\mathrm {d}\theta ^2}\nonumber \\&\quad +\,k_2^2\left( \frac{\mathrm {d} w_1(\theta )}{\mathrm {d}\theta }\right) ^2 +\frac{\mathrm {d}^2w_0(\theta )}{\mathrm {d}\theta ^2}\!+\!k_1\frac{\mathrm {d}w_1(\theta )}{\mathrm {d}\theta }\!=\!0. \end{aligned}$$

(3)

On the basis of (2) and (3), we gain the monotonicity condition

$$\begin{aligned} \frac{\mathrm {d} w_1(\theta )}{\mathrm {d} \theta }\leqslant 0,\quad \forall \theta \in [\overline{\theta },\underline{\theta }]. \end{aligned}$$

(4)

Suppose, next, that both the local incentive compatibility and monotonicity conditions hold. Then it must be the case that all the innovator’s incentive compatibility conditions hold. To see this result, without loss of generality, suppose that $\theta <\hat{\theta }$. By integrating the local incentive compatibility condition (1) and using the monotonicity condition (4), we can obtain

$$\begin{aligned} w_{0}(\hat{\theta })-w_{0}(\theta )\leqslant (w_1(\theta )-w_1(\hat{\theta }))\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) . \end{aligned}$$

That is to say

$$\begin{aligned} \pi _m(\theta ,\theta )\geqslant \pi _m(\theta ,\hat{\theta }),\quad \forall \theta ,\hat{\theta }\in [\overline{\theta },\underline{\theta }]. \end{aligned}$$

On the other hand, if $\theta \geqslant \hat{\theta }$, by integrating the local incentive compatibility condition (1) and using the monotonicity condition (4), we also can obtain

$$\begin{aligned} w_{0}(\theta )-w_{0}(\hat{\theta })\geqslant (w_1(\hat{\theta })-w_1(\theta ))\left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) . \end{aligned}$$

That is to say

$$\begin{aligned} \pi _m(\theta ,\theta )\geqslant \pi _m(\theta ,\hat{\theta }),\quad \forall \theta ,\hat{\theta }\in [\overline{\theta },\underline{\theta }]. \end{aligned}$$

This result establishes the equivalence between the monotonicity condition together with the local incentive compatibility condition and the full set of the innovator’s incentive constraints.

Differentiating $\pi _m(\theta ,\theta )$ with respect to $\theta$ yields

$$\begin{aligned} \displaystyle \frac{\mathrm {d} \pi _m(\theta ,\theta )}{\mathrm {d} \theta }= & {} \left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \frac{\mathrm {d}w_1(\theta )}{\mathrm {d}\theta } \\&+\,k_2^2w_1(\theta )\frac{\mathrm {d} w_1(\theta )}{\mathrm {d}\theta }+\frac{\mathrm {d} w_0(\theta )}{\mathrm {d}\theta }+k_1w_1(\theta )\nonumber \\= & {} k_1w_1(\theta )\geqslant 0. \end{aligned}$$

The individual rationality constraint is equivalent to

$$\begin{aligned} \pi _m(\underline{\theta },\underline{\theta })\geqslant 0. \end{aligned}$$

The constraint is binding under the optimal mechanism because the marketer can reap the redundant profit under the optimal incentive mechanism, so that $\pi _m(\underline{\theta },\underline{\theta })=0$. $\square$

Proof of Proposition 3

Because

$$\begin{aligned} \displaystyle \frac{\partial \pi _m(\theta ,\theta )}{\partial \theta }=k_1w_1(\theta ), \end{aligned}$$

we can derive

$$\begin{aligned} \pi _m(\theta ,\theta )=\int ^{\theta }_{\underline{\theta }}\frac{\partial \pi _m(\theta ,\theta )}{\partial \theta }-\pi _m(\underline{\theta },\underline{\theta })=\int ^{\theta }_{\underline{\theta }}k_1w_1(x)\mathrm {d}x. \end{aligned}$$

Combining the definition of $\pi _m(\theta ,\theta )$ yields

$$\begin{aligned} w_0(\theta )= & {} \int ^{\theta }_{\underline{\theta }}k_1w_1(x)\mathrm {d}x- \left( k_1\theta +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }\right) \nonumber \\&w_1(\theta )-\frac{1}{2}k_2^2w_1^2(\theta ). \end{aligned}$$

By substituting the fixed wage into the objective function, we can derive

$$\begin{aligned} \Pi _m^D= & {} \int ^{\overline{\theta }}_{\underline{\theta }} \left[ k_1\theta \!+\!k_2^2w_1(\theta )\! +\!l_1e -\frac{1}{2}e^2-\frac{1}{2}k_2^2w_1^2(\theta )\right. \\&\left. +\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1\!-\!\alpha }{\alpha }w_1(\theta ) -\int ^{\theta }_{\underline{\theta }}k_1w_1(x)\mathrm {d}x\right] f(\theta )\mathrm {d}\theta . \end{aligned}$$

We can use the first-order condition $\frac{\partial \Pi _m^D}{\partial e}=0$ to yield the first-best effort level $e^D=l_1$. Substituting it into the objective function and ignoring the monotonicity constraint in Lemma 1, the marketer’s problem can be written as unconstrained optimization problem:

$$\begin{aligned} \displaystyle \max _{0\leqslant w_1(\theta )\leqslant 1} \Pi _m^D= & {} \int ^{\overline{\theta }}_{\underline{\theta }} \left[ k_1\theta +\frac{1}{2}l_1^2+(k_2^2-k_1h(\theta ) \right. \\&+\frac{\sqrt{3}\sigma _1}{\pi } \ln \frac{1-\alpha }{\alpha }w_1(\theta ) \\&\left. -\frac{1}{2}k_2^2w_1^2(\theta )\right] f(\theta )\mathrm {d}\theta . \end{aligned}$$

The first-order variation and the second-order variation of the marketer’s expected profit are derived as

$$\begin{aligned} \displaystyle \delta \Pi _m^D= & {} \int ^{\overline{\theta }}_{\underline{\theta }}\left[ k_2^2-k_1h(\theta )+\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }-k_2^2w_1(\theta )\right] \\&f(\theta )[\delta w_1(\theta )]\mathrm {d}\theta , \end{aligned}$$

and

$$\begin{aligned} \displaystyle \delta ^2\Pi _m^D=-\int ^{\overline{\theta }}_{\underline{\theta }}k_2^2f(\theta )[\delta w_1(\theta )]^2\mathrm {d}\theta . \end{aligned}$$

If $\alpha <1/2$, there are two cases: (i) $\underline{\theta }\leqslant \theta \leqslant \theta _{12}$ and (ii) $\theta _{12}<\theta \leqslant \overline{\theta }$ (here the threshold $\theta _{12}$ satisfies $\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }-k_1 h(\theta )=0$). For case (i), we can obtain $w_1^D=1+\frac{1}{k_2^2}(\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }- h(\theta ))$. For case (ii), we can obtain $w_1^D(\theta )=1$. On the other hand, if $\alpha \geqslant 1/2$, there are two cases: (iii) $\underline{\theta }\leqslant \theta \leqslant \theta _{11}$ and (iv) $\theta _{11}<\theta \leqslant \overline{\theta }$ (here the threshold $\theta _{11}$ satisfies $\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }-k_1 h(\theta )+k_2^2=0$). For case (iii), we can obtain $w_1^D(\theta )=0$. For case (iv), we can obtain $w_1^D=1+\frac{1}{k_2^2}(\frac{\sqrt{3}\sigma _1}{\pi }\ln \frac{1-\alpha }{\alpha }- h(\theta ))$. Following the determinate optimal incentive commission rate $w_1^D(\theta )$, the corresponding optimal fixed payment $w_0^D(\theta )$ and the optimal effort levels $e_1^D$, $e^D$ can be obtained immediately. The proof of the proposition is complete. $\square$

Proof of Lemma 2

Similar to the Proof of Lemma 1. $\square$

Proof of Proposition 4

Similar to the Proof of Proposition 3.

$\square$

Proof of Proposition 5

The result is derived directly by comparing the contracts shown in Propositions 1–4. $\square$

Proof of Propositions 6–8

The result is derived directly by comparing the marketer’s profits shown in Corollaries 1–2. $\square$