Divisional performance measurement and transfer pricing for intangible assets

Abstract

This paper examines the effectiveness of three transfer pricing methodologies for an intangible asset that is developed through bilateral, sequential investment. In general, a royalty-based transfer price that can be renegotiated provides better investment incentives than either a non-negotiable royalty-based transfer price or a purely negotiated transfer price, and in some cases induces first-best investment. This result contrasts with previous research that finds that the inability to limit renegotiation of initial contracts reduces investment efficiency. Further, I examine how tax transfer pricing rules inform optimal internal transfer prices when the firm decouples internal and external transfer prices.

Appendix A

Proof of Lemma 1

The firm’s central office chooses first-best levels of investment to maximize firm profit:

$$ (I_1^{\ast},I_2^{\ast})\in\mathop{\hbox{argmax}}\limits_{I_1,I_2}\; \{(1-t)[v(I_1)+w(I_2)-I_1-I_2]- h[{\hat \alpha}(v(I_1)+w(I_2))-I_1]\}. $$

Then I ^*₁ , I ^*₂ solve the following first order conditions:

$$ v^{\prime}(I_1)=\frac{1-t-h}{1-t-h{\hat \alpha}},$$

(21a)

$$ w^{\prime}(I_2)=\frac{1-t}{1-t-h{\hat \alpha}}. $$

(21b)

(A2) and the strict concavity of v(·) and w(·) guarantee an interior solution. The RHS of (21a) is decreasing in h and the RHS of (21b) is increasing in h. Then by decreasing marginal returns, I ^*₁ is increasing in h and I ^*₂ is decreasing in h.

Proof of Proposition 1

Assume that the investments are quasi-independent. Then first-best investments, I ^*₁ , I ^*₂ , satisfy

$$ v^{\prime}(I_1^{\ast})=\frac{1-t-h}{1-t-{\hat \alpha}h},\quad w^{\prime}(I_2^{\ast})=\frac{1-t}{1-t-{\hat \alpha}h}. $$

Under a uniform royalty-based transfer pricing scheme, divisional managers choose I ^u₁ and I ^u₂ to solve

$$ v^{\prime}(I_1^u)=\frac{1}{\hat \alpha},\quad w^{\prime}(I_2^u)=\frac{1}{1-{\hat \alpha}}. $$

(A3) and the strict concavity of v(·) and w(·) guarantee an interior solution. Since

$$ \frac{1}{\hat \alpha}> \frac{1-t-h}{1-t-{\hat \alpha}h}\quad \hbox {and}\quad \frac{1}{1-{\hat \alpha}}> \frac{1-t}{1-t-{\hat \alpha}h} $$

decreasing marginal returns imply that I ^*₁ >I ^u₁ and I ^*₂ >I ^u₂ .

Proof of Corollary to Proposition 1

Lemma 1 shows that with quasi-independent assets, I ^*₁ is increasing in h and I ^*₂ (I ₁) is decreasing in h. Under a uniform royalty based transfer pricing system, both divisions underinvest, as shown in Proposition 1. Further, both divisions’ investments are independent of h, as can be seen from their first order conditions:

$$ v^{\prime}(I_1)=\frac{1}{\hat \alpha}, \quad w^{\prime}(I_2)=\frac{1} {1-{\hat \alpha}}.$$

The result immediately follows.

Proof of Proposition 2

When v(I)≡ w(I) and h=0, the firm’s objective is

$$ \mathop{\hbox{max}}\limits_{I_1,I_2} \{[v(I_1)+v(I_2)-I_1-I_2](1-t)\}. $$

Since the problem is symmetric and marginal returns are decreasing, firm-wide profit is optimized when I ₁=I ₂. The divisions will choose I ₁ and I ₂ to solve

$$ v^{\prime}(I_1)=\frac{1-t}{\alpha-{\hat \alpha}t},\quad w^{\prime}(I_2)=\frac{1-t}{1-\alpha-({1-\hat \alpha})t}. $$

Since v(I)=w(I), the two divisions will choose the same level of investment when they face the same marginal cost, i.e.

$$ \frac{1-t}{\alpha-{\hat \alpha}t}=\frac{1-t} {1-\alpha-({1-\hat \alpha})t}. $$

(22)

(22) implies that the optimal decoupled royalty rate, α^*, is

$$ \alpha^{\ast}=\frac{1-t}{2}+{\hat \alpha}t. $$

Then when ${\hat \alpha}=\frac{1}{2},\alpha^{\ast}=\frac{1}{2}$ and there is no gain from decoupling.

Proof of Proposition 3

The available negotiation surplus will be maximized if I ₂(I ₁) is chosen such that

$$ I_2^{\ast}(I_1)\in\mathop{\hbox{argmax}}\limits_{I_2}\{M(I_1, I_2)(1-t-{\hat \alpha}h)-I_1(1-t-h)-I_2(1-t)\}. $$

Division 2 will choose I ₂(I ₁) so that

$$ \hat{I}_2(I_1)\in\mathop{\hbox{argmax}}\limits_{I_2}\{M(I_1, I_2)[(1-\beta)-(1-{\hat \alpha})t)]-I_2(1-t)\}. $$

Then Division 2 will invest efficiently if the post-negotiation royalty rate, β, solves

$$ 1-t-{\hat \alpha}h=1-\beta-(1-{\hat \alpha})t, $$

which implies $\beta={\hat \alpha}(t+h)$.

Given that Division 2 will invest efficiently ex-post, Division 1 chooses I ₁ such that

$$ \hat{I_1}\in\mathop{\hbox{argmax}}\limits_{I_1}\{\gamma [M(I_1,I_2^{\ast}(I_1)) (1-t-{\hat \alpha}h)-I_2^{\ast}(I_1)(1-t)]-I_1(1-t-h)\}, $$

whereas firm-wide profit is maximized when I ₁ is chosen such that

$$ I_1^{\ast}\in\mathop{\hbox{argmax}}\limits_{I_1}\{M(I_1,I_2^{\ast}(I_1))(1-t-{\hat \alpha}h)-I_1(1-t-h)-I_2^{\ast}(I_1)(1-t)\}.$$

Then since $\gamma< 1, \hat{I_1}< I_1^{\ast}$ under negotiated transfer pricing. ²⁰

Proof of Proposition 4

Investments are quasi-independent

If the royalty scheme is carried out, Division 2’s payoff from investing I ⁰₂ is

$$ [v(I_1)+w(I_2^0)](1-\alpha-{(1-\hat \alpha)}t)-I_2^0(1-t). $$

(23)

Let ${\tilde{\alpha}}(I_1)$ be the value of α, conditional on I ₁, that makes (23) equal to zero. Then

$$ {\tilde{\alpha}}(I_1)=1-(1-{\hat \alpha})t-\frac{I_2^0(1-t)} {v(I_1)+w(I_2^0)}. $$

(24)

Then for a given level of I ₁, any $\alpha> {\tilde{\alpha}}(I_1)$ will give Division 2 a negative payoff if he invests I ⁰₂ . Further, if $\alpha>{\tilde{\alpha}}(I_1)$, Division 2 cannot improve his payoff by investing I ₂>I ⁰₂ . To see why, note that Division 2’s first order condition is

$$ w^{\prime}(I_2)(1-\alpha-({1-\hat \alpha})t)-(1-t)=0. $$

(25)

Then because of the strict concavity of w(·), his marginal return from investing more than I ⁰₂ is only positive if

$$ w^{\prime}(I_2^0)(1-\alpha-({1-\hat \alpha})t)-(1-t)> 0. $$

(26)

Replacing α in (26) with ${\tilde{\alpha}}(I_1)$ from (24), we get

$$ w^{\prime}(I_2^0)\left[\frac{I_2(1-t)} {v(I_1)+w(I_2^0)}\right]-(1-t)> 0. $$

(27)

But (27) implies

$$ I_2^0\cdot w^{\prime}(I_2^0)> v(I_1)+w(I_2^0), $$

(28)

which is never true because w(·) is strictly concave and strictly increasing, which implies that I ₂· w′(I ₂)<w(I ₂) for all I ₂. Then at $\alpha={\tilde{\alpha}}(I_1)$, Division 2 will never invest more than I ⁰₂ and will only invest I ⁰₂ if (23) is non-negative.

Now, suppose that α is set equal to ${\tilde{\alpha}}(I_1^{\ast})$ but I ₁<I ^*₁ . Plugging ${\tilde{\alpha}}(I_1^{\ast})$ from (24) into (23) gives us Division 2’s payoff from investing I ⁰₂ :

$$ \frac{I_2(1-t)[v(I_1)+w(I_2^0)]} {v(I_1^{\ast})+w(I_2^0)}-I_2^0(1-t). $$

(29)

This expression will always be negative if I ₁<I ^*₁ since v(I ₁) is increasing in I ₁. Then if $\alpha={\tilde{\alpha}}(I_1^{\ast})$, (23) will always be negative (and Division 2 will not invest) unless Division 1 invests at least I ^*₁ .

Division 1 will not invest more than I ^*₁ since under the royalty scheme, he maximizes his payoff by choosing $\hat{I}_1< I_1^{\ast}$ and by decreasing marginal returns, his payoff is decreasing in I ₁ for $I_1>\hat{I}_1$. He will be willing to invest exactly I ^*₁ as long as his payoff from doing so is positive. At $\alpha={\tilde{\alpha}}(I_1^{\ast})$, Division 1’s status quo payoff if he invests I ^*₁ is

$$ [v(I_1^{\ast})+w(I_2^0)]({\tilde{\alpha}}(I_1^{\ast})-{\hat \alpha}(t+h))-(1-t-h)I_1^{\ast}= $$

(30a)

$$ v(I_1^{\ast})(1-t-\hat\alpha h)-I_1^{\ast}(1-t-h) + w(I_{2}^{o})(1-t-\hat\alpha h)-I_{2}^{o}(1-t)> 0. $$

(30b)

Since (30b) corresponds to the firm’s objective function and is always positive for I ₁≤ I ^*₁ , I ₂ ≤ I ^*₂ (I ₁) (because of (A2)), Division 1 will invest I ^*₁ when $\alpha=\tilde{\alpha}(I_1^{\ast})$. In equilibrium, the internal royalty rate will be renegotiated to $\alpha=\hat{\alpha}(t+h)$, as shown in the proof to Proposition 3, so that Division 2 will invest I ^*₂ (I ^*₁ ) and first best will be achieved.

Investments are substitutes: (M₁₂<0)

For this case, I will show that I ₁ = 0 at $\alpha=\hat{\alpha}(t+h)$ and that I ₁>I ^*₁ at $\alpha=1-(1-\hat{\alpha})t$ and then infer the result by the intermediate value theorem.

First consider the case of $\alpha=\hat{\alpha}(t+h)$. Division 1’s payoff under the royalty agreement will be zero and Division 2 will be unwilling to renegotiate after Division 1 invests because I ₂ will be efficient, leaving no gain from bargaining. Therefore Division 1 will choose I ₁ = 0.

Now consider the case of $\alpha=1-(1-\hat{\alpha})t$. In this case, Division 2’s payoff from investment is zero under the royalty agreement, so he will choose I ₂ = 0 unless the agreement is renegotiated. Then Division 1’s marginal payoff is

$$ (1-\gamma)\frac{\partial}{\partial I_1}M(I_1,0)+\gamma \frac{\partial}{\partial I_1}M(I_1,I_2^{\ast}(I_1))=\frac{1-t-h}{ 1-t-\hat{\alpha}h}. $$

(31)

When γ=1, Division 1 will choose I ₁=I ^*₁ . When γ=0, Division 1’s marginal payoff is

$$ \frac{\partial}{\partial I_1}M(I_1,0)=\frac{1-t-h}{1-t-\hat{\alpha}}=\frac{\partial}{\partial I_1}M(I_1^{\ast},I_2^{\ast}(I_1^{\ast})).$$

(32)

Suppose I ₁<I ^*₁ and (32) holds. Then it follows that $\frac{\partial}{\partial I_1}M(I_1,0)>\frac{\partial}{\partial I_1}M(I_1^{\ast},0)>\frac{\partial}{\partial I_1}M(I_1^{\ast},I_2^{\ast}(I_1^{\ast}))$, a contradiction. The first inequality holds because of decreasing marginal returns and the second because the investments are substitutes. Therefore I ₁>I ^*₁ if the investments are substitutes at γ = 0. Note that by the implicit function theorem,

$$ \hbox{sgn}(I_1^{\prime}(\gamma))=\hbox{sgn}\left(-\frac{\partial}{\partial I_1}M(I_1,0)+\frac{\partial}{\partial I_1}M(I_1,I_2^{\ast}(I_1))\right). $$

(33)

The RHS (33) is negative when the investments are substitutes, so I ^′₁ (γ)<0.

Then when $\alpha=1-(1-\hat{\alpha})t$, we have I ₁=I ^*₁ at γ = 1, I ₁>I ^*₁ at γ = 0 and I ^′₁ (γ)<0, indicating that I ₁>I ^*₁ . Recall that I ₁ = 0 at $\alpha=\hat\alpha(t+h)$. Then by the intermediate value theorem, and given Assumption 1, there exists a value of α that implements first best, i.e. I ₁=I ^*₁ , when the investments are substitutes.

Proof of Proposition 5

When the investments are strict complements, Division 1’s first order condition under renegotiable royalty-based transfer pricing is

$$\begin{aligned} &\left\{\frac{\partial}{\partial I_1}M(I_1,I_2(I_1|\alpha)) +\frac{\partial}{\partial I_2} M(1_1,I_2(I_1|\alpha)I_2^{\prime}(I_1)\right\}[\alpha-\hat{\alpha}(t+h)- \gamma(1-t-\hat{\alpha}h)] \\ &\quad+\gamma\left[\frac{\partial}{\partial I_1}M(I_1,I_2^{\ast}(I_1))(1-t-\hat{\alpha}h)+I_2^{\prime}(I_1)(1-t)\right]=(1-t-h) \end{aligned} $$

(34)

and under negotiation, it is

$$ \gamma\frac{\partial}{\partial I_1}M(I_1,I_2^{\ast}(I_1))(1-t-\hat{\alpha}h)=(1-t-h).$$

(35)

When

$$ \alpha>\hat{\alpha}(t+h)+\gamma(1-t-\hat{\alpha}h),$$

(36)

the RHS of (34) is greater than the RHS of (35) for a given I, implying that when (36) holds, I ₁ is higher under renegotiable royalty-based transfer pricing than under negotiation. We know from Proposition 3 that Division 1 always underinvests under negotiated transfer pricing. This result, combined with Assumption 1, implies that it is always possible to choose a value of α such that (36) holds and Division 1 invests more efficiently under the renegotiated royalty-based system than under negotiation.

(Note that the RHS of (36) is always less than 1. To see why, notice that the RHS of (36) is increasing in γ and $\hat{\alpha}$, and that at $\gamma=\hat{\alpha}=1$, it is equal to one. Then since γ<1 by assumption, the RHS of (36) is less than one.)