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The optimal focus of transfer prices: pre-tax profitability versus tax minimization

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Abstract

This paper studies transfer prices influencing managerial decisions and determining corporate taxes in a multinational firm. Common sense suggests that the transfer price decision should be made to maximize the firm’s after-tax profit and thus achieve the optimal trade-off between pre-tax profitability and tax minimization. Based on a model of a decentralized firm facing asymmetric information with respect to operations, I examine why this conclusion does not hold in general. In particular, I demonstrate that a policy of negotiated transfer pricing, under which the divisions exploit their superior information but select the transfer price to maximize the firm’s pre-tax profit, is the firm’s optimal organizational choice if the high-tax division’s productivity is high. With respect to the firm’s discretion over the transfer price, I identify situations where the firm’s optimal policy choice does not depend on the arm’s length range and where less discretion increases the firm’s profitability.

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Notes

  1. See, for example, Anthony and Govindarajan (2000, p. 201) or Tang (2002, p. 42) for the functions of (international) transfer prices and their empirical prevalence.

  2. The figures relate to those firms using the same transfer price for both management and tax purposes.

  3. See Hyde and Choe (2005) for a model of a firm that deliberately fails to comply with the tax rules.

  4. See Keuschnigg and Devereux (2013) for a similar assumption.

  5. With a slight abuse of notation, \(\theta \) is used for both the random variable and its realization.

  6. In terms of transfer pricing methods for tax purposes as described in 26 CFR § 1.482-3 or OECD (2010, § II) the assumed range matches with the comparable uncontrolled price method, the resale price method, and the cost plus method based on budgeted costs. See Ernst & Young (2010, p. 13) for the prevalence of these methods.

  7. The case study in Cools and Slagmulder (2009) suggests that firms might eliminate price negotiations to substantiate their tax compliance efforts. Here, I assume that tax compliance has no effect on the policies.

  8. The model refers to a constant transfer price. With such a simple contract, the central office cannot design a truth-telling mechanism to extract the divisions’ knowledge.

  9. See Czechowicz et al. (1982), Davis (1994), Granfield (1995), Durst (2002), and Cools and Slagmulder (2009) for confirmation of this argument.

  10. See 26 USC §§ 1.6038A, 1.6038C of the US Internal Revenue Code and OECD (2010) for the (statutory) disclosure requirements and the associated penalties. The PATA Transfer Pricing Documentation Package (Pacific Association of Tax Administrators 2011) is suggestive of the information regularly provided to tax authorities. During tax audits, tax authorities even request access to the firm’s electronic information system and operational personnel (Ernst & Young 2010, p. 14).

  11. Czechowicz et al. (1982, p. 59) report a corresponding share of firms of 84 %, Ernst & Young (2001, p. 6) of 77 %, and Ernst & Young (2003, p. 17) of 80 %.

  12. Due to the linear setting of the model, the joint trade decision can equivalently be interpreted as bilateral negotiations or as one division setting the quantity and the other accepting or rejecting this decision. The sequence of the trade decision and the transfer price agreement does not play a role either.

  13. Given internal trade, any transfer price not equal to \(p^{*}\) as given by (13) is inefficient because it does not minimize taxes. Therefore, one may wonder whether NTP or ATP benefit from revising or postponing the transfer price decision after the trade decision is made; see Martini (2011) for an analysis in this direction. The answer to this question is no, which is due to the adverse effect on the divisions’ trade decisions.

  14. Figure 3 is based on the parameter setting \(\omega _d/\omega _u=2\) and \((1 - \tau _d)/(1 - \tau _u) = 0.824\) and thereby corresponds to scenario A from Table 1.

  15. According to OECD (2013), the tax ratio varies between 0.696 and 1.0 for the OECD member countries; see column “Combined corporate income tax rate” for 2013 in Table II.1 “Basic (non-targeted) corporate income tax rates”. Concentrating on the G7 states, it varies between 0.791 and 0.967 for relations involving the United States.

  16. Numerical examples contain rounded values. The corporate tax rates in Table 1 are taken from OECD (2013).

  17. There are two transfer prices implying an expected profit shift of 12.5I under ATP. I chose the one inducing the higher expected after-tax profit.

  18. As shown in Lemma 5 in “Appendix 2”, the inclusion of CP does not interfere with this conclusion.

  19. According to Lemma 6 in “Appendix 2”, CP does not benefit from narrowing the arm’s length range. Hence, NTP also dominates CP for both considered degrees of discretion.

  20. The inclusion of CP does not make Proposition 5 obsolete as CP is not an optimal policy choice if the difference in the tax levels becomes sufficiently small. This is because the performance of CP is merely driven by profit shifting.

  21. The Kalai–Smorodinsky solution goes for the Pareto-efficient profit allocation inducing equal divisional pre-tax profits relative to the divisions’ maximal pre-tax profits. Accordingly, the negotiated transfer price solves \(\pi _u(\theta , p)\big /\pi _u \left( \theta , \min \{\overline{p}, \overline{p}_0\}\right) = \pi _d(\theta , p)\big /\pi _d ( \theta , \max \{\underline{p}, \underline{p}_0\}) \) with \(\pi _d(\theta , \overline{p}_0)=0\) and \(\pi _u(\theta , \underline{p}_0)=0\). In this sense, the Kalai–Smorodinsky solution averages the transfer prices \(\max \{\underline{p}, \underline{p}_0\}\) and \(\min \{\overline{p}, \overline{p}_0\}\).

  22. Again, narrowing the arm’s length range does not reverse the dominance of NTP over CP; see footnote 19.

  23. See, for example, Rosenmüller (2000, § 8) for the details.

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Acknowledgments

For helpful comments, I thank two anonymous reviewers, Stefan Reichelstein (the editor), and seminar participants at the universities of Bern, Frankfurt, Graz, Hannover, Mannheim, Tübingen, and Würzburg and at the annual congresses of the European Accounting Association (EAA) and the German Economic Association of Business Administration (GEABA).

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Correspondence to Jan Thomas Martini.

Appendices

Appendix 1: Benchmark case

The benchmark decisions maximize the firm’s after-tax profit given the cost-reduction probability \(\theta \), the investment level I, and the arm’s length range \([\underline{p}, \overline{p}]\). In Sects. 3 and 4, we have \([\underline{p}, \overline{p}]=[k - \omega _u I, k + \omega _d I]\); in Sect. 5, the arm’s length range is some interval not necessarily equal to or a subset of \([k - \omega _u I, k + \omega _d I]\).

The decision on internal trade maximizes

$$\begin{aligned} {[}\pi (\theta ) - t(\theta , p) ] q = (2\theta - 1) (\omega _u + \omega _d) I q - \left( \sum _{i \in \{u, d\}} \tau _i (2\theta - 1) \omega _i I - (\tau _d - \tau _u) (p - k) \right) q, \end{aligned}$$
(12)

where the probability \(\theta \) and the transfer price p are given and known. The expression \(q^*(\theta , p)\) denotes the corresponding maximizer. The optimal transfer price \(p^*\) maximizes (12) evaluated for \(q = q^*(\theta , p)\).

Lemma 2

In the benchmark case, the decision on internal trade is

$$\begin{aligned} q^*(\theta , p) =&\left. {\left\{ \begin{array}{ll} 1 &\quad \hbox {if} \quad \pi (\theta ) - t(\theta , p) \ge 0\\ 0 &\quad \hbox {otherwise} \end{array}\right. }\right\} \end{aligned}$$

and maximizes the firm’s after-tax profit for a given cost-reduction probability \(\theta \) and transfer price p. The transfer price choice minimizes firm-wide taxes per unit through a maximal profit shift to the low-tax division and equals

$$\begin{aligned} p^*&\left. {\left\{ \begin{array}{ll} = \overline{p} &{} \hbox { if }\quad \tau _u<\tau _d \\ = \underline{p} &{} \hbox { if }\quad \tau _u>\tau _d\\ \in \left[ \underline{p}, \overline{p}\right] &{} \hbox { if }\quad \tau _u=\tau _d \end{array}\right. }\right\} . \end{aligned}$$
(13)

Internal trade is optimal for the firm if both divisions and thus the firm show a nonnegative pre-tax profit per unit. In the event that one division’s pre-tax profit is positive and the other’s is negative, it must be assessed whether the tax on the positive profit is offset by the firm’s pre-tax profit plus the tax benefit from the negative profit. Thus, a nonnegative pre-tax profit for the firm is not necessary for the optimality of internal trade. The optimal transfer price minimizes the firm’s taxes per unit, (2), by shifting the firm’s pre-tax profit as much as possible to the low-tax division.

Appendix 2: Centralized Planning

In contrast to the hypothetical benchmark case, there is information asymmetry between the corporate and the divisional levels under CP, NTP, and ATP as to the cost-reduction probability \(\theta \). For CP, this is the only difference from the benchmark case.

The objective of the central office’s trade decision is the firm’s expected after-tax profit, that is, the expectation of (12) with respect to the random cost-reduction probability \(\theta \):

$$\begin{aligned} {\text {E}} \left( \left[ \pi (\theta ) - t(\theta , p) \right] q \right) = (\tau _d - \tau _u) (p - k) q. \end{aligned}$$
(14)

In expectation, the firm’s pre-tax profit vanishes due to the symmetric effect of the investment on production costs. In turn, this means that any expected after-tax profit is the result of profit shifting. Expression \(q_{{{\rm c}}}(p)\) denotes the corresponding maximizer, and the transfer price \(p_{{\rm c}}\) maximizes (14) for \(q = q_{{\rm c}}(p)\), given the general arm’s length range \([\underline{p}, \overline{p}]\).

Lemma 3

Given Centralized Planning (CP), the decision on internal trade maximizes the firm’s expected after-tax profit for a given transfer price \(p\) and is equal to

$$\begin{aligned} q_{{\rm c}}(p) = \left. {\left\{ \begin{array}{ll} 1 &\quad \hbox { if }\quad (\tau _d - \tau _u) (p - k) \ge 0\\ 0 &\quad \hbox {otherwise} \end{array}\right. }\right\} . \end{aligned}$$

The transfer price choice is \(p_{{\rm c}} = p^*\) and thus minimizes firm-wide taxes per unit through a maximal profit shift to the low-tax division.

As information asymmetry is the only difference between the benchmark case and CP, the trade decision under CP is equal to the benchmark decision evaluated for the expected cost-reduction probability \({\text {E}}(\theta )=0.5\), that is, \(q_{{\rm c}}(p) = q^*(0.5, p)\). The link between the benchmark case and CP is even stronger with respect to the chosen transfer price, as the optimality of the benchmark transfer price \(p^*\) does not depend on the cost-reduction probability and hence also prevails under CP.

Compared to NTP and ATP, the advantage of CP is that it is congruent with the firm’s goal of after-tax profit maximization with respect to both the transfer price and the trade decision. Its weakness is that it does not allow the central office to exploit the divisions’ information about the cost-reduction probability \(\theta \).

Including CP in the firm’s policy choice leads to an extension of Proposition 3:

Lemma 4

Let the arm’s length range \([\underline{p}, \overline{p}]=[k-\omega _u I, k+\omega _d I]\). Given unequal tax rates, the firm’s expected after-tax profit ...

  1. 1.

    ... under NTP is higher than (equal to) that under CP if and only if \({{\rm TR}} \mathrel {> (=)} (7 \, {{\rm PR}}-1)/(9 \, {{\rm PR}}+1)\).

  2. 2.

    ... under ATP is higher than (equal to) that under CP if and only if \({{\rm TR}} \mathrel {> (=)} (3 - \sqrt{2}/2 - (\sqrt{2} - 1)/(2 \, {{\rm PR}})\).

Given equal tax rates, the firm’s expected after-tax profit under CP does not reach the level of NTP and ATP.

The relations identified in Lemma 4 are depicted in Fig. 4 showing that the optimality of CP requires sufficiently high differences in the tax levels or, equivalently, sufficiently low values of the tax ratio \({{\rm TR}}\). Tax ratios exceeding \((3-\sqrt{2})/2 = 0.793\) even exclude CP as an optimal policy irrespective of the divisions’ productivities.

In this context, it is worth noting that the tax ratio can be interpreted not only as a result of taxes but also as a result of minority shares. From the central office’s perspective, both the taxes levied on the firm’s shares of divisional profits and the profits distributed to minority shareholders not belonging to the firm represent charges on divisional profits. Refer to Table 1 for an example and suppose that there are minority shareholders external to the firm holding a total of 25 % of the high-tax division’s shares. In conjunction with the US corporate tax rate of 39.1 %, this yields a total rate of \(\tau _h = 0.25 + (1-0.25) \cdot 0.391 = 0.544\) and reduces \({{\rm TR}}\) from 0.824 to 0.618. Accordingly, the ranges of \({{\rm TR}}\) given in footnote 15 start at 0.348 and 0.395, respectively, when allowing for minority shareholders.

Returning to a general arm’s length range \([\underline{p}, \overline{p}]\), we have the following addition to Lemma 1.

Lemma 5

The firm’s expected after-tax profit under CP exceeds neither that under NTP nor that under ATP if tax rates are equal.

It is true because the performance of CP solely depends on profit shifting which becomes irrelevant if there is no differential taxation.

Finally, Proposition 6 says that ATP does not benefit from narrowing the arm’s length range. This property equally applies to CP as both policies imply centralized transfer price decisions:

Lemma 6

Narrowing the arm’s length range does not increase the firm’s expected after-tax profit under CP.

Appendix 3: Proofs

Proof of Proposition 1

The proof refers to axiomatic bargaining theory, according to which any symmetric bargaining solution satisfies at least the axioms of feasibility, individual rationality, Pareto efficiency, covariance with permutations, and covariance with positive affine transformations of utility.Footnote 23 The last of these axioms allows us to concentrate on the divisions’ pre-tax profits.

For \(\pi (\theta ) \ge 0\) or, equivalently, \(\theta \ge 0.5\), the set of transfer prices inducing internal trade is \([k - (2\theta -1) \omega _u I, k + (2\theta -1) \omega _d I]\); all of these transfer prices are accepted as arm’s length prices. The pairs of divisional profits corresponding to these prices are given by the set

$$\begin{aligned} \left\{ \left( \pi _u(\theta , p), \pi _d(\theta , p)\right) : p \in \left[ k - (2\theta -1) \omega _u I, k + (2\theta -1) \omega _d I \right] \right\} . \end{aligned}$$
(15)

For any other transfer price, there is no internal trade, and profits drop to zero. Hence, the set in (15) consists of all pairs of profits, which are feasible, individually rational, and Pareto efficient.

The bargaining problem implied by (15) exhibits transferability of utility at rate 1. To see this, conclude from \(\pi _u(\theta , p) + \pi _d(\theta , p) = \pi (\theta )\) that the choice of the transfer price allows the divisions to transform one unit of U’s into one unit of D’s pre-tax profit and vice versa. Moreover, given the range of transfer prices in (15), this transformation is feasible for any individually rational pair of pre-tax profits satisfying \(\pi _u(\theta , p) + \pi _d(\theta , p) = \pi (\theta )\).

It is well known from bargaining theory that any symmetric bargaining solution satisfying the above-mentioned minimal set of axioms implies equal surpluses with respect to the players’ status-quo points for both players if utility is transferable at rate 1. Because the status-quo point is zero, the negotiated transfer price, \(p_n(\theta )\), is the unique solution to \(\pi _u(\theta , p) = \pi _d(\theta , p)\). \(\square \)

Proof of Proposition 2

The optimizer \(p_a = k\) for identical tax rates is derived in the text. The same transfer price is optimal for \(\tau _u \not = \tau _d\) if \(I = 0\), because without the cost-reduction investment the arm’s length range collapses to market price k.

For \(\tau _u \not = \tau _d\) and \(I > 0\), I concentrate on the case \(\tau _u < \tau _d\), because the proofs for \(\tau _u < \tau _d\) and \(\tau _u > \tau _d\) are symmetric. The following lemma allows us to focus on the price range \([k, k + \omega _d I]\). \(\square \)

Lemma 7

The optimal transfer price under ATP, \(p_a\), satisfies

$$\begin{aligned} p_a \in \left. {\left\{ \begin{array}{ll} {[}k, k + \omega _d I] &{} \hbox {if}\quad \tau _u \le \tau _d \\ {[}k - \omega _u I, k] &{} \hbox {if}\quad \tau _u \ge \tau _d\\ \end{array}\right. }\right\} . \end{aligned}$$

Proof

The cases \(\tau _u = \tau _d\) and \(I = 0\) are discussed above. The proof for \(\tau _u > \tau _d\) and \(I > 0\) follows from that for \(\tau _u < \tau _d\) and \(I > 0\) by symmetry, so I concentrate on the latter case. First assume \(\omega _u = 0\). Then, the arm’s length range is \([k, k + \omega _d I]\), and the lemma holds. To show \(p_a \ge k\) for \(\omega _u > 0\), it is sufficient to verify that the firm’s expected after-tax profit, (7), increases in the transfer price over \([k - \omega _u I, k]\). An inspection of the form of the profit function reveals that it can only be constant, linear, or quadratic over this range. The corresponding derivative reads

$$\begin{aligned} (k - p) \frac{\omega _u + \omega _d}{2 \omega _u^2 I} - \left[ (k - p) \frac{\tau _u \omega _u + \tau _d \omega _d}{2 \omega _u^2 I} - (\tau _d - \tau _u) \left( \frac{1}{2} - \frac{k - p}{\omega _u I} \right) \right] . \end{aligned}$$

Consequently, the monotonicity property follows from the fact that both the derivative for \(p = k - \omega _u I\), that is, \((1 - \tau _d) (\omega _u + \omega _d) / (2\omega _u)\), and the derivative for \(p = k\), that is, \((\tau _d - \tau _u)/2\), are positive. \(\square \)

Assuming \(\omega _d > 0\), an inspection of the functional form of the firm’s expected after-tax profit given by (7) shows that it is quadratic and strictly concave over \([k, k + \omega _d I]\). The corresponding derivative is

$$\begin{aligned} (k - p) \frac{\omega _u + \omega _d}{2 \omega _d^2 I} - \left[ (k - p) \frac{\tau _u \omega _u + \tau _d \omega _d}{2 \omega _d^2 I} - (\tau _d - \tau _u) \left( \frac{1}{2} - \frac{p - k}{\omega _d I} \right) \right] . \end{aligned}$$
(16)

The derivative for \(p = k\) is \((\tau _d - \tau _u)/2\) and thus positive, whereas the derivative for \(p = k + \omega _d I\) equals \(-(1 - \tau _u)(\omega _u + \omega _d)/(2\omega _d)\) and is therefore negative. Hence, \(p_a\) is the unique root of (16) or, equivalently, the unique solution of (9); see (10) for the explicit value. Now assume \(\omega _d = 0\). The arm’s length range is then given by \([k - \omega _u I, k]\). \(p_a = k\) directly follows from Lemma 7 and is also covered by (10). \(\square \)

Proof of Proposition 3 and Lemma 4

The firm’s expected after-tax profits under CP, NTP, and ATP read

$$\begin{aligned} (\tau _h - \tau _l) \omega _h I,\,\frac{(2 - \tau _u - \tau _d) (\omega _u + \omega _d) I}{8},\hbox { and } \frac{(1 - \tau _l)^2 \omega ^2 I}{4 [(1 - \tau _l) (\omega _u + \omega _d) + (\tau _h - \tau _l) \omega _h]}. \end{aligned}$$
(17)

Evaluating these expressions for \(\tau _u=\tau _d\) reveals that the expected after-tax profits per unit of investment are identical and positive under NTP and ATP, whereas the profit under CP is zero. This implies that the central office’s optimal investments and the corresponding expected after-tax profits under NTP and ATP are the same and positive. The corresponding investment and expected after-tax profit under CP are zero.

For unequal tax rates, interpret the expressions in (17) as linear functions of I and express the pairwise comparisons of their slopes in terms of PR and TR. I take the second part of Lemma 4 as an example. The firm’s expected after-tax profit under ATP is higher than (equal to) that under CP if and only if

$$\begin{aligned} \left( \frac{3 + \sqrt{2}}{2} + \frac{\sqrt{2} + 1}{2\,{{\rm PR}}} - {{\rm TR}} \right) \left[ {{\rm TR}} - \left( \frac{3 - \sqrt{2}}{2} - \frac{\sqrt{2} - 1}{2\,{{\rm PR}}} \right) \right] \mathrel {>(=)} 0. \end{aligned}$$
(18)

The factor in parentheses is positive due to \({{\rm PR}} \ge 0\) and \({{\rm TR}} \in (0,1)\), and hence the sign of the factor in brackets determines the sign of the product. For Proposition 3, the expression corresponding to the left-hand side of (18) is \((1 - {{\rm TR}})({{\rm TR}} - 1/{{\rm PR}})\); for the first part of Lemma 4, it is \({{\rm TR}} - (7\,{{\rm PR}} - 1)/(9\,{{\rm PR}} + 1)\). Similar to the case of equal tax rates, the dominance relations between the firm’s expected after-tax profits before investment costs hold for all positive investment levels. Hence, they carry over to the firm’s maximal expected after-tax profits after investment costs. \(\square \)

Proof of Lemmas 1 and 5

For part 1 of Lemma 1 and for Lemma 5, realize that the firm’s expected after-tax profit is nonnegative under NTP and ATP, whereas it is zero under CP for \(\tau _u=\tau _d\). The equality of the profits under NTP and ATP trivially holds if no transfer price from \([k - \omega _u I, k + \omega _d I]\) is accepted or if \(I=0\) holds, because both NTP and ATP then show zero profit. For the opposite case, \(\tau _u = \tau _d\) allows us to concentrate on the firm’s pre-tax profit. Then, the only relevant role of the transfer price is its effect on the decentralized trade decision. Under NTP, the divisions agree on an arm’s length price from \([k - (2\theta -1) \omega _u I, k + (2\theta -1) \omega _d I]\) for a given cost-reduction probability \(\theta \) and thus maximize the firm’s pre-tax profit subject to the constraint imposed by the arm’s length range; confer Proposition 1. Any transfer price from this range implies the same decision on internal trade and thereby the same pre-tax profit. Under ATP, the central office induces the same trade decisions and thus the same pre-tax as well as after-tax profit as the divisions under NTP by choosing the arm’s length price closest to \(p_a=k\). As the relations between the firm’s expected after-tax profits under CP, NTP, and ATP equally hold for all investment levels, the relations carry over to the firm’s maximal expected after-tax profits after investment costs. Parts 2 and 3 of Lemma 1 follow from the discussion in the text. \(\square \)

Proof of Proposition 4

The quantiles corresponding to the interquantile ranges, that is, the limits \(\underline{p}\) and \(\overline{p}\) of the arm’s length range, are referred to as \(p_{\frac{1-\delta}{2}}\) and \(p_{\frac{1+\delta}{2}}\), where parameter \(\delta \in [0, 1]\) reflects the degree of discretion. We shall concentrate on \(I, \delta >0\) to prevent the arm’s length range from collapsing.

The case of equal tax rates is covered by part 1 of Lemma 1. The proof for unequal tax rates can be structured as follows. First, distinguish between (1) \(\omega _u<\omega _d\), (2) \(\omega _u=\omega _d\), and (3) \(\omega _u>\omega _d\), where the third case follows from the first by symmetry. Second, divide case 1 into (a) \({{\rm PR}}>1\) and (b) \({{\rm PR}}<1\). Third, divide case 1.a into (i) \(p_{0.5}>p_a\), (ii) \(p_{0.5}=p_a\), and (iii) \(p_{0.5}<p_a\). Fourth, divide case 1.a.iii into (A) \({{\rm TR}}>1/{{\rm PR}}\), (B) \({{\rm TR}}=1/{{\rm PR}}\), and (C) \({{\rm TR}}<1/{{\rm PR}}\). Throughout this proof, I concentrate on cases 1.a.iii.C, 1.b, and 2, for which \(\hbox {ATP} \succ \hbox {NTP}\) holds by Proposition 3. The remainder of the proof can be provided on request.

The Nash transfer price under NTP is denoted \(p_N(\theta , \delta )\); it is the arm’s length price closest to \(p_n(\theta )\) as given in Proposition 1. Similarly, the Nash transfer price under ATP, \(p_A(\delta )\), is the arm’s length price closest to \(p_a\) as given in Proposition 2.

Under NTP in case 1, there are three ranges for \(\delta \). For high degrees of discretion, that is, \(\delta \ge \delta _k\), negotiation outcomes are not affected according to Lemma 1, that is, \(p_N(\theta , \delta )=p_n(\theta )\) for all \(\theta \ge 0.5\) with \(\delta _k\) solving \(p_{\frac{1 - \delta}{2}} = k\). For intermediate and low degrees of discretion, that is, \(\delta < \delta _k\), the minimal cost-reduction probability inducing internal trade raises from 0.5 to \(\theta _{1, n}(\delta )\) with \(\theta _{1, n}(\delta )\) defined as the solution of \(\pi _d[\theta , p_{\frac{1 - \delta}{2}}]=0\). In addition to the quantity effect, the reduction of discretion causes a price effect for success probabilities \(\theta \in [\theta _{1, n}(\delta ), \theta _{2, n}(\delta ))\) with \(\theta _{2, n}(\delta )\) defined as the solution of \(p_{\frac{1 - \delta}{2}} = p_n(\theta )\). In this range of the cost-reduction probability, \(p_{\frac{1 - \delta}{2}} > p_n(\theta )\) and thus \(p_N(\theta , \delta ) = p_{\frac{1 - \delta}{2}}\) are true. For low degrees of discretion, that is, \(\delta < \delta _1\) where \(\delta _1\) is the solution of \(p_{\frac{1 + \delta}{2}} = p_n(1)\), there is an additional price effect occurring for high success probabilities, that is, \(\theta > \theta _{3, n}(\delta )\) where \(\theta _{3, n}(\delta )\) solves \(p_{\frac{1 + \delta}{2}} = p_n(\theta )\). For such high cost-reduction probabilities, transfer price \(p_n(\theta )\) is not feasible anymore, that is, \(p_n(\theta ) > p_{\frac{1 + \delta}{2}}\), and thus \(p_N(\theta , \delta ) = p_{\frac{1 + \delta}{2}}\). The relations \(0< \delta _1 < \delta _k \le 1\) and \(0.5 \le \theta _{1, n}(\delta )<\theta _{2, n}(\delta )<\theta _{3, n}(\delta ) \le 1\) are verified easily. Consequently, the firm’s expected after-tax profit for \(\delta \in [\delta _1, \delta _k]\) is \(\int _{\theta _{1, n}(\delta )}^{\theta _{2, n}(\delta )} \sum _{i \in \{u, d\}} (1 - \tau _i) \pi _i(\theta , p_{\frac{1 - \delta}{2}}) d\theta + \int _{\theta _{2, n}(\delta )}^{1} \sum _{i \in \{u, d\}} (1 - \tau _i) \pi _i[\theta , p_n(\theta )] d\theta \). For \(\delta \le \delta _1\), the second summand of the firm’s expected after-tax profit becomes \(\int _{\theta _{2, n}(\delta )}^{\theta _{3, n}(\delta )} \sum _{i \in \{u, d\}} (1 - \tau _i) \pi _i[\theta , p_n(\theta )] d\theta + \int _{\theta _{3, n}(\delta )}^{1} \sum _{i \in \{u, d\}} (1 - \tau _i) \pi _i(\theta , p_{\frac{1 + \delta}{2}}) d\theta \).

Under ATP in case 1.a.iii, the restrictions imposed by the arm’s length range do not bear on the transfer price if discretion is high, that is, \(p_A(\delta )=p_a\) for \(\delta \ge \delta _a\). The threshold \(\delta _a>0\) is the lowest degree of discretion for which \(p_a\) is at arm’s length and solves \(p_{\frac{1 + \delta}{2}} = p_a\); this is an application of Lemma 1. For low degrees of discretion, the highest arm’s length price restricts the optimal transfer price choice, that is, \(p_A(\delta )=p_{\frac{1 + \delta}{2}}\) for \(\delta \le \delta _a\). The firm’s expected after-tax profit for \(\delta \le \delta _a\) therefore amounts to \(\int _{\theta _{1, a}(\delta )}^1 \sum _{i \in \{u, d\}} (1 - \tau _i) \pi _i[\theta , p_{\frac{1 + \delta}{2}}] d\theta \) with \(\theta _{1, a}(\delta )\) as the minimal cost-reduction probability inducing internal trade, that is, \(\pi _d[\theta _{1, a}(\delta ), p_{\frac{1 + \delta}{2}}]=0\).

Regarding the comparison of NTP and ATP in case 1.a.iii.C, we have \(\hbox {ATP} \succ \hbox {NTP}\) for \(\delta \ge \delta _k\) because \(\delta _a < \delta _k\) holds in case 1.a.iii. A further reduction of discretion does not increase the firm’s expected after-tax profit under ATP; see Proposition 6. Yet, according to the following lemma, profitability under NTP increases in case 1.a.iii.C until the degree of discretion falls to \(\delta _1\).

Lemma 8

Given (1) the arm’s length range defined as the interquantile range of the triangular distribution with lower limit \(k - \omega _u I\), mode \(k\), and upper limit \(k + \omega _d I\), (2) NTP and the Nash bargaining solution, and (3) \({{\rm TR}}<1/{{\rm PR}}<1\), the firm’s expected after-tax profit decreases in \(\delta \) over the range \([\delta _1, \delta _k]\). The thresholds \(\delta _1\) and \(\delta _k\) are defined as the solutions of \(p_{\frac{1 + \delta}{2}} = p_n(1)\) and \(p_{\frac{1 - \delta}{2}} = k\) for \(\omega _u < \omega _d\) and of \(p_{\frac{1 - \delta}{2}} = p_n(1)\) and \(p_{\frac{1 + \delta}{2}} = k\) for \(\omega _u > \omega _d\), respectively.

Proof

The proof focuses on case 1; case 3 follows by symmetry. The firm’s expected after-tax profit for \(\delta \in [\delta _1, \delta _k]\) is defined above. The sign of its derivative with respect to \(\delta \) is the sign of \((1 - \tau _d)/(1 - \tau _u) - \omega _u/\omega _d\). For \(\omega _u < \omega _d\), the sign is negative if and only if \({{\rm TR}}<1/{{\rm PR}}\). \(\square \)

Given these monotonicity properties, \(\hbox {ATP} \succ \hbox {NTP}\) equally holds for \(\delta \in [\delta _1, \delta _k]\), because the difference in the firm’s expected after-tax profits under ATP and NTP is positive for \(\delta =\delta _1\). The difference is still positive for \(\delta \in (0, \delta _1]\) as it is zero for \(\delta =0\), see Proposition 5, and is strictly convex over \([0, \delta _1]\).

In case 1.b, D becomes the low-tax division, implying \(p_a \le k<p_{0.5}\) and \(p_A(\delta )=p_{\frac{1 - \delta}{2}}\) for \(\delta \le \delta _a\), where \(\delta _a\) now solves \(p_{\frac{1 - \delta}{2}}=p_a\) and satisfies \(\delta _a \ge \delta _k\). \(\hbox {ATP} \succ \hbox {NTP}\) holds for \(\delta \ge \delta _k\), because the firm’s expected after-tax profit for transfer price \(p_A(\delta )\) is not lower than for price \(k\), see Proposition 2, which in turn is higher than for price \(p_n(\theta )\). For \(\delta \in (0, \delta _k]\), the minimal success probabilities inducing internal trade under ATP and NTP coincide, \(p_A(\delta )=p_N(\theta , \delta )\) for \(\theta \in [\theta _{1, n}(\delta ), \theta _{2, n}(\delta )]\) and \(p_A(\delta )<p_N(\theta , \delta )\) for \(\theta \in [\theta _{2, n}(\delta ), 1]\). Consequently, \(\hbox {ATP} \succ \hbox {NTP}\) for \(\delta \in (0, \delta _k]\), which at the same time confirms part 3 of Proposition 5.

Case 2 is a limiting case of cases 1.a.iii.C and 1.b: \(p_n(\theta ) = p_N(\theta , \delta ) = p_{0.5} = k, p_A(\delta ) = \min \{p_a, p_{\frac{1 + \delta}{2}}\}\) for \(\tau _u<\tau _d\), and \(p_A(\delta )=\max \{p_a, p_{\frac{1 - \delta}{2}}\}\) for \(\tau _u>\tau _d\). Moreover, \(p_A(\delta )\) is the unique maximizer of the firm’s expected after-tax profit over all (constant) transfer prices from the arm’s length range; see Proposition 2. \(\hbox {ATP} \succ \hbox {NTP}\) then holds as \(p_A(\delta ) \not = k\) for \(\delta >0\).

As the relation between the firm’s expected after-tax profits under NTP and ATP equally holds for all positive investment levels and no investment means no profits, the relations carry over to the firm’s maximal expected after-tax profits after investment costs. \(\square \)

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Martini, J.T. The optimal focus of transfer prices: pre-tax profitability versus tax minimization. Rev Account Stud 20, 866–898 (2015). https://doi.org/10.1007/s11142-015-9321-3

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