Abstract
This study examines how a strategic tax auditor affects a multinational firm’s transfer pricing in a tax compliance game. Our model uses a divisionalized firm, in both a lowtax and a hightax country, that decides to implement a transferpricing regime with either one or two sets of books. After observing its unit costs, the firm reports a compliant or noncompliant tax transfer price. In a regime with one set of books, the single transfer price coordinates the quantity decision and determines the tax payments. In a regime with two sets, different transfer prices serve those tasks. In contrast to previous studies, our analysis incorporates a strategic tax auditor, who observes the tax transfer price and decides whether to audit the firm. Realworld regulations suggest larger penalties for detected noncompliance under a twosetsofbooks transferpricing regime. Our analysis identifies the mixed strategy equilibria and examines how variations in the tax regulation—the tax rate difference and the penalty difference—affect the firm’s tax aggressiveness. We show that a firm acts less tax aggressively with a higher tax rate difference. Additionally, the model predicts that the firm either increases or decreases the probability of keeping one set of books for a smaller penalty difference.
Introduction
Transfer prices are necessary for computing divisional profits in a multinational firm whenever its divisions engage in intrafirm trade. As multinationals can use divisional profit for managerial performance evaluation, transfer pricing affects internal decisions. Moreover, divisional profits determine a multinational’s tax liability in its different countries. Research shows that multinationals keep either two sets of books (TSB) or one set of books (OSB).^{Footnote 1} The TSB transferpricing regime uses an internal transfer price that coordinates the quantity decision and a tax transfer price that determines the tax liability, respectively. Thus TSB allow the multinational to optimize the tax liability and the internal decisions. In contrast, the OSB transferpricing regime uses the same transfer price for those tasks and thus limits the multinational’s flexibility to optimize both tasks.
To reduce their tax liability, multinationals often use tax transfer prices to shift profit from high to lowtax countries (e.g. Blouin et al. 2018; Clausing 2003; De Simone et al. 2017; Jacob 1996). The tax transfer prices might not comply with tax regulation. As profit shifting deprives tax authorities of a large amount of tax income, the authorities have their auditors rigorously examine multinationals’ transfer pricing for noncompliance (OECD, 2015).^{Footnote 2} However, as tax authorities are resource constrained (e.g., Hoopes et al. 2012), tax auditors cannot scrutinize each multinational’s transfer pricing. Instead, the auditors seek to effectively deploy their available resources (OECD 2015, p. 9); that is, they strategically decide whether to audit an multinational’s transfer pricing.
When a tax auditor challenges the tax transfer price, many multinationals fear negative consequences from keeping TSB instead of OSB.^{Footnote 3} Although the international transferpricing literature acknowledges these different consequences, the specific transferpricing regime choice of the multinational while it is considering the possibility of a tax audit remains largely unexplored. This omission is problematic, because it restricts scholars’ and tax regulators’ understanding of multinationals’ profitshifting incentives. In this study, we examine how a strategic tax auditor affects a multinational’s transferpricing regime in a tax compliance game.
We study an multinational with an upstream division in a lowtax country and a downstream division in a hightax country. The upstream division produces an intermediate product and transfers it to the downstream division. Our model comprises three decisionmaking stages. In the first stage, the multinational decides whether to implement an OSB or a TSB transferpricing regime (implementation decision). In the second, after experiencing operating conditions (i.e., high or low unit costs of the intermediate product), the multinational determines the transfer price(s). The multinational chooses a compliant or a noncompliant tax transfer price (compliance decision). In the third stage, a strategic tax auditor observes the tax transfer price and decides whether to conduct an audit, a decision that factors in the audit costs.
During an audit, the tax auditor evaluates the multinational’s transfer pricing. With OSB, the tax transfer price follows the management view,^{Footnote 4} thereby exhibiting economic substance. In contrast, the use of an internal transfer price different from the tax transfer price under TSB may undermine the economic substance of the tax transfer price (Cools & Emmanuel, 2006; Narayanan & Smith, 2010).^{Footnote 5} The lack of economic substance indicates that the multinational’s primary objective for keeping TSB is tax minimization.
Many countries reduce the penalty for detected noncompliance when the multinational shows economic substance, whereas they increase the penalty if the the primary objective is tax minimization. For example, in Spain, a tax auditor can reduce or eliminate the penalty when the multinational keeps OSB (KPMG 2012, p. 202). Australia levies a penalty of 25%, which decreases to 10% when the multinational demonstrates economic substance and increases to 50% when the tax auditor shows that the dominant purpose is tax minimization (EY, 2012). In New Zealand, the penalties vary between 20% and 150% (EY, 2012), with the applied rate depending on the degree of intent to avoid tax payments in the multinational’s gross negligence. In Hong Kong, the tax auditor scales the penalty upward or downward according to the nature of the omission and the amount of understatement of profits (EY, 2012). In sum, these realworld regulations suggest higher penalty factors for detected noncompliance as a negative consequence of keeping TSB. We incorporate this penalty difference into our model.
In our analysis, the multinational faces a tradeoff between flexibility and the level of penalties. With TSB, it uses the internal transfer price to affect the downstream division’s quantity decision. In contrast, the tax transfer price under OSB limits the multinational’s possibilities of affecting the quantity decision; that is, the multinational is less flexible. Beyond flexibility, the multinational also considers the level of penalties. With low unit costs, the multinational (hereafter “lowcost multinational”) has an incentive to mimic a multinational with high unit costs (hereafter “highcost multinational”) and to use a noncompliant tax transfer price. If the tax auditor detects a noncompliant tax transfer price, the penalty under TSB is higher than under OSB. In other words, with the use of TSB, higher flexibility goes hand in hand with higher penalties.
After identifying the mixed strategy equilibria, we find that the penalty difference determines the firststage implementation decision about using OSB or TSB. For a large penalty difference between OSB and TSB, the advantage of OSB is high. In this case, the multinational never implements TSB as a pure strategy but instead randomizes between OSB and TSB. While the lowcost multinational may benefit from the lower level of penalties under OSB, this advantage is irrelevant for the highcost multinational, which always acts compliantly. Consequently, the multinational never implements OSB as a pure strategy. If the advantage of OSB is small, an equilibrium exists in which the multinational always implements TSB, due to its greater flexibility.
The secondstage decision is a choice between compliance and noncompliance. We find that, when an OSB regime is in place, a lowcost multinational always reports noncompliantly. The lowcost multinational’s compliance decision under a TSB regime depends on the level of the tax auditor’s audit costs. For low audit costs, the tax auditor wants to conduct audits more frequently. Thus, ceteris paribus, the probability of detecting noncompliance increases: anticipating the higher detection probability, the lowcost multinational reports compliantly with a higher probability.
Using the equilibrium strategies, we derive empirically testable predictions for tax aggressiveness. As various definitions of tax aggressiveness exist (Hanlon and Heitzman, 2010), in our model, we define tax aggressiveness as noncompliant reporting under either TSB or OSB. We consider noncompliant reporting under TSB a more taxaggressive transfer pricing than that under OSB. Our findings show that a higher tax rate difference between the countries decreases the multinational’s tax aggressiveness. Because profit shifting is especially beneficial for a high tax rate difference, less tax aggressiveness appears counterintuitive. Our finding, however, is due to the presence of the strategic tax auditor. Given that a high tax rate difference causes high penalties for detected noncompliance, the tax auditor has strong incentives for conducting an audit. The multinational anticipates the stronger audit incentives and counteracts them by less taxaggressive transfer pricing.
In addition, our model predicts that the multinational either increases or decreases the probability of keeping OSB for a smaller penalty difference. The multinational’s choice in the first stage depends on its compliance decision in the second stage. Because a smaller penalty difference results in a smaller OSB advantage, an increasing probability of an multinational’s keeping OSB for a decreasing penalty difference might appear surprising at first glance. The finding stems from the presence of the strategic tax auditor, who incorporates a higher penalty for detected noncompliance under OSB and the multinational’s incentives for switching toward more TSB in his or her audit decision, thereby increasing audit incentives. For an intermediate level of audit costs, the multinational counteracts the stronger audit incentives by using more OSB in the first stage, because the OSB advantage still remains.
For our main analysis, we assume that the multinational chooses the transferpricing regime before observing the operating conditions. In the short term, because of implementation, user training, and other organizational issues, the multinational does not revise the transferpricing regime according to operating conditions (Martini et al. 2012). Nevertheless, for comparison, we also study a model in which the multinational chooses the transferpricing regime after observing the operating conditions. Our analysis shows that our findings persist in the model with the alternative timing.^{Footnote 6}
Our model builds on the costbased transfer pricing setting established by Baldenius et al. (2004) while introducing the following adaptations. (1) The multinational chooses to keep either OSB or TSB. (2) The multinational can either incur high or low unit costs, so that the lowcost multinational has an incentive to choose a noncompliant tax transfer price. (3) The tax auditor strategically decides whether to audit the multinational. These adaptations allow us to study the multinational’s implementation and compliance decision when the multinational considers a potential tax audit.
Our paper contributes to the vast literature studying international tax transfer pricing that incorporates internal decisionmaking.^{Footnote 7} Research typically takes the transferpricing regime as given; that is, either OSB or TSB is in place (Baldenius et al. 2004; Choe and Hyde, 2007; Hyde & Choe, 2005; Narayanan & Smith, 2010). An exception is Nielsen and RaimondosMøller (2012), who investigate whether OSB or TSB is preferable under certain circumstances. However, they do not consider the presence of a strategic tax auditor. In contrast, rather than assuming the dominance of a specific transferpricing regime, we endogenize the multinational’s implementation decision.
Our paper adds to the international tax transferpricing literature on tax audits. Kant (1988), Smith (2002), Hyde and Choe (2005), and Choe and Hyde (2007) study the impact of a penalty for noncompliance on multinational’s transfer prices. However, they neither examine the presence of a strategic tax auditor nor consider whether the multinational keeps OSB or TSB.
Our paper also contributes to the strategic coordination literature that examines the benefits that multinationals accrue from keeping OSB. Schjelderup and Sorgard (1997), Arya and Mittendorf (2008), and Dürr and Göx (2011) illustrate that, under imperfect competition, multinationals gain benefits from keeping OSB. In their studies, OSB serves as a commitment device for softening competition in external markets. While this strand of the literature assumes that the competitors observe the multinational’s transferpricing regime, we do not make a similar assumption for the tax auditor. Instead, and in line with previous research (Bärsch et al. 2019; Davis, 1994; Tang, 1993),^{Footnote 8} we assume that the tax auditor observes the multinational’s transferpricing regime during a tax audit. We complement the strategic coordination literature by showing that the multinational keeps OSB as part of the equilibrium strategy in response to the presence of a strategic tax auditor.
The paper proceeds as follows. Section 2 describes the model. Section 3 presents the multinational’s internal and tax transfer prices. Section 4 identifies and describes the mixed strategy equilibria. Section 5 depicts comparative statics. Section 6 shows that our main findings do not depend on the timing of the game. Section 7 concludes.
Model description
We study a multinational operating an upstream division in a lowtax country and a downstream division in a hightax country, where transfer prices evaluate intrafirm trade. In the lowtax country, an income tax rate t prevails, whereas the hightax country taxes income at a rate t + h, with 0 ≤ t, h ≤ 1 and t + h ≤ 1. The parameter h captures the tax rate difference between the countries. We assume taxation in terms of the separate entity approach and that each division has additional income, so that the divisional aftertax income is always positive.
In the first stage, the multinational chooses whether to implement an OSB or a TSB transferpricing regime (implementation decision). With OSB, the multinational uses a single transfer price to evaluate the intrafirm trade internally and to calculate the tax liability; that is, the internal transfer price p_{i} equals the tax transfer price p_{r}. For TSB, the multinational decouples its transferpricing decisions and uses two different transfer prices; that is, p_{i}≠p_{r}.
The upstream division makes an intermediate product that is transformed into the final product by the downstream division, which faces monopolistic market conditions for the final product. No external market for the intermediate product exists. The upstream division faces either high unit costs c_{H} with probability β or low unit costs c_{L} with probability 1 − β for producing the intermediate product, where 0 < β < 1 and 0 ≤ c_{L} < c_{H}. While the probability β is common knowledge, only the multinational observes the cost realization after the implementation decision. We label the multinational with low (high) unit costs as a lowcost (highcost) multinational.
In the second stage, after the realization of the unit costs, the multinational sets the tax transfer price using a costplus method, which comports with the OECD transfer pricing guidelines and the monopolistic setting. Under this transfer pricing method, unit costs plus an appropriate markup fulfill the arm’s length principle. Thus the upper bound for the tax transfer price of a lowcost multinational is given by \(\underline {p_{r}}=c_{L}+m_{L}\), where m_{L} ≥ 0 captures the accepted markup. The appropriate markup for the highcost multinational is m_{H}, yielding an upper bound \(\overline {p_{r}}=c_{H}+m_{H}\). We assume m_{H} and m_{L} such that \(\overline {p_{r}}>\underline {p_{r}}\). For convenient notation, we assume that the lower bound of the arm’s length range is p_{rL} (p_{rH}) for the lowcost (highcost) multinational, where \(0\leq p_{rL} \leq p_{rH} < \underline {p_{r}} < \overline {p_{r}}\) holds. Thus the arm’s length ranges are \(\left [p_{rL},\underline {p_{r}}\right ]\) and \(\left [p_{rH},\overline {p_{r}}\right ]\) for the lowcost and highcost multinational, respectively. The multinational chooses either a compliant tax transfer price or a noncompliant tax transfer price that does not belong to the costspecific arm’s length range (compliance decision). In addition, with TSB, the multinational determines the internal transfer price.
The multinational evaluates the downstream division on the basis of pretax divisional profit π^{D},^{Footnote 9} so that the downstream division uses the internal transfer price for the quantity decision. Without loss of generality, the downstream division’s costs for transforming the intermediate product into the final product are equal to zero. Due to monopolistic market conditions, the revenue function for the final product is \(R(q)=\left (a\frac {1}{2} q\right )q\), where q denotes the quantity. Thus the downstream division determines the quantity according to
In the third stage, the tax auditor observes the tax transfer price and decides to conduct an audit. The tax auditor is located in the hightax country, that is, the home country of the downstream division.^{Footnote 10} We assume that the tax auditor maximizes the additional income that he or she generates for the tax authority while facing personal audit costs K_{a} if he or she conducts an audit.
In line with empirical findings (Cools et al. 2008; Cools & Slagmulder, 2009) and the extensive documentation requirements imposed on multinationals, we assume that, if a tax audit occurs, the tax auditor observes the realized unit costs, the transferpricing regime (i.e., OSB or TSB), and, for TSB, the internal transfer price.
If noncompliance is detected, the tax auditor enforces a compliant transfer price, where \(\underline {p_{r}}\) (\(\overline {p_{r}}\)) is the enforced transfer price for a lowcost (highcost) multinational. The tax auditor asks the multinational to pay the previously unpaid taxes, which are the difference between the tax payment that the tax auditor determines using the enforced transfer price p_{a} and the tax transfer price p_{r}. Moreover, the tax auditor levies an additional penalty, captured by a linear penalty factor δ ∈{δ_{OSB},δ_{TSB}} applied to the previously unpaid taxes (Yitzhaki, 1974).^{Footnote 11} In line with realworld regulations (EY 2003, 2012; KPMG 2012), we assume 1 ≤ δ_{OSB} < δ_{TSB}. Thus, depending on its transferpricing regime, the multinational faces the following payment.
Hereafter, we refer to S as the penalty.^{Footnote 12} Thus the tax auditor’s payoff is as follows.
After the third stage, the multinational obtains the following global aftertax profit:
with \(c\in \left \{c_{L},c_{H}\right \}\). The multinational maximizes its global aftertax profit and incorporates tax savings due to the tax rate difference and the possibly resulting penalty.
Figure 1 shows the timing of the game.
Internal and tax transfer prices
This section shows that the multinational reports \(p_{r} \in \{\underline {p_{r}}, \overline {p_{r}}\}\) if the reservation price a is sufficiently large. Furthermore, it demonstrates how the internal transfer price p_{i} is adjusted for tax payments and tax audit risk.
The multinational may choose an arbitrary tax transfer price p_{r} > 0. A tax transfer price \(p_{r} >\overline {p_{r}}\) is evidence of noncompliance for the tax auditor, even without a tax audit. Because obvious noncompliance with tax regulation is beyond dispute, we assume that the tax auditor punishes obvious noncompliance without facing substantial audit costs. Thus neither the highcost nor the lowcost multinational reports \(p_{r} > \overline {p_{r}}\).
The highcost multinational minimizes the tax payment with a tax transfer price \(\overline {p_{r}}\). Therefore the highcost multinational uses \(\overline {p_{r}}\) with TSB. The highcost multinational using OSB considers a tradeoff between tax savings and quantity distortion. Given q = a − p_{r}, the optimal tax transfer price fulfills
rearranging yields
Despite the tradeoff, for a sufficiently large reservation price a, the multinational chooses \(\overline {p_{r}}\) as the tax transfer price—unit costs plus markup. When choosing the quantity, the downstream division considers both the tax transfer price and the reservation price. By internalizing the markup as costs, the downstream division distorts the quantity downward (standard double marginalization problem). Because the downstream division internalizes the reservation price as the multinational does, the negative quantity distortion is constant in a (see Eq. 5). As a large reservation price results in a large quantity, which in turn causes a high marginal benefit from the tax savings, a large a makes the tax savings more important than the quantity distortion. Consequently, the multinational seeks a high tax transfer price and thus implements \(\overline {p_{r}}\).
The lowcost multinational may choose a noncompliant tax transfer price; that is, \(p_{r}>\underline {p_{r}}\). As the tax auditor anticipates that a highcost multinational will always report \(\overline {p_{r}}\), reporting a tax transfer price in the range \((\underline {p_{r}},\overline {p_{r}})\) immediately identifies the lowcost multinational as noncompliant. Therefore the noncompliant lowcost multinational chooses \(p_{r} = \overline {p_{r}}\) for both OSB (for sufficiently high a) and TSB. If, instead, the lowcost multinational decides not to mimic the highcost multinational and reports compliantly, to minimize the tax payment, the lowcost multinational uses the highest compliant arm’s length price \(\underline {p_{r}}\) under TSB. Likewise, as with the highcost multinational, a sufficiently large reservation price ensures that \(\underline {p_{r}}\) is optimal for a compliant lowcost multinational that keeps OSB. Lemma 1 summarizes:
Lemma 1
Assume a sufficiently large reservation price a.

1.
A highcost multinational reports the compliant tax transfer price \(p_{r} = \overline {p_{r}}\) for TSB and OSB.

2.
A noncompliant lowcost multinational reports \(p_{r} = \overline {p_{r}}\) for TSB and OSB.

3.
A compliant lowcost multinational reports \(p_{r} = \underline {p_{r}}\) for TSB and OSB.
Proof
See Appendix. □
With TSB, to maximize global aftertax profits, the multinational additionally determines an internal transfer price. The compliant multinational uses taxadjusted unit costs as the internal transfer price (see Baldenius et al. 2004). This transfer price induces the downstream division to make the optimal quantity decision. In the noncompliance case, the lowcost multinational also considers the costs following a potential tax audit. Thus when the lowcost multinational considers a strategic tax auditor, the noncompliant lowcost multinational uses tax and auditadjusted unit costs to induce the optimal quantity decision. Lemma 2 summarizes:
Lemma 2
Given that the multinational has installed a TSB regime in the first stage, and the tax auditor audits with probability η, the multinational determines the internal transfer price as follows.
A noncompliant lowcost multinational adopts tax and auditadjusted unit costs as internal transfer price p_{iL1}:
In case of compliance, the high and lowcost multinationals adopt taxadjusted unit costs c_{H} and c_{L} as internal transfer prices p_{iH} and p_{iL2}, respectively:
and
Proof
See Appendix. □
The game tree in Fig. 2 displays those strategies (for the multinational and the tax auditor) that are not dominated by another strategy. The tax auditor never audits a tax transfer price \(p_{r} = \underline {p_{r}}\) because \(p_{r} = \underline {p_{r}}\) indicates a compliant lowcost multinational.
Transfer pricing regimes and compliance
Our model is a tax compliance game, with the audit decision depending on the tax transfer price. Pure strategy equilibria exist for extremely high or low audit costs in combination with low or high penalties for a detected noncompliance. In such cases, the multinational selects TSB or OSB in the first stage and then either always or never chooses a compliant tax transfer price in the second stage. The tax auditor either never or always audits the multinational. Given that pure strategy equilibria cannot explain why compliance and noncompliance and nontrivial tax audit strategies appear simultaneously in reality, we concentrate our analysis on mixed strategy equilibria.
In our model, randomization may appear at three stages. First, the multinational may randomize between TSB and OSB. We denote the corresponding implementation probability of TSB (OSB) by τ (1 − τ). Second, after observing the unit costs, the lowcost multinational chooses the noncompliant tax transfer price \(\overline {p_{r}}\) with probability λ_{j}, \(j\in \left \{TSB,OSB\right \}\), and the compliant tax transfer price \(\underline {p_{r}}\) with probability 1 − λ_{j}. We refer to λ_{j} as the noncompliance probability. Third, after observing the tax transfer price \(\overline {p_{r}}\), the tax auditor decides whether to audit the multinational (probability η) or not (probability 1 − η).
The strategies τ, λ_{TSB}, λ_{OSB}, and η constitute an equilibrium if the following conditions hold.

1.
First Stage: Implementation
$$ \tau\in \arg\max_{\acute{\tau}} E_{t=0}\left[{\varPi}\left( \acute{\tau},\lambda_{TSB},\lambda_{OSB},\eta\right)\right] $$ 
2.
Second Stage: Compliance
$$ \lambda_{TSB}\in \arg\max_{\acute{\lambda}_{TSB}} E_{t=2}\left[\left.{\varPi}\left( \acute{\lambda}_{TSB},\eta\right)\right\text{TSB}\right] \text{~and} $$$$ \lambda_{OSB}\in \arg\max_{\acute{\lambda}_{OSB}} E_{t=2}\left[\left.{\varPi}\left( \acute{\lambda}_{OSB},\eta\right)\right\text{OSB}\right] $$ 
3.
Third Stage: Audit
$$ \eta\in \arg\max_{\acute{\eta}} E_{t=4}\left[\left.{\varPi}^{TA}\left( \tau,\lambda_{TSB},\lambda_{OSB},\acute{\eta}\right)\right p_{r}=\overline{p_{r}}\right] $$
In a mixed strategy equilibrium, each player—in our model, the multinational and the tax auditor—has to be indifferent between all the pure strategies that the player plays with positive probability. The multinational’s strategy comprises the implementation decision in the first stage and the compliance decision in the second. Randomization in both stages requires indifference between OSB and TSB and between compliance and noncompliance. The tax auditor, however, has only the audit probability for inducing indifference. Therefore the multinational randomizes at either the implementation or the compliance stage.^{Footnote 13} Proposition 1 exhibits the mixed strategy equilibria of our model.
Proposition 1
Assume a sufficiently large reservation price a. The following three mixedstrategy equilibria exist.

1.
Equilibrium I: For \(\delta _{OSB} \geq \overline {\delta }_{OSB}\) and K_{a} < K_{a2}(η_{I}), the multinational always implements TSB. After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational reports the noncompliant tax transfer price \(\overline {p_{r}}\) (compliant tax transfer price \(\underline {p_{r}}\)) with probability λ_{TSB,I} (1 − λ_{TSB,I}). The tax auditor audits \(\overline {p_{r}}\) with audit probability η_{I}.

2.
Equilibrium II: For K_{a} < K_{a1}(δ_{OSB}), the multinational implements TSB (OSB) with probability τ_{II} (1 − τ_{II}). After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational reports the compliant tax transfer price \(\underline {p_{r}}\) under TSB and the noncompliant tax transfer price \(\overline {p_{r}}\) under OSB. The tax auditor audits \(\overline {p_{r}}\) with probability η_{II}.

3.
Equilibrium III: For \(K_{a1}(\delta _{OSB}) < K_{a} < K_{a2}\left (\eta _{III}\right )\), the multinational chooses TSB (OSB) with probability τ_{III} (1 − τ_{III}). After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational chooses the noncompliant tax transfer price \(\overline {p_{r}}\) under TSB and under OSB. The tax auditor audits \(\overline {p_{r}}\) with probability η_{III}.
Proof
All proofs, equilibrium probabilities, and thresholds appear in the Appendix. □
Figure 3 depicts the findings of Proposition 1 and shows that the equilibrium in our model is unique if the penalty factor δ_{OSB} is sufficiently low. For a fixed penalty factor δ_{TSB}, a low penalty factor δ_{OSB} corresponds to a large penalty difference. In such a case, the multinational randomizes between TSB and OSB. After the realization of the unit costs, the highcost multinational always reports compliantly while the audit costs determine the compliance decision of the lowcost multinational under TSB. In other words, both TSB and OSB can be part of the equilibrium strategy.
This finding comports with both empirical and anecdotal evidence from Klassen et al. (2017) and Springsteel (1999). We observe that the equilibrium—where the multinational randomizes between OSB and TSB in the first stage and the lowcost (highcost) multinational always reports a noncompliant (compliant) tax transfer price (equilibrium III)—appears if δ_{OSB} is low and K_{a} is high. This finding is intuitive: a low penalty factor, under OSB together with high audit costs, implies weak audit incentives. Therefore the lowcost multinational chooses TSB with noncompliance instead of TSB with compliance. For a small penalty difference, the deterministic implementation of TSB and random compliance (equilibrium I) coexists with equilibria II and III.
The deterministic implementation of TSB (equilibrium I) is an equilibrium strategy for a small penalty difference. The intuition for this finding is as follows: with the deterministic implementation of TSB, the quantity is optimal, and the lowcost multinational randomizes between compliance and noncompliance. For a small penalty difference, the disadvantage of TSB is small. In contrast, for a large penalty difference, the multinational prefers to deviate to OSB in the first stage so that the deterministic implementation of TSB does not occur.
When the multinational randomizes between TSB and OSB in the first stage, the highcost multinational always chooses compliance in the second stage, and the lowcost multinational conditions its compliance decision in the second stage on the audit costs. This randomization occurs in equilibria II and III. In the case of low audit costs, the multinational expects frequent tax audits. In the second stage, the lowcost multinational thus chooses OSB with the noncompliant tax transfer price \(\overline {p_{r}}\) or TSB with the compliant tax transfer price \(\underline {p_{r}}\). With the noncompliant use of OSB, the lowcost multinational realizes tax savings. The compliant use of TSB minimizes tax payments within the legal boundaries and yields the optimal quantity decision. With high audit costs, tax audits are infrequent, and, for the lowcost multinational, reporting the noncompliant tax transfer price dominates in the second stage. While OSB implies lower penalties if noncompliance is detected for the lowcost multinational, both types of the multinational benefit from separating internal and tax transfer prices under TSB. Consequently, neither OSB nor TSB dominates in the first stage.
Furthermore, Proposition 1 shows that we do not observe an equilibrium in which the lowcost multinational reports a compliant tax transfer price under OSB because, given the quantity distortion under OSB, any equilibrium with OSB and a compliant tax transfer price as part of a mixed strategy implies a strictly lower payoff for the lowcost multinational than with TSB and compliance.
Effects of increases in tax rate difference and penalties
Equipped with the equilibrium strategies of both the multinational and the tax auditor, we analyze how variations in the tax regulation affect the multinational’s decisions on quantity, compliance, and the implementation of transferpricing regimes. The regulatory parameters of interest are the tax rate difference and the penalty factors for detected noncompliance.
Tax rate difference
For each equilibrium determined in Section 4, Proposition 2 shows how the internal transfer price and the noncompliance or implementation probability change with an increase in the tax rate difference h.^{Footnote 14}
Proposition 2
Assume a sufficiently large reservation price a. An increase in the tax rate difference h

1.
decreases the internal transfer price under TSB in all equilibria;

2.
decreases the probability of a noncompliant tax transfer price in equilibrium I (λ_{TSB,I});

3.
increases the probability of using TSB in equilibrium II (τ_{II}), where the lowcost multinational uses a compliant tax transfer price with TSB;

4.
decreases the probability of using TSB in equilibrium III (τ_{III}), where the lowcost multinational uses a noncompliant tax transfer price with TSB.
Proof
See Appendix. □
Proposition 2, part 1, states that the multinational boosts the quantity for a higher tax rate difference whenever the multinational uses TSB. A higher tax rate difference increases the multinational’s profitshifting benefits. With TSB the tax transfer price is already set to optimally exploit the tax rate difference. Thus the multinational exploits the increasing tax rate difference by selling a larger quantity. Because the lowcost noncompliant multinational that implements TSB uses tax and auditadjusted unit costs as the internal transfer price (see Lemma 2), the lowcost noncompliant multinational additionally incorporates the tax auditor’s reaction to a higher tax rate difference. Nevertheless, we show that the lowcost noncompliant multinational also lowers the internal transfer price for an increasing tax rate difference.
The noncompliance and implementation results in Proposition 2, parts 2 to 4, have an instructive interpretation in terms of tax aggressiveness. In our model, we define tax aggressiveness as noncompliant reporting under either TSB or OSB. We consider noncompliant reporting under TSB more tax aggressive transfer pricing than noncompliant reporting under OSB. An increasing tax rate difference implies less frequent noncompliance in equilibria I and II, and a shift from TSB with noncompliance to less tax aggressive OSB with noncompliance in equilibrium III for the lowcost multinational. As better profitshifting opportunities induce less taxaggressive decisions by the multinational, this result appears counterintuitive.
The appealing intuition that a higher tax rate difference induces more tax aggressiveness due to more beneficial profitshifting holds true when no tax audit is considered. For example, Baldenius et al. (2004, p. 600) show that the incremental gain of TSB is relatively large for a high tax rate difference. However, with a strategic tax auditor, the multinational also considers potential penalties and audit incentives. In this case, as the penalty depends on the previously unpaid taxes, an increasing tax rate difference directly increases the penalty for detected noncompliance. Additionally, in a setting with TSB, an increasing tax rate difference leads to a larger quantity, which also increases the penalty for detected noncompliance. Therefore the tax auditor obtains a higher income when he or she detects noncompliance in a tax audit; that is, a higher penalty implies stronger audit incentives for the tax auditor. In equilibrium, to counteract the stronger audit incentives, the multinational reports less tax aggressively.
Our findings comport with the empirical finding of Chan and Chow (1997), who show that high tax rate differences are not crucial for inducing noncompliant transfer prices. Their work demonstrates that the tax auditor is aware of an multinational’s profit shifting incentives. Thus, accounting for the strategic interaction with the tax auditor, the multinational is less tax aggressive with an increasing tax rate difference. Furthermore, our result is consistent with the finding of Hoopes et al. (2012) that a stricter tax enforcement is associated with less tax aggressiveness.
Penalty factors
For each equilibrium determined in Section 4, Proposition 3 shows how the internal transfer price and the noncompliance or implementation probability react to changes in the penalty factors for detected noncompliance under OSB and TSB.
Proposition 3
Assume a sufficiently large reservation price a. An increase in the penalty factor for detected noncompliance under OSB (TSB), δ_{OSB} (δ_{TSB}) has the following effects.

1.
Neither δ_{OSB} nor δ_{TSB} affects the internal transfer prices in equilibria I and II or the internal transfer price of the highcost multinational in equilibrium III. In equilibrium III, the internal transfer price of the lowcost multinational under TSB increases (decreases) in δ_{OSB} (δ_{TSB}).

2.
An increasing δ_{OSB} (δ_{TSB}) does not affect (decreases) the probability that the lowcost multinational chooses a noncompliant tax transfer price λ_{TSB,I} in equilibrium I.

3.
An increasing δ_{OSB} (δ_{TSB}) increases (does not affect) the probability of using TSB τ_{II} in equilibrium II, where the lowcost multinational uses a compliant tax transfer price.

4.
The probability of using TSB τ_{III} in equilibrium III, where the lowcost multinational uses a noncompliant tax transfer price, decreases in δ_{OSB} for \(K_{a} \in (K_{a1},{K_{a}^{c}})\)^{Footnote 15} and increases in δ_{OSB} for \(K_{a} \in ({K_{a}^{c}}, K_{a2}(\eta _{III}))\). The probability τ_{III} decreases in δ_{TSB}.
Proof
See Appendix. □
We start with the intuition for Proposition 3, part 1. Because penalty factors are irrelevant when the multinational reports compliantly, penalty factors do not affect the internal transfer price with TSB and compliance in equilibria I, II, or III. When the lowcost multinational keeps TSB with noncompliance in equilibrium I, the audit probability is such that the expected penalty equals the tax savings that the lowcost multinational obtains through noncompliance. Consequently, for the lowcost multinational, the internal transfer prices with TSB and compliance or noncompliance are equal in equilibrium I and neither δ_{OSB} nor δ_{TSB} affects the internal transfer price with TSB and noncompliance.
When the lowcost multinational keeps TSB with noncompliance in equilibrium III, the penalty factors influence the internal transfer price via the audit adjustment. First, we consider the effect of δ_{OSB}. The penalty factor δ_{OSB} affects the audit adjustment only via the audit probability η_{III}. We show that η_{III} increases in δ_{OSB}, and thus the internal transfer price increases in δ_{OSB}. The intuition is as follows. TSB with noncompliance becomes more attractive to the lowcost multinational when δ_{OSB} increases. The tax auditor counters with a higher audit probability because TSB with noncompliance implies a higher penalty than for OSB with noncompliance.
Second, we consider the effect of δ_{TSB}. The penalty factor δ_{TSB} affects the audit adjustment both directly and via the audit probability η_{III}. The direct effect increases the audit adjustment and, in turn, the internal transfer price. Because η_{III} decreases in δ_{TSB}, the effect via the audit probability works in the opposite direction. The intuition is as follows. An increase in δ_{TSB} makes TSB with noncompliance less attractive for the lowcost multinational and induces the tax auditor to audit less frequently. Our results show that the decrease in the audit probability overcompensates for the direct effect; thus, in equilibrium III, the internal transfer price of the lowcost multinational decreases with δ_{TSB}.
Part 2 of Proposition 3 is intuitive: because a higher penalty for a detected noncompliance under TSB decreases the benefit of using a noncompliant tax transfer price with TSB, the lowcost multinational reduces the noncompliance probability λ_{TSB,I}.
As the lowcost multinational reports the compliant tax transfer price under TSB in equilibrium II, changes in the TSB penalty factor δ_{TSB} do not affect the implementation probability for TSB. An increasing penalty factor δ_{OSB} reduces the attractiveness of OSB and, in an intuitively appealing way, induces the multinational to shift from OSB to TSB in equilibrium II (see Proposition 3, part 3).
An increase in the penalty factor δ_{OSB} ambiguously affects the implementation probability in equilibrium III, where the lowcost multinational uses a noncompliant tax transfer price under both the TSB and OSB transferpricing regimes. Proposition 3, part 4, states that the multinational implements TSB less frequently for an intermediate level of audit costs; that is, \(K_{a}\in (K_{a1},{K_{a}^{c}})\) with \({K_{a}^{c}}<K_{a2}(\eta _{III})\), when δ_{OSB} increases. In this case, ceteris paribus the lowcost multinational has a stronger incentive to implement the more taxaggressive TSB with noncompliance. The tax auditor anticipates the multinational’s incentive while deciding whether to audit the multinational. Additionally, the higher penalty factor directly increases the tax auditor’s audit incentives. As an advantage of OSB still remains, the multinational reacts to the tax auditor’s stronger audit incentives by increasing the probability of keeping OSB for an intermediate level of audit costs. For audit costs above \({K_{a}^{c}}\), the tax auditor’s audit incentives are sufficiently weak so that the lowcost multinational’s incentive to switch towards more taxaggressiveness by keeping TSB hardly affects the audit decision. Therefore a higher penalty for detected noncompliance under OSB induces the multinational to refrain from the use of OSB.
In equilibrium III, an increase in the penalty factor δ_{TSB} makes TSB with noncompliance less attractive for the lowcost multinational, and thus, before the realization of the unit costs, the multinational implements OSB more frequently.
In sum, either an increasing penalty factor δ_{OSB} or a decreasing penalty factor δ_{TSB} cause a smaller penalty difference. Proposition 3 states that a decreasing penalty difference ambiguously affects the implementation of the transferpricing regime: for intermediate audit costs and a smaller penalty difference caused by an increase of δ_{OSB}, the probability of OSB increases, if TSB with noncompliance of the lowcost multinational is the alternative action in equilibrium, that is, in equilibrium III. In all other cases of equilibrium III and in equilibrium II, the probability of OSB decreases for a smaller penalty difference.
Our findings relate to those of Klassen et al. (2017), who find that the enforcement level does not affect multinationals’ transferpricing focus if the focus is either “tax minimization” or “lack of disputes.” In our model, we interpret (1) the enforcement level as the size of the penalty factors for detected noncompliance under OSB or TSB, (2) “tax minimization” as a noncompliant tax transfer price for the lowcost multinational, and (3) the “lack of disputes” as a compliant tax transfer price. We show that the penalty factors affect the implementation decision, which is exogenous in Klassen et al. (2017). For exogenous implementation decisions, where OSB and TSB coexist, the compliance decisions and thus the transferpricing focuses are deterministic. Put differently, the penalty factors do not affect multinationals’ compliance. By considering both decisions, we show in Proposition 3 that larger penalty factors can influence multinationals’ implementation decisions, thereby affecting their compliance and thus the transferpricing focus.
Alternative sequence of events
Thus far in this paper, the multinational has chosen its transferpricing regime, OSB or TSB, before knowing its unit costs. This timing assumes that it does not continually revise its decision to implement OSB or TSB to adjust for changes in the shortterm operating conditions, such as unit costs. Nevertheless, multinationals may reconsider their decisions. For example, a new executive team might revise the previous team’s decisions, or changes in tax regulation might induce an multinational to adjust its accounting system. Thus the multinational might be able to choose its transferpricing regime after observing the unit costs. This section discusses the potential impact of this alternative sequence of events on our findings.^{Footnote 16}
The model with the alternative sequence of events qualitatively yields the same results as those in Propositions 1, 2, and 3. All three mixedstrategy equilibria still occur for the lowcost multinational, and no further mixedstrategy equilibria appear. Because the lowcost multinational still trades off flexibility and the level of penalties, this finding is intuitive. For the highcost multinational, no advantage of OSB exists. Consequently, when the highcost multinational chooses its transferpricing regime after observing the realized unit costs, it always implements TSB. As the tax auditor cannot observe the multinational’s unit costs prior to an audit, he or she audits the high tax transfer price with positive probability.
In contrast to the model with the previous timing of the game, only the lowcost multinational adapts its implementation decision as response to a varying tax rate difference or changes in the penalty factors for OSB and TSB. Nevertheless, in the model with the previous timing of the game, the lowcost multinational’s incentives cause the comparative static results, meaning that the results qualitatively carry over to the model with the alternative sequence of events. The lowcost multinational becomes less tax aggressive in response to a higher tax rate difference. Additionally, for an intermediate level of audit costs, the lowcost multinational keeps OSB more frequently when the penalty factor for OSB increases. In sum, our results show that the timing of the game is immaterial for obtaining our main findings.
Conclusion
This paper examines how a strategic tax auditor affects a multinational’s transferpricing regime choice—TSB or OSB—and compliance decision in a tax compliance game. The multinational faces a tradeoff between flexibility and the level of penalties. Our analysis identifies the mixed strategy equilibria and shows that the level of the tax auditor’s audit costs and the penalty factors for detected noncompliance determine the multinational’s equilibrium behavior. Specifically, our findings illustrate that OSB is part of the multinational’s equilibrium strategy whenever the penalty difference is large. Put differently, strategic considerations in a tax compliance game are a potential reason for multinationals to implement OSB.
Our analysis shows that a higher tax rate difference reduces multinationals’ tax aggressiveness. This result stems from the presence of the strategic tax auditor. A high tax rate difference in particular yields a high taxsavings potential for the multinational, allowing it to benefit from shifting profits to the lowtax country. At the same time, a high tax rate difference corresponds to a high penalty for detected noncompliance. The tax auditor incorporates the penalty into the audit decision, making the audit incentives strong. To counteract the stronger audit incentives, the multinational increasingly refrains from tax aggressiveness.
Furthermore, our analysis illustrates that, for an intermediate level of audit costs, the multinational increases the probability of keeping OSB when the penalty advantage for detected noncompliance under OSB decreases. Specifically, the OSB advantage decreases for an increasing OSB penalty factor. Thus the multinational’s incentives for keeping noncompliant TSB increase. As with the tax rate difference finding, because the penalty for detected noncompliance under OSB increases, the tax auditor’s incentives increase. The stronger audit incentives greatly diminish the multinational’s incentives for keeping noncompliant TSB. Therefore the multinational increases the probability of keeping OSB.
Our paper adds to the literature on international transfer pricing by enhancing the theoretical understanding of how a strategic tax auditor and tax regulation—in the form of the tax rate difference and the penalty factors for noncompliant reporting—affect a multinational’s transferpricing decisions and resulting tax aggressiveness. A promising extension of our research would be to investigate how the presence of a strategic tax auditor affects a multinational’s decisions in more general intrafirm relationships. For example, in a setting where the divisions decide on upfront investments to either enhance revenues or decrease unit costs, the transferpricing regime affects both the investment incentives and the compliance decision. Consequently, the tax auditor’s incentives potentially affect the divisions’ investment decisions.
Notes
While previous studies discuss the fear of negative consequences, they assume the transferpricing regime is exogenous (e.g., Baldenius et al. 2004; Halperin and Srinidhi1991; Johnson 2006; Narayanan & Smith 2010; Nielsen & RaimondosMøller 2012; Reineke & WeiskirchnerMerten 2021; Shunko et al. 2014; Smith 2002).
Alignment with the management view enhances the multinational’s defensibility of its transfer pricing (EY, 2003).
Martini (2015) notes that unrelated firms would not use two different prices for an interfirm trade. Therefore a tax auditor may doubt the economic substance of an multinational’s keeping TSB.
Section 6 analyzes an alternative sequence of events.
Sansing (2014) provides a comprehensive overview of international transfer prices and their functions.
In Germany, e.g., a firm must provide all documents related to a specific transaction to the tax auditor, even if these documents do not relate to tax accounts (§90 AO).
Other studies assume that the divisions maximize their aftertax profits. This assumption is also ad hoc in the transfer pricing setting. Baldenius et al. (2004) explicitly highlight this fact. For further discussion of the advantages of pretax versus aftertax profit maximization for divisional performance measurement, see Nielsen and RaimondosMøller (2012).
We do not consider tax audits in the lowtax country, which anticipates the multinational’s incentives to shift profits to it, so that profit shifting does not deprive it of tax income. Therefore the lowtax country cannot generate additional tax income by auditing the multinational’s transfer pricing.
While we assume that the tax auditor enforces the upper bound of the arm’s length range, in some countries (e.g., the U.S.), the enforced transfer price is the median of the arm’s length range. By enforcing the median instead of the upper bound, the tax auditor additionally punishes a noncompliant multinational. If a country enforces the median of the arm’s length range, we assume that this additional punishment is included in the penalty factor δ.
A transferpricing adjustment by the tax auditor in the hightax country leads to double taxation. We assume that the lowtax country does not pay any refunds that may result from double taxation agreements.
We do not consider knifeedge cases where the multinational randomizes at both stages.
Alternative possibilities for modeling the tax rate difference exist. Nevertheless, because all results hold for the entire range of t and h, our specification is without loss of generality when we conduct comparative statics with respect to h.
See Appendix for the threshold \({K_{a}^{c}}\).
All formal claims and proofs for the alternative sequence of events are in the internet Appendix. The internet appendix is available at https://ssrn.com/abstract=3904634.
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Acknowledgements
We thank Eva Eberhartinger, Christian Hofmann (discussant), Clemens Löffler, Zoltán NovotnyFarkas, Thomas Pfeiffer, Christian Riegler, Georg Schneider, Caren SurethSloane (discussant), and participants in the annual VHB tax group meeting 2020 and the EAA Virtual Congress 2021 for helpful comments. We are grateful to Harald Amberger, Oliver Dürr (discussant), Michael Ebert, Joachim Gassen, Ulrich Schäfer, Andreas Scholze, Robert Ullmann, and Alfred Wagenhofer, and participants in the ARFAWorkshop 2017, the EAA Annual Congress 2017, the Annual VHB Conference 2017, and the Accounting Research Workshop 2017 for helpful comments on an earlier version of this paper. We are also grateful to Stefan Reichelstein (the editor) and two anonymous reviewers for their constructive suggestions. This paper is a substantially revised version of an earlier draft, which was coauthored by Marcel Haak, whose contribution we gratefully acknowledge.
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Appendix
Appendix
Proof of Lemma 1
Figure 4 depicts the multinational’s possible strategies.
Note that \(p_{r}>\overline {p_{r}}\) is an unambiguous signal of tax evasion (punished without audit costs) so that the multinational never chooses \(p_{r}>\overline {p_{r}}\).
For \(p_{r}\leq \overline {p_{r}}\), a highcost multinational is compliant. To minimize its tax payments, the highcost multinational keeping TSB sets \(p_{r}=\overline {p_{r}}\).
Under OSB, a highcost multinational maximizes its expected profit determining a transfer price:
for sufficiently large a so that a highcost multinational keeping OSB chooses \(p_{r}=\overline {p_{r}}\).
A noncompliant lowcost multinational keeping TSB that reports a \(p_{r}\in \left (\underline {p_{r}}, \overline {p_{r}}\right )\) reveals itself as a noncompliant lowcost multinational. By choosing \(p_{r}=\overline {p_{r}}\), a lowcost multinational mimics a highcost one. Thus a noncompliant lowcost multinational keeping TSB sets \(p_{r}=\overline {p_{r}}\).
A noncompliant lowcost multinational keeping OSB chooses a transfer price \(\underline {p_{r}}< p_{r}\leq \overline {p_{r}}\). A noncompliant lowcost multinational maximizes its expected profit by determining a transfer price, where the tax auditor audits with probability η:
\(\left .\frac {d^{2} {\varPi }}{d {p_{r}^{2}}}\right _{q=ap_{r}}\) is negative for \(\delta _{OSB}<\frac {1t+h}{2\eta (t+h)}\). Thus, for a \(\delta _{OSB}>\frac {1t+h}{2\eta (t+h)}\), the firstorder condition (FOC) determines a local minimum, and the multinational prefers a corner solution, that is, \(p_{r}\in \left \{\underline {p_{r}},\overline {p_{r}}\right \}\). With \(p_{r}=\underline {p_{r}}\), the multinational is compliant. Thus a noncompliant lowcost multinational keeping OSB sets \(p_{r}=\overline {p_{r}}\) for \(\delta _{OSB}>\frac {1t+h}{2\eta (t+h)}\). For \(\delta _{OSB}<\frac {1t+h}{2\eta (t+h)}\) the FOC determines a local maximum. For a sufficiently large a, the multinational’s FOC determines a \(p_{r}>\overline {p_{r}}\) so that a noncompliant lowcost multinational keeping OSB sets \(p_{r}=\overline {p_{r}}\) for \(\delta _{OSB}<\frac {1t+h}{2\eta (t+h)}\).
A compliant lowcost multinational keeping TSB minimizes its tax payments by choosing \(p_{r}=\underline {p_{r}}\).
Under OSB a compliant lowcost multinational maximizes its expected profit determining a transfer price:
for sufficiently large a. Thus a compliant lowcost multinational keeping OSB chooses \(p_{r}=\underline {p_{r}}\).
Proof of Lemma 2
Note that TSB allows the multinational to disentangle its internal from its tax transfer price. The multinational’s profit with unit costs c_{j} and j = H,L is as follows.
The noncompliant lowcost multinational considers the consequences resulting from a tax transfer price \(\overline {p_{r}}\) in a tax audit:
Thus the FOC for p_{i} determines a local maximum:
The FOC for a compliant multinational reduces to
Thus the FOC for p_{i} determines a local maximum for the compliant highcost multinational:
Thus the FOC for p_{i} determines a local maximum for the compliant lowcost multinational:
Proof of Proposition 1
We examine each equilibrium. We identify the tax auditor’s audit and the multinational’s randomization probability. In addition, we determine the parameter constellations for K_{a} and δ_{OSB} so that the identified probabilities constitute an equilibrium in mixed strategies.
Proof of equilibrium I
With TSB in place, the lowcost multinational randomizes between compliance and noncompliance if and only if the expected profits from both strategies are the same; that is, \(\eta {\varPi }_{3}^{}+ (1\eta ) {\varPi }_{4}^{}={\varPi }_{5}^{} \):
The tax auditor incurs costs for conducting an audit. Thus the tax auditor audits with probability
The multinational wants to deviate to OSB with \(\overline {p_{r}}\) in t = 0 when \(\overline {\delta }_{OSB}\leq \delta _{OSB}\). The following equation determines \(\overline {\delta }_{OSB}\):
When the tax auditor observes a high tax transfer price, he or she is indifferent between audit and no audit if \( \beta {\varPi }_{1}^{TA}+ (1\beta )\lambda _{TSB}{\varPi }_{3}^{TA}=0 \)
λ_{TSB,I} is smaller than 1 if and only if K_{a} < K_{a2}(η_{I}), where
In sum, for \(\delta _{OSB} \geq \overline {\delta }_{OSB}\) and K_{a} < K_{a2}(η_{I}), the multinational always implements TSB. After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational reports the noncompliant (compliant) tax transfer price \(\overline {p_{r}}\) (\(\underline {p_{r}}\)) with probability λ_{TSB,I} (1 − λ_{TSB,I}). The tax auditor audits \(\overline {p_{r}}\) with audit probability η_{I}.
Proof of equilibrium II
The multinational randomizes the strategies OSB and TSB if
for sufficiently large a.
η_{II} is always smaller than 1 because:
π_{6} < π_{1}, and for a sufficiently large prohibitive price a, \({\varPi }_{9}^{} > {\varPi }_{5}^{}>0\) holds true.
The lowcost multinational might have an incentive to deviate to TSB with the noncompliant tax transfer price \(\overline {p_{r}}\). This deviation occurs if and only if
For a sufficiently large a, the term \(\left [h\eta _{II}\delta _{TSB}(t+h)\right ]\) becomes negative so that \(a\left [h\eta _{II}\delta _{TSB}(t+h)\right ]\) is negative. Therefore the lowcost multinational does not want to deviate to noncompliant TSB.
When the tax auditor observes a high tax transfer price, he or she is indifferent between audit and no audit if:
τ_{II} is positive if and only if K_{a} ≤ K_{a1}(δ_{OSB}) with
In sum, for K_{a} < K_{a1}(δ_{OSB}), the multinational implements TSB (OSB) with probability τ_{II} (1 − τ_{II}). After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational reports the compliant tax transfer price \(\underline {p_{r}}\) under TSB and the noncompliant tax transfer price \(\overline {p_{r}}\) under OSB. The tax auditor audits \(\overline {p_{r}}\) with probability η_{II}. __
Proof of equilibrium III
The multinational randomizes the strategies OSB and TSB if
where
For sufficiently large a, B < 0 and B^{2} − C > 0. Thus the multinational is indifferent between OSB and TSB for
For a sufficiently large a, η_{III} is smaller than 1.
The multinational might have an incentive to deviate to TSB and then choose the compliant tax transfer price \(\underline {p_{r}}\) in the case of low unit costs. This deviation occurs if and only if
For a sufficiently large a, the term \(\left [\eta _{III}\delta _{OSB}(t+h)h\right ]\) becomes negative so that \(a\left [\eta _{III}\delta _{OSB}(t+h)h\right ]\) is negative. Therefore the lowcost multinational never deviates to TSB with a compliant tax transfer price.
When the tax auditor observes a high tax transfer price, he or she wants to randomize between conducting and not conducting an audit if:
τ_{III} is positive and smaller than 1 if and only if K_{a1}(δ_{OSB}) < K_{a} < K_{a2}(η_{III}), where K_{a1} is defined in Eq. 11 and
In sum, for \(K_{a1}(\delta _{OSB}) < K_{a} < K_{a2}\left (\eta _{III}\right )\), the multinational chooses TSB (OSB) with probability τ_{III} (1 − τ_{III}). After the realization of the unit costs, the highcost multinational reports the compliant tax transfer price \(\overline {p_{r}}\) and the lowcost multinational chooses the noncompliant tax transfer price \(\overline {p_{r}}\) under TSB and under OSB. The tax auditor audits \(\overline {p_{r}}\) with probability η_{III}. __
Proof of Proposition 2
An increasing tax rate difference

1.
decreases the internal transfer price (see Lemma 2) under TSB in all equilibria:
$$ \begin{array}{@{}rcl@{}} &&\frac{d p_{iH}}{d h}=(1)\frac{1}{(1th)^{2}}\left[(1t)m_{H}\right]<0,\\ &&\frac{d p_{iL2}}{d h}=(1)\frac{1}{(1th)^{2}}\left[(1t)m_{L}\right]<0,\\ &&\frac{d p_{iL1}\left( \eta_{I}\right)}{d h}=(1)\frac{1}{(1th)^{2}}\left[(1t)m_{L}\right]<0,\\ &&\frac{d p_{iL1}\left( \eta_{III}\right)}{d h}=\frac{(1t)}{(1\beta) (1ht)^{2}}\cdot\\ &&\left[a (\beta 1) (c_{L}\overline{p_{r}}) (\delta_{OSB}\delta_{TSB}) (h+t1)\right.\\ &&+\overline{p_{r}} (\overline{p_{r}}c_{L}) (\delta_{OSB}\delta_{TSB}) (h+t1)\\ &&+\beta (c_{H} \delta_{TSB}(1t)(c_{H}2\overline{p_{r}}) +\overline{p_{r}} (c_{L} (\delta_{OSB}\delta_{TSB}) (h+t1)\\ &&\left.\left.\left.+\overline{p_{r}} (\delta_{TSB} h\delta_{OSB}(h+t1))\right)\right)\right]\cdot\\ &&\left[(a (\delta_{TSB}\delta_{OSB}) (1ht)+c_{L} \delta_{TSB} (t1)\right.\\ &&+\overline{p_{r}} (\delta_{TSB} h\delta_{OSB} (h+t1)))^{2}\\ &&\left.+\frac{\delta_{TSB}^{2} (t1)^{2} \left( \beta (c_{H}c_{L}) (c_{H}+c_{L}2 \overline{p_{r}})+(c_{L}\overline{p_{r}})^{2}\right)}{\beta 1}\right]^{\frac{1}{2}} \end{array} $$\(\frac {d p_{iL1}\left (\eta _{III}\right )}{d h}\) is negative for sufficiently high a.

2.
decreases the probability of a noncompliant tax transfer price in equilibrium I (see Proposition 1):
$$ \begin{array}{@{}rcl@{}} \frac{\partial \lambda_{TSB,I}}{\partial h} = \frac{(1)}{(t+h)^{2}} \frac{K_{a}}{(1\beta)\delta_{TSB}(ap_{iL1})(\overline{p_{r}}\underline{p_{r}})}<0, \\ \frac{\partial \lambda_{TSB,I}}{\partial p_{iL1}}=\frac{K_{a} (ap_{iL1})^{2}}{(1\beta)(t+h)\delta_{TSB}(\overline{p_{r}}\underline{p_{r}})}>0. \end{array} $$Therefore
$$ \frac{d\lambda_{TSB,I}}{d h}=\underbrace{\frac{\partial \lambda_{TSB,I}}{\partial h}}_{<0}+\underbrace{\frac{\partial \lambda_{TSB,I}}{\partial p_{iL1}}}_{>0}\underbrace{\frac{d p_{iL1}}{d h}}_{<0}<0. $$ 
3.
increases the probability of using TSB in equilibrium II (see Proposition 1):
$$ \frac{d\tau_{II}}{d h}=\frac{K_{a}}{(1\beta)(t+h)^{2}(a\overline{p_{r}})\delta_{OSB}(\overline{p_{r}}\underline{p_{r}})}>0. $$ 
4.
decreases the probability of using TSB in equilibrium III (see Proposition 1):
$$ \begin{array}{@{}rcl@{}} \frac{\partial \tau_{III}}{\partial h}\!&=&\!(1)\frac{K_{a}}{(1  \beta)(t + h)^{2}(\overline{p_{r}}  \underline{p_{r}})\left[\delta_{TSB}(a  p_{iL1})  \delta_{OSB}(a  \overline{p_{r}})\right]} <0.\\ \frac{\partial \tau_{III}}{\partial p_{iL1}}\!&=&\!\delta_{TSB}\left[\delta_{TSB}(a  p_{iL1})  \delta_{OSB}(a  \overline{p_{r}})\right]^{2} \left[(1  \beta)(t + h)(\overline{p_{r}}  \underline{p_{r}})\right]^{1}\cdot \\ &&\underbrace{\left[K_{a}\delta_{OSB}(a\overline{p_{r}})(1\beta)(t+h)(\overline{p_{r}}\underline{p_{r}})\right]}_{>0 \text{ for } K_{a}>K_{a1}} >0. \end{array} $$In sum,
$$ \frac{d\tau_{III}}{d h}=\underbrace{\frac{\partial \tau_{III}}{\partial h}}_{<0}+\underbrace{\frac{\partial \tau_{III}}{\partial p_{iL1}}}_{>0}\underbrace{\frac{d p_{iL1}}{d h}}_{<0}<0. $$
Proof of Proposition 3
An increase in δ_{OSB} or δ_{TSB} affects

1.
the internal transfer prices (see Lemma 2) as follows:
$$ \begin{array}{@{}rcl@{}} \frac{d p_{iH}}{d \delta_{OSB}}&=&0, \frac{d p_{iH}}{d \delta_{TSB}}=0, \frac{d p_{iL2}}{d \delta_{OSB}}=0, \frac{d p_{iL2}}{d \delta_{TSB}}=0, \\ \frac{d p_{iL1}(\eta_{I})}{d \delta_{OSB}}&=&0, \frac{d p_{iL1}(\eta_{I})}{d \delta_{TSB}}=0, \\ \frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \delta_{OSB}}&=&0, \frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \eta_{III}}=\frac{t+h}{1th}\delta_{TSB}(\overline{p_{r}}\underline{p_{r}})>0, \\ \frac{d\eta_{III}}{d\delta_{OSB}}&=&\frac{(1th)(a\overline{p_{r}})}{\delta_{TSB}(\overline{p_{r}}\underline{p_{r}})(t+h)}\frac{\eta_{III}}{\sqrt{B^{2}C}}>0, \end{array} $$In sum,
$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\!\frac{d p_{iL1}\left( \eta_{III}\right)}{d \delta_{OSB}}=\underbrace{\frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \delta_{OSB}}}_{=0}+\underbrace{\frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \eta_{III}}}_{>0}\underbrace{\frac{d \eta_{III}}{d \delta_{OSB}}}_{>0}>0. \\ &&\!\!\!\!\!\frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \delta_{TSB}}=\frac{t+h}{1th}\eta_{III}(\overline{p_{r}}\underline{p_{r}})>0, \\ &&\!\!\!\!\!\frac{d\eta_{III}}{d\delta_{TSB}}=\frac{1}{\sqrt{B^{2}C}}\left[\frac{1}{2} \frac{(2)(1t)^{2}}{(1\beta)\delta_{TSB}^{3}(\overline{p_{r}}\underline{p_{r}})^{2}(t+h)^{2}}\right.\\ &&\!\!\!\!\!\times\left[\beta(c_{H}c_{L})(c_{H}+c_{L}2\overline{p_{r}})+(c_{L}\overline{p_{r}})^{2}\right] \\ &&\!\!\!\!\!\left.+\frac{\eta_{\mathit{III}}\left[a(1  t  h)(\delta_{\mathit{TSB}}  2\delta_{\mathit{OSB}})  c_{L}\delta_{\mathit{TSB}}(1  t) + \overline{p_{r}}h\delta_{\mathit{TSB}} + 2\overline{p_{r}}\delta_{\mathit{OSB}}(1  t  h)\!\right]}{\delta_{TSB}^{3}(\overline{p_{r}}\underline{p_{r}})(t+h)}\right]\\&&\!\!\!\!\!<0, \end{array} $$for sufficiently large a. In sum,
$$ \frac{d p_{iL1}\left( \eta_{III}\right)}{d \delta_{TSB}}=\underbrace{\frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \delta_{TSB}}}_{>0}+\underbrace{\frac{\partial p_{iL1}\left( \eta_{III}\right)}{\partial \eta_{III}}}_{>0}\underbrace{\frac{d \eta_{III}}{d \delta_{TSB}}}_{<0}, $$where \(\frac {d p_{iL1}\left (\eta _{III}\right )}{d \delta _{TSB}}\) is negative for sufficiently large a.

2.
the probability that the lowcost multinational chooses a noncompliant tax transfer price λ_{TSB,I} in equilibrium I (see Proposition 1) as follows.
$$ \frac{d \lambda_{TSB,I}}{d\delta_{OSB}}=0, \frac{d \lambda_{TSB,I}}{d\delta_{TSB}}=\frac{K_{a}}{(1\beta)(t+h)\delta_{TSB}^{2}(\overline{p_{r}}\underline{p_{r}})(ap_{iL1})}<0. $$ 
3.
the probability of using TSB τ_{II} in equilibrium II (see Proposition 1) as follows:
$$ \frac{d \tau_{II}}{d \delta_{OSB}}=\frac{K_{a}}{(1\beta)(t+h)(a\overline{p_{r}})\delta_{OSB}^{2}(\overline{p_{r}}\underline{p_{r}})}>0, \frac{d \tau_{II}}{d \delta_{TSB}}=0. $$ 
4.
the probability of using TSB τ_{III} in equilibrium III (see Proposition 1) as follows.
$$ \begin{array}{@{}rcl@{}} \frac{d \tau_{III}}{d \delta_{OSB}}=\frac{a\overline{p_{r}}}{(1\beta)(t+h)(\overline{p_{r}}\underline{p_{r}})(\delta_{TSB}(ap_{iL1})\delta_{OSB}(a\overline{p_{r}}))^{2}}\cdot \\ \left[K_{a}\left( 1+\frac{\eta_{III}}{\sqrt{B^{2}C}}\right)K_{a2}(\eta_{III})\left( 1+\frac{\eta_{III}}{\sqrt{B^{2}C}}\frac{\delta_{OSB}(a\overline{p_{r}})}{\delta_{TSB}(ap_{iL1})}\right)\right]. \end{array} $$\(\frac {d \tau _{III}}{d \delta _{OSB}}\) is monotonically increasing in K_{a}, negative for K_{a} = K_{a1} and positive for K_{a} = K_{a2}(η_{III}). Thus a \({K_{a}^{c}}\in (K_{a1},K_{a2}(\eta _{III}))\) exists so that \(\frac {d \tau _{III}}{d \delta _{OSB}}\) equals zero. Therefore \(\frac {d \tau _{III}}{d \delta _{OSB}}\) is negative (positive) for \(K_{a}\in (K_{a1},{K_{a}^{c}})\) (\(K_{a}\in ({K_{a}^{c}},K_{a2}(\eta _{III}))\)).
$$ \frac{\partial \tau_{III}}{\partial \delta_{TSB}}=\frac{(K_{a1}K_{a})(ap_{iL1})}{(1\beta)(t+h)(\overline{p_{r}}\underline{p_{r}})\left[\delta_{TSB}(ap_{iL1})\delta_{OSB}(a\overline{p_{r}})\right]^{2}}, $$which is negative because equilibrium III occurs for K_{a} ≥ K_{a1}. As shown in the Proof of Proposition 2, \(\frac {\partial \tau _{III}}{\partial p_{iL1}}>0\). In sum, for sufficiently large a,
$$ \frac{d\tau_{III}}{d \delta_{TSB}}=\underbrace{\frac{\partial \tau_{III}}{\partial \delta_{TSB}}}_{<0}+\underbrace{\frac{\partial \tau_{III}}{\partial p_{iL1}}}_{>0}\underbrace{\frac{d p_{iL1}}{d \delta_{TSB}}}_{<0}<0. $$
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Reineke, R., WeiskirchnerMerten, K. & Wielenberg, S. When do firms use one set of books in an international tax compliance game?. Rev Account Stud (2022). https://doi.org/10.1007/s11142021096679
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DOI: https://doi.org/10.1007/s11142021096679
Keywords
 Transfer pricing
 Two sets of books
 One set of books
 Strategic tax auditor
JEL Classification
 H26
 H87
 M42