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Area Conferences 2013

CESifo Area Conference on

Public Sector Economics11–13 April

Interest-rate Manipulation and Debt Shifting by Multinationals

Dirk Schindler and Guttorm Schjelderup

Interest-rate Manipulation and Debt Shifting by

Multinationals∗

Dirk Schindler†

Norwegian School of Economics, NoCeT, and CESifo

Guttorm Schjelderup‡

Norwegian School of Economics, NoCeT, and CESifo

February 08, 2013

Abstract

We examine the tax-engineering strategies of multinationals that simultane-

ously engage in using internal debt shifting and in shifting profits by manipulating

interest rates on internal debt to reduce their overall tax payments. Given our

specification of concealment costs, interest manipulation does not affect any real

choices concerning output and sales in markets, while internal debt fosters real

investment. We point out that the interplay of internal debt and interest manipu-

lation in concealment cost functions matters for the sensitivity of tax engineering to

governmental policies. In particular, if the tax-engineering strategies increase each

others concealment costs, the tax sensitivity of internal debt will be reduced, and

restricting thin capitalization can aggravate the problem of (more harmful) profit

shifting.

Keywords: Multinational enterprises, profit shifting, debt shifting, concealment

costs

JEL classification: H25, F23, D21

∗This paper benefitted from suggestions by Thomas Gresik, Andreas Haufler, Søren Bo Nielsen, andparticipants at the IIPF conference in Dresden and in the OFS seminar in Oslo. Financial support fromthe Deutsche Forschungsgemeinschaft (SCHI 1085/2-1) and the Research Council of Norway is gratefullyappreciated.

†Norwegian School of Economics, Department of Accounting, Auditing and Law, Helleveien 30, 5045Bergen, Norway; email: Dirk.Schindler@nhh.no; phone +47-55959628, fax +47-55959320.

‡Department of Finance and Management Science, Norwegian School of Economics, Helleveien 30,5045 Bergen, Norway; email: Guttorm.Schjelderup@nhh.no; phone: + 47-55959238, fax +47-55959350.

1 Introduction

It is well known that the management of multinationals have at least two methods to save

tax payments. First, they can overinvoice sales to subsidiaries in high-tax countries and

underinvoice transactions to low-taxed subsidiaries so that income is shifted from high-

to low-tax countries. Second, management can set up a tax-efficient financing structure

by allocating debt across countries so that the net benefits of interest tax deductions

in high-tax countries exceed the net costs of the corresponding tax payments in low-tax

countries.1

The theoretical literature on tax-engineering activities of multinationals has so far

analyzed debt shifting in isolation from transfer pricing decisions (see Devereux, 2007).

In reality, any manager or headquarters of a multinational firm must decide both on its

leverage and its transfer prices. This paper sets itself apart from the existing literature

by studying how management behaves when it must jointly decide on its debt-shifting

and transfer-pricing strategy. In doing so, we focus on the manipulation of interest rates

on internal debt as a special form of transfer pricing.

We show that management decisions critically depend on how management perceive

the costs of debt shifting and interest manipulation. In the unlikely event that manage-

ment faces no restrictions on how internal interest rates are set, the use of (excessive)

debt is less desirable, since one can achieve the objective of tax savings by a single in-

strument. In general, however, tax authorities have strong incentives to guard the tax

base by auditing multinationals; so, interest-rate manipulation comes at a cost. Like-

wise, tax authorities are aware of that multinationals may rely too much on debt in

high-tax countries, and countries like Germany and the U.S. have therefore implemented

thin-capitalization rules that are costly to avoid.

Circumventing transfer-price regulation and thin-capitalization rules induces costs

from hiring some experts for hiding the transactions or for finding tax loop-holes. Usually,

these concealment costs increase above average with the amount of profits or debt shifted.

As it will turn out in this paper, another crucial feature is the interplay of the tax-

engineering strategies in each other concealment costs. On the one hand, it may be that

the instruments are ‘mutually abetting’ (i.e., there are positive spill-over effects) if, for

example, knowledge acquired by higher profit shifting can be used for increased debt

shifting, as well. On the other hand, they can be ‘mutually impeding’ if the activities

enforce each others (marginal) concealment costs. The latter appears if profit shifting

gives thin-capitalization rules more bite, for example by reducing (book value of the)

firm’s assets or if reduced profits by larger debt shifting make it more complicated to

hide transfer pricing.

1Hines (1999) and Gresik (2001) provide surveys of the literature on transfer pricing, whereas Mintzand Weichenrieder (2010) provide a survey of the literature on tax-efficient financing structures formultinationals.

2

We generate the following findings. First, while shifting income to low-tax countries

by interest-rate manipulation and debt shifting have in common that each activity reduces

tax payments of affiliates in high-tax countries, larger debt shifting will increase domestic

investment, whereas manipulation of interest rates does not have any real effect. Second,

if the two tax-engineering measures affect each others’ concealment costs positively (i.e.,

for ‘mutual impediment’), their tax sensitivity turns ambiguous. For example, if profit

shifting is increased due to a tax-rate increase, larger profit shifting increases (marginal)

costs of debt shifting, mitigating the responsiveness of debt shifting to increases of the af-

filiate’s tax rate. Third, the effects of regulations to protect tax bases can turn ambiguous

and come at unexpected costs. For ‘mutual impediment,’ tightening thin-capitalization

rules can increase debt shifting, in case the induced reduction in profit shifting will trig-

ger a large decrease in concealment costs of debt shifting. Even worse, tighter effective

thin-capitalization rules can also lead to more profit shifting, when a reduction in debt

shifting substantially reduces concealment costs of profit shifting. In the latter case, the

desired increase in tax revenue might not realize, and real investment will decrease at the

same time.

For showing these results, we depart from a standard model of debt shifting (e.g.,

Mintz and Smart, 2004; Schindler and Schjelderup, 2012), but allow the management

to overinvoice the interest rate on internal debt (i.e., allow for transfer pricing). The

headquarters of a multinational firm decides both on the financing structure of its affiliates

and on the manipulation of interest rates on internal debt. In its decision making, the

headquarters balances tax savings and costs of external as well as internal debt, and

concealment costs of interest-rate manipulation, respectively.

This paper also contributes to a well-known puzzle in the empirical literature on tax

engineering. Pak and Zdanowicz (2001) claim that the volume of profit shifting in U.S.

multinationals has been equal to 18% of total reported corporate profits in 2000. Clausing

(2003) and Bernard et al. (2006) confirm the importance of transfer pricing in the U.S.

Bartelsman and Beetsma (2003), analyzing transfer pricing among OECD countries, point

out that 65% to 87% of the (potential) additional tax revenue, stemming from a unilateral

tax increase and being available in absence of profit shifting, is lost due to profit shifting

by transfer pricing. Hence, profit shifting matters and is highly tax sensitive. For debt

shifting, empirical evidence also provides highly significant effects, but these effects are

surprisingly low in value. The semi-elasticity of internal debt lies between 0.69 and 1.3 in

the very most studies (see, e.g., Desai et al., 2004; Buttner and Wamser, 2007; Buttner

et al., 2009, Møen et al., 2011). Buttner and Wamser (2007, p. 25) conclude that

“...our findings suggest that the implied magnitude of tax-revenue losses is rather modest

even for wholly-owned firms. To conclude, our findings are indicative for substantial

costs of adjusting the capital structure for means of profit-shifting.” One explanation

for adjustment costs, offered in the (empirical) literature, are binding thin-capitalization

3

rules. However, empirical evidence shows that, though effective, thin-capitalization rules

still allow for sufficient leeway to circumvent them in parts or even largely (see Buttner

et al., 2012, and in particular Weichenrieder and Windischbauer, 2008).

Our results could (partly) explain modest effects of tax-rate differentials on debt

shifting by negative cost effects from profit shifting making debt shifting less profitable if

the tax-engineering instruments are mutually impeding in the concealment cost function.

The rest of the paper is organized as follows. Section 2 describes the basic model and

introduces the concealment cost functions. In section 3, we derive the optimal use of debt

policy and of interest-rate manipulation, and analyze the implications of tax engineering

on real investment of the multinational firm. Section 4 examines the tax sensitivity of

debt shifting and of profit shifting, while section 5 analyzes the effectiveness and spill-over

effects of regulation to protect tax bases. Section 6 offers some concluding remarks.

2 The Model

We set up a model of a multinational firm (henceforth MNC) that has its headquarters

(henceforth HQ) located in any country p ∈ {1, n}. The MNC can invest in affiliates in

n countries. These affiliates are assumed for simplicity to be price takers and they are

wholly owned. Each affiliate i employs Ki units of real capital that is used to produce

xi = F (Ki) units of a homogenous good whose output price is normalized to unity. The

production function F (Ki) exhibits positive and decreasing returns to capital (i.e., FK > 0

and FKK < 0). We shall further assume that world markets for real and financial capital

are integrated and that capital is perfectly mobile. Each country is small and cannot

influence interest rates and the market interest rate is exogenously given by r > 0.

To finance its investments in an affiliate in country i, the HQ can use equity Ei and

debt Di. Debt can be further broken down into external debt(DE

i

)and internal debt(

DIi

), where internal debt is obtained by borrowing from related affiliates. We define Ki

as the total (real) capital employed by affiliate i and let bEi = DEi /Ki be the external

debt-to-asset ratio. In a similar fashion, bIi = DIi /Ki is the internal debt-to-asset ratio,

and we define the overall leverage ratio (bi) of the MNC by bi = bEi +bIi =(DE

i +DIi

)/Ki.

Within the MNC, it must be the case that the sum of market interest payments on internal

borrowing and lending is zero across all affiliates, that is,

∑i

r ·DIi =

∑i

bIi · r ·Ki = 0. (1)

The MNC can shift income to affiliates in low-tax countries by under- or overinvoicing

intra-firm transactions. We model this by allowing the firm to deviate from the market

interest rate by levying a surcharge ri on the market interest rate in affiliate i. The total

interest costs of internal debt are then r+ ri, and the amount of profit shifted away from

4

affiliate i is given by

Pi = ri · bIi ·Ki. (2)

The sum of shifted profits across all affiliates can now be written as

∑i

ri · bIi ·Ki = 0. (3)

Most of the literature on debt and tax-efficient financing structures assumes that there

are costs per unit of capital associated with the use of debt that are convex both in ex-

ternal and in internal debt.2 External debt is useful in order to discipline local managers

from lax management and “empire-building” strategies. However, as the leverage ratio

goes up, the risk of bankruptcy increases and may cause bankruptcy costs, or induce a

debt-overhang situation, in which profitable investment is not undertaken. Furthermore,

excessively higher external debt may also be associated with a higher premium due to

informational asymmetries. Consequently, there is an optimal leverage ratio bEi for exter-

nal debt in the absence of taxes.3 Increasing external debt from a leverage ratio bEi < bEi

will decrease leverage costs, whereas any increase for bEi ≥ bEi will cause positive marginal

costs of (external) leverage.

Costs pertaining to internal debt may be related to the use of lawyers and accountants

to avoid that such transactions are restricted by thin-capitalization or controlled-foreign-

company rules (see, e.g., Fuest and Hemmelgarn, 2005).4 We shall also argue that the

costs of using internal debt are influenced by the level of profit income shifted by ma-

nipulating interest rates. This is so, because the income of any given affiliate may be

lowered by a combination of debt shifting and interest-rate manipulation. It is the sum

of these transactions that reduces the relevant year’s-end book equity (due to low profits)

which then gives thin-capitalization rules more bite if larger amounts of internal debt

shall be used. Furthermore, the resulting combination of low profits and high leverage

may arouse suspicion by the tax authorities and induce them to control compliance with

thin-capitalization rules more closely. Accordingly, we define CD = CD(bEi , b

Ii , Pi) as the

cost of debt.

From the discussion above, it follows that the costs and benefits of internal and

2See for example Mintz and Smart (2004), Fuest and Hemmelgarn (2005), and Schindler andSchjelderup (2012).

3See Hovakimian et al. (2004) and Aggrawal and Kyaw (2010) for recent overviews on costs andbenefits of external debt. To focus on the interplay of internal debt and profit shifting and to keep themodel simple, we neglect overall bankruptcy costs on the parent level. The latter would set an incentiveto shift external debt internationally, as well; see Huizinga et al. (2008).

4Thin-capitalization rules are in place in many countries such as Germany, the U.K, and the U.S.,and also apply to foreign subsidiaries. See, e.g., Gouthiere (2005) for a description of several EU andnon-EU countries’ rules. Controlled-foreign-company rules are in place, e.g., in the US and Germanyand they deny tax-exemption of passive income in the home country of the MNC, provided that taxavoidance is suspected (see Ruf and Weichenrieder, 2012).

5

external debt differ. Internal debt could be seen as tax-favored equity, since it does

neither affect the risk of bankruptcy nor reduce any informational asymmetry.5 We shall

therefore assume that the cost function of debt is additively separable in external and

internal leverage, that is CD(bEi , b

Ii , Pi) = CE(b

Ei ) + CI(b

Ii , Pi), as long as external credit

markets are perfect except for allowing for taxation as well as for costs of financial distress

and bankruptcy.

As dating back to Mintz and Smart (2004) and Fuest and Hemmelgarn (2005), and as

being common standard in the debt-shifting literature since then, we assume that agency

costs of debt expand convexly in leverage, but proportionally in real capital employed.

For internal debt, designing strategies to avoid anti-avoidance regulation (particularly,

working around thin-capitalization rules) and asking for experts’ advice should become

more difficult and above-average expensive for higher leverage ratios, whereas it is not

obvious why the size of the firm (i.e., the amount of capital employed) should feature

increasing or decreasing returns in the concealment cost function.

As no compliance with thin-capitalization rules is necessary if an affiliate is not bor-

rowing internal debt, we assume that there are no concealment costs in internal debt

for bIi ≤ 0; no matter how many profits are shifted by interest-rate manipulation, i.e.,

CI(0, Pi) = 0. While the costs of internal debt are otherwise positively affected by the

total amount of profit shifting (∂CI/∂Pi > 0), it turns out that the effect on the marginal

costs of internal leverage (∂2CI/[∂bIi ∂Pi]) from an increase in income shifted is not ob-

vious. We shall for the time being not impose any restrictions on this cross derivative

∂2CI/(∂bIi ∂Pi).

Formally, the properties applied to the cost function of debt can be summarized as:

Assumption 1 External credit markets are assumed to be perfect except for the debt

tax shield and financial distress costs. The debt cost function is additively separable,

CD(bEi , b

Ii , Pi) = CE(b

Ei ) + CI(b

Ii , Pi), and exhibits the properties

CE(bEi ) > 0 with C

′E(b

Ei ) > 0, C

′′E(b

Ei ) > 0 if bEi > bEi ,

C′E(b

Ei ) ≤, C

′′E(b

Ei ) > 0 if bEi ≤ bEi ,

CI(bIi , Pi) > 0 with

∂CI(bIi , Pi)

∂bIi> 0,

∂2CI(bIi , Pi)

∂(bIi )2

> 0 if bIi > 0,

∂CI(bIi , Pi)

∂Pi

> 0,∂2CI(b

Ii , Pi)

∂P 2i

> 0 if bIi > 0,

CI(bIi , Pi) = 0 with

∂CI(bIi , Pi)

∂bIi=

∂CI(bIi , Pi)

∂Pi

= 0 ∀Pi if bIi ≤ 0.

Concealment costs related to profit shifting by interest-rate manipulation are given

5Indeed, Gertner et al. (1994) point out that internal debt does not show the properties of externaldebt and that it should rather be seen as equity. Stonehill and Stitzel (1969) and Chowdhry and Coval(1998, pp. 87) qualify internal debt as “tax-preferred equity”, supporting this view.

6

by the cost function CP (Pi, bIi ). Inspired by the literature on tax evasion (cf. Allingham

and Sandmo, 1972; Yitzhaki, 1974), we assume that these costs depend on the total level

of profit shifting, Pi = ri · bIi · Ki, and are convex in the level of income shifted (Pi).

The concealment costs can be seen as real resource costs due to the use of lawyers and

accountants, and they may also include expected penalties imposed if illegal interest-rate

manipulation is detected and fined by the tax authorities. In the latter case, for example,

our cost function would imply that the detection probability as well as the fines increase

in the amount of shifted profits. Furthermore, we shall assume that the concealment

costs of profit shifting depend on the level of internal debt used. The intuition for the

latter is that it is more costly to hide (illegal)6 profit shifting if the level of debt shifting

is very high, i.e., if taxable profits are low already. For example, this would imply that

such an affiliate of a MNC is significantly less profitable than a comparable domestic

firm and that increases the likelihood of a close auditing by the tax authority, cf. OECD

(2010) and Gresik and Osmundsen (2008) for the so-called ‘comparable-profit method’

under the arm’s-length regulation. Consequently, ∂CP

∂bIi> 0. Once more, the effect on

marginal costs in profit shifting is ambiguous, either because interest-rate manipulation

and internal debt can reinforce concealment costs, or because of positive spill-over effects

by enhanced knowledge in hiding tax engineering. Hence, ∂2CP

∂Pi∂bIi≷ 0.

Note that it is relatively easy to determine deviations from the correct arm’s-length

interest rate in our model, because we assume a uniform world-market interest rate and

the absence of any risk issues. In reality, interest rates on different securities differ based

on differences in issues such as maturity, solvency or the currency of notation. Such differ-

ences will make it difficult for regulators to determine the correct arm’s-length price and

give rise to a trade-off between (efficiency) costs from falsely claiming abusive interest-rate

manipulation (for strict arm’s-length rules) and generating regulatory loopholes for a lax

regulation (trying to take into account non-abusive reasons for interest-rate differentials).

For focusing on the interplay of debt shifting and profit shifting by interest-rate manip-

ulation and on its implications for regulatory measures, we will neglect such additional

complexity in order to keep the model tractable.

Put together, we assume the concealment costs of interest-rate manipulation to be

given by a (convex) cost function CP (Pi, bIi ) if Pi > 0, and zero otherwise. Formally, this

is summarized by

6Manipulation of interest rates for the purpose of shifting profit income is according to most OECDcountries’ legislation an illegal activity.

7

Assumption 2 The cost function of profit shifting exhibits

CP (Pi, bIi ) > 0 with

∂CP (Pi, bIi )

∂Pi

> 0,∂2CP (Pi, b

Ii )

∂P 2i

> 0 if Pi > 0,

∂CP (Pi, bIi )

∂bIi> 0,

∂2CP (Pi, bIi )

∂(bIi )2

> 0 if Pi > 0,

CP (Pi, bIi ) = 0 with

∂CP (Pi, bIi )

∂Pi

=∂CP (Pi, b

Ii )

∂bIi= 0 if Pi ≤ 0.

The HQ maximizes its share in global profits after corporate taxation. In the next

section, we investigate how the MNC invests, structures its debt, and shifts income to

low-taxed affiliates.

3 Optimal Investments

Net global profits of the MNC are given by

Π =∑i

[πi − ti · πt

i

], (4)

where πi is economic profit in subsidiary i, πti is taxable profit, and ti is the corporate tax

rate in country i. We shall assume that the tax-exemption principle is in place and that

debt is tax deductible.7 Economic profit is given by revenue minus user costs of capital

and profit shifting,

πi = F (Ki)− [r + CE(bEi ) + CI(b

Ii , Pi)] ·Ki − Pi − CP (Pi, b

Ii ). (5)

Following OECD tax codes, we assume that costs of equity are not tax deductible.

Hence, taxable profit differs from true economic profit in that only interest expenses re-

lated to borrowing costs, shifted profits and costs of borrowing are tax deductible. In

defining taxable profit, we assume that costs per unit of capital associated with both

external and internal borrowing are tax deductible. Parts of these costs are often asso-

ciated with informational asymmetries between investors and managers of the firm, or

illegitimate action from the point of view of the tax authority. One could argue that

these costs should not be tax deductible. It is straightforward to show by examination

of the equations to follow that even if they were not deductible, it would not affect our

results.

7The exemption principle is used in many OECD countries and in the European Union and impliesnon-taxation of repatriated dividends at the level of the holding. Therefore, it does not matter, in whichcountry the HQ is located. The tax exemption principle is given by the Parent-Subsidiary Directive inthe European Union. Repatriation taxes and strategies to evade these are analyzed, e.g., in Altshulerand Grubert (2003).

8

Taxable profit income can, after some manipulations, be written as

πti = F (Ki)− [rbEi + (r + ri)b

Ii + CE(b

Ei ) + CI(b

Ii , Pi)] ·Ki − CP (Pi, b

Ii ), (6)

where capital invested in country i is financed either by debt Di = DIi +DE

i or by equity

Ei, so that Ki = DIi +DE

i + Ei.

3.1 Profit Shifting and Debt Shifting

The HQ maximizes the value of the MNC after corporate taxes, neglecting any effect

that personal taxes may have. This is in line with most of the literature on MNCs and is

also a reasonable assumption, since MNCs often either are owned by many institutional

investors, or shareholders located in different countries.8 The optimization problem of

the firm can be seen as a two-tier process: First, it chooses its optimal debt-to-asset ratio

and the optimal interest rate on internal debt for any given value of real investment Ki.

Second, the firm decides on how much real capital to use and therefore how much of

the final good to produce in each country. Taking real investment Ki as fixed initially,

the firm’s optimal tax-planning behavior is found by maximizing equation (4). Inserting

for equations (5) and (6), collecting terms, and taking into account the constraints on

internal lending and on profit shifting, that is, equations (1) and (3), the maximization

problem can be written as

maxbEi ,bIi ,ri

Π =∑i

{(1− ti)

[F (Ki)− CP (Pi, b

Ii )]

(7)

− Ki

[r − tir(b

Ei + bIi ) + (1− ti)

(CE(b

Ei ) + CI(b

Ii , Pi)

)+ (1− ti)rib

Ii

]}s.t.

∑i

r · bIi ·Ki = 0 (λ) s.t.∑i

ri · bIi ·Ki = 0 (η),

where λ and η are the associated Lagrangian parameters for internal debt and transfer

pricing, respectively.

8It can be shown that from the viewpoint of a shareholder in a MNC, maximizing profits of the MNCafter global corporate taxation and maximizing the net pay-off on equity investment after opportunitycosts and personal (income) taxes, yield identical results under mild assumptions. For example, ifcorporate taxes cannot be deducted against personal income tax and if the personal tax rate on dividendsand interest income is the same, it is straightforward to show that maximizing the value of the firm tothe owner and maximizing corporate profits coincide. These restrictions are fulfilled for a wide rangeof real world tax codes: the classical corporate taxation system (e.g., in the U.S.), the German systemsince 2009 (“Abgeltungssteuer”), where interest income, dividends and capital gains are taxed at 25%and deductions for corporate taxes are not possible, and the Norwegian shareholder tax, introduced in2006.

9

Optimal manipulation of interest rates. Maximizing (7) with respect to ri, we

obtain

η − (1− ti) ≤ (1− ti)

(∂CP

∂Pi

+∂CI

∂Pi

Ki

)∀ i, (8)

The left hand side is the net marginal benefit of profit shifting and it should be equal to

or less than the after-tax marginal concealment cost of interest-rate manipulation. The

Lagrangian parameter η gives the shadow value of an additional unit of profit income

shifted and can be shown to be equal to η = maxi(1 − ti). We shall for convenience let

country 1 be the country with the lowest tax rate so that by definition η ≡ (1− t1). The

first-order conditions in (8), then, imply that, for internal debt, each affiliate i > 1 pays

a (positive) surcharge on the market interest rate in order to shift profits into affiliate 1

located in the lowest-tax country.

Tax efficient financing structure. The first-order condition for external debt (bEi ) is

given by

C′E(b

Ei ) =

ti1− ti

· r > 0 ∀ i. (9)

Equation (9) states that the value of the debt tax shield should be exploited up until the

point where the associated costs of using external debt equals the marginal value of the

tax shield. The positive value of the debt tax shield implies that the optimal leverage

ratio of external debt in the presence of taxation (bE∗i ) is higher than the optimal leverage

ratio in absence of taxation(bEi

), that is, bE∗

i > bEi .

Deriving and rearranging the first-order condition for internal leverage bIi , we obtain

(ti − λ)r = (1− ti)

(∂CI

∂bIi+

∂CP

∂bIi

1

Ki

), (10)

where we have used that either equation (8) holds with equality, or that ri = 0.

The left hand side of equation (10) is the net marginal benefit of debt shifting. It

should be equal to the tax-adjusted marginal cost of concealing debt and profit shifting.

The bracket on the left hand side of (10) consists of the marginal value of interest deduc-

tions, ti, minus the the shadow cost of lending given by the Lagrangian multiplier λ. It

is straightforward to show that λ = mini ti = t1, since we have defined country 1 as the

lowest-tax country. The implication is that, for minimizing tax payments on interest in-

come from internal debt, lending activities should be conducted from the affiliate located

in the country with the lowest rate of tax, which in our case is country 1.

3.2 Optimal Real Investment

After determining the optimal degree of leverage and the interest rate on internal debt,

the HQ derives the effective cost of capital (evaluated at a tax-efficient financial structure

10

with optimal bE∗i and bI∗i and for optimal transfer price r∗i ). The effective rental rate of

capital can be shown to be equal to

reffi = r − tibE∗i r + (1− ti)CE(b

E∗i )− (ti − t1) b

I∗i r + (1− ti)CI(b

I∗i , P ∗

i )

+ [(1− ti)− (1− t1)]︸ ︷︷ ︸=−(ti−t1)

bI∗i r∗ + (1− ti)CP (P∗i , b

I∗i )

1

Ki

. (11)

In what follows, it is useful to derive the following conditions9

∂reffi

∂ri= − (ti − t1) b

I∗i + (1− ti)b

I∗i

(∂CI

∂Pi

Ki +∂CP

∂Pi

)= 0, (12)

∂reffi

∂Ki

= − 1

Ki

[(1− ti)CP (P

∗i , b

I∗i )

1

Ki

− (ti − t1) bI∗i r∗i

]. (13)

Inserting for the optimal values of debt and the rental rate of capital into the maxi-

mization problem, we can express the MNC’s maximization problem with respect to its

use of capital by

maxKi

∑i

((1− ti)F (Ki)− reffi (Ki) ·Ki

),

where, after applying equations (12) and (13), the corresponding first-order condition is

given by

F iK =

r

1− ti− ti

1− tirbE∗

i + CE(bE∗i )−

(ti − t11− ti

)rbI∗i + CI(b

I∗i , P ∗

i ). (14)

Equation (14) shows that the use of external and internal debt has a direct effect on

the user cost of capital, fostering real investment due to the tax deductibility of debt.

Interest-rate manipulating, in contrast, has no direct effect on the user cost of capital.

We summarize this as

Lemma 1 Thin capitalization reduces effective capital costs and increases real invest-

ment. Manipulating the interest rate on internal debt affects the investment decision only

indirectly via the interplay with internal debt in the concealment cost functions.

From Lemma 1 follows that manipulating interest rates does not have any effect on

the real activity of firms, if internal debt and profit shifting do not affect each others’

concealment costs. However, as seen from equations (11) and (10), manipulating interest

rates affects the user cost of capital as well as the tax sensitivity of internal debt if

concealment costs of debt shifting and profit shifting also depend on the level of abusive

internal interest expenses and internal debt, respectively. This is analyzed in the next

sections.

9In deriving these results, we have used equation (8) twice.

11

4 The Tax Sensitivity of Debt and of Profit Shifting

In this section, we examine how transfer pricing as well as debt shifting are affected by a

change in tax rates. Totally differentiating the first-order condition (9), yields

dbEidti

=r

(1− ti)2 · C ′′E(b

Ei )

> 0. (15)

Equation (15) shows that an increase in the tax rate of country i will induce the

MNC to use more external debt since the value of the debt tax shield has risen. This tax

sensitivity is completely independent from the level of internal debt shifting and from the

level of profit shifting.

To facilitate a discussion on how the transfer price (ri) and the internal debt-to-

asset ratio (bIi ) are affected by a tax increase, we must make assumptions on how the

marginal cost of internal leverage is affected by profit shifting, that is, on the sign of

∂2CI/(∂bIi ∂Pi). We assume that the effects of one activity on concealment costs of the

other activity are qualitatively symmetric, that is, we assume sign{∂2CI/(∂bIi ∂Pi)} =

sign{∂2CP/(∂bIi ∂Pi)}. If higher profit shifting (debt shifting) makes debt shifting (profit

shifting) more/less expensive, more debt shifting (profit shifting) will vice versa in-

crease/decrease concealment costs of profit shifting (debt shifting).

The sign of this cross-derivative is ambiguous in principle. It can be positive if profit

shifting and internal debt mutually lead to higher marginal costs of using internal debt.

For example for earnings-stripping rules, which restrict the amount of tax-deductible

(internal) interest expenses, interest-rate manipulation and higher profit shifting will di-

rectly give more bite to these rules and make debt shifting marginally more expensive. For

specific thin-capitalization rules, restricting the internal leverage, higher profit shifting re-

duces profits and book equity at the end of the year so that the combination of low profits

and high leverage ratios makes thin-capitalization rules more binding and might induce

tax authorities to closer auditing – which will also increase marginal concealment costs.

Similarly, higher debt shifting will decrease profits, and lower profitability compared to

domestic firms might trigger more detailed auditing of compliance with arm’s-length reg-

ulation. The cross derivative may, however, be negative as well, due to pure economies of

scale. This may happen if a firm has acquired special skills in concealing profit-shifting

activities due to the sheer volume of such transactions and can use these skills for debt

shifting as well (and vice versa).

We introduce the following definition to have a simple wording:

Definition 1 Mutual abetment (impediment) in concealment cost functions implies that

debt shifting and profit shifting, respectively, mitigate (bolster) marginal concealment costs

of the other tax-engineering activity, i.e., ∂2CI

∂bIi ∂Pi, ∂2CP

∂bIi ∂Pi< 0 ( ∂2CI

∂bIi ∂Pi, ∂2CP

∂bIi ∂Pi> 0).

Then, we can state

12

Proposition 1 The tax sensitivity of the internal debt-to-asset ratio and of total profit

shifting is enforced if the instruments are mutually abetting, but turns ambiguous if the

tax-engineering strategies are mutually impeding:

(a) If ∂2CI

∂bIi ∂Pi< 0, we have dbIi /dti > 0 and dPi/dti > 0.

(b) If ∂2CI

∂bIi ∂Pi> 0, we have dbIi /dti ≷ 0 and dPi/dti ≷ 0.

The effect on manipulating the interest rate ri for internal debt is ambiguous in any case,

i.e., dri/dti ≷ 0.

Part (a) of Proposition 1 states that, for mutual abetment, the initially positive effect

of a tax-rate increase on both the internal debt-to-asset ratio and total profit shifting will

be boosted by the interplay in concealment costs. The reason for the additional effect is as

follows: on the margin, an increased use of debt shifting (profit shifting) makes it cheaper

to shift income by interest-rate manipulation (debt) all else equal. This leads to more use

of both instruments; both debt shifting and profit shifting will increase unambiguously,

hence.

Part (b) of Proposition 1 shows, however, that a rise in the tax rate in country i needs

not lead to a higher internal debt-to-asset ratio and to more transfer pricing if there is

mutual impediment. Still, a higher tax rate ceteris paribus induces both higher internal

debt-to-asset ratios and larger profit shifting. All else equal, however, increasing one

activity makes the other activity marginally more costly due to mutual impediment in

concealment costs. Thus, the total effect turns ambiguous and the outcome then depends

on the strength of the offsetting indirect cost effects. Hence, after a tax-rate increase, it

could even be optimal to reduce internal debt, say, in order to enable a better exploitation

of the profit-shifting channel.

Note that the effect on the optimal interest-rate manipulation ri is ambiguous for any

specification of concealment costs. In general, a higher tax rate induces more total profit

shifting Pi = ri · bIi · Ki, but also a higher leverage bIi for more debt shifting. Since the

higher internal leverage simultaneously increases profit shifting, less mark-up ri on the

interest rate is required to ensure the optimal amount of shifted profits, all else equal. If

there is a strong increase in internal leverage bIi , it might become necessary to decrease the

mark-up ri to keep the optimal profit shifting Pi in check. The cross effects on marginal

costs additionally enforce or mitigate the former ambiguous effect.

Proposition 1 follows from equations (16) and (17), which are obtained from differen-

tiating the first-order conditions (8) and (10) for tax rate changes (see the Appendix):

dbIidti

=(1− t1)

[(∂2CP

∂P 2i

+ ∂2CI

∂P 2iKi

)Pi −

(∂2CI

∂bIi ∂PiKi +

∂2CP

∂bIi ∂Pi

)bIi

](1− ti)2SOC

{> 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,

(16)

13

where SOC > 0 is the second-order condition. The less attractive profit shifting (i.e.,

interest-rate manipulation) is due to increasing marginal concealment costs (see the first

two terms in the squared brackets), the higher is the tax sensitivity of internal debt.

In case of mutually abetting (impeding) tax-engineering strategies, this effect is fostered

(hampered) by the fact that a larger use of internal debt facilitates (complicates) profit

shifting by reducing (raising) marginal concealment costs of the latter; see the last two

terms in the squared bracket of equation (16).

dPi

dti=

dridti

bIiKi +dbIidti

riKi (17)

=(1− t1)

[(∂2CI

∂(bIi )2Ki +

∂2CP

∂(bIi )2

)bIi −

(∂2CI

∂bIi ∂PiKi +

∂2CP

∂bIi ∂Pi

)Pi

](1− ti)2SOC

{> 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,

where SOC > 0 is the second-order condition, again. The interpretation is analogous

to equation (16). The tax sensitivity of total profit shifting increases with the costliness

to expand internal debt-to-asset ratios marginally. Mutual abetment in the concealment

cost functions will foster the tax sensitivity of profit shifting further; mutual impediment

will hamper it, however.

In sum, from Proposition 1 follows that the empirical findings of a highly significant,

but surprisingly low tax sensitivity of internal debt (e.g., Buttner and Wamser, 2007;

Møen et al., 2011) could partly be explained by having available two tax-engineering

instruments, which are mutually impeding in their concealment cost functions so that

higher profit shifting makes debt shifting less profitable. If so, marginal costs of profit

shifting should be low and, in order to fit to the empirical facts fully, the cost function of

internal debt should be rather ‘steep’ – advocating for high profit shifting. Low costs of

conducting profit shifting could then not only explain high levels of shifted profits, but

also a low tax sensitivity of internal debt.

These conjectures are not implausible, since higher profit shifting reduces firms’ assets,

giving thin-capitalization rules more bite and making it costlier to circumvent them.

Moreover, as long as monitoring profit shifting is rather focused on deviations from the

correct arm’s length price (i.e., on ri in our model), the effect of debt shifting on marginal

costs of profit shifting can be low indeed. In any case, our results call for empirical research

on the shape of the concealment cost functions.

5 Effectiveness of Regulation to Protect Tax Bases

After having seen that the tax sensitivity of tax engineering can turn ambiguous due

to the interplay of debt shifting and profit shifting in concealment cost functions, the

next step is to analyze, how this interplay affects the effectiveness of tax-base protection

14

measures such as thin-capitalization rules and anti-mispricing legislation.

For doing so, we rewrite the concealment cost function of internal debt as CI =

CI(bIi , Pi, σi), where σi is a parameter measuring the tightness of thin-capitalization rules

in country i. A higher σi (i.e., tighter thin-capitalization rules) indicates that circumvent-

ing thin-capitalization rules becomes more difficult, viz., costlier. This fits to empirical

findings in Weichenrieder and Windischbauer (2008) and Buttner et al. (2012), who show

that both German and international thin-capitalization rules are effective, but have some

leeway. We also assume that tighter thin-capitalization rules will increase marginal costs

of internal debt for both debt shifting and profit shifting, i.e., ∂2CI

∂bIi ∂σi> 0 and ∂2CI

∂Pi∂σi> 0.

For example, a reduced safe harbor threshold bIi = DIi /K

bi gives thin-capitalization rules

more bite, no matter whether debt shifting increases the amount of internal debt DIi or

whether profit shifting reduces the book value of capital employed Kbi . Perfectly binding

thin-capitalization rules with no leeway to work around would correspond to the limiting

case in our approach, where ∂2CI

∂bIi ∂σi→ ∞ at bIi = bIi .

Furthermore, we rewrite the concealment costs of profit shifting as CP = CP (bIi , Pi, αi),

where αi is a parameter which indicates how strict arm’s-length regulation is in country

i. An increase in αi then represents higher costs to comply with arm’s-length regulations

or higher fines if profit shifting is detected. Since both larger overpricing ri and a higher

leverage bIi increase profit shifting, a higher fine for or better monitoring of profit shifting,

say, will increase expected marginal concealment costs, viz., ∂2CP

∂bIi ∂αi, ∂2CP

∂Pi∂αi> 0.

From differentiating the first-order conditions (8) and (10) and doing comparative

statics for tighter thin-capitalization rules (see the Appendix), it follows

dbIidσi

= −∂2CI

∂bIi ∂σi

[∂2CP

∂P 2ibIiKi +

∂2CI

∂P 2ibIiK

2i

](1− ti)2SOC

(18)

+

∂2CI

∂Pi∂σiKi

[∂2CI

∂bIi ∂PibIiKi +

∂2CP

∂bIi ∂PibIi

](1− ti)2SOC

{< 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,

dPi

dσi

=dridσi

bIiKi +dbIidσi

riKi (19)

=

[∂2CI

∂Pi∂σi

(∂2CI

∂(bIi )2Ki +

∂2CP

∂(bIi )2

)bIiKi − ∂2CI

∂bIi ∂σi

(∂2CI

∂bIi ∂PiKi +

∂2CP

∂bIi ∂Pi

)bIiKi

]−SOC

{< 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,

where SOC > 0 from the second-order conditions once more.

Tighter thin-capitalization regulation makes internal debt costlier ( ∂2CI

∂bIi ∂σi> 0) and

this standard effect matters more if the use of internal debt is more attractive, i.e., the

more expensive is profit shifting (∂2CP

∂P 2ibIiKi+

∂2CI

∂P 2ibIiK

2i > 0); see the first line in equation

(18). As long as tighter thin-capitalization rules also increase marginal costs of profit

shifting, the effectiveness of thin-capitalization rules is fostered for mutual abetment

15

in concealment costs. The marginal cost increase of profit shifting ( ∂2CI

∂Pi∂σi> 0) leads

ceteris paribus to a reduction in shifting activities, but this implies that debt shifting

becomes even more expensive ( ∂2CI

∂bIi ∂Pi, ∂2CP

∂bIi ∂Pi< 0); see the second line in equation (18).

For mutual impediment, the second line, however, shows that a perverse effect could

occur: the induced reduction in profit shifting will relax the costs of working around

thin-capitalization rules ( ∂2CI

∂bIi ∂Pi, ∂2CP

∂bIi ∂Pi> 0), all else equal, and it could be optimal to

increase internal debt even if the drop in profit shifting delivers a substantial cost saving

in debt shifting.

Even more important, the issue of the interplay in concealment costs pops up in the

sensitivity of total profit shifting as well. The first term in the second line of equation (19)

shows that profit shifting is reduced in order to relax the effect of thin-capitalization rules

on marginal costs of internal debt. For mutual abetment, ∂2CI

∂bIi ∂Pi< 0, the former effect

is accompanied by a further decrease in profit shifting, since a reduction in internal debt

makes profit shifting marginally more expensive; see the second term in the second line.

For mutual impediment, however, the total effect on profit shifting is ambiguous: all else

equal, less internal debt makes concealing profit shifting less costly. Hence, manipulating

interest rates becomes more attractive, and it can well happen that restricting debt

shifting by thin-capitalization rules will worsen the problem of profit shifting.

We summarize this as

Proposition 2 Introducing tighter thin-capitalization rules will decrease both debt shift-

ing and profit shifting if these strategies are mutually abetting with respect to their conceal-

ment costs. However, in case of mutual impediment, stricter thin-capitalization regulation

has unintended effects and either thin capitalization or profit shifting can increase.

Turning to profit-shifting regulation, we find from doing comparative statics for the

parameter αi (see the Appendix)

dbIidαi

=− ∂2CP

∂bIi ∂αi

[∂2CP

∂P 2ibIi +

∂2CI

∂P 2ibIiKi

]+ ∂2CP

∂Pi∂αi

[∂2CI

∂bIi ∂PibIiKi +

∂2CP

∂bIi ∂PibIi

](1− ti)2SOC

{< 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,(20)

dPi

dαi

=dridαi

bIiKi +dbIidαi

riKi (21)

=

[∂2CP

∂Pi∂αi

(∂2CI

∂(bIi )2Ki +

∂2CP

∂(bIi )2

)bIi − ∂2CP

∂bIi ∂αi

(∂2CI

∂bIi ∂PiKi +

∂2CP

∂bIi ∂Pi

)bIi

]−SOC

{< 0 if ∂2CI

∂bIi ∂Pi< 0,

≷ 0 if ∂2CI

∂bIi ∂Pi> 0,

where SOC > 0.

All else equal, profit-shifting regulation will decrease internal debt-to-asset ratios the

more this regulation is fostering marginal costs of internal debt and the more internal

debt is used, see the first term in equation (20). For mutual abetment, the decrease in

profit shifting following a regulation-induced increase in marginal costs ( ∂2CP

∂Pi∂αi> 0), will

16

reduce the use of internal debt further, since debt shifting is becoming more expensive,

as well; see the second term in (20). However, for mutual impediment, the reduction in

profit shifting relaxes concealment costs of debt shifting and fosters the use of internal

debt, ceteris paribus. We end up with an ambiguous effect once more, and it can well

happen that reducing profit shifting will aggravate the problem of thin capitalization.

For profit shifting, we see from the first term in the second line of equation (21) that an

increase in its concealment costs will decrease this shifting activity. The standard effect

is enforced if profit shifting and debt shifting are mutually abetting in concealment costs.

As stricter regulation increases marginal costs of debt shifting ( ∂2CP

∂bIi ∂αi> 0), internal debt

will decrease and by that becomes profit shifting even more expensive ( ∂2CI

∂bIi ∂Pi, ∂2CP

∂bIi ∂Pi< 0)

as the second term in the second line shows. However, the now well-known ambiguity

is back for mutual impediment, as here the decrease in internal debt relaxes marginal

concealment costs of profit shifting, all else equal.

We summarize as

Proposition 3 Stricter regulation to prevent profit shifting will decrease both debt shift-

ing and profit shifting if the strategies are mutually abetting in concealing their effects.

However, for mutual impediment, tighter profit-shifting regulation has unintended effects

and either thin capitalization or profit shifting can increase.

From Propositions (2) and (3), it follows that fighting against tax engineering might

come at higher costs than thought at first sight. In particular, a potential, unintended

increase in profit shifting following fiercer thin-capitalization rules appears unattractive,

since more profit shifting does not affect the real economy, whereas a decrease in debt

shifting will be accompanied by less real investment (see Lemma 1). For assessing the

likelihood of such a case, it turns out once more that some empirical research on the

shape of concealment costs functions is in order.

6 Conclusions

We set up a model, in which the headquarters of a multinational simultaneously decides

on debt shifting by internal debt and on profit shifting by manipulating interest rates

on internal debt in all its affiliates. We show that the two strategies for tax engineering

do not directly interfere with each other and that interest-rate manipulation does not

affect the optimality condition for real investment. However, it turns out that internal

debt and manipulation of interest rates are connected via their interplay in concealment

costs and that this interplay crucially affects the sensitivity of tax-engineering strategies

to variations in tax rates and in regulation to protect tax bases.

From our analysis follows that the stylized facts of large effects of tax-rate differentials

on transfer pricing, but only modest ones on internal debt can (partly) be due to a mutual

17

impediment in concealment costs: A tax-induced increase in profit shifting (internal

debt) will increase marginal costs of debt shifting (interest-rate manipulation), making

the other instrument for tax engineering costlier, i.e., mitigating its tax sensitivity. Our

conjecture would be that concealment costs of internal debt are rather steep and highly

affected by profit shifting (e.g., by decreasing book values of equity and giving thin-

capitalization rules more bite). The marginal impact on concealment costs of profit

shifting should be rather low on the contrary. We believe that empirical research on the

shape of concealment costs is worth while and necessary. Since the literature is silent on

the interplay of strategies in the cost functions, we support the suggestion by Buttner

and Wamser (2007) that one has to study the costs of adjusting the capital structure

more closely.

Better knowledge of concealment costs is particularly in order for designing thin-

capitalization rules and anti-mispricing regulation. Given that there were mutual imped-

iment, these measures to protect tax bases may cause very unintended effects and come at

high costs. Stricter thin-capitalization rules can aggravate the problem of profit shifting

for example. This is odd, since both strategies reduce domestic tax revenue, but debt

shifting at least increases domestic real investment by relaxing the initial distortion from

denying tax deductibility for costs of equity, whereas profit shifting only shifts resources

to other countries.

18

AAppendix

Thefirst-order

conditionof

external

debt(9)isfullyseparab

lefrom

theother

decisions.

Hence,wecanneglect

itan

dfocuson

theother

twoconditions(8)an

d(10).Denotethetigh

tnessof

anti-profit-shiftingregu

lation

incountryibyparam

eter

αian

dthetigh

tnessof

thin-cap

italizationrulesin

countryibyparam

eter

σi.Thefirst-order

conditionsforinternal

debt(10)

andforinterest-rateman

ipulation

pricing(8)canbetran

sformed

into

(ti−

t 1)r

1−t i

−∂C

I

∂bI i

−∂C

P

∂bI i

1 Ki

=0,

(22)

t i−t 1

1−t i

−∂C

P

∂Pi

+K

i∂C

I

∂Pi

=0,

(23)

wherewemad

eusof

λ=

t 1an

dη=

1−t 1.

Totally

differentiatingtheseexpressionslead

sto

[ ∂2C

I

∂(b

I i)2

+∂2C

I

∂bI i∂Pi

r iK

i+

∂2C

P

∂(b

I i)2

1 Ki

+∂2C

P

∂bI i∂Pi

r i

] dbI i

+

[ ∂2C

I

∂bI i∂Pi

bI iK

i+

∂2C

P

∂bI i∂Pi

bI i

] dr i

=1−t 1

(1−t i)2rdt i−

∂2C

P

∂bI i∂αi

1 Ki

dαi−

∂2C

I

∂bI i∂σi

dσi,

[ ∂2 CP

∂P

2 i

r iK

i+

∂2C

P

∂bI i∂Pi

+∂2C

I

∂P

2 i

r iK

2 i+

∂2C

I

∂bI i∂Pi

Ki] d

bI i+

[ ∂2 CP

∂P

2 i

bI iK

i+

∂2C

I

∂P

2 i

bI iK

2 i

] dr i

=1−t 1

(1−t i)2dt i−

∂2C

P

∂Pi∂αi

dαi−

∂2C

I

∂Pi∂σi

Kidσi.

andcollectingterm

sresultsin (

∂2C

I

∂(b

I i)2+

∂2C

I

∂bI i∂Pir iK

i+

∂2C

P

∂(b

I i)2

1 Ki+

∂2C

P

∂bI i∂Pir i

∂2C

I

∂bI i∂PibI iK

i+

∂2C

P

∂bI i∂PibI i

∂2C

P

∂P

2 ir iK

i+

∂2C

P

∂bI i∂Pi+

∂2C

I

∂P

2 ir iK

2 i+

∂2C

I

∂bI i∂PiK

i∂2C

P

∂P

2 ibI iK

i+

∂2C

I

∂P

2 ibI iK

2 i

)(dbI i

dr i

) =

(1−t

1

(1−t

i)2r

1−t

1

(1−t

i)2

) dt i−( ∂

2C

P

∂bI i∂αi

1 Ki

∂2C

P

∂Pi∂αi

) dαi−

(∂2C

I

∂bI i∂σi

∂2C

I

∂Pi∂σiK

i

) dσi.

(24)

19

Thesecond-order

conditionim

plies

SOC

=

[ ∂2C

I

∂(b

I i)2

+∂2C

I

∂bI i∂Pi

r iK

i+

∂2C

P

∂(b

I i)2

1 Ki

+∂2C

P

∂bI i∂Pi

r i

][ ∂2C

P

∂P

2 i

bI iK

i+

∂2C

I

∂P

2 i

bI iK

2 i

](25)

−[ ∂2 C

P

∂P

2 i

r iK

i+

∂2C

P

∂bI i∂Pi

+∂2C

I

∂P

2 i

r iK

2 i+

∂2C

I

∂bI i∂Pi

Ki][

∂2C

I

∂bI i∂Pi

bI iK

i+

∂2C

P

∂bI i∂Pi

bI i

] >0.

ApplyingCramer’srule,wefinally

receiveforinternal

debt

dbI i

dt i

=(1

−t 1)[ ∂

2C

P

∂P

2 irb

I iK

i+

∂2C

I

∂P

2 irb

I iK

2 i−

∂2C

I

∂bI i∂PibI iK

i−

∂2C

P

∂bI i∂PibI i

](1

−t i)2SOC

{ >0,

if∂2C

I

∂bI i∂Pi<

0,

≷0,

if∂2C

I

∂bI i∂Pi>

0,(26)

dbI i

dσi

=−

∂2C

I

∂bI i∂σi

[ ∂2C

P

∂P

2 ibI iK

i+

∂2C

I

∂P

2 ibI iK

2 i

] +∂2C

I

∂Pi∂σiK

i

[ ∂2C

I

∂bI i∂PibI iK

i+

∂2C

P

∂bI i∂PibI i

](1

−t i)2SOC

{ <0,

if∂2C

I

∂bI i∂Pi<

0or

∂2C

I

∂Pi∂σi=

0,

≷0,

if∂2C

I

∂bI i∂Pi>

0,(27)

dbI i

dαi

=−

∂2C

P

∂bI i∂αi

[ ∂2C

P

∂P

2 ibI i

+∂2C

I

∂P

2 ibI iK

i] +∂2C

P

∂Pi∂αi

[ ∂2C

I

∂bI i∂PibI iK

i+

∂2C

P

∂bI i∂PibI i

](1

−t i)2SOC

{ <0,

if∂2C

I

∂bI i∂Pi<

0.

≷0,

if∂2C

I

∂bI i∂Pi>

0.(28)

Theeff

ects

onman

ipulatingtheinterest

ratesfortran

sfer

pricingare

dr i dt i

=(1

−t 1)

(1−t i)2SOC

[ ∂2C

I

∂(b

I i)2

+∂2C

P

∂(b

I i)2

1 Ki

−( ∂2

CP

∂P

2 i

+∂2C

I

∂P

2 i

Ki) r i

rKi−( ∂

2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) (r−

r i)] ≷

0,(29)

dr i

dσi

=−1 SOC

[ ∂2C

I

∂Pi∂σi

Ki

( ∂2C

I

∂(b

I i)2

+∂2C

P

∂(b

I i)2

1 Ki

) −∂2C

I

∂bI i∂σi

( ∂2C

P

∂bI i∂Pi

+∂2C

I

∂bI i∂Pi

Ki) −

∂2C

I

∂bI i∂σi

( ∂2C

P

∂P

2 i

+∂2C

I

∂P

2 i

Ki) r i

Ki

+∂2C

I

∂Pi∂σi

( ∂2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) r iK

i] ≷0,

(30)

20

dr i

dαi

=−1 SOC

[ ∂2C

P

∂Pi∂αi

( ∂2C

I

∂(b

I i)2

+∂2C

P

∂(b

I i)2

1 Ki

) −∂2C

P

∂bI i∂αi

1 Ki

( ∂2C

P

∂bI i∂Pi

+∂2C

I

∂bI i∂Pi

Ki) −

∂2C

P

∂bI i∂αi

( ∂2C

P

∂P

2 i

+∂2C

I

∂P

2 i

Ki) r i

+∂2C

P

∂Pi∂αi

( ∂2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) r i

] ≷0.

(31)

Theeff

ects

oninterest-rateman

ipulation

aream

biguou

sin

allcases,

since

thelevelof

shiftedprofits

also

dep

endson

internal

leverage

bI i.For

exam

ple,ahigher

taxrate

inducesmoreprofitshifting(firstterm

sin

squared

bracket

inequation(29)).

How

ever,since

ahigher

internal

leverage

also

increasesprofitshiftingPi=

r i·bI i

·Ki,thereisalso

anegativeeff

ecton

theinterest

rate

r i(thefirstterm

inbrackets

inthesquared

bracket

in(29)).

Finally,wehavethecrosseff

ects

onmarginal

costs,

whichcanenforceor

reduce

theform

ereff

ects

(see

last

term

insquared

bracket).

Focusingon

totalprofitshiftingPi,theeff

ects

simplify

andbecom

eas

expected.Since

totalprofitshiftingisgivenbyPi=

r i·b

I i·K

i,

wefind:

dPi

dt i

=dr i dt ibI iK

i+

dbI i

dt ir iK

i=

1−t 1

(1−t i)2SOC

[( ∂2C

I

∂( b

I i)2K

i+

∂2C

P

∂( b

I i)2

) bI i−

( ∂2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) Pi]{

>0,

if∂2C

I

∂bI i∂Pi<

0,

≷0,

if∂2C

I

∂bI i∂Pi>

0,(32)

dPi

dσi

=dr i

dσi

bI iK

i+

dbI i

dσi

r iK

i=

−1 SOC

[ ∂2C

I

∂Pi∂σi

( ∂2C

I

∂(b

I i)2K

i+

∂2C

P

∂(b

I i)2

) bI iK

i−

∂2C

I

∂bI i∂σi

( ∂2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) bI iK

i]{<

0,if

∂2C

I

∂bI i∂Pi<

0,

≷0,

if∂2C

I

∂bI i∂Pi>

0,(33)

dPi

dαi

=dr i

dαi

bI iK

i+

dbI i

dαi

r iK

i=

−1 SOC

[ ∂2C

P

∂Pi∂αi

( ∂2C

I

∂(b

I i)2K

i+

∂2C

P

∂(b

I i)2

) bI i−

∂2C

P

∂bI i∂αi

( ∂2C

I

∂bI i∂Pi

Ki+

∂2C

P

∂bI i∂Pi

) bI i

]{<

0,if

∂2C

I

∂bI i∂Pi<

0.

≷0,

if∂2C

I

∂bI i∂Pi>

0.(34)

21

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