Optimal Taxation of Multinational
Enterprises: A Ramsey Approach∗Sebastian Dyrda

Guangbin Hong

Joseph B. Steinberg

University of Toronto

University of Toronto

University of Toronto

September 14, 2023
Abstract
What is the optimal design of the international corporate tax system? We revisit thisclassic question in a multi-country general equilibrium model that incorporates three
key features of the modern globalized economy: multinational production; intangible
capital; and international profit shifting. Our model’s competitive equilibrium is ineﬀicient due to an externality that arises from international spillovers in intangible
investment. In the absence of profit shifting, there is little, if anything, a Ramsey
planner can do with corporate income taxes to improve the allocation of intangible investment across countries. However, profit shifting allows the planner to use corporate
income taxes to internalize the externality and achieve an eﬀicient allocation of intangible investment. To quantitatively investigate the properties of the Ramsey planner’s
optimal policy in a more realistic setting, we extend our model to an environment withfirm heterogeneity and selection into multinational production. Without spillovers, itwould be optimal to shut down profit shifting as much as possible. With spillovers,
it would be optimal to allow MNEs to continue to shift profits, and if the planner isrestricted to Pareto-improving policies, it would be optimal to allow even more profit
shifting than under the status quo.
Keywords: Multinational enterprise; intangible capital; profit shifting; corporate taxation; optimal policy; Ramsey problem.
JEL Codes: E6, F23, H25, H27∗

Steinberg:
joseph.steinberg@utoronto.ca; Hong:g.hong@mail.utoronto.ca; Dyrda:
sebastian.dyrda@utoronto.ca. Address: University of Toronto, Max Gluskin House, 150 St. George Street,Toronto, ON M5S 3G7, Canada. In preparation for the 100th Carnegie-Rochester-NYU Conference on
Public Policy: International Policy Coordination and Competition. Acknowledgments: We thank V.V.
Chari and Yena Park for comments on early drafts of the paper, seminar participants at the Seoul
National University and University of Toronto, Ana Maria Santacreu for her excellent discussion at theCarnegie-Rochester-NYU Conference, and the editors for their invaluable suggestions.

11Introduction

How should the international corporate tax system be organized? What economic tradeoffs do policymakers face when designing this system? The public finance literature haslong studied these questions, but globalization and the rising importance of multinational
enterprises (MNE) have the potential to alter the answers dramatically. The modern global
economy is increasingly dominated by large MNEs that invest heavily in intangible capital,
which makes them more productive at home and abroad and allows them to shift profits to
subsidiaries in low-tax countries. In this paper, we study optimal corporate tax policy usinga model that puts these phenomenona front and center.
We study the problem of a cooperative global Ramsey planner in a multi-country neoclassical growth model with three key ingredients: multinational production; nonrival intangible
capital; and international profit shifting. In modeling the first two ingredients, we emphasize international spillovers that rise to an externality—intangible investment in one country
leads to more output and also more intangible investment abroad—which implies that theallocation of intangible investment across countries in a competitive equilibrium is ineﬀicient. In the theoretical part of the paper, we study the extent to which the Ramsey planner
can use corporate income taxes to improve the allocation of intangible investment in theface of this spillover. In the absence of profit shifting, corporate income taxes have little, ifany, utility; at best, they can improve one country’s intangible investment eﬀiciency whileworsening others’. In the presence of profit shifting, however, the planner can use corporate
income taxes to not only improve the eﬀiciency of intangible investment in all countries, but
in fact to fully correct the externality and achieve a Pareto-optimal allocation of intangible
investment across countries.
In the quantitative part of the paper, we analyze the Ramsey planner’s problem in aricher environment with heterogeneous firms and selection into multinational production. In
this environment, the planner can achieve welfare gains by cutting corporate income taxes,
which boosts MNEs’ incentives to invest in intangible capital and makes worldwide allocationof resources more eﬀicient. In the absence of a spillover externality, it would be optimal to
shut down profit shifting as much as possible. However, if such an externality is present,
MNEs should be allowed to shift profits, and the optimal level of profit shifting is increasingin the size of the externality. If the planner is constrained to Pareto-improving tax reforms
that at least weakly benefit all regions, it would actually be optimal to increase the amountof profit shifting by MNEs relative to the status quo.
Our theoretical analysis builds on Chari, Nicolini, and Teles (2023) (henceforth CNT)
who study the cooperative Ramsey problem in a multi-country Backus, Kehoe, and Kydland2

(1994) environment. We extend their analysis in two ways to capture the three ingredients
described above. First, we incorporate multinational production and nonrival intangible
capital as in McGrattan and Prescott (2009, 2010) (henceforth MP). Each country has arepresentative MNE that produces a distinct intermediate good, which can be sold abroad byexporting domestic output as in Backus et al. (1994), but also by producing locally via foreign
aﬀiliates. MNEs produce their products using local rival factors and nonrival intangible
capital, which is in turn produced by employing workers in MNEs’ home countries. We
depart from MP by assuming that there are spillovers in the production of intangible capital:
an increase in one country’s usage of foreign intangible capital makes its own investments
more effective (Javorcik, 2004; Bitzer and Kerekes, 2008). Second, we incorporate the theoryof transfer pricing and profit shifting that we developed in Dyrda, Hong, and Steinberg(2022) (henceforth DHS). MNEs’ foreign aﬀiliates pay licensing fees to use intangible capital
according to arms-length transfer pricing rules. Normally, these fees are paid to MNEs’
domestic parent companies, but MNEs can shift the rights to this licensing income to a tax
haven by paying a convex cost.
We begin our theoretical analysis by studying the eﬀiciency properties of our environment. Several of the conditions that characterize an eﬀicient allocation are familiar from
CNT. For example, the marginal rate of substitution between labor and consumption should
equal the marginal product of labor, and the social return to tangible capital should beequated across countries. In our environment, though, an additional condition characterizes
the optimal allocation of intangible investment. This condition states that allocating an
additional unit of labor to producing intermediate goods should yield the same worldwide
utility gains as allocating it to producing intangible investment, highlighting the externality
created by spillovers in this investment. When MNEs from one country invest in additional
intangible capital, that increases the production of their foreign aﬀiliates in other countries,
but it also makes these other countries’ MNEs better at producing intangible capital. The
first country’s MNEs internalize the first effect, but not the second one.
We proceed to characterize competitive equilibria with distortive taxes under three scenarios. First, a scenario without transfer pricing or profit shifting in which MNEs’ foreign
aﬀiliates use intangible capital free of charge as in McGrattan and Waddle (2020). Second,a scenario in which foreign aﬀiliates pay licensing fees to use intangible capital but profit
shifting is not allowed. Third, our main scenario is with both transfer pricing and profit
shifting. Our baseline tax system includes a set of standard instruments that are used tofinance government expenditures: taxes on consumption, labor income, exports and imports,
and corporate income. In the free transfer and transfer pricing scenarios, the allocation of3intangible investment is always ineﬀicient regardless of the tax structure. In the former, corporate income taxes affect intangible investment, but do so in a way that improves eﬀiciency
in one country while worsening it in others. In the latter, corporate income taxes are even lesseffective at improving the eﬀiciency of intangible investment—in fact, they have no effect atall. In the profit shifting scenario, however, the situation is markedly different. Here, raising
corporate income taxes increases the gains from profit shifting, which in turn increases thereturns on intangible investment, leading MNEs to do more of it. Raising corporate income
taxes worldwide—or reducing taxes in the tax haven, if that is possible—improves the eﬀiciency of intangible investment in all countries, unambiguously moving the economy closer tothe Pareto frontier. In fact, we show corporate income taxes can be used to fully internalize
the spillover externality and achieve an eﬀicient allocation of intangible investment.
We then formalize the Ramsey problem in our economy and equip the planner with arich set of instruments including consumption, labor income, trade and corporate taxes. The
Ramsey planner has two goals in our framework. The first goal is standard: minimizing
distortions associated with financing exogenous government expenditures. The second goalis new and unique to our environment: to internalize the spillover externality and eﬀiciently
allocate intangible capital across countries. We first argue that with a benchmark set of
instruments, the planner can not implement a Pareto-eﬀicient allocation in any of the scenarios. Notably, this includes the profit shifting scenario, where it is possible to achieve aneﬀicient allocation of intangible investment. This is due to the tension between achieving astatically-eﬀicient allocation of intangible investment and a dynamically-eﬀicient allocationof tangible investment. We then demonstrate that if we equip the planner with an additional
instrument, a tax (or subsidy) on tangible capital income, the planner can achieve bothgoals in the profit shifting scenario and implement a Pareto-optimal allocation. In the freetransfer and transfer pricing scenarios, however, even a planner equipped with this additional
instrument can not restore eﬀiciency. In the profit shifting scenario, where Pareto-optimality
can be attained, implementing the Ramsey allocation requires generically non-zero corporate
income taxes that differ across countries and non-zero capital income taxes to correct intertemporal distortions. Thus, corporate income tax harmonization is not optimal, and theclassic result of Chamley (1986) and Judd (1985) does not hold.
In our quantitative analysis, we dig deeper into the tension between correcting the spillover
externality in intangible investment and attaining dynamic eﬀiciency in tangible investment
in a richer model with firm heterogeneity and selection into multinational activity. This
allows us to account for the fact that while MNEs represent a small fraction of the total
number of firms in the economy (and those that shift profits an even smaller fraction), they4

account for a large fraction of production and an even larger fraction of intangible investment.
Firms in this version of the model differ in productivity and must pay fixed costs to export,
larger fixed costs to establish foreign aﬀiliates in other productive countries, and even larger
fixed costs to establish aﬀiliates in the tax haven in order to shift profits. Following DHS, we
divide the world into five regions: two high-tax rich regions, North America and Europe; alow-tax rich region that includes Ireland, Switzerland, and other profit-shifting destinations
that nevertheless have productive, diversified economies; a tax haven that includes small
countries like Bermuda and the Bahamas where much of the economy is centered aroundprofit shifting; and the rest of the world, which has relatively high taxes and a much lower
standard of living than the other four regions. We then calibrate the model to reproduce
salient facts about production, trade, multinational activity, and international profit shifting
under the current international corporate tax regime. We then solve numerically for theRamsey planner’s optimal tax regime in each of the three scenarios described above. We also
consider the problem of a constrained Ramsey planner who is restricted to Pareto-improving
policies that all countries prefer, at least weakly, to the current tax regime.
The unconstrained planner’s solution highlights four takeaways. First, it would be optimal
to shut down profit shifting completely in the absence of a spillover externality. Whenspillovers are present it would be optimal to allow MNEs to continue to shift profits, although
the optimal level of profit shifting would be lower relative to the status quo. Second, dynamic
eﬀiciency plays an important role in allocating intangible investment eﬀiciently. Corporate
income taxes boost intangible investment through the profit shifting channel but also reducetangible investment, which in turn drags down intangible investment as well. Our results
show that the second channel is stronger than the first: intangible investment moves in
the opposite direction as corporate income taxes in equilibrium. Third, spillovers allow theplanner to achieve larger welfare gains by enacting larger corporate income tax cuts. Fourth,
the unconstrained planner would benefit high-tax regions at the expense of the low-tax region.
This is due to the small Pareto weight the planner puts on the low-tax region, but also because
reducing profit shifting requires raising this region’s corporate tax rate, which discourages its
MNEs from investing in intangible capital.
The constrained planner’s solution differs in two key ways. First, it would never be
optimal for the constrained planner to shut down profit shifting, regardless of whether there isa spillover externality. This is because reducing profit shifting hurts the low-tax region, which
the constrained planner is not allowed to do. In fact, it would be optimal for the constrainedplanner to actually increase the level of profit shifting that occurs in equilibrium relative
to the status quo, especially when there is a spillover externality. Second, the constrained5

planner is limited to smaller tax reforms that generate commensurately smaller welfare gains
for North America and the rest of the world, the two largest regions on which the planner’s
objective puts the most weight.

2

Related LiteratureThis paper is closely related to two main strands of literature. First, we contribute to theliterature on the optimal design of the international tax system, which has a long tradition anda variety of approaches. Notable papers are Feldstein and Hartman (1979), Gordon (1986),
Keen and Wildasin (2004) and more recently Lyon and Waugh (2018), Costinot, RodríguezClare, and Werning (2020), Costinot and Werning (2022), Hosseini and Shourideh (2022),
and Chari, Nicolini, and Teles (2023). In an influential paper Keen and Wildasin (2004) argue
that in a static trade model where countries have distinct government budget constraints, theproduction eﬀiciency result established by Diamond and Mirrlees (1971) fails to hold. Recent
work by Chari et al. (2023) challenges this result, however. They develop a dynamic trademodel based on Backus, Kehoe, and Kydland (1994) and use it to examine how countries
should coordinate their fiscal and trade policies when government spending is financed by
distortionary taxes. They show, among other things, that a Ramsey planner can achieve
production eﬀiciency, and that free flow of goods, services, and capital across borders is
optimal.
Our paper builds upon CNT, extending their model in two dimensions. First, we incorporate multinational production and nonrival intangible capital as in McGrattan and Prescott(2009, 2010). Second, we incorporate the theory of transfer pricing and profit shifting that
we developed in Dyrda et al. (2022). Third, we model international spillovers of nonrival
intangible capital investment (Javorcik, 2004). The intangible spillovers create an externality, which implies that a competitive equilibrium in our framework will generally not be
eﬀicient event absent distortionary taxes. Thus, a Ramsey planner in our framework will
aim to minimize distortions associated with financing government expenditure and to correct
the externality. It leads to an interesting trade-off between ensuring production eﬀiciency
and allocating intangible capital eﬀiciently, which is absent in CNT. One potential resolution of this trade-off is to include a capital income taxation on top of the other standard
taxes we consider, which breaks the Chamley-Judd (Judd, 1985; Chamley, 1986) result in
our framework. Again, this is in contrast to CNT.
Second, we contribute to the line of research studying the macroeconomic and policy
implications of intangible capital and technology transfer. McGrattan and Prescott (2009,
6

2010) incorporate multinational production and technology capital into a standard, neoclassical growth framework and show that this channel substantially increases the gains toopenness to FDI. Holmes, McGrattan, and Prescott (2015) use a version of this model toquantify the impact of China’s quid pro quo policy and show that it has had a significant
effect on global innovation and welfare. In Dyrda et al. (2022), we develop a framework to
study the aggregate implications of taxing multinational enterprises that shift profits to tax
havens by transferring ownership of nonrival intangible capital. Santacreu (2023) develops
an Armington trade model of innovation and international technology licensing and uses it
to rationalize the bilateral royalty payments across countries observed in the data. Other
studies in this strand of the literature are Yang and Maskus (2001), Glass and Saggi (2002),
Benhabib et al. (2017), and Mandelman and Waddle (2020). The vast majority of papers inthis line of research concentrate on a positive analysis. In contrast, in this paper, we embed the theory of intangibles and profit shifting we developed in DHS into a multi-country
neoclassical growth model and use this framework to conduct normative analysis. Our study
is the first to analyze an optimal policy in a framework where global production, nonrival
intangible capital, and profit shifting are explicitly modeled.
Perhaps the most similar paper to our is Quadrini and Ríos Rull (2023), also published inthis volume, who study international tax competition in a model with multinational production, intangible capital, and profit shifting. There are numerous differences in our approaches
to modeling and quantifying these forces, but the primary distinction between between theirpaper and ours is that we ask a normative question—what should the optimal international
corporate tax system look like if governments could agree to coordinate—whereas they ask
a positive question—what system emerges in equilibrium if such coordination is impossible.
We view their paper and ours as highly complementary.

3

Theoretical analysis

We begin with a theoretical treatment of the implications of multinational production, intangible capital, and profit shifting for optimal corporate income taxation. Our starting point isa Backus et al. (1994) multi-country growth model with distortionary taxation as studied by
CNT. To account for these key features of the modern global economy, we first incorporate avariation of MP’s theory of multinational production with nonrival intangible capital with aninternational spillover externality, add then add profit shifting as in DHS. We first describe
the environment and characterize its Pareto frontier. We then describe the market structure
and characterize a competitive equilibrium with distortionary taxes. We conclude with an7

analysis of the problem of a cooperative global Ramsey planner who has access to corporate
income taxes in addition to the standard set of distortionary taxes.

3.1 Preferences and technology
There are I countries indexed by i and j. Each country has a representative household with∞preferences over sequences of consumption, {cit }∞t=0 , and labor, {hit }t=0 , given by
Ui =

∞
X

β t ui (cit , hit ) .

(1)

t=0

The utility function, ui , satisfies the usual properties and the endowment of time is normalized
to be one. We use uic,t and uih,t to denote the marginal utility of country i’s household withrespect to consumption and labor.
Each country produces a distinct intermediate good. Different from Backus et al. (1994),
intermediate goods can be produced both at home as well as in each foreign country. The
technology according to which country i’s good is produced in country j is given by
yijt = F ij (zit , kijt , lijt ) ,(2)

where kijt and ℓijt are tangible capital and labor, which are rival and specific to the country
in which production takes place, and zit is country i’s intangible capital, which is nonrivaland can be used in all production locations simultaneously. We assume that domesticallyproduced intermediates (i.e., country i’s intermediate produced in i, yiit ) can be exported,
while intermediates produced abroad (i.e., yijt for j ̸= i) must be used domestically.1 We
assume F ij has constant return to scale, following Chari et al. (2023). The marginal products
of intangible capital, tangible capital, and labor in producing country i’s good in country j
ijijij
are denoted by Fz,t, Fk,t, and Fl,t
, respectively.
Intermediate goods are used to produce nontradable final goods according to a constantreturns-to-scale technology
qit = Gi (q1it , ..., qIit , q̂1it , ..., q̂Iit ) .

(3)

The first I elements are domestically-produced intermediates (which are imported whenj ̸= i), while the last I − 1 elements, accented with hats, are foreign intermediates produced
1

We do not impose any assumption about the substitutability between yiit and yijt at this stage, but
we assume they are imperfect substitutes in our quantiative analysis. Our theoretical results are robust toallowing for export platforms.

8

locally in country i. The marginal product of a domestically-produced intermediate fromcountry j in producing country i’s final good is denoted by Gij,t := ∂Gi /∂qjit . Similarly,
the marginal product of a locally-produced foreign intermediate from country j is Giĵ,t :=∂Gi /∂ q̂jit . Tangible capital follows the standard law of motion
xit = kit+1 − (1 − δ) kit ,(4)

where δ is the depreciation rate.
We assume that intangible capital, zit , is produced according to the technology
zit = H i (ℓz1t , . . . , ℓzIt ) .

(5)This technology depends on the vector of labor inputs used to produce intangible capital by
all countries, not just the home country. We refer to litz as research labor. This technology
captures, in a parsimonious way, the technology spillover effects. As the intangible capitalcan be used by all foreign subsidiaries to product the intermediate goods locally, this can beviewed as a vertical spillover from intermediate good production by FDI to intangible capital
production by the host country, which is in line with the empirical literature on productivity
spillovers of FDI (Javorcik, 2004). We denote the marginal product of an additional unit ofz.
research labor in country j in producing intangible capital in country i as Hji := ∂H i /∂ljti
2
If Hj > 0 for j ̸= i, then the spillover effect is positive.
For the sake of notation, define an allocation for country i aso∞n
Ai ≡ cit , hit , litz , zit , {ljit , kjit , qjit }∀j , {q̂jit }∀j̸=i

t=0

and let the vector of global allocations be denoted A = {Ai }Ii=1 . A feasible allocation vector
satisfies several resource constraints. The resource constraints for intermediate goods are
yiit = qiit +

X

qijt

(6)

j̸=i

yijt = q̂ijt∀j̸=i .

(7)

Final goods are used for private consumption, cit , public consumption, git , and tangible
2
In a quantitative model, we relax the assumption that every country has subsidiaries in all other locations. There, the location choice is endogenous, and the spillover effect in country i depends only on theintangible investment of countries with subsidiaries in i.

9investment, xit . Thus, the resource constraint for final goods is(8)

cit + gi + xit = qit
and the market clearing condition for tangible capital is
kit =

I
X

(9)

kjit ,

j=1

which says that the stock of tangible capital in country i must be split between the productionof domestic intermediates and locally-produced foreign intermediates. The market clearing
condition for labor is
I
X
hit =
ljit + litz .
(10)
j=1

3.2 Pareto frontier
Let ω = (ω1 , . . . , ωI ) denote a vector of Pareto weights and define AP (ω) as the allocation
vector that maximizes global welfare weighted by ω:
A (ω) = arg max
P

A

( I
X
i=1

ωi

∞
X

)
t i

β u (cit , hit )subject to (2)–(10)

(11)

t=0


The Pareto frontier is then defined as AP = AP (ω) : ω ∈ RI+ . Allocations on the Pareto
frontier satisfy the following five eﬀiciency conditions.3
First is the no intratemporal wedges condition:
uic,t
1
1
− i = i ii = i ji
uh,tGi,t Fl,t
G Fl,t∀i , ∀j̸=i .

(12)

ĵ,t

This condition states that the marginal rate of substitution between consumption and leisureshould equal the marginal rate of transformation of labor into final goods. Note that thiscondition holds for all goods produced in location i, both domestic and foreign.
Second is the no intertemporal wedges condition that links the marginal rate of substitution between consumption across periods with the marginal product of capital:
uic,t
ji
ii
= (1 − δ) + Giĵ,t+1 Fk,t+1= (1 − δ) + Gii,t+1 Fk,t+1
i
βuc,t+1
3

We relegate all the detailed derivations to the Appendix A.

10∀i , ∀j̸=i .

(13)

Note that this condition, too, holds for all goods produced in location i, both domestic andforeign.
Third is static production eﬀiciency, which states that the marginal rate of technicalsubstitution between any pair of imported goods (m, n) should be equated across countries:
Gin,t
Gnn,t unc,t
= m m
i
Gm,t
Gm,t uc,t∀i , ∀m,n̸=i .

(14)

Fourth, we have dynamic production eﬀiciency, which states that the social return totangible capital should be equated across countries:

Gij,t
ii
=
(1 − δ) + Gii,t+1 Fk,t+1
i
Gj,t+1

Gjj,t ujc,tj
Gj,t+1βujc,t+1!
∀i , ∀j̸=i .

(15)

If an allocation satisfies both static and dynamic production eﬀiciency we say it is simply
production eﬀicient.
Before moving on to the fifth and most important eﬀiciency condition, it is useful tocompare the first four to the ones in CNT. Their Pareto frontier is characterized by similar
conditions, but the addition of multinational production and nonrival intangible capital alters
several of them. Specifically, the intra- and intertemporal wedge conditions (12) and (13)hold across for all goods produced in each country i, both foreign and domestic (hence thesecond equalities in both conditions). This implies that the marginal rates of technologicaltransformation between labor and tangible capital are equated across all goods produced in
ni
F mi
Fk,t
country i, both foreign and domestic: Fk,tmi = F ni for all m, n.
l,tl,tThe last Pareto-eﬀiciency condition, eﬀiciency of intangible investment, describes how
intangible investment should be allocated across countries:
Nonrivalry effect

z
}|
{
j
j
ij
X uc,t Gî,t Fz,t

ii
Fl,t
=1+
ii
i
Fz,t
Hi,t
ui Gi F ii
j̸=i c,t i,t z,t
!
#
" j
ji
jj
k uj Gj F kj
X Hi,tX Hi,tGiĵ,t Fz,tujc,t Gjj,t Fz,tc,t k̂,t z,t++ i i ii +
.
i
ii
i
c
ii
Hi,t
Gii,t Fz,tuc,t Gi,t Fz,t
Hi,t
ui,t
Gii,t Fz,tj̸=ik̸=i,j
{z
}
|

(16)

Spillover effect: ΣitThis condition states that allocating labor to producing the domestic intermediate good, qiit ,should yield the same worldwide utility gains as allocating it to producing intangible capital,
zit . The left-hand side is the marginal rate of technical substitution between research labor11and productive labor in producing the domestic good. It consists of two parts: the marginalii
ii
rate of technical substitution between productive labor and intangible capital, Fl,t
/Fz,t
, and
i
the marginal rate of transformation between research labor and intangible capital, Hi,t
. Holding fixed the latter, the greater the marginal rate of technical substitution between productive
labor and intangible capital, the lower the ratio of labor to intangible capital. Thus, loosely,
speaking, the greater the right-hand side of (16), the more intangible-intensive country ought
to be.
To unpack what the right-hand side of this condition represents, it’s helpful to first shutij
ujc,t Gj Fz,tjdown the spillover effect, i.e. set Hi,t
= 0 ∀j ̸= i, so that Σit = 0. The term ui Gî,ti
ii
c,t i,t Fz,trepresents the increase in country j’s utility that comes from the additional output of country
i’s good in country j generated by an increase zit , measured relative to the increase incountry i’s utility that comes from increased domestic production. This effect is due solely
to nonrivalry, and as we will see, it will be internalized by the market in a competitive
equilibrium.
Now add the spillover effect, which we denote by Σit . Each part of the spillover effectrepresents a different channel through which increased intangible investment in country iaffects welfare around the world by increasing other countries’ intangible capital. Consider
Hj
first the part in parentheses multiplied by Hi,ti , which itself consists of two terms. The termi,tji
Gi Fz,tĵ,tii
Gii,t Fz,treflects the increase in the home country’s utility caused by the increase in locallyuj Gj F jj

j,t z,tproduced foreign goods. The term uc,ti Gi F ii reflects the increase in foreign countries’ utilityc,t i,t z,tcaused by the increase in their own domestic production. The second part multiplied bykHi,trepresents the marginal impact of additional intangible capital on utility in each foreigniHi,tcountry j due to increased output of locally-produced foreign goods from each third countryk≠ i, j.
Having laid out and described all of the conditions that characterize the Pareto frontier,
we are now in a position to formally define an eﬀicient allocation:

Definition 1 (Eﬀicient allocation) An allocation A is eﬀicient, i.e., A ∈ AP , if it satisfies the following conditions:
1. feasibility constraints for intangible capital, tangible capital and labor: (5), (9), (10).
2. resource constraints for intermediate and final goods, (6),(7), (8);3. no intratemporal and intertemporal wedges, (12), (13);
4. static and dynamic production eﬀiciency, (14), (15); and5. eﬀicient allocation of intangible investment, (16).

12

3.3 Market arrangements and competitive equilibrium
We now specify the market arrangements in our economy and define a competitive equilibrium. As in a standard Backus et al. (1994) environment, households in each country
choose consumption, labor supply, tangible investment, and holdings of international bonds
to maximize lifetime utility subject to budget constraints, and each country has a competetive
final-good producer. As in McGrattan and Waddle (2020), each country also has a representative multinational intermediate-good producer (MNE) that chooses intangible investmentin the home country and rival factor inputs in each production location around the world tomaximize global profits. As in DHS, multinationals use transfer pricing to account for theincome generated by intangible capital, and can shift the rights to this income to a tax havenby paying a cost. Finally, as in CNT, each country has a government that finances public consumption using proportional taxes on consumption, labor income, and international
trade, as well as a tax on corporate income, which is the focus of our analysis.
3.3.1

Fiscal policy

Public consumption, git , is exogenous. The taxes on consumption and labor income arem, on goods
denoted by τitc and τith , respectively. Trade taxes consist of a bilateral import tax, τijtx
shipped from country j to country i, and an analogous export tax, τijt . The corporate income
tax is denoted by τitp . It applies to all taxable income earned by MNEs operating in country i
regardless of their home country. We denote the sequence of government spending for a given
all countries as G = {Gi }oIi=1 .
country i as Gi = {git }∞t=0 and a vector of such sequences for n∞
 x m, τjit ∀j̸=i
We denote a collection of taxes for a given country i by Ti = τitp , τith , τitc , τijt
t=0

and the vector of worldwide tax policies by T = {Ti }Ii=1 .

3.3.2

Final goods producers

Final goods producers choose inputs of intermediates to maximize profits period by period:
max{qjit }∀j ,{q̂jit }∀j̸=i

pit qit −XXmpjit qjit −1 + τjitp̂jit q̂jit

j

(17)

j̸=i

subject to (3), where pit denotes the price of the final good in country i, pjit denotes the priceof domestically-produced intermediate goods imported from country j, and p̂jit denotes theprice of locally-produced foreign intermediates.

13

3.3.3

Intermediate goods producers

Each country has a representative MNE with a subsidiary in each of its production locations
around the world. The MNE’s objective is to maximize the total profits generated by all of
its subsidiaries. We describe this problem in three steps: first, in a standard environment
without transfer pricing or profit shifting; second, with transfer pricing but without profit
shifting; and third, with profit shifting as modeled in DHS.
Before describing these scenarios and the MNE’s problem in full, it is helpful to first
define the simpler problem of maximizing the profits in a single production location taking
as given the MNE’s level of intangible capital, zit , as this problem and the associated equilibrium objects are the same in all three scenarios. For the domestic parent division (i.e., thesubsidiary that produces yiit ), this problem is given by"
πiit (zit ) =max{liit ,kiit ,qijt }Ij=1(1 − τitp ) piit qiit +X#x)pijt qijt − wit liit − δpit kiit − rit kiit
(1 − τijt

j̸=i

(18)
subject to (2) and (6). Note that depreciation of tangible capital, δpit kit , is tax-deductable,
while the remainder of the cost of renting tangible capital, rit kit , is not. This is consistent
with standard accounting practices and implies that demand for tangible capital is decreasingin the corporate tax rate. The analogous problem for foreign divisions (i.e., that produce
yijt , j ̸= i) is
πijt (zit ) =maxq̂ijt ,lijt ,kijt


1 − τjtp [p̂ijt q̂ijt − wjt lijt − δpjt kijt ] − rjt kijt

(19)

subject to (2) and (7). Having defined these objects, we can now describe the MNE’s problem
in full.
Free transfer scenario (FT). In this scenario, all of an MNE’s foreign subsidiaries use
intangible capital at no cost, just as in McGrattan and Waddle (2020). The MNE’s problem
is to choose intangible capital to maximize its worldwide dividends after taxes:(
dit = maxzzit ,lit

πiit (zit ) − (1 − τitp ) wit litz +X)
πijt (zit )

(20)

j̸=i

subject to (5). Note that the cost of intangible capital, wit ℓzit , is tax-deductable.
Transfer pricing scenario (TP). In this scenario, a foreign subsidiary in country j of an
14

MNE from country i pays a licensing fee of ϑijt to the parent division in the MNE’s homecountry for the rights to use each unit of the MNE’s intangible capital. We assume thatlicensing fee income and costs are tax deductable. The MNE’s worldwide profits in this
scenario can be written as
)
(
X


πijt (zit ) + (τjtp − τitp )ϑijt zit
dit = max
πiit (zit ) − (1 − τitp )wit litz +(21)zzit ,litj̸=isubject to (5). Note that the terms ϑijt enter both as income for the parent division, which
is taxed at a rate of τitp , and as costs for foreign divisions, which are deducted from thesepdivisions at rates of τijt
. For now, we leave the parameterization of these licensing fees
unspecified, but it is worth stating that the appropriate arms-length licensing fee that would
prevail if an MNE’s foreign aﬀiliates rented intangible capital in a competitive market wouldij
be equal to the aﬀiliates’ marginal revenue products of intangible capital, i.e., ϑijt = p̂ijt Fz,t.
We will adopt this specification later on.
Profit Shifting scenario (PS). In this scenario, firms can choose to sell a fraction λ oftheir intangible capital to a tax haven that taxes corporate income at a rate of τTp H . This
sale occurs at a markdown φ below the intangible capital’s market price, which is given byPνit := Ij=1 ϑijt . Note that this object includes licensing fees that the parent “pays” to itself.
The sale also incurs a cost C(λ) in proportion to νit . We follow DHS and assume that thiscost function is given by
C (λ) ≡ λ − (1 − λ) log(1 − λ),(22)which implies that C(0) = 0, limλ→1 C(λ) = 1, and that C is increasing and convex over[0, 1). The tax haven then collects a fraction λ of the worldwide licensing fees generated by
the MNE’s intangible capital, while the parent division collects the remainder. The MNE’s
profits in this scenario are
"


#
X
dit = max
πiit (zit ) + (1 − τitp ) −wit litz + (1 − λit )ϑijt zit + λit (φ − C(λit ))νit zit − ϑiit zitzzit ,lit ,λit+Xπijt (zit ) − (1 − τjtp )ϑijt zit



j̸=i

+ (1 − τTp Ht )λit (1 − φ)νit zit

(23)

j̸=i

subject to (5). The first line contains the parent division’s after-tax profits, which now include: licensing fee income from the portion of intangible capital that is retained,P(1 − λit ) j̸=i ϑijt (zit ); a licensing fee payment to the tax haven on the portion of the intangible capital that is sold, λit ϑiit (zit ); income from selling intangible capital to the tax
15

haven, λit φνit (zit ); and the cost of profit shifting, λit C(λit )νit (zit ). The second line contains
productive foreign aﬀiliates’ after-tax profits, which are the same as in the transfer pricing
scenario; and the after-tax income of the aﬀiliate in the tax haven, which consists solely of
the licensing fees this aﬀiliate collects minus the cost of purchasing the intangible capitalfrom the parent division. The dividends of the the MNE’s aﬀiliates in productive countries
are the same as in the transfer pricing scenario.
3.3.4

Households

The household’s problem is formulated in the same way as in CNT. The household in country
i chooses sequences of consumption, labor, and tangible capital, {cit , hit , kit }∞t=0 , to maximize
lifetime utility (1) subject to the law of motion for capital (4) and a lifetime budget constraint,
∞
X




Qt (1 + τitc ) pit cit + pit (kit+1 − (1 − δ + rit )kit ) − 1 − τith wit hit = ai0 ,

(24)

t=0

where Qt is the intertemporal price in units of the numeraire in period 0. The right-handside represents the present value of the household’s lifetime wealth in period 0,
ai0 = Ri0 + Q−1 bi0 + 1 + rf fi0 + Vi0 ,(25)

where Ri0 = (1 − δ + ri0 )ki0 , initial holdings of domestic public debt are denoted by Q−1 bi0 ,and 1 + rf fi0 denotes initial net holdings of international bonds. Vi0 denotes the initialstock value of the domestic multinational, which is given by
Vi0 =

∞
X

Qt dit .

(26)

t=0

We assume that interest parity holds and world capital markets are fully integrated, so thatfthe return on one-period bonds, rt+1:= Qt /Qt+1 , is the same in all countries.
3.3.5

Equilibrium conditions

The market-clearing conditions for labor, tangible capital, and final goods have already been
defined in (10), (9), and (8). In addition to these conditions, the government’s budget
constraint must hold. The government in each country i collects the following stream of16

revenues:
REVi =∞
X"
Qt τitc pit cit + τith wit hit + τitpπ̂iit +

t=0

X

!
π̂jit

m
+ τjitj̸=iXxpjit qjit + τijt

X

j̸=i

#
pijt qijt

j̸=iwhere π̂jit denotes the corporate tax base of the MNE from country i’s aﬀiliate in country j.
Then the government budget constraint can be written as
REVi =∞
X

Qt pit git + Q−1 bi0

(27)

t=0

The last condition we need to complete a description of a competitive equilibrium is thebalance of payments. This condition is different in each of the three scenarios defined above.
Free transfer scenario (FT). In the absence of transfer pricing and profit shifting, thebalance of payments for country i is
∞
X

Qt

t=0

X
j̸=i

(pijt qijt − pjit qjit ) +

∞
X

Qt

t=0

X


(dijt − djit ) = − 1 + rf fi0

(28)

j̸=i

where dijt are dividends paid to country i by its MNE’s aﬀiliate in country j, which in this
scenario are simply dijt = πijt . The first term on the left is net exports of intermediate goods
and the second is net factor payments. The right is the (negative of the) financial account.
Transfer pricing scenario (TP). Licensing fees enter the balance of payments in two
ways: indirectly, through the dividend payments of foreign MNEs, and directly, as net exports of intangible capital services. In the transfer pricing scenario the balance of payments
is
∞
X
t=0

Qt

X

[pijt qijt − pjit qjit + ϑijt zit − ϑjit zjt + dijt − djit ] = −(1 + rf )fi0

(29)

j̸=i

Note that in this case, foreign aﬀiliate dividends are dijt = πijt − (1 − τjtp )ϑijt (zit ). Notealso that licensing fees affect net exports and net foreign payments differently. The fullvalue of the licensing fees shows up on the left, while the only the after-tax portion shows
up on the right. Thus, transfer pricing affects net exports of intermediate goods, either
through quantities or prices, in equilibrium. Specifically, countries that export relatively
large amounts of intangible capital services see their net exports of intermediate goods fall.4
4

The United States, which consistently runs a trade surplus in services and a deficit in goods, exemplifies

17

Profit shifting scenario (PS). In the profit shifting scenario we need to adjust net exports
of intangible capital services to account for the fact that the parent division takes in fewer
licensing fees and now pays some out. We also need to include the sale of intangible capitalto the tax haven in the financial account. The balance of payments now becomes
∞
X

"
Qt

X

t=0

=−

#
(pijt qijt − pjit qjit + dijt − djit + (1 − λit ) ϑijt zit − ϑjit zjt ) − λit ϑiit zit

j̸=i
∞
X


Qt φλit νit zit − 1 + rf fi0

(30)

t=0

We also need to include the tax haven’s balance of payments. The tax haven’s exports ofPintangible capital services are Ii=1 λit νit zit . It makes net foreign payments in the form ofP
dividends repatriated to MNEs’ home countries, which are equal to Ii=1 λit νit zit (1 − τTp H ).
P
Its financial account includes purchases of intangible capital in the amount of Ii=1 λit φνit zit .
And it imports intermediate goods from other countries for consumption in the amount ofPIj̸=T H pjT Ht qjT Ht . Hence its balance of payments can be written as
∞
X
t=0

Qt

I
X
i=1

λit τTp H νit zit −I
X!
pjT Ht qjT Ht

=

∞
X

Qt

t=0

j̸=T HI
X

λit φνit zit

(31)

i=1

We are now ready to define a competitive equilibrium.
First, for the sakeno∞ of notation
denote a collection of prices for country i as Pi := pit , wit , {pijt }∀j , {p̂ijt }∀j̸=i
and denote
t=0
a vector of worldwide prices as P. Then we can define a competitive equilibrium as follows.
Definition 2 Given government expenditures and tax policies, (G, T ), and initial conditions,{fi0 , bi0 , ki0 , Q−1 }Ii=1 , a competitive equilibrium is an allocation and prices, (A, P), such thatin each country, households maximize utility subject to their budget constraints; final goods
producers and MNEs maximize profits; goods and factor market clearing conditions hold;government budget constraints hold; and the balance of payments holds.

3.4 Competitive equilibrium characterization
Now that we have defined a competitive equilibrium in our environment, we turn now to
its characterization and a comparison to the Pareto frontier. All derivations and proofs arerelegated to Appendix B.2 for brevity.
this effect.

18

Manipulating the first-order conditions of the household’s problem and the firms’ problems, we obtain counterparts to the no intratemporal wedges condition (12) and no intertemportal wedges condition (13):uic,t(1 + τitc ) 1(1 + τitc )
1


− i =
=
ji
i
ii
h
h
i
uh,t
1 − τit Gi,t Fl,t
1 − τit G Fl,t∀i , ∀j̸=i ,

(32)

ĵ,t

and
 i ii
uic,t(1 + τitc ) 
p

1
+
1
−
τ
G
F
−
δ
=
i,t k,t+1
it+1c
βuic,t+1
1 + τit+1
i
  i ji
(1 + τitc ) h
p

=
F
−
δ
1
+
1
−
τ
G
it+1
ĵ,t k,t+1c
1 + τit+1∀i , ∀j̸=i .

(33)

The first-order conditions of the final good producers can be combined to yield counterparts
to the static eﬀiciency condition (14) and dynamic eﬀiciency condition (15):xm(1 − τnit) (1 + τmit
) Gin,tpnt Gnn,t=mx(1 + τnit) (1 − τmit
) Gim,tpmt Gmm,t∀i , ∀m,n̸=i ,

(34)

and
m
1 + τjit+1x
1 − τjit+1x
1 − τjitm

1 + τjit

 i
Gij,t 
p
ii
1
+
1
−
τ
G
F
−
δ
=
i,t+1
k,t+1
it+1
Gij,t+1
Gjj,t Gjj,t+1p
1 + 1 − τjt+1

jj
Gjj,t+1 Fk,t+1−δ

∀i , ∀j̸=i .

(35)

As in our characterization of the Pareto frontier in section 3.2, these conditions are similar to
those found in CNT. Most notably, perhaps, (34) says that, just as in CNT, static eﬀiciencyxm
can be achieved if export subsidies are used to offset import tariffs, i.e., τijt
= −τijtforall i and j. There are two primary differences, however. First, as in the characterizationof the Pareto frontier, (32) and (33) hold for locally-produced foreign intermediate goodsp
as well as domestic intermediates. Second, and more important, corporate tax rates τit+1
enter the dynamic equilibrium conditions (33) and (35). In CNT, the planner can achieve
dynamic eﬀiciency and no intertemporal wedges without corporate income taxes by setting
consumption taxes to be constant over time. Here, if corporate income taxes are non-zero, thisis not the case even if consumption taxes are constant. As we will see, this creates a tension
between achieving dynamic eﬀiciency and an eﬀicient allocation of intangible investment. It
is useful to establish the following lemma before moving on to the last equilibrium condition.
19

Lemma 1 In any competitive equilibrium in which static eﬀiciency (14) holds, we have
i Gi
uct
j,t
= 1, ∀i,j .
j
uct Gjj,t

The last condition we need to characterize the equilibrium governs the allocation of labor
to production of domestic intermediates, liit , and intangible capital, litz . This condition, which
we obtain by manipulating the MNE’s first-order conditions, is different in each of the three
scenarios described above.
Free transfer scenario (FT). In the absence of transfer pricing and profit shifting, theallocation of labor to these two activities is characterized byijii
X (1 − τjtp )p̂ijt Fz,t
Fl,t
=1+
i
ii
ii
Hi,t
Fz,t
(1 − τitp )piit Fz,t
j̸=i

(36)

The right-hand side represents the additional after-tax profits that MNEs earn in foreign affiliates relative to the domestic parent from additional intangible capital. The higher (lower)
the domestic (foreign) corporate tax rate, the higher the ratio of intangible capital to productive labor. This is because foreign aﬀiliates’ returns to intangible investment are taxed at
different rates than rate at which the cost of intangible investment is deducted in the homecountry. It is important to note that this expression differs from (16), which describes an
eﬀicient allocation of intangible investment across countries, due to the externality created by
the nonrivalry of intangible capital in an intuitive way. The social planner cares about how
much additional utility each country gains from additional intangible investment in countryi, whereas MNEs care about the additional profits they earn.
In order to shed further light on how an equilibrium allocation of intangible investment differsfrom an eﬀicient one (and to define the Ramsey planner’s problem later on), it is helpful towrite (36) solely in terms of policies and allocations:
ii
X
Fl,t
=1+
i
ii
Hi,t Fz,t
j̸=iij
ujc,t Gjî,t Fz,t!

ii
uic,t Gii,t Fz,t
 i i !
x1 − τjtp 1 − τjit
Gjt uc,tpm(1 − τit ) 1 + τjit
Gjjt ujc,t(37)

Comparing this with (16), we can now see clearly that firms in the free transfer scenario do
not internalize the spillover externality. While the first term in (16) that captures the utility
gains from increased global production of country i’s good are present, all of the terms thatinvolve Hji for j ̸= i that capture the gains from greater production of foreign goods are
missing.
20

To more sharply characterize the role of corporate income taxes in the equilibrium allocationxof intangible investment, suppose that static eﬀiciency holds, so that (1 − τjit
)Gijt uic,t is equalmto (1 + τjit
)Gjjt ujc,t , and that taxes on corporate income are zero in all countries. Thenthe right-hand side of (37) is lower than the right-hand side of (16) in all countries. Thisimplies that the marginal rate of transformation between productive labor and intangible
capital is lower than in a Pareto-optimal allocation for all countries; all countries do too
little intangible investment. More generally, for any vector of corporate tax rates {τitp }Ii=1 ,it must be that for at least one country, the second term in parentheses in (37) is less thanone. For this country, the marginal rate of transformation between productive labor andintangible is unambiguously lower than in a Pareto-optimal allocation; this country does too
little intangible investment. This implies that it is impossible to achieve an eﬀicient allocationof intangible investment in the free transfer scenario using corporate income taxes as long as
static eﬀiciency holds.
Transfer pricing scenario (TP). In the transfer pricing scenario, without putting further
structure on the licensing fees, ϑijt (zit ), this condition becomes
"
#

ij
ii
X (1 − τjtp )p̂ijt Fz,tFl,tτjtp − τitp ϑijt
=1+
+
.
i
ii
ii
ii
Hi,t
Fz,t
(1 − τitp )piit Fz,t(1 − τitp )piit Fz,t
j̸=i

(38)

The second term in parentheses is decreasing in the domestic corporate income tax rateand increasing in the foreign tax rate. Thus, the effects of corporate income taxes on themarginal rate of transformation between productive labor and intangible investment are now
ambiguous. This is because some of the intangible income generated by an MNE’s foreign
aﬀiliates are now taxed at the home instead of abroad.
To provide a sharper characterization, from this point forward we assume that licensing fees
are set according to the arms-length principle, which states that transfer prices should be
set to the prices that would prevail in a competitive secondary market. In such a market,
foreign aﬀiliates would choose to rent intangible capital so that the cost equalled the marginal
ij
benefit, i.e., the marginal revenue product of intangible capital. Thus, we set ϑijt = p̂ijt Fz,t.
Under this assumption, the equilibrium condition (38) above simplifies toijii
X p̂ijt Fz,t
X
Fl,t
=
1
+
=
1
+
i
ii
ii
Hi,t
Fz,t
piit Fz,t
j̸=i
j̸=i

ij
ujc,t Gjî,t Fz,tiii
uic,t Gi,t
Fz,t

!

x
) Gijt uic,t(1 − τjitm(1 + τjit) Gjjt ujc,t!
(39)

Now, none of the corporate income taxes enters at all. This is because all of the worldwide
income generated by an MNE’s intangible capital is ultimately taxed in the MNE’s home21

country at the same rate at which the cost of this capital is deducted. Here, if static eﬀiciency
holds, all countries do too little intangible investment regardless of how corporate income
taxes are set across countries, and the planner has no ability at all to improve the allocationof intangible investment using these taxes.
Profit shifting scenario (PS). Maintaining our assumption that lincensing fees are set
according to the arms-length principle, in the scenario with both transfer pricing and profit
shifting, this condition changes to!
λit (1 − φ) (τitp − τTp Ht )1 − C (λit ) +
(40)
1+
ii
piit Fz,t(1 − τitp )
j̸=i
"
!# 
ij !

x
X ujc,t Gjî,t Fz,t(1 − τjit) Gijt uic,tλit (1 − φ) (τitp − τTp Ht )= 1+
1 − C (λit ) +.
iimuic,t Gii,t Fz,t
(1 + τjit) Gjjt ujc,t(1 − τitp )
j̸=i
|
{z
}

ii
Fl,t
=
i
ii
Hi,t
Fz,t

ij
X p̂ijt Fz,t

p
Ω(τit
)

The first term in square brackets is the same as (39) for the transfer pricing scenario. The
second term, Ω(τitp ), is the gain from profit shifting per unit of intangible capital, where λit
endogenous and given by the formula (B.29) derived in Appendix B.1. Note that only the
corporate tax rate of an MNE’s home country and the tax haven’s tax rate matter; the rates
at which foreign aﬀiliates profits are taxed still do not show up. The marginal rate of technicalsubstitution between productive labor and intangible capital now unambiguously increases
with the home country’s tax rate as in the free transfer scenario, but it also decreases with thetax haven’s tax rate. Additionally, Ω(τitp ), which represents the per-unit net gain from profit
shifting, is weakly greater than one, with strict inequality when τitp > τTp H . This indicatesthat MNEs do more intangible investment in the profit shifting scenario as we showed in
DHS.
Since the spillover term Σit in (16) is strictly positive, this result implies that each country’s
intangible investment level can be improved to the Pareto-optimal level by increasing its
corporate income tax rate, which increases the returns to profit shifting and thereby increases
the incentive to invest in intangible capital. In other words, profit shifting helps competitive
firms internalize the effects of the spillover externality. If it were possible, lowering the tax
haven’s tax rate to further increase profit shifting would accomplish the same thing, and we
consider this possibility in our quantitative analysis. More importantly, Ω(τitp ) is increasing
and approaches infinity as τitp approaches one. Intuitively, this increases the gain from profit
shifting relative to the cost, and the latter goes to zero as τitp approaches one due to taxdeductability. We formalize this result in the following lemma.
22Lemma 2 We can establish the following properties of the return on profit shifting, Ω(τitp ):1. Ω(τitp ) = 1 when τitp = τTp Ht ;2. Ω(τitp ) > 1 when τitp > τTp Ht ; and3. Ω(τitp ) is increasing in τitp , with Ω(τitp ) → ∞ as τitp → 1.
This lemma implies that in the profit shifting scenario, corporate taxes can always be set
high enough to achieve an eﬀicient level of intangible investment in each country. This is instark contrast to the free transfer and transfer pricing scenarios, where it is impossible toachieve an eﬀicient allocation of intangible investment using corporate income taxes. This is
our first main result, which we formalize in the proposition below.
Proposition 1 (Intangible investment eﬀiciency in competitive equilibrium) Consider
competitive equilibria that satisfy the static eﬀiciency condition (14). In both the free transfer and the transfer pricing scenarios, the competitive equilibrium allocation associated withany tax vector T always has an ineﬀicient allocation of intangible investment. In the profit
shifting scenario, the competitive equilibrium allocation features an eﬀicient level of intangible
investment in each country if Ω(τitp ) is such that (40) is equivalent to (16); otherwise it is
ineﬀicient.
Note that it is not necessarily the case that there exists a Pareto eﬀicient competitive
equilibrium in the profit shifting scenario, i.e., a competitive equilibrium that satisfies all of
the other eﬀiciency conditions (12)–(15) as well as (16). The purpose of this proposition is tohighlight the fact that the only scenario in which it is even possible to attain Pareto eﬀiciency
is the profit shifting scenario; in the free-transfer and transfer-pricing scenarios, where (16)cannot be satisfied, attaining Pareto eﬀiciency in a competitive equilibrium is impossible.
It is useful to contrast these results with the case in which there are no spillovers iniintangible capital production, i.e., zit = H i (ℓitz ) so that Hj,t
= 0 for all j ̸= i. In this case, theintangible allocation condition in the free transfer scenario (37) is the same as the Paretooptimal condition (16) with zero corporate income taxes if trade taxes are set to ensurestatic eﬀiciency. In the free transfer scenario, where corporate income taxes do not enter(39) at all, intangible investment is eﬀicient under static eﬀiciency for any corporate income
taxes at all. The difference is perhaps the most striking in the profit shifting scenario. In
the presence of spillovers, improving the eﬀiciency of intangible investment dictates raising
corporate income taxes to increase the gains from profit shifting. In the absence of spillovers,on the other hand, eﬀiciency can be achieved under static eﬀiciency by eliminating profit
23

shifting by setting corporate income taxes zero (or, alternatively, to any value below the tax
haven’s tax rate), in which case (40) collapses to the transfer pricing scenario’s version (39).
The following remark formalizes this point.
Remark 1 If there are no spillovers in intangible capital production, then a competitive
equilibrium has an eﬀicient allocation of intangible investment in the following circumstances:• In the free transfer scenario: under static eﬀiciency and zero corporate income taxes,xor more generally when corporate income taxes are set so that (1 − τjtp )(1 − τjit
)Gijt uic,tmis equal to (1 − τitp )(1 + τjit
)Gjjt ujc,t .
• In the transfer pricing scenario: under static eﬀiciency for any corporate income taxes.
• In the profit shifting scenario: under static eﬀiciency with either zero corporate income
taxes or when each country’s corporate income tax is equal to the tax haven’s tax rate.
The final piece of our characterization is a description of the conditions required for an
allocation to be implementable as a competitive equilibrium with an appropriately-chosen
tax vector. The following proposition summarizes these conditions. They are essentially thesame as in CNT and thus merit no further discussion.
Proposition 2 (Necessary conditions for implementation) Given a government spending vector G and initial conditions {ki0 , bi0 , fi0 }Ii=1 , an allocation A and a collection of period-0
prices {pi0 }Ii=1 the resource constraints (2)–(10) and the following implementability condition:∞Xβ uic,t cit − uih,t hit =
t

t=0

uic0
(1 + τi0c )



 fi0
bi0
Ri0 + Q−1
+ 1 + rfpi0
pi0

, ∀i = 1, . . . , I.

(41)

where Ri0 = (1 − δ + ri0 ) ki0 .

3.5 Corporate taxes and intangible investment in an example
Before moving on to the Ramsey planner’s problem, we use an example economy with parameterized production technologies to provide a sharper characterization of how corporate
income taxes affect a given country’s intangible capital investment in partial equilibrium,taking all other prices and quantities as given. In order to do so we depart from the constantreturns-to-scale assumption that we imposed in Section 3.1. Suppose now that the production
technology for intermediate goods is
α γlijt ,
F ij (zit , kijt , lijt ) = Aj (Nj zit )ϕ kijt

24

ϕ+α+γ <1(42)and that the production technology for intangible capital iszit = H i (litz ) = Bi litz

(43)

as in DHS. Note here that since we are taking other countries’ intangible investments asgiven, we can ignore spillovers for the moment; implicitly, we are summarizing the effects ofall of these spillovers by the constant Bi . Under these assumptions, we can solve for eachcountry’s equilibrium level of intangible capital in closed form in each of the three scenarios
described above.
Free transfer scenario (FT). The equilibrium intangible investment in this scenario is

1−γ−α
 1−γ−α−ϕX (1 − τjtp )(τit ) Λiit +zit = p r̂jt (τjt ) Λijt  r̂|it{z
}
(1
−
τ
)
it


j̸=i |p{z}↘ in τitwhere Λijt := ϕ wBitihp̂ijt Aj Njϕ



γ
wjt

(44)

p
p
↗ in τit, ↘ in τjtγ αrjt +pjt δ1
α i 1−γ−ααh (1−τ p )(r +p δ) i 1−γ−α
jt

jt

jt
and r̂jt (τjt ) := rjt +(1−τ.
pjt )pjt δ
The impact of domestic corporate income taxes is ambiguous in this scenario because there
are two opposing effects. The first effect, which is negative, operates through domestic tangible capital. Since tangible capital costs are not fully tax-deductable, an increase in τitp reduces
domestic tangible investment, which in turn lowers the return to intangible capital deployed
domestically. The second effect, which is positive, operates through multinational production. The costs of intangible investment are deducted from the taxes of an MNE’s domestic
parent division, so when τitp rises the after-tax return to intangible capital deployed abroadincreases relative to the after-tax costs. Note that if there were no tangible capital—or,
alternatively, tangible capital costs were fully tax deductable—the first effect would vanish
because r̂jt (τjtp ) = 1, and intangible investment would unambiguously increase with the domestic tax rate. On the other hand, foreign corporate income taxes unambiguously reduce
domestic intangible investment, as both effects above operate in the same direction. Higher
foreign taxes reduce foreign subsidiaries’ tangible investment, and also reduce the after-tax
return to intangible capital deployed abroad relative to the after-tax cost.

25

Transfer pricing scenario (TP). In this scenario, the solution for zi is

1−γ−α
 1−γ−α−ϕXzit = r̂it (τit ) Λiit +r̂jt (τjt ) Λijt | {z }| {z }
j̸=ip↘ in τit(45)p↘ in τjt

Here, where all of an MNE’s worldwide intangible income is booked in the home country
and taxed at the home country’s tax rate, the only effect of corporate taxes on intangible
investment operates through the tangible investment channel. An increase in corporate
income taxes, either at home or abroad, reduces an MNE’s tangible capital in that locationand thereby its return to intangible investment. Note that in this scenario, if there were notangible capital or its costs were fully deductable, corporate taxes would have no effect onintangible investment at all.
Profit shifting scenario (PS). The solution for zi in this scenario is








1−γ−α
 1−γ−α−ϕXλit (1 − φ) (τitp − τTp Ht ) zit = r̂it (τit ) Λiit +r̂jt (τjt ) Λijt  1 − C (λit ) +| {z }(1 − τitp ) | {z }


j̸
=
i
p
|
{z
}
↘ in τit↘ in τ p
jt

p
Ω(τit
):↗ in τit , ≥1

(46)
The first part in square brackets is the same as in the transfer pricing scenario. The second
part, Ω(τitp ), is familiar from the characterization above. As mentioned, it is increasing in thedomestic tax rate, and so it operates in a qualitatively similar fashion to the second effectin the free transfer scenario. When the domestic tax rate rises, the after-tax income of the
MNE’s subsidiary in the tax haven rises relative to the after-tax cost of intangible investment
at home. Thus, the effect of domestic tax rates is now ambiguous again. Also as mentioned,Ω(τitp ) is greater than one as long as τitp > τTp H , which means that intangible investment ishigher in the profit shifting scenario than in the transfer pricing scenario as in DHS.
This example highlights the fact that corporate income taxes affect MNEs’ intangible investment through their negative effects on tangible investment as well as their positive effects
on the returns to deploying intangible capital in foreign aﬀiliates. Although increasing corporate income taxes can help improve the allocation of intangible investment in the presenceof profit shifting, it may not be optimal to do so if the first effect dominates. This caveat isparticularly relevant if some firms do not engage in multinational production; for these firms,
corporate taxes affect intangible investment only through the tangible investment channel.
26

Our quantitative model, in which the vast majority of firms are not MNEs, takes this into
account.

3.6 Cooperative Ramsey planner’s problem
Given that competitive equilibria are generally ineﬀicient in our environment, how should
governments coordinate to set taxes? What trade-offs do they face in doing so? To formalize these issues, we now study the problem of a cooperative global Ramsey planner whochooses the best allocation that can be supported with prices and policies as a competitive
equilibrium. We first characterize the Ramsey problem in the environment described above,
where competitive equilibria are not Pareto optimal in all three scenarios. We then equip
the planner with an additional fiscal instrument, a tax on tangible capital income, and showthat the Ramsey allocation can be implemented in the profit shifting scenario, but not in thefree transfer or transfer pricing scenarios.
The Ramsey problem is to choose an allocation, A, a vector of prices, P, and a vector oftax policies, T to maximize a weighted sum of utilities across countries,
I
X

ωiU i

(47)

i=1

subject to the restriction that (A, P) is the competitive equilibrium associated with T . Ultimately, though, what the planner cares about is the allocation, so we follow CNT andanalyze the relaxed Ramsey problem, which consists of choosing an allocation A and periodzero prices and policies to maximize (47) subject to the implementability condition (41) andthe resource constraints (2)-(10).5 As in CNT, we also impose a restriction that each country
must be allowed to keep an exogenous value of initial wealth Wi measured in units of utility.6
Definition 3 (Relaxed Ramsey problem) Given a vector of Pareto weights ω, the relaxed Ramsey problem is defined as
max
A

( I
X
i=1

ωi

∞
X

t i

β v cit , hit ; φi)− φ Wi0 (τi0c )isubject to (2)–(10)t=05

Of course, in order to demonstrate that the solution to the relaxed Ramsey problem can in fact be
implemented as a competitive equilibrium, one has to construct policies that, together with the allocation,
constitute a competitive equilibrium. In particular, one has to show that the balance of payments condition(28) (or (29), or (30) depending on the scenario) is satisfied. See CNT for the details of this argument.
6This is a technical condition that restricts the planner’s ability to confiscate initial wealth.

27

where
v i cit , hit ; φi := ui (cit , hit ) + φi uic,t cit − uih,t hit



uic0
bi0
c
f fi0
Wi0 (τi0 ) :=Ri0 + Q−1
+ 1+r
(1 + τi0c )
pi0
pi0

(48)
(49)

and φi is the multiplier on country i’s implementability condition (41). The solution is
denoted by AR (ω). The set of Ramsey allocations is defined as AR = AR (ω) : ω ∈ RI+ .
Ramsey allocations are characterized by the following conditions, with details shown inAppendix B.4. First, we have the planner’s versions of the intra- and intertemporal wedge
conditions:
i
vc,t
1
1
− i = i ii = i ji
vh,t
Gi,t Fl,t
G Fl,t∀i ∀j̸=i

(50)

ĵ,t

i

 
vc,t
ji
i
ii
i
= (1 − δ) + Gi,t+1 Fk,t+1 = (1 − δ) + Gĵ,t+1 Fk,t+1
i
βvc,t+1∀i ∀j̸=i .

(51)

Second, we have the planner’s versions of static and dynamic eﬀiciency conditions, i.e. counterparts of (B.30) and (B.34). The static eﬀiciency states that for any imported goods (m, n)
in location i we havenGnn,t vc,t
Gin,t
= m m
∀i ∀m,n̸=iGim,tGm,t vc,t
The dynamic eﬀiciency of the Ramsey planner reads as:

Gij,t
ii
(1 − δ) + Gii,t+1 Fk,t+1=
i
Gj,t+1j
Gjj,t vc,tj
Gjj,t+1 βvc,t+1!
∀jFinally, we have a condition that describes how the planner allocates intangible investment
across countries:
ij !
j
ii
Gjî,t Fz,tX vc,t
Fl,t
= 1+
(52)
ii
i
Fz,t
Hi,t
v i Gi F ii
j̸=i c,t i,t z,t
!
!
ji
j
jj
j
k v j Gj F kj
X Hi,tX Hi,tGiĵ,t Fz,tvc,tGjj,t Fz,tc,t k̂,t z,t+ i i ii +
.
+
i
ii
i
c
ii
Hi,t
Gii,t Fz,tvc,t Gi,t Fz,t
Hi,t
vi,t
Gii,t Fz,tk̸=i,jj̸=i28

Notice that:i= uic,t 1 + φi 1 + σticc − σtich
vc,t


i
= uih,t 1 + φi 1 + σtihh − σtihc
vh,twhere σticc and σtihh are the own elasticities of the marginal utilities of consumption and labor,and σtich and σtihc are the cross elasticities. In general, these elasticities depend on allocationsand vary across countries and over time, which means that the ratios of the derivatives of v i
do not necessarily coincide with marginal rates of substitution. Importantly, if countries areasymmetric, it is generally the case that
j
j
i
i
−vc,t
/vh,t
−vc,t
/vh,t
̸=
−uic,t /uih,t−ujc,t /ujh,t

and

j
j
i
i
−vc,t
/vc,t+1−vc,t/vc,t+1̸=−uic,t /uic,t+1−ujc,t /ujc,t+1

and

j
vc,t
ujc,t
̸= i .
i
vc,t
uc,t

(53)

Beyond that, there is little we can say without putting further structure on the problem.
Implementations, given the tax instruments T , may have intra- and/or intertemporal wedges,
i.e., they may not satisfy one or more of (50) and (51) or may violate dynamic eﬀiciency(15). As we have discussed above, in the profit shifting scenario, the planner can achieve aneﬀicient allocation of intangible investment using corporate income taxes. Unless countries
are symmetric, however, this requires non-zero corporate income taxes that differ acrosscountries, which leads to intertemporal wedges and violates dynamic eﬀiciency. The tension
between these two margins implies that in general, the planner must make sacrifices on both.
In the free transfer and transfer pricing scenarios, where it is altogether impossible to achieve
an eﬀicient allocation of intangible investment, it is not clear that eliminating intra- andintertemporal wedges is the second-best option.
3.6.1

Adding taxes on tangible capital income

One way to make further progress is to equip the planner with an additional policy instrument: a tax on tangible capital income, τitk . In this case, the planner can implement Ramsey
allocations in the profit shifting scenario, but not in the free transfer or transfer pricing. Only
two equilibrium conditions are affected by capital income taxes. The intertemporal wedge
condition becomesuic,t(1 + τitc ) pkii 1 + (1 − τit+1=)(1 − τit+1)(Gii,t Fk,t+1− δ) ,ic
βuc,t+1
1 + τit+1

29

(54)

and the dynamic eﬀiciency condition now says thatm
1 + τjit+1x
1 − τjit+1x
1 − τjitm

1 + τjit

 i
Gij,t pkii
1 + (1 − τit+1) 1 − τit+1
Gi,t+1 Fk,t+1−δ .
i
Gj,t+1(55)

is the same across countries.
To support a Ramsey allocation in the profit shifting scenario, first choose the trade taxesmx
to be offsetting, i.e. τijt
= −τijt. This ensures static eﬀiciency, i.e. that (34) coincides withx
(1+τ m ) (1−τjit)(14). It also implies that the term 1−τjit+1
in (55) is equal to one. Then, as we showxm
( jit+1) (1+τjit)x
in the Appendix following arguments by CNT, that one can set τijt
so that the balance of
payments holds.
Now, compare (32) with (50), (54) with (51), and (55) with (15). It is clear that given a
set of corporate income taxes, the taxes on consumption, labor, and capital income that eliminate intratemporal wedges, eliminate intertemporal wedges, and ensure dynamic production
eﬀiciency in Ramsey equilibrium are given by−uic,t /uih,t(1 + τitc )
;
=
i
i
−vc,t
/vh,t
1 − τith−uic,t /βuic,t+1(1 + τitc )
;
=
i
i
c
−vc,t
/βvc,t+1
1 + τit+1

τitk = −τitp .

(56)

We are left to determine how to set corporate income taxes to achieve a Ramsey allocationof intangible investment. For the free transfer case (37) using result in Lemma 1 under static
eﬀiciency we have:
ii
X
Fl,t
=
1
+
i
ii
Hi,t
Fz,t
j̸=i

ij
ujc,t Gjî,t Fz,tii
uic,t Gii,t Fz,t!

1 − τjtp
1 − τitp

(57)

Now comparing (57) with (52) and following the argument laid out in Proposition 1 weconclude the Ramsey planner can not implement an eﬀicient allocation in general. For thetransfer pricing using Lemma 1 in (39) we get:
ii
X
Fl,t
=1+
i
ii
Hi,t Fz,t
j̸=iij
ujc,t Gjî,t Fz,t
i
ii
uc,t
Gii,t Fz,t!
(58)

Now comparing (58) with (52) it’s immediate that the Ramsey planner does not have any
instruments to correct the externality from the spillover. Thus an eﬀicient can not be implemented in this case. Finally, for the profit shifting case we have, using Lemma 1, that (40)30

becomes:
"
ii
X
Fl,t
=
1
+
i
ii
Hi,t
Fz,t
j̸=i

ij
ujc,t Gjî,t Fz,tii
uic,t Gii,t Fz,t!#Ω(τitp ).

(59)

where Ω(τitp ) is the gain from profit shifting per unit of intangible capital as defined in(40). Now, compare (59) with (52). Using the properties of Ω(τitp ) summarized in Lemma
2 and invoking mean value theorem it’s immediate that there exists τ̂itp that implements
Ramsey allocation of intangible capital. This together with other taxes as specified in (56)
implements Ramsey allocation as a competitive equilibrium. We summarize this discussion
in the following proposition.7 The remark that follows provides some additional, and we
think quite interesting, context about how the Ramsey problem in our environment departs
from the classic Chamley-Judd result that holds in CNT.
Proposition 3 (Ramsey allocations with tangible capital income taxes) A Ramsey
planner equipped with tangible capital income taxes in addition to the instruments T specified
in section 3.3.1 can implement an eﬀicient allocation in profit shifting case, but not in freetransfer and transfer pricing cases.
Remark 2 It is important to note that the tax system described above that supports a Ramsey
allocation in the profit shifting scenario has non-zero taxes on capital income. As stated in(56), tangible capital income taxes should be chosen to offset corporate income taxes, and(59) says that as long as countries marginal utilities differ, corporate income taxes should be
non-zero and differ across countries. Hence, tangible capital income taxes are non-zero as
well. This is in sharp contrast to CNT, where the classic Chamley-Judd result holds wherezero taxes on capital income are optimal.
Remark 3 It is also important to note that, as is widely known in the literature on Ramsey
taxation, there are other sets of instruments that the planner could use to implement an eﬀicient allocation. In our context, the most obvious is an intangible investment subsidy, which
the planner could use to correct the spillover externality regardless of the presence of transfer
pricing or profit shifting. In fact, many countries do subsidize research and development andother forms of intangible investment. In this paper, we focus on corporate taxation because
this is where the current efforts at cooperation between international policymakers are; there
is widespread discussion among policymakers about corporate tax harmonization and shutting
down profit shifting, but there is little, if any, discussion about coordinating R&D subsidies.
7

The proof containing additional details is relegated to the Appendix C.

31

Remark 4 Note that we are not restricting the planner’s ability to set date-zero taxes, see thedefinition of T . Thus, by setting the initial taxes τi0c or τi0k the Ramsey planner can effectivelyimake the implementability condition non-binding. This implies vj,t= uij,t for j = c, h andthe planner can implement an eﬀicient allocation as defined in Definition 1. Importantly,
if we restrict planner’s ability to set initial taxes and therefore make the implementability
iconstraint binding our Proposition 3 still holds. In that case, we generically have vj,t
̸= uij,t
for j = c, h and the Ramsey allocation is constrained eﬀicient.

4

Quantitative analysis

We now turn to our quantitative analysis. First, we describe the main differences between
our quantitative model and the theoretical model described in the previous section. Second,
we discuss our calibration strategy. Third, we study the optimal policies of cooperative global
Ramsey planners who choose corporate income taxes in all countries to maximize worldwide
welfare.

4.1 Overview of the quantitative model
Our quantitative model builds closely on DHS. Each region i (the term “region” better describes the way we take the model to the data than “country”) has a continuum of firmsthat differ in productivity, produce differentiated varieties of intermediate goods, and compete monopolistically as in Chaney (2008). Exporting, multinational production, and profit
shifting now require firms to pay fixed costs, and so only a subset of firms choose to engage
in these activities in equilibrium. Final-goods producers in each region aggregate the set ofavailable varieties into a single nontradable final good. Households and the government aremodeled in the same way as in the theory section. Below, we briefly describe the main differences between the quantitative model and the theory, suppressing time subscripts where
possible for the sake of brevity. See Appendix D for a full description of the quantitativemodel.
There are two significant differences between the quantitative model in this paper and
DHS. The first is the international spillover in intangible investment, which creates an externality that is not present in DHS and significantly complicates the computation of themodel’s equiliibrium. The second is the addition of tangible capital, which changes the effects of corporate income taxes on intangible investment. In DHS, an increase in domestic
corporate income taxes has no effect on non-MNEs’ intangible investment because this investment is tax deductable, and actually increases MNEs’ intangible investment by reducing
32

foreign subsidiaries’ relative tax rates. The addition of tangible capital makes these effects
more complicated. Tangible capital costs are not fully tax deductable, which means thattangible investment is decreasing in the domestic corporate income tax rate. All else equal, alower tangible capital stock reduces the desired level of intangible investment. Consequently,
the overall effect of corporate income taxes on MNEs’ intangible investment ambiguous asshown in the theory section above, and corporate income taxes now unambiguously reduce
non-MNEs’ intangible investment.
Demand for intermediates. The final good in each region is a constant-elasticity-of-substitution
aggregate of all of the varieties available in that region:

qit =

" I Z
X
j=1

ρ
# ρ−1

qjit (ω)

ρ−1
ρ

dω

,

(60)

Ωjit

where ω indexes varieties, Ωjit is the set of varieties from j available in i, and ρ is the elasticity
of substitution between varieties. This implies a standard CES demand function,1

pjit (ω) = pit qitρ qjit (ω)− ρ ,1(61)

where the aggregate price index pit is given by the usual ideal price index formula.
Production technology. A firm from region i with productivity a(ω) produces output inregion j according to the constant-returns-to-scale technology
yijt (ω) = σij Aj a(ω) (nj zit (ω))ϕ kijt (ω)α ℓijt (ω)1−ϕ−α(62)

where Aj is region j’s aggregate total factor productivity, ϕ is the share of intangible capital,
α is the share of tangible capital, and σij captures barriers to FDI as in McGrattan andWaddle (2020). We assume that σij ∈ [0, 1] and that σii = 1. We assume constant returns toscale here, rather than decreasing returns as in the theory section, because we have moved
to a monopolistic-competition framework.
Research & development. As in the theory section, firms hire workers in their homecountries to produce intangible capital. We model spillovers from other countries’ R&D
efforts in a slightly different fashion. We assume now that the productivity of domestic R&D
labor is augmented by the total amount of intangible capital deployed by foreign MNEs’
local aﬀiliates. This differs from the setup in the theory section in two ways. First, only
foreign MNEs with local aﬀiliates create spillovers; the R&D done by foreign firms without33

operations in region i has no effect on that region’s R&D productivity. Second, spillovers
are created by foreign intangible capital, not the workers employed abroad to produce it.
This is without loss of generality, but makes programming the model slightly simpler. The
intangible capital production technology is specified as follow:
zit (ω) = H iℓzit (ω),

!

XZ
j̸=i

zjt (ω ′ ) dω ′R"P:= Ai ℓzit (ω)

Ωjit

j̸=i

P

j̸=i

Ωjit

zjt (ω ′ ) dω ′Ω̄jiz̄j (ω ′ ) dω ′

R

#υ
,

(63)

where ω indexes the domestic firm, ℓzit (ω) is the firm’s R&D labor, Ωjit is the mass of firms
from region j ̸= i with aﬀiliates in region i, and ω ′ indexes these foreign firms. The parameterυ controls the degree of spillovers. When υ = 0 there are no spillovers as in DHS. The higher
υ, the greater the spillovers. We normalize the spillover term in square brackets to one in
the benchmark equilibrium calibrated under the current tax code. This facilitates calibration
by ensuring that the benchmark equilibrium is the same with and without spillovers. Bars
denote equilibrium objects in the benchmark. It is important to point out that the spillover
effect greatly increases the computation burden relative to the model in DHS. The R&D
technology (63) depicts a fixed-point problem that needs to be solved each time one tries
a new candidate price vector in trying to satisfy the equilibrium conditions. Holding prices
fixed, one needs to iterate on firms’ R&D decisions until the term in brackets above converges.
Trade and foreign direct investment. Firms can sell their products to foreign countries
through exporting and/or multinational production. Firms based in region i must pay aF
fixed cost κX
i for each country to which they export, and a fixed cost κi for each country
in which they operate a foreign aﬀiliate. Fixed costs are denominated in units of the homecountry’s labor. Each unit of goods shipped abroad incurs an iceberg transportation cost ξij .
As in McGrattan and Waddle (2020) and Garetto et al. (2019), exports and locally-produced
products are treated as distinct varieties; firms can simultaneously export to, and produce
locally for, the same foreign country. We use JitX (ω) ⊆ I \ {i} to denote a firm’s export
destinations and JitF (ω) ⊆ I \ {i} to denote its portfolio of foreign aﬀiliates. The firm’s
resource constraints can then be written as follows:yiit (ω) = qiit (ω) +Xξij qijt (ω)

(64)

X (ω)
j∈Jit

yijt = q̂ijt (ω), j ∈ JitF (ω)
where qijt (ω) and q̂ijt (ω) represent exports and locally-produced products, respectively.

34

(65)

Transfer pricing. Transfer pricing works in the same way as in the theory section. An
aﬀiliate in region j of a firm based in region i with intangible capital z pays a licensing
fee ϕpijt (ω)yijt (ω)/zit (ω) ≡ ϑijt (ω)zit (ω), where ϑijt (ω) is the the firm’s marginal revenue
product of intangible capital in j. The total amount of licensing fees across the firm’s portfolio
Pof foreign aﬀiliates is νit (ω)zit (ω) ≡ j∈J F (ω)∪{i} ϑijt (ω)zit (ω). Note that this includes theitlicensing fee the parent corporation “pays” itself for the use of its own intangible capital.
Profit shifting. Profit shifting works in basically the same way as in the theory section,
except that we allow firms to shift profits to the productive region with the lowest tax rate,
denoted by LT , as well as an unproductive, zero-population tax haven, denoted by T H. Suppose a firm from region i sells a fraction λi,LT,t (ω) of its intangible capital to the low-tax region
and a fraction λi,T H,t (ω) to the tax haven. Its aﬀiliate in the low-tax region receives licensP
ing fees of λi,LT,t (ω) j∈J F (ω)∪{i}\{LT } ϑijt (ω)zit (ω) and its aﬀiliate in the tax haven receivesitP
λi,T H,t (ω) j∈J F (ω)∪{i} ϑijt (ω)zit (ω). The firm’s domestic parent corporation receives the reitPmaining fees, [1 − λi,LT,t (ω) − λi,T H (ω)] j∈J F (ω) ϑijt zit (ω). The total variable profit shiftingitcost is Ci,LT (λi,LT,t (ω)) + Ci,T H (λi,T H,i (ω)), where Cij (λ) = [λ + (1 − λ) log(1 − λ)] ψij . Setting up an aﬀiliate in the tax haven also requires a fixed cost κTi H . See DHS for a detailed
treatment of our quantitative theory of profit shifting.
Tax treatment of capital expenditures. Like McGrattan and Waddle (2020), we follow
standard accounting practices in modeling the tax treatment of firms’ expenses associatedwith intangible and tangible capital. As in the theory section, R&D expenditures and depreciation of tangible capital are tax-deductible, but other tangible capital costs are notdeductable. As before, this makes corporate income taxes have an ambiguous effect on
MNEs’ intangible investment. But most firms in the quantitative model are not MNEs, andthese accounting rules imply that corporate income taxes unambiguously reduce non-MNEs’
intangible investment.

4.2 Calibration
We follow the same calibration procedure as in DHS, so we describe this procedure only briefly
here. We first partition to world into five regions: North America (NA), Europe (EU), thelow-tax region (LT), the rest of the world (RW), and the tax haven (TH). The low-tax region
includes includes Belgium, Ireland, Hong Kong, the Netherlands, Singapore, and Switzerland. The tax-haven region includes Luxembourg, small European countries and territories
like Cyprus, Malta, and the Isle of Man, and a number of Caribbean countries. Data for eachregion are obtained by aggregating or averaging country-level data as appropriate. We then35

choose parameter values for each region so that, under the current international tax regime,
the model’s equilibrium matches data on production, international trade, foreign direct investment, and profit shifting. The key parameters and the data moments that disciplinethem are listed below; Table 1 reports the values of these parameters and moments.
• population (ni ), which is taken directly from the data;• aggregate TFP (Ai ), which is identified by real GDP per capita;• corporate tax rates (τip ), which are taken from Tørsløv et al. (2022);• labor income tax rates (τil ) are set to 22.4% as in McGrattan and Waddle (2020);• the intangible share (γ), which is identified by the fraction of income generated byforeign MNEs’ local aﬀiliates that accrues to intangible capital;• trade costs (κFi and ξij ), which are identified by export participation and bilateral tradeflows;• FDI costs (κFi and σi ), which are identified by the fraction of firms that engage in
multinational production and the fraction of each region’s gross value added that is
produced by local aﬀiliates of foreign MNEs;• and profit-shifting costs (κi,T H , ψi,LT , and ψi,T H ), which are identified by the numberof firms in each high-tax region with aﬀiliates in the tax haven region, the aggregate
amount of profits shifted from each high-tax region to the low-tax region, and theaggregate amount of profits shifted from high-tax regions to the tax-haven region.
We do not attempt to calibrate the spillover parameter, υ, by matching moments, as
identifying it from macro data is challenging. Our specification of the R&D technology (63)normalizes the spillover effect to one in the calibrated benchmark equilibrium, which ensuresthat this equilibrium matches the data regardless of the strength of the spillover effect. Our
focus is not on how spillovers affect the world economy under the current system, but rather
on how they shape the optimal corporate income tax system, In our counterfactual policy
experiments, we consider three values of υ: zero (no spillovers, 0.1 (weak spillovers), and 0.2(strong spillovers). The weak spillover calibration implies a similar effect of FDI on domestic
firms’ productivity as estimated by Javorcik (2004).

36

4.3 Experiments
Having laid out the model and its calibration, we turn now to our quantitative optimal
policy experiments. We restrict attention here to long-run steady states; we do not study
the transition dynamics that would follow policy changes. We allow the planner to choose
corporate income taxes τip in each country, letting the taxes on labor income adjust to ensurethat the government’s budget in each region clears. We first consider a cooperative global
Ramsey planner that seeks to maximize a weighted sum of the welfare of consumers in
all regions. We then study a constrained global planner that seeks to maximize the same
objective, but is restricted to Pareto improvements relative to the benchmark equilibrium.
We solve each planner’s problem six times: (a) without spillovers, holding fixed the tax
haven’t tax rate τTp H ; (b) with weak spillovers, holding τTp H fixed; (c) with strong spillovers,
holding τTp H fixed; (d) without spillovers, allowing the planner to choose τTp H as well as theother regions’ corporate tax rates; (e) with weak spillovers, allowing the planner to choose
τTp H ; and (f) with strong spillovers, allowing the planner to choose τTp H .
It is important to state explicitly that we focus our experiments on optimal policies inlong-run stationary equilibria; we do not model the transition dynamics from the status
quo to the new stationary equilibria associated with these policies. As we will show, thesepolicies would entail cutting corporate income taxes, which would stimulate investment inboth tangible and intangible capital, and the dynamics of these investments could havesignificant implications for welfare. However, modeling the dynamics of multinational firms’
decisions about intangible investment and profit shifting would be an extremely challenging
computational task, especially in the presents of spillovers. Accomplishing this task would
represent a major contribution to the literature, and we leave it for future research.
4.3.1

Unconstrained planners

The unconstrained global planner maximizes worldwide welfare by choosing a vector of corporate income tax rates τ = (τ1p , . . . , τIp ). This problem is given formally by
(
max
τ

X

)
ωi ui (c∗i (τ ), h∗i (τ )) ,

(66)

i∈I

where c∗i (τ ) and h∗i (τ ) denote region i’s consumption and labor allocations in the steadystate equilibrium associated with the policy vector τ , and ωi is the Pareto weight on region i.
We assume that these weights are proportional to each region’s real GDP in the benchmark

37

calibrated equilibrium.8 Note that in versions (c) and (d) of this experiment, the choice
vector τ also includes the tax haven’s tax rate, τTp H . The optimal corporate tax rates andwelfare implications are shown in Table 2. Table 3 shows additional results about the effectsof these policy changes on public finances and macro variables in the case where the tax
haven’s tax rate is fixed, and Table 4 shows additional results in the case where the plannerchooses the tax haven’s tax rate.
There are four main takeaways. First, and most important, it is optimal to shut down
profit shifting in the absence of spillovers, but profit shifting should remain active when
spillovers are present, just as the theory shows. In the no-spillover calibration, the unconstrained planner sets high enough tax rates in the low tax region and, if possible, the tax
haven to completely eliminate profit shifting. In the calibrations with spillovers, on the other
hand, the planner reduces profit shifting but still allows it to operate to both the low tax
region and the tax haven, even then she can choose the latter’s tax rate. The optimal amountof profit shifting is higher in the strong-spillover calibration than in the weak-spillover calibration, just as one would expect. In both cases, the optimal amount of profit shifting ishighest in the rest of the world and lowest in Europe.
Second, lowering corporate tax rates increases intangible investment, which means thatthe intertemporal distortion channel is stronger than the profit shifting channel. Recall thatthe effect of corporate taxes on intangible investment is ambiguous from equation (46). On
the one hand, higher corporate taxes increase the return to profit shifting, which also increases
the return to intangible investment. On the other hand, they reduce tangible investment,
which also reduces the returns to intangible investment. The results show that tangible
investment reacts strongly to corporate tax changes, and that intangible investment moves
in the same direction in equilibrium. Thus, in order to stimulate intangible investment, theplanner needs to cut corporate income taxes.
Third, spillovers allow the planner to achieve larger welfare gains by cutting high-tax
regions’ corporate taxes more aggressively. The planner cuts corporate taxes in all threehigh-tax regions regardless of the strength of the spillover effect, but the optimal tax cuts
are smallest in the no-spillover calibration and largest in the strong-spillover calibration.
Fourth, the unconstrained planner would always benefit the three high tax regions at theexpense of the low-tax region. This is because the planner puts more weight on the high tax
regions as they have larger economies, but also because it would be optimal to reduce profit
8

This is similar to the standard approach in which Pareto weights are related to country wealth. Alternatively, one could assume the planner’s objective is weighted by population, which would put much moreweight on the rest of the world and incentivize the planner to redistribute resources towards this poorer
region.

38

shifting (even in the presence of spillovers). This requires the planner to raise the low-tax
region’s corporate tax rate, which discourages its firms from investing in intangible capital.
In the next set of experiments, we restrict the planner to Pareto-improving tax reforms.

4.4 Constrained global planner
The constrained global planner has the same objective function as the unconstrained global
planner, but is constrained to choose a Pareto-improving policy vector that leaves no region
worse off relative to the status-quo equilibrium. This constraint can be written asui (c∗i (τ ), h∗i (τ )) ≥ ui (c∗i (τ̄ ), h∗i (τ̄ )) , i ∈ I,

(67)

where τ̄ is the status-quo policy vector, and c∗i (τ̄ ) and h∗j (τ̄ ) are the associated allocations inthe status-quo stationary equilibrium. Note that this problem can be formulated equivalently
in the language of the relaxed Ramsey problem (3) with Wi > 0 chosen appropriately. The
results of this set of experiments are shown in Tables 5–7.
There are two takeaways from this second set of experiments. First, it is never optimal forthe constrained planner to shut down profit shifting, regardless of the presence of spillovers.
As discussed above, reducing the amount of profits that are shifted from high-tax regions
to the low-tax region reduces the latter’s welfare, which the constrained planner cannot do.
Therefore, the constrained planner always sets corporate taxes in the low-tax region below thehigh-tax regions’ tax rates. In all six versions of this experiment, the amount of profits shiftedto the low-tax region actually rise. Perhaps more interestingly, the constrained planner nevershuts down profit shifting to the tax haven, even in the absence of spillovers. In experiments(d)–(f), where the contrained planner can choose the tax haven’s tax rate, she sets it well
below the high-tax regions’ tax rates.
Second, the constrained planner would choose smaller corporate tax cuts, especially inEurope, and would generally achieve more modest welfare gains. North America and the restof the world would always gain less than under the unconstrained planner’s optimal policy.
With no spillovers or weak spillovers, Europe would also gain less, but with strong spillovers
Europe would actually gain more. The low-tax region would always be indifferent between
the status quo and the constrained planner’s optimal policy.

39

5

Conclusion

We study optimal corporate income taxation in an environment that emphasizes three closely
related features of the modern globalized economy: multinational enterprises, intangible
capital, and profit shifting. In our model, MNEs use nonrival intangible capital to produce
output in many locations around the world as in McGrattan and Prescott (2009, 2010), andthey can use transfer pricing to shift the income generated by this capital to subsidiaries
in foreign tax havens as in Dyrda et al. (2022). We assume that there are international
spillovers in intangible investment, which creates an externality that implies competitive
equilibria without distortionary taxes are ineﬀicient.
In our theoretical analysis, we show that in the absence of profit shifting it is impossible fora Ramsey planner to use corporate income taxes to correct this externality, but in the absenceof profit shifting corporate income taxes can be used to implement an eﬀicient allocation of
intangible investment. Under profit shifting, higher corporate income taxes increase thereturn to intangible investment because the costs are tax-deductable, and the planner canuse this to her advantage to internalize the spillover externality. However, there is a tension
between achieving a statically-eﬀicient allocation of intangible investment and a dynamicallyeﬀicient level of tangible investment. If the planner is equipped with taxes on tangible capitalincome as well, we show that she can implement a Pareto-optimal allocation.
In our quantitative analysis, we study the Ramsey planner’s problem in a richer versionof our model with asymmetric countries, firm heterogeneity, selection into multinational
activity. We find that the optimal policy entails cutting corporate income taxes to boost
intangible investment, especially in the presence of a spillover externality, and raising taxes
on labor income to restore fiscal balance. Consistent with the theory, it would be optimal
to shut down profit shifting completely in the absence of a spillover externality, but profit
shifting should be allowed to operate, albeit at a lower level, if such an externality is present.
When the planner is constrained to Pareto improvements that leave all regions weakly better
off, the optimal policy would actually cause the level of profit shifting to increase relative tothe status quo.

40

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43

Table 1: Calibration
Statistic or parameter value(a) Assigned parameters and target moments
Population (NA = 100)
Real GDP (NA = 100)
Corporate tax rate (%)
Labor tax rate (%)
Foreign MNEs’ VA share (%)
Total lost profits ($B)
Lost profits to TH (%)
Imports from… (% GDP)
North America
EuropeLow tax
Row(b) Calibrated parameter values
TFP (Ai )Prod. dispersion (ηi )Utility weight on leisure (ψi )Fixed export cost (κXi )Variable FDI cost (σi )Fixed FDI cost (κFi )Cost of shifting profits to LT (ψiLT )Cost of shifting profits to TH (ψiT H )Fixed FDI cost to TH (κTi H )Variable trade cost from…
North America
EuropeLow tax
RoW

North
America

EuropeLow-tax

RoW

Tax haven

100
100
22.5
22.4
11.12
143
66.4

92
80.78
17.3
22.4
19.82
216
44.5

11
14.57
11.4
22.4
28.73
–
–

1,323
297.10
17.4
22.4
9.55
257
71.1

–
–
3.3
–
–
–
–

–
1.70
0.35
6.15

1.28
–
2.98
7.96

1.77
12.39
–
6.78

1.74
3.78
0.59
–

–
–
–
–

.00
4.30
1.46
2.5e-3
0.46
2.56
3.73
2.46
0.13

0.90
4.32
1.49
5.2e-3
0.55
2.27
0.42
1.37
0.08

1.43
4.87
1.51
1.5e-3
0.52
0.65
–
–
–

0.28
4.15
1.47
2.1e-2
0.55
12.70
2.73
2.05
0.75

–
–
–
–
–
–
–
–
–

–
1.87
2.00
2.19

3.25
–
1.59
2.56

3.45
1.69
–
2.96

2.12
1.35
1.58
–

–
–
–
–

Notes: Population and real GDP from World Bank WDI. Corporate tax rate from Tørsløv et al. (2022). Foreign MNEs’
VA share from OECD AMNE database. Fractions of firms with foreign aﬀiliates from Compustat. Lost profits from
Tørsløv et al. (2022). Imports/GDP from WIOD. Dashes (–) represent “not applicable.”44

Table 2: Unconstrained planner’s problem: optimal corporate tax rates and welfare implications
North
America

EuropeLow tax

Rest of
world

Tax
haven

18.71
-3.79
0.092

13.64
-3.66
0.116

17.61
6.21
-1.051

17.87
0.47
0.063

3.30
0.00
–

(b) Weak spillovers, τT H fixed
Corp. tax rate (%)
15.52
Chg. in corp tax rate (p.p.)-6.98
Welfare (% chg.)
0.097

9.96
-7.34
0.051

13.76
2.36
-0.730

16.69
-0.71
0.118

3.30
0.00
–

(c) Strong spillovers, τT H fixed
Corp. tax rate (%)
14.36
Chg. in corp tax rate (p.p.)-8.14
Welfare (% chg.)
0.224

5.06
-12.24-0.0999.40-2.00-0.462

15.01
-2.39
0.242

3.30
0.00
–

(d) No spillovers, planner chooses τT H
Corp. tax rate (%)
20.21
Chg. in corp tax rate (p.p.)-2.29
Welfare (% chg.)
0.142

15.82
-1.48
0.224

19.78
8.38
-1.179

20.55
3.15
0.218

20.50
17.20
–

(e) Weak Spillovers, planner chooses τT H
Corp. tax rate (%)
18.60
12.04
Chg. in corp tax rate (p.p.)
-3.90
-5.26
Welfare (% chg.)
0.161
0.123

17.02
5.62
-0.925

18.88
1.48
0.234

18.57
15.27
–

(f) Strong Spillovers, planner chooses τT H
Corp. tax rate (%)
15.10
8.54
Chg. in corp tax rate (p.p.)
-7.40
-8.76
Welfare (% chg.)
0.190
0.067

10.29
-1.11
-0.468

16.89
-0.51
0.303

15.20
11.90
–

Variable(a) No spillovers, τT H fixed
Corp. tax rate (%)
Chg. in corp tax rate (p.p.)
Welfare (% chg.)

Notes: Table shows corporate tax rates and welfare under unconstrained planner’s optimal
policy. Panel (a): υ = 0, planner cannot choose τT H . Panel (b): υ = 0.1, planner cannot choose
τT H . Panel (c): υ = 0.2, planner cannot choose τT H . Panel (d): υ = 0, planner can choose
τT H . Panel (e): υ = 0.1, planner can choose τT H . Panel (f): υ = 0.2, planner can choose τT H .

45

Table 3: Unconstrained planner’s problem: public-finance and macro implications
North
America

EuropeLow tax

Rest ofworld(a) No spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-17.54
0.54
1.88
2.07
2.50
0.38
-1.11
-0.40
-0.63

-20.50
0.31
1.61
1.86
2.41
0.52
-0.95
-0.33
-0.51

27.46
0.00
-1.36
-3.36
-2.98
-0.30
0.97
-0.58
-0.40

3.51
0.83
-0.30
-0.25
-0.23
1.12
0.15
-0.11
0.16

(b) Weak spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-32.60
0.40
3.49
3.61
5.15
2.22
-2.04
-0.62
-1.22

-42.73
0.16
3.36
3.46
5.37
2.33
-1.93
-0.59
-1.22

0.36
0.18
0.07
-1.41
0.04
1.43
0.12
-0.61
-0.65

-4.06
0.84
0.33
0.34
1.24
3.34
-0.20
-0.15
-0.01

(c) Strong spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-38.11
0.41
4.05
4.20
7.14
4.80
-2.37
-0.50
-1.31

-71.53
0.02
5.62
5.42
9.55
4.82
-3.22
-0.89
-2.20

-30.77
0.35
1.70
0.66
3.77
3.71
-0.84
-0.60
-1.02

-14.50
0.81
1.18
1.20
3.60
6.06
-0.68
-0.08
-0.20

Variable

Notes: Table shows macroeconomic effects of unconstrained planner’s optimal policy. Panel (a):
υ = 0, planner cannot choose τT H . Panel (b): υ = 0.1, planner cannot choose τT H . Panel (c):
υ = 0.2, planner cannot choose τT H .

46

Table 4: Unconstrained planner’s problem: public-finance and macro implications. continued
North
America

EuropeLow tax

Rest ofworld(d) No spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)

-8.74
0.00
0.92
1.52
1.26
-1.12
-0.54
-0.21
-0.21

-5.42
0.00
0.42
0.96
0.66
-1.21
-0.30
-0.09
0.02

45.58
0.00
-2.31
-4.45
-4.73
-1.84
1.52
-0.50
-0.16

22.68
0.00
-1.93
-1.43
-2.30
-0.35
0.99
0.21
0.87

(e) Small Spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-16.93
0.03
1.79
2.28
2.73
0.38
-1.04
-0.28
-0.52

-29.49
0.00
2.32
2.64
3.51
0.51
-1.35
-0.36
-0.77

25.06
0.12
-1.23
-3.02
-2.56
-0.31
0.87
-0.51
-0.34

11.46
0.08
-0.98
-0.60
-0.70
1.51
0.48
0.14
0.55

(f) Large Spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-34.01
0.10
3.61
3.93
6.07
3.23
-2.10
-0.49
-1.17

-50.95
0.00
3.99
4.11
6.81
3.08
-2.29
-0.58
-1.44

-22.86
0.41
1.28
0.35
2.56
2.48
-0.60
-0.54
-0.87

-1.80
0.30
0.11
0.39
1.81
4.44
-0.11
0.08
0.23

Variable

Notes: Table shows macroeconomic effects of unconstrained planner’s optimal policy. Panel (d):υ = 0, planner can choose τT H . Panel (e): υ = 0.1, planner can choose τT H . Panel (f): υ = 0.2,
planner can choose τT H .

47

Table 5: Constrained planner’s problem: optimal corporate tax rates and welfare implications
North
America

EuropeLow tax

Rest of
world

Tax
haven

19.32
-3.18
0.051

16.32
-0.98
0.000

10.32
-1.08
0.000

17.46
0.06
0.007

3.30
0.00
–

(b) Weak spillovers, τT H fixed
Corp. tax rate (%)
18.05
Chg. in corp tax rate (p.p.)-4.45
Welfare (% chg.)
0.075

15.48
-1.82
0.024

9.10
-2.30
0.000

17.22
-0.18
0.038

3.30
0.00
–

(c) Strong spillovers, τT H fixed
Corp. tax rate (%)
14.04
Chg. in corp tax rate (p.p.)-8.46
Welfare (% chg.)
0.095

15.64
-1.66
0.107

7.46
-3.94
0.000

14.94
-2.46
0.100

3.30
0.00
–

(d) No spillovers, planner chooses τT H
Corp. tax rate (%)
18.74
Chg. in corp tax rate (p.p.)-3.76
Welfare (% chg.)
0.086

16.43
-0.87
0.024

10.77
-0.63
0.000

18.51
1.11
0.035

5.77
2.47
–

(e) Weak Spillovers, planner chooses τT H
Corp. tax rate (%)
17.20
15.87
Chg. in corp tax rate (p.p.)
-5.30
-1.43
Welfare (% chg.)
0.088
0.039

9.62
-1.78
0.000

17.84
0.44
0.058

5.27
1.97
–

(f) Strong Spillovers, planner chooses τT H
Corp. tax rate (%)
14.88
15.61
Chg. in corp tax rate (p.p.)
-7.62
-1.69
Welfare (% chg.)
0.135
0.122

7.62
-3.78
0.000

15.57
-1.83
0.119

6.45
3.15
–

Variable(a) No spillovers, τT H fixed
Corp. tax rate (%)
Chg. in corp tax rate (p.p.)
Welfare (% chg.)

Notes: Table shows corporate tax rates and welfare under constrained planner’s optimal policy.
Panel (a): υ = 0, planner cannot choose τT H . Panel (b): υ = 0.1, planner cannot choose τT H .
Panel (c): υ = 0.2, planner cannot choose τT H . Panel (d): υ = 0, planner can choose τT H .
Panel (e): υ = 0.1, planner can choose τT H . Panel (f): υ = 0.2, planner can choose τT H .

48

Table 6: Constrained planner’s problem: public-finance and macro implications
North
America

EuropeLow tax

Rest ofworld(a) No spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-15.07
0.75
1.61
1.73
2.21
0.70
-0.93
-0.29
-0.56

-6.25
1.01
0.49
0.48
0.85
0.47
-0.27
-0.14
-0.18

-12.70
1.11
0.68
0.71
1.17
0.45
-0.38
-0.16
-0.25

0.48
1.08
-0.04
-0.07
0.09
1.00
0.02
-0.07
0.02

(b) Weak spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-21.03
0.68
2.25
2.39
3.25
1.26
-1.30
-0.38
-0.77

-11.41
0.99
0.90
0.89
1.62
1.07
-0.50
-0.19
-0.31

-22.55
1.18
1.20
1.25
2.16
1.33
-0.65
-0.24
-0.45

-1.10
1.10
0.09
0.06
0.54
1.82
-0.05
-0.08
0.01

(c) Strong spillovers, τT H fixedCorporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-39.49
0.43
4.23
4.31
6.46
3.30
-2.48
-0.77
-1.50

-11.10
1.16
0.86
0.79
2.29
2.86
-0.49
-0.25
-0.22

-36.21
1.26
1.93
1.91
3.85
2.82
-1.07
-0.41
-0.71

-14.87
0.89
1.25
1.18
2.77
3.72
-0.70
-0.29
-0.36

Variable

Notes: Table shows macroeconomic effects of constrained planner’s optimal policy. Panel (a): υ = 0,
planner cannot choose τT H . Panel (b): υ = 0.1, planner cannot choose τT H . Panel (c): υ = 0.2,
planner cannot choose τT H .

49

Table 7: Constrained planner’s problem: public-finance and macro implications, continued
North
America

EuropeLow tax

Rest ofworld(d) No spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-17.47
0.59
1.88
2.06
2.47
0.40
-1.11
-0.38
-0.64

-5.02
0.87
0.40
0.48
0.71
0.22
-0.22
-0.16
-0.12

-8.81
1.09
0.47
0.52
0.87
0.20
-0.26
-0.20
-0.17

7.21
1.02
-0.61
-0.55
-0.62
0.89
0.34
-0.04
0.26

(e) Weak Spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-24.74
0.52
2.65
2.82
3.68
1.16
-1.56
-0.51
-0.92

-8.72
0.92
0.68
0.74
1.32
1.02
-0.38
-0.21
-0.22

-18.24
1.16
0.98
1.01
1.79
0.95
-0.54
-0.28
-0.36

2.99
1.02
-0.26
-0.22
0.09
1.61
0.14
-0.09
0.15

(f) Strong Spillovers, planner chooses τT H
Corporate tax revenue (% chg.)
Lost profits (benchmark = 1)
Labor income tax rate (p.p. chg.)
Tangible capital (% chg.)
Domestic intangible capital (% chg.)
Foreign MNE intangible capital (% chg.)
Employment (% chg.)
Real GDP (% chg.)
Consumption (% chg.)-35.46
0.36
3.79
3.94
5.78
2.78
-2.24
-0.68
-1.31

-10.72
1.01
0.83
0.85
2.18
2.62
-0.48
-0.23
-0.20

-34.59
1.28
1.85
1.85
3.58
2.27
-1.02
-0.41
-0.68

-10.75
0.74
0.90
0.91
2.20
3.15
-0.51
-0.23
-0.22

Variable

Notes: Table shows macroeconomic effects of constrained planner’s optimal policy. Panel (d): υ = 0,
planner can choose τT H . Panel (e): υ = 0.1, planner can choose τT H . Panel (f): υ = 0.2, planner
can choose τT H .

50

Appendix
A

Pareto Frontier

In this section, we derive the optimality conditions for allocations in our economy for twocases. We first present the benchmark case with no spillovers from non-rival intangible
capital. We then add the spillovers and compare the optimality conditions. The Pareto
frontier is characterized by a solution to the following problem
max

∞
I X
X

At


Ni β

t


u

i

i=1 t=0

subject to
F ii (zit , kiit , liit ) = qiit +cit hit
,
Ni Ni
X



qijt

(PF-NR)

∀i

j̸=i
ij

F (zit , kijt , lijt ) = q̂ijt∀j̸=i ∀i

Gi (q1it , ..., qIit , q̂1it , ..., q̂Iit ) = cit + git + xit∀ixit = kit+1 − (1 − δ) kit
H i (litz ) = zit ∀iX
hit =ljit + litz
j

kit =

X

kjit

j

k0

given

where the set of constraints can be simplified tocit + git +

X
j

Gi

kjit+1 − (1 − δ)Xkjit =j

q1it , ..., F ii H i (litz ) , kiit , liit −

X



z
z
) , kIit , lIit
) , k1it , l1it , ..., F Ii H I (lIt
qijt , ..., qIit , F 1i H 1 (l1t

j̸=i

51



!

and then the first-order conditions of the problem (PF-NR) are
cit : 0 = β t uic,t − λit

(A.1)

ji
ljit : 0 = β t uih,t + λit Giĵ,t Fl,t

(A.2)

ii
liit : 0 = β t uih,t + λit Gii,t Fl,tiiilitz : 0 = β t uih,t + λit Gii,t Fz,t
Hl,t
+

X

(A.3)
ij
i
λjt Gjî,t Fz,t
Hl,t

(A.4)

j̸=i

qjit : 0 = −λjt Gjj,t + λit Gij,t

(A.5)

ji
kjit+1 : 0 = (1 − δ) λit+1 − λit + λit+1 Giĵ,t+1 Fk,t+1

(A.6)

ii
kiit+1 : 0 = (1 − δ) λit+1 − λit + λit+1 Gii,t+1 Fk,t+1(A.7)

and thus we get from (A.1), (A.3) and (A.2) we get the no intratemporal wedge condition
uic,t
1
1
− i = i ii = i ji
uh,tGi,t Fl,t
G Fl,t∀i ∀j̸=i

(A.8)

ĵ,t

which equalizes the marginal rate of substitution between consumption and labor with themarginal rate of technical substitution between final consumption good and labor input. The
second equality implies that
ji
Fl,t
Gii,t
= ii
(A.9)
Giĵ,tFl,tNext, from (A.1), (A.7) and (A.6) we get

 
uic,t
ji
i
ii
i
= (1 − δ) + Gi,t+1 Fk,t+1 = (1 − δ) + Gĵ,t+1 Fk,t+1
βuic,t+1∀i ∀j̸=i(A.10)and the second equality implies thatji
Fk,t+1
Gii,t+1=ii
Giĵ,t+1
Fk,t+1(A.11)

which put together with (A.9) implies
ji
Fl,t
ji
Fk,t+1

=

52

ii
Fl,t
ii
Fk,t+1(A.12)

i.e. the marginal rate of technological transformation between tangible capital and labor
is equalized across goods produced domestically by all MNEs operating in location i. An
implication of that is symmetric tax treatment of these two goods. Now, toward static
eﬀiciency, we have that for any imported goods (m, n) in location i we have from (A.5) and(A.1) that
Gnn,t unc,t
Gin,t
=
m
Gim,t
Gm
m,t uc,t(A.13)and note that the right-hand side is independent on i, thus the ratio of marginal rates
of technical substitution for any pair of intermediate, imported goods are equated acrosscountries. Now toward dynamic eﬀiciency we have from (A.5) in two consecutive periods andusing (A.7) that!
Gjj,t λjt
Gij,t
1

=
ii
Gij,t+1(1 − δ) + Gii,t+1 Fk,t+1
Gjj,t+1 λjt+1
implying using (A.1) that

Gij,t
ii
(1 − δ) + Gii,t+1 Fk,t+1=
i
Gj,t+1!

Gjj,t ujc,tGjj,t+1 βujc,t+1(A.14)and note that the right-hand side is independent of i. Toward labor-intangible condition,from (A.3), (A.4) and (A.1) we getiiiuic,t Gii,t Fl,t
= Hl,tii
uic,t Gii,t Fz,t
+

X

!
ij
ujc,t Gjî,t Fz,tj̸=iwhich rearranged yields
ijii
X ujc,t Gjî,t Fz,t
Fl,t
=1+
ii
i
Fz,t
Hl,t
ui Gi F ii
j̸=i c,t i,t z,t(A.15)

This concludes the characterization of the Pareto frontier for the non-rival case.

A.1 With the externality
Suppose now thatzit = H i (ℓz1 , . . . , ℓzI ) = H i

ℓzi ,

X
j̸=i

53

!
ℓzj

The notation Hki is shorthand for ∂H i (·)/∂ℓzkt . Then the FOC for intantible labor becomes"litz : 0 = β t uih,t +λitii
iGii,t Fz,t
Hi,t
+

X

#
k
Gik̂,t Fztki Hi,t

k̸=i

"

X j
X j kj
ij
jj j
k
i
+
λt Gjî,t Fz,tHi,t+ Gjj,t Fz,tHi,t +
Gk̂,t Fzt Hi,t
j̸=i

k̸=j

Note that in the case of two countries, this collapses tolitz : 0 = β t uih,t +λit

h

ii
i
k
Gii,t Fz,tHi,t+ Giĵ,t Fztji Hi,t

i

i
h
ij
jj j
i
+λjt Gjî,t Fz,tHi,t+ Gjj,t Fz,tHi,tCombining with the FOCs for consumption and labor as before, we getiii= Hi,tuic,t Gii,t Fl,t

+

X

ii
uic,t Gii,t Fz,t
+

j
Hi,t



X

!
ij
ujc,t Gjî,t Fz,t

j̸=i
jj
ji
uic,t Giĵ,t Fz,t+ ujc,t Gjj,t Fz,t



j̸=i

+

XX

kj
k j
Hi,t
uc,t Gjk̂,t Fz,tj̸=i k̸=jiii
as before:
and then by Hi,t
Now divide both sides by uic,t Gii,t Fz,t
ii
Fl,t
=
ii
i
Fz,t
Hi,t

+

1+

j̸=i
j
X Hi,tj̸=i+ij
X ujc,t Gjî,t Fz,t

i
Hi,t

!

ii
uic,t Gii,t Fz,tji
Giĵ,t Fz,tjj
ujc,t Gjj,t Fz,t
+ i i ii
ii
Gii,t Fz,tuc,t Gi,t Fz,tk uj Gj F kj
X X Hi,tc,t k̂,t z,tj̸=i k̸=i,ji
Hi,t
uci,t Gii,t F i iz,t

54

!

#

B

Competitive equilibrium

B.1 Derivation of Balance of Payment Conditions
Start with the budget constraint of the household
∞
X




Qt (1 + τitc ) pit cit + pit (kit+1 − (1 − δ + rit )kit ) − 1 − τith wit hit = ai0

t=0


P∞
where ai0 = Q−1 bi0 + 1 + rf fi0 + Vi0 and Vi0 =t=0 Qt dit . The government budget
constraint reads as
!
#
"
∞
∞
X
X
X
X
X
m
x
π̂jit + τjit
pjit qjit + τijt
pijt qijt .
Qt pit git +Q−1 bi0 =Qt τitc pit cit + τith wit hit + τitp π̂iit +
t=0

t=0

j̸=i

j̸=i

j̸=i

Combining the two budget constraints, we can obtain
ai0 =

∞
X

Qt pit cit + git + [kit+1 − kit ] − wit hit − Rit kit"

t=0− τitpπ̂iit +

X

!
π̂jit

j̸=i

m
+ τjitXj̸=ixpjit qjit + τijt

X#!
pijt qijt+ P−1 bi0

(B.1)

j̸=i

We now drive the balance of payment conditions for the three scenarios.
Free transfer
In this scenario, we have the headquarter corporate tax base π̂iit as:π̂iit = piit qiit +

X


x
z
1 − τijt
pijt qijt − wit liit − pit δkiit − wit lizj̸=i

and the final good market clearing condition as
Git = cit + (kit+1 − (1 − δ) kit ) + git .

55

(B.2)

Plugging these into equation (B.1), we have∞X

Qt pit Git =

t=0

∞
X

"
Qt

piit qiit +

t=0

+

#
z
pijt qijt − wit liit − pit δkiit − wit liz

j̸=i

+
∞
X

X

X


p

1 − τjt π̂ijt − rjt kijt

− rit kiit

!



j̸=i

"

X
p

Qt wit hit + pit δkit + Rit kit + τitt=0m
π̂jit + τjit

j̸=i

X

#


pjit qjit + 1 + rf fi0 .

j̸=iAnd the labor market clearing condition
X

hit =

ljit + litzjimplies∞X

Qt pit Git =

∞
X

"

+

∞
X
t=0

piit qiit +

Qt

t=0

t=0

"
Qt

X

#
pijt qijt − pit δkiit − rit kiit +p1 − τjt π̂ijt − rjt kijt

j̸=i

j̸=i

X

X

X
p

ljit + pit δkit + Rit kit + τitj̸=imπ̂jit + τjit

j̸=i

X



!
+

#


pjit qjit + 1 + rf fi0 .

j̸=iNow, using the zero profit condition of the final good producer
pit Git (·) = piit qiit +Xp̂jit q̂jit +

j̸=i

X


m
1 + τjit
pjit qjitj̸=iand the market clearing condition of tangible capital
kit =

X

kjit

j

we have
∞
X
t=0

Qt

X

(pjit qjit − pijt qijt ) =

∞
X

"

+

∞
X

−

Qt

t=0

j̸=i

Qt wit

+ 1+r

f



#
p̂jit q̂jit − pit δkiit − rit kiit +

j̸=i

"

t=0

X

X
j̸=i

fi0 .
56

1 − τjtp

π̂ijt − rjt kijt

j̸=i

!
ljit

X

+ pit δXjkjit + Rit

X
j

X
p

kjit + τit

j̸=i

#
π̂jit



!

Finally, using the definition of foreign aﬀiliate’s corporate tax base π̂ijt
π̂ijt = p̂ijt q̂ijt − wjt lijt − pjt δkijt
and the definition of aﬀiliate dividend dijt
dijt = (1 − τjt ) π̂ijt − rjt kijt
we can derive the BoP condition as
∞
X

Qt

X

t=0


(pijt qijt − pjit qjit + dijt − djit ) = − 1 + rf fi0

(B.3)

j̸=i

with the condition that
∞
X
t=0

Qt

"
X X
i

#
(dijt − djit ) = −X
1 + rf fi0

i

j̸=i

Towards expressing the condition in terms of only allocations and policy instruments,
first note that with the intermediate good producer’s FOCs,
ij
wjt = p̂ijt Fl,trjt = 1 − τjtp

ijp̂ijt Fk,t
− pjt δ

we have
ij
dijt = 1 − τjtp [p̂ijt q̂ijt − wjt lijt − δpjt kijt ] − rjt kijt = 1 − τjtp p̂ijt Fz,tzit .
Also, with the final good producer’s FOCs
xpiit = 1 − τijt
pijt
piit = pit Gii,tm
1 + τijt
pjit = pit Gij,tp̂ijt = pjt Gjî,t57

we can then restate the BoP condition as∞
X

Qt pit

t=0

X
j̸=i

p
ji
ij
−
 + 1 − τjtp jt Gj Fz,tzit − (1 − τitp ) Giĵ,t Fz,tzjt
xm
pit î,t1 − τijt
1 + τijt
Gii,t qijt

Gij,t qjit

where we will show later that!

= − 1 + rf fi0x(1 − τjit) uic,t Gij,t
pjt
=
m
pit
(1 + τjit) ujc,t Gjj,ttogether with the household’s intertemporal condition
Qt pit
Rit+1=1+
Qt+1 pit+1
pit+1


ii
ii
where Rit = (1 − τitp ) piit Fk,t
− pit δ = pit (1 − τitp ) Gii,t Fk,t
− pit δ . We thus have the BoP
condition as
X
1

p
ii
1 + (1 − τit ) Gii,t Fk,t
− δ j̸=iΠt
t=0 s=0∞X

1 − τjtp

Gij,t qjit

Gii,t qijt

1 + τijtx
1 − τijt!

−mx
 (1 − τjit
) uic,t Gij,t j ij
ji
G F zit + (1 − τitp ) Giĵ,t Fz,tzjt
m
(1 + τjit) ujc,t Gjj,t î,t z,t−= 1 + rf

(B.4)
 fi0
.
pi0

Transfer pricing
The BoP condition in the transfer pricing can be obtained by following the same procedure.
The only difference is the definition of corporate tax bases. In this scenario, we haveπ̂iit = piit qiit +

X

X

x
z
1 − τijt
pijt qijt − wit liit − pit δkiit − wit liz+ϑijt zit

j̸=i

j̸=i

and
π̂ijt = p̂ijt q̂ijt − wjt lijt − pjt δkijt − ϑijt zit .
With these definitions, the BoP condition in this scenario can be written as
∞
X
t=0

Qt

X


(pijt qijt − pjit qjit + dijt − djit + ϑijt zit − ϑjit zjt ) = − 1 + rf fi0

j̸=i

58

(B.5)

with the condition that
∞
X
t=0

Qt

"
X X
i

#
(dijt − djit ) = −X
1 + rf fi0 .

i

j̸=i

We can easily show that, in the transfer pricing scenario, dijt = 0, ∀j ̸= i given constantreturn-to-scale production function F . Following the steps in the free transfer scenario, the
BoP condition can be restated as
∞
X

X
1

p
ii
Πt
1 + (1 − τit ) Gii,t Fk,t
− δ j̸=i
t=0 s=0

Gij,t qjit

Gii,t qijt

1 + τijt!

x
1 − τijt

−mx(1 − τjit) uic,t Gij,t j ij
jiijj Gî,t Fz,t zit + Gĵ,t Fz,t zjt
m(1 + τjit ) uc,t Gj,t

= 1 + rf

−

(B.6)

 fi0
pi0

Profit shifting
The BoP condition in the profit shifting can be obtained by following the same procedure,
as well. There are two differences: the definition of corporate tax bases and the final goodmarket clearing condition. The corporate tax bases in this scenario are defined asπ̂iit = piit qiit +

X

X
X

x
z
1 − τijt
pijt qijt −wit liit −pit δkiit −wit lizϑijt zit +(φλit − C (λit ) − λit )ϑijt zit
+

j̸=i

j̸=i

j

and
π̂ijt = p̂ijt q̂ijt − wjt lijt − pjt δkijt − ϑijt zit .
And the final good’s market clearing condition, due to costly profit shifting, is modified
as
Git = cit + (kit+1 − (1 − δ) kit ) + git + C (λit )Xϑijt zit .

jWith these changes, we can derive the BoP condition in the profit shifting scenario, of
non-tax-haven countries, as
∞
X
t=0

"
Qt

X

#
(pijt qijt − pjit qjit + dijt − djit + (1 − λit ) ϑijt zit − ϑjit zjt ) − λit ϑiit zit =

j̸=i

−

∞
X
t=0

"
Qt φλitX#ϑijt zit − 1 + rf fi0 . (B.7)

j

For the tax haven, we assume that it imports intermediate goods from other countries for59the balance of payments
∞
X

I
X

Qt

t=0

λit τTp H νit zit −

i=1

X

!
pjT Ht qjT Ht

=

∞
X

Qt

I
X

t=0

j̸=T Hφλit νit zit .

i=1

Summing up the BoP of all countries, we have
∞
X
t=0

Qt

"
X X
i

#!
=−

(dijt − djit ) − (1 − τTp H ) λit νit zitX
1 + rf fi0 .

i

j̸=i

We can again show that dijt = 0, ∀j ̸= i, T H. We can restate the BoP condition in the
profit shifting scenario as
"
X
1


ii
Πt
1 + (1 − τitp ) Gii,t Fk,t−δ
t=0 s=0
j̸=i

∞
X

x
) uic,t Gij,t j ij
(1 − τjitjiijj Gî,t Fz,t zit + Gĵ,t Fz,t zjt
m(1 + τjit ) uc,t Gj,t

Gij,t qjit

Gii,t qijt

1 + τijtx
1 − τijt−m!

+ (1 − φ) λit−xX (1 − τjit
) uic,t Gij,t
j

ij
i
j
j Gĵ,t Fz,t zit
m(1 + τjit ) uc,t Gj,t

(B.8)#
= 1 + rf

B.2 Characterization of the competitive equilibrium
We show how to derive the conditions that characterize the competitive equilibrium here.
Start from the FOCs characterizing households’ choices. We have the intratemporal consumptionlabor condition asiuc,t(1 + τitc ) pit

− i =
uh,t
1 − τith wit

and then from FOCs of the intermediate producers thatjiii
wit = piit Fl,t
= p̂jit Fl,t

and then from the FOC of the final producer that
piit
pit
p̂jit
Giĵ,t =pit
Gii,t =

60

 fi0
.
pi0

which put together gives
uic,t(1 + τitc ) 1(1 + τitc )
1


− i =
=
ji
i
ii
h
h
i
uh,t
1 − τit Gi,t Fl,t
1 − τit G Fl,t∀i , ∀j̸=i

(B.9)

ĵ,t

Now consider the intertemporal FOC of the household, which reads as


uic,t
Rit+1
(1 + τitc )β 1 +=
cuic,t+1pit+1
1 + τit+1
and we have that


ji
ii
Rit = (1 − τitp ) piit Fk,t
− pit δ = (1 − τitp ) p̂jit Fk,t
− pit δthus we have
 i ii
uic,t(1 + τitc ) 
p

=
1
+
1
−
τ
G
F
−
δ
i,t
k,t+1
it+1
c
βuic,t+1
1 + τit+1
i
  i ji
(1 + τitc ) h
p

F
−
δ
1
+
1
−
τ
G
=
it+1
ĵ,t k,t+1c
1 + τit+1∀i , ∀j̸=i(B.10)For the final goods producers and two imported goods (m, n) we have thatmGin,t
(1 + τnit) pnit=imGm,t(1 + τmit ) pmit

then notice that we have from the condition for the intermediate goods producer that
1
pnit
=
x
(1 − τnit )
pnnt
1
pmit
=
x
(1 − τmit )pmmt
and thus combining these equations we havemx
) Gin,t) (1 + τmit
pnnt
(1 − τnit=mxi(1 + τnit ) (1 − τmit ) Gm,t
pmmt

61

and notice that the RHS of the equation is the same for all i. This is equivalent result toChari, Nicolini, Teles (2022) equation (17). Using the final good producer’s FOC, we havexm
(1 − τnit) (1 + τmit
) Gin,tpnt Gnn,t=mx(1 + τnit
pmt Gm) (1 − τmit
) Gim,tm,t∀i , ∀m,n̸=i(B.11)

Toward the dynamic production eﬀiciency, we have from the final good producer’s problem:
piit
pit

m
1 + τijt
pijt
jGi,t =
pjt
Gii,t =

and from the intermediate good producer’s problem:
xpijt
piit = 1 − τijt
x
pjjt = 1 − τjit
pjit
From the final good producer’s problem, for the same good intertemporally we have
piit
pit
p
ii,t+1
Gii,t+1 =
pi,t+1m
1 + τijt
pijt
jGi,t =pjtm
1 + τijt+1
pijt+1j
Gi,t+1 =pjt+1Gii,t =We then haveGii,tpi,t+1 piit=iGi,t+1pit pii,t+1m
1 + τijt
pijt
pj,t+1=jm
pjt 1 + τijt+1
pijt+1
Gi,t+1Gji,tThen using the intermediate good producer’s problem
xpiit = 1 − τijt
pijt

62

we havexm· piit1 + τijt/ 1 − τijt
pj,t+1=jmx
pjt 1 + τijt+1 / 1 − τijt+1 · piit+1
Gi,t+1
Gji,t

Hence


m
x
1 + τijt/ 1 − τijt
pit Gii,t=mx
pi,t+1 Gii,t+1
1 + τijt+1/ 1 − τijt+1
Gji,t+1 pj,t+1
Gji,t

pjt

(B.12)

Now we derive the intertemporal price ratio from the household’s problem. Note that
kit+1 :

0 = −λi Qt pit + Qt+1 λi [pit+1 + Rit+1 ]kjt+1 :

0 = −λj Qt pjt + Qt+1 λj [pit+1 + Rjt+1 ]we have


pit
Qt+1
Rit+1
=
1+
pit+1
Qt
pit+1


Qt+1
Rjt+1
pjt
=
1+
pjt+1
Qt
pjt+1
and

ii
Rit = (1 − τitp ) piit Fk,t
− pit δ
jj
Rjt = 1 − τjtp pjjt Fk,t
− pjt δTherefore,Gji,t Qt+1
Gji,t+1 Qt



Rjt+1
1+
pjt+1






 i
m
x
Gi,t
1 + τijt/ 1 − τijt
Qt+1
Rit+1


=
1
+
m
x
Qt
pit+1 Gii,t+1
1 + τijt+1/ 1 − τijt+1

and further, we haveGji,tGji,t+1p
1 + 1 − τjt+1



jj
−δ
Gjj,t+1 Fk,t+1



=

 i
x
 i

G
1 − τijt+1
pii
 i i,t 1 + 1 − τit+1
Gi,t+1 Fk,t+1−δxm
1 + τijt+1 1 − τijt Gi,t+1m
1 + τijt

63(B.13)

Toward the labor-intangible condition, in the free transfer scenario we haveii
liit : wit = piit Fl,t
X 1 − τjtp

ij
i
ii
i
Hl,t
+
litz : wit = piit Fz,tp̂ijt Fz,tHl,tp
(1 − τit )j̸=i

and combining these equations, we getijii
X 1 − τjtp p̂ijt Fz,t
Fl,t
=1+
i
ii
ii
Hl,t
Fz,t
(1 − τitp ) piit Fz,tj̸=iand notice that from the FOCs of the final goods producers we havepiit = pit Gii,tp̂ijt = pjt Gjî,tthus we haveijii
X 1 − τjtp pjt Gjî,t Fz,t
Fl,t
=
1
+
i
ii
ii
Hl,t
Fz,t
(1 − τitp ) pit Gii,t Fz,t
j̸=i
"
#

ij
X ujc,t Gji,t Fz,t1 − τjtp uic,t pjt=1+
ui Gi F ii (1 − τitp ) ujc,t pit
j̸=i c,t i,t z,t(B.14)

Further, we havemm
(1 + τjit)pjit(1 + τjit
)pjjt
=
i
x
Gj,t
(1 − τjit )Gij,t
pjjt
pjt = j .
Gj,t

pit =

We can then rewrite the above condition asij
ii
X ujc,t Gji,t Fz,t
Fl,t
=
1
+
i
ii
Hl,t
Fz,t
ui Gi F ii
j̸=i c,t i,t z,t

"

#

x
1 − τjtp (1 − τjit
) uic,t Gij,tm(1 − τitp ) (1 + τjit
) ujc,t Gjj,t(B.15)The labor-intangible capital condition is different in the transfer pricing and profit shifting
scenarios. In the transfer pricing scenario, the FOCs with respect to liit and litz for the64intermediate good producer isiiliit : wit = piit Fl,t
X 1 − τjtp

ij
i
i
ii
i
i
+
Hl,t
+
ϑijt Hl,tlitz : wit = piit Fz,tp̂ijt Fz,tHl,t− ϑijt Hl,tp(1 − τit )
j̸=i
j̸=i
X

Following the same steps as in the free transfer scenario, we can obtain:
#
"


ij
ii
X ujc,t Gjî,t Fz,tFl,t1 − τjtp uic,t pjt
τjtp − τitp ϑijt
=1+
+
i
ii
ii
ii
Hl,t
Fz,t
uic,t Gii,t Fz,t(1 − τitp ) ujc,t pit(1 − τitp ) piit Fz,tj̸=i

(B.16)

ij
When setting ϑijt = p̂ijt Fz,t, we can rewrite equation (B.16) as

"
#


ij
ij
ii
X ujc,t Gjî,t Fz,tFl,t1 − τjtp uic,t pjt
τjtp − τitp p̂ijt Fz,t
=1+
+
p
j
i
ii
i
i
ii
ii
Hl,t
Fz,t
u
G
F
(1
−
τ
)
(1 − τitp ) piit Fz,t
u
p
c,t
z,t
i,t
it
it
c,t
j̸=i
#
"


j
j
ij
ij
X ujc,t Gjî,t Fz,t1 − τjtp uic,t pjt uc,t Gî,t Fz,t τjtp − τitp uic,t pjt
+
=1+
ii
ii
uic,t Gii,t Fz,t(1 − τitp ) ujc,t pit uic,t Gii,t Fz,t(1 − τitp ) ujc,t pit
j̸=i
!
ij
X ujc,t Gjî,t Fz,tuic,t pjt
=1+
ui Gi F ii ujc,t pit
j̸=i c,t i,t z,t!
ijx
X ujc,t Gjî,t Fz,t(1 − τjit) uic,t Gij,t(B.17)
=1+
j
j
i
i
ii
m
u
G
F
(1
+
τ
)
u
G
c,t
z,t
i,t
jit
c,t j,tj̸=iIn the profit shifting senario, the FOCs with respect to liit and litz for the intermediate
good producer isiiliit : wit = piit Fl,t
h
X
ii
i
i
i
litz : wit = piit Fz,tHl,t+ (1 − λit ) Hl,tϑijt (zit ) − λit Hl,tϑiit (zit )i− Ci (λit ) Hl,t

X

j̸=i
i
ϑijt (zit ) + φλit Hl,t

j

X
j

i
ϑijt (zit )X (1 − τTp Ht )1 − τjtp ii
iiip̂ijt Fz,t Hl,t − Hl,t ϑijt (zit ) +λit (1 − φ) Hl,tϑijt (zit )+pp(1 − τit )(1 − τit )jj̸=i,T H
X

65

ij
Setting ϑijt = p̂ijt Fz,t, we can derive the labor-intangible condition as"ij
ii
X ujc,t Gjî,t Fz,t
Fl,t
= 1+
i
ii
Hl,t
Fz,t
ui Gi F ii
j̸=i c,t i,t z,tx(1 − τjit) uic,t Gij,tm(1 + τjit) ujc,t Gjj,t!# λi (1 − φ) (τitp − τTp Ht )1 − C (λit ) +.
(1 − τitp )(B.18)

Lastly, when intangible capital is rival, we have the FOCs as:iiliit : wit = piit Fl,t
z
ii
i
liit
: wit = piit Fz,tHl,tzwhere liit
is the amount of worker headquarter i employs to produce the rival intangible
capital for itself. Then we have the labor-intangible capital equilibrium condition as
ii
Fl,t
= 1.
ii
i
Fz,t
Hl,t

B.3 The Example
We solve for the optimal intangible capital investment with specific functional forms imposed
in Section 3.5.
Free transfer
With the functional forms imposed, we can derive the following FOCs for the intermediate
good firm using equation (20):α γ−1liit : wit = piit γAi (Ni zit )ϕ kiit
liit

(B.19)

α γ−1lijt : wjt = p̂ijt γAj (Ni zit )ϕ kijtlijt
∀j ̸= i
ϕ α−1 γpkiit : rit = 1 − τjt piit αAi (Ni zit ) kiit liit − pit δ


α−1 γ
kijt : rjt = 1 − τjtp p̂ijt αAj (Nj zit )ϕ kijt
lijt − pjt δX 1 − τjtpijilitz : wit = piit Fz,tAi +p̂ijt Fz,tAip
(1 − τit )j̸=ixqijt : piit = 1 − τijt
pijt
∀j ̸= i(B.20)(B.21)(B.22)(B.23)(B.24)

From the FOCs of labor and tangible capital inputs, equations (B.19), (B.20), (B.21),

66

(B.22), we can getliit =

kiit =

γ
wit
γ
wit

1−α 
 1−γ−αγ
 1−γ−α(1 − τitp ) α
rit + (1 − τitp ) pjt δ(1 − τitp ) α
rit + (1 − τitp ) pjt δα 1−γ−αpiit Ai (Ni zit )1−γ
 1−γ−αϕpiit Ai (Ni zit )ϕ1 1−γ−α1
 1−γ−α(B.25)(B.26)

and
lijt =
kijt =

γ
wjt
γ
wjt

1−α
 1−γ−αγ
 1−γ−αα! 1−γ−α 11 − τjtp α
ϕ 1−γ−αp̂ijt Aj (Nj zit )rjt + 1 − τjtp pjt δ1−γ! 1−γ−α 11 − τjtp α
ϕ 1−γ−α

p̂
A
(N
z
)
ijt
j
j
it
rjt + 1 − τjtp pjt δ(B.27)(B.28)

Plugging these into the FOC of litz , equation (B.23), we can obtain the optimal intangible
capital as:1−γ−α! 1−γ−α−ϕ
X 1 − τjtpr̂jt (τjt ) Λijt
zitF T =(1 − τitp )jwhere we defineAiΛjt ≡ ϕwitp̂ijt Aj Njϕandr̂jt (τjt ) ≡

γ
wjt

γ 

α
rjt + pjt δ1α  1−γ−α! α
1 − τjtp (rjt + pjt δ) 1−γ−αrjt + 1 − τjtp pjt δTransfer pricing
The maximization problem of the transfer pricing case is specified in equation (21). Optimallabor and tangible capital inputs can be derived from the same FOCs as in the free transfercase. The FOC for litz with transfer pricing is different:"
liz : 0 = (1 − τitp ) piit Fztii Ai − wi + Ai

X#ϑijt +j̸=iX

1 − τjtp

p̂ijt Fztij Ai − Ai ϑijt



j̸=i

Plugging in optimal labor and tangible inputs using equations (B.25), (B.26), (B.27),

67

(B.28), we can derive the optimal intangible capital zit as
X

zitT P =1−γ−α! 1−γ−α−ϕr̂jt (τjt ) Λijt

jProfit shifting
The maximization problem of the profit shifting case is specified in equation (23). As before,
we obtain the optimal labor and tangible capital inputs can be derived from the same set of
FOCs. In the profit shifting case, firms need to decide the share of intangible capital to shift
to the tax haven λit , in addition to intangible capital investment zit . The FOC with respectto λit is
λit : 0 = − (1 − τitp ) zit

hiX′ϑijt · −1 − Ci (λit ) + φ + (1 − τTp Ht ) zit
ϑijt · [1 − φ]

X
j

j

Then
′

Ci (λit ) =(τitp − τTp Ht )(1 − φ)(1 − τit )

and


′ −1

λit = Ci(τitp − τTp Ht )(1 − φ)(1 − τit )
(B.29)The FOC wrt to litz is"litz : 0 = (1 − τitp ) piit Fztii Ai − wit + Ai
+ φλit+X

X− λit ϑiit (zit ) + (1 − λit )



j̸=i

!#

ϑijt (zit ) − Ci (λit )Xϑijt (zit )jϑijt (zit )j

1 − τjpXp̂ijt Fztij Ai − ϑijt (zit )j̸=i,T H+ (1 − τTp Ht ) λit (1 − φ) Ai

Xϑijt (zit )j

Plugging in optimal labor and tangible inputs using equations (B.25), (B.26), (B.27),

68

(B.28), we can derive the optimal intangible capital zit in this case as

zitP S =X!r̂jt Λijtj

= zitT P ·
λit (1 − φ) (τitp − τTp Ht )1 − Ci (λit ) +(1 − τitp )λit (1 − φ) (τitp − τTp Ht )1 − Ci (λit ) +(1 − τitp )1−γ−α! 1−γ−α−ϕ1−γ−α 1−γ−α−ϕB.4 Derivation of Ramsey Equilibria
The relaxed Ramsey problem is characterized by a solution to the following problem
max
At

I
X

ωi

i=1

∞
X


β t v i cit , hit ; φi − φi Wi0 (τi0c )

t=0subject to the following constraints

F ii (zit , kiit , liit ) =qiit +

X

qijt

∀i

j̸=i
ij

F (zit , kijt , lijt ) =q̂ijt∀j̸=i ∀i

Gi (q1it , ..., qIit , q̂1it , ..., q̂Iit ) =cit + git + xit∀ixit =kit+1 − (1 − δ) kitzz
H i (l1t, ..., lIt) =zit ∀iXhit =ljit + litz
j

kit =

X

kjit

j

k0

given

where
Wi0 (τi0c ) =uic0(1 + τi0c )



 fi0
bi0
Ri0 + Q−1
+ 1 + rf
pi0
pi0

Ri0 = (1 − δ + ri0 ) ki0
v i cit , hit ; φi = ui (cit , hit ) + φi uic,t cit − uih,t hit

69and the set of constraints can be simplified tocit + git +X

kjit+1 − (1 − δ)

X

j

Gi

kjit =jX

q1it , ..., F ii H i (litz ) , kiit , liit −zz) , kIit , lIit
) , k1it , l1it , ..., F Ii H I (lIt
qijt , ..., qIit , F 1i H 1 (l1tj̸=iand φi is the multiplier on the implementability condition
∞
X
t=0



β uic,t cit − uih,t hit =t

uic0(1 + τi0c )



 fi0
bi0
Ri0 + Q−1
+ 1 + rfpi0
pi0

, ∀i = 1, . . . , I.

If the implementability condition is held in i, then we have φi = 0 and v i (cit , hit ; φ) =ui (cit , hit ). More generally, the first-order conditions of this problem for periods t > 0 areicit : 0 = β t vc,t
− λit

(B.30)

jiiljit : 0 = β t vh,t
+ λit φi Giĵ,t Fl,t

(B.31)

i
ii
liit : 0 = β t vh,t
+ λit Gii,t Fl,t"iiiilitz : 0 = β t vh,t
+ λit Gii,t Fz,t
Hi,t
+

X

(B.32)#kGik̂,t Fztki Hi,t
+

k̸=i

"

ij
jj j
i
λjt Gjî,t Fz,tHi,t+ Gjj,t Fz,tHi,t +

X#k
Gjk̂,t Fztkj Hi,t

(B.33)

k̸=jqjit : 0 = −λjt Gjj,t + λit Gij,t(B.34)

jikjit+1 : 0 = (1 − δ) λit+1 − λit + λit+1 Giĵ,t+1 Fk,t+1(B.35)

ii
kiit+1 : 0 = (1 − δ) λit+1 − λit + λit+1 Gii,t+1 Fk,t+1(B.36)For the initial period t=0, the first-order condition for ci0 is
i
ci0 : 0 = vc,t− λit + φi

∂Wi0
∂ci0

(B.37)

Optimal conditions when t > 0
We can first get the no intratemporal wedge condition from equations (B.30)-(B.32) as
i
vc,t
1
1
− i = i ii = i ji
vh,t
Gi,t Fl,t
G Fl,tĵ,t70∀i ∀j̸=i!

Next, from equations (B.30), (B.36) and (B.35), we can get the no intertemporal wedge
condition as
i

 
vc,t
ji
ii
i
i
F
=
(1
−
δ)
+
G
=
(1
−
δ)
+
G
F
i,t+1 k,t+1i
ĵ,t+1 k,t+1
βvc,t+1∀i ∀j̸=i

Now, towards static eﬀiciency, for any imported goods (m, n) in location i we have fromequations (B.30) and (B.34) thatnGin,t
Gnn,t vc,t
=
m
Gim,t
Gm
m,t vc,t∀i ∀m,n̸=iTowards dynamic eﬀiciency, from equations (B.34) and (B.36) we can derive

Gij,t
ii
=
(1 − δ) + Gii,t+1 Fk,t+1
i
Gj,t+1j
Gjj,t vc,tj
Gjj,t+1 βvc,t+1!
∀jFinally, for the intangible eﬀiciency, from equations (B.30), (B.32) and (B.33) we havejijiiGjî,t Fz,tX vc,t
Fl,t
=1+
ii
i
Fz,t
Hi,t
v i Gi F ii
j̸=i c,t i,t z,t
" j
!
#
ji
j
jj
k v j Gj F kj
X Hi,tX Hi,tGiĵ,t Fz,tvc,tGjj,t Fz,tc,t k̂,t z,t++ i i ii +
i
ii
i
c
ii
Hi,t
Gii,t Fz,tvc,t Gi,t Fz,t
Hi,t
vi,t
Gii,t Fz,tk̸=i,jj̸=iC

Proofs

Proof of Lemma (1).
From the two FOCs of static eﬀiciency characterizing the Pareto frontier
cit : 0 = β t uic,t − λitqjit : 0 = −λjt Gjj,t + λit Gij,t
we can immediately obtain
λjt Gjj,t = λit Gij,tujc,t Gjj,t = uic,t Gij,t71∀ihenceuic,t Gij,tujc,t Gjj,t= 1.

Proof of Lemma (2).
We prove the properties of the return to profit shifting, Ω(τitp ), here. We first show thatΩ(τitp ) > 1 if τitp > τTp Ht , and Ω(τitp ) = 1 if τitp = τTp Ht . Start from the FOC with respect to λitof the firm’s problem in the profit shifting scenario:
λit : 0 = − (1 − τitp ) zit

X

h
i
X
′
ϑijt · −1 − Ci (λit ) + φ + (1 − τTp Ht ) zit
ϑijt · [1 − φ] .

jjThen, we have′Ci (λit ) =(τitp − τTp Ht )(1 − φ) .
(1 − τit )Assuming that C (λ) = λ − (1 − λ) log(1 − λ), we can solve for λit analytically as(1 − φ)(τitp − τTp Ht )λit = 1 − exp −1 − τitp

.

Then, we can show that when τitp > τTp Ht , λit > 0 andΩ(τitp ) = 1 − Ci (λit ) +λit (1 − φ) (τitp − τTp Ht )(1 − τitp )′= 1 − Ci (λit ) + λit Ci (λit ) > 1.
The inequality comes from the fact that C (λ) is an increasing and convex function in λ.
It is straightfoward to see that when τitp = τTp Ht , λit = 0 and Ω(τitp ) = 1. We then show thatΩ(τitp ) is increasing in τitp
∂λit ∂λit ′∂λit∂Ω (τitp )′′′= −C (λit ) p +p
p C (λit ) + λit C (λit )
∂τit
∂τit
∂τit
∂τitp
∂λit
′′
= λit C (λit ) p∂τit
1 − τTp Ht= λit (1 − φ)
> 0.
(1 − τitp )2
Proof of Proposition (1). We start by proving in the transfer pricing scenario, all compet72

itive equilibria that satisfy static eﬀiciency will have ineﬀicient intangible capital investment.
The competitive equilibrium condition governing intangible capital in this scenario reads
ijii
X ujc,t Gji,t Fz,t
Fl,t
=1+
i
ii
Hl,t
ui Gi F ii
Fz,tj̸=i c,t i,t z,t" i i #x1 − τjit
G uct
 jt
m
1 + τjit Gjjt ujct

Note that static eﬀiciency condition requires no trade tax wedges, i.e.

x(1−τjit)= 1, ∀j̸=i ,m(1+τjit )G i uic,t
and that Gjt= 1, ∀j̸=i . Hence, when static eﬀiciency is satisfied, the condition abovejj
jt uc,tbecomes:ij
ii
X ujc,t Gjî,t Fz,t
Fl,t
=1+
.
(C.1)
i
ii
Hl,t
Fz,t
ui Gi F ii
j̸=i c,t i,t z,tThe same condition on the Pareto frontier isij
ii
X ujc,t Gjî,t Fz,t
Fl,t
=1+
+ Σit
i
ii
Hl,t
Fz,t
ui Gi F ii
j̸=i c,t i,t z,t(C.2)

with Σit > 0 in presence of positive spillovers. We have thus shown that in the transfer
pricing scenario when the static eﬀiciency condition is met, the marginal rate of technicalsubstitution between labor and intangible capital is always smaller in the competitive equilibrium. Hence, the competitive equilibrium intangible capital allocation is ineﬀicient. We now
show that there is a set of corporate income taxes {τitp }∀i that achieves eﬀicient intangible
capital allocation in the profit shifting scenario. The competitive equilibrium condition of
intangible capital, with static eﬀiciency imposed, in this scenario isij
ii
X ujc,t Gji,t Fz,tFl,t=1+
· Ω(τitp )
i
ii
i
i
ii
Hl,t Fz,tu G F
j̸=i c,t i,t z,twhere Ω(τitp ) = 1 − Ci (λit ) +p
λit (1−φ)(τit−τTp Ht )(1−τitp )pcapital amounts to find τit that satisfies(C.3). Achieving eﬀicient allocation of intangible

ij
ij
ii
X ujc,t Gji,t Fz,tX ujc,t Gjî,t Fz,t
Fl,t
p
=1+
· Ω(τit ) = 1 ++ Σit , ∀i
i
ii
Hl,t
Fz,t
ui Gi F ii
ui Gi F ii
j̸=i c,t i,t z,tj̸=i c,t i,t z,t73

which can be re-written asij
X ujc,t Gjî,t Fz,tΩ(τitp ) =

j̸=i

!−1

ii
uic,t Gii,t Fz,tij
X ujc,t Gjî,t Fz,t
j̸=iii
uic,t Gii,t Fz,t

!
+ Σit

>1

(C.4)

We have shown in Lemma 2 that Ω(τitp ) is a continuous and increasing in τitp , with Ω(τitp ) =0 when τitp = τTp Ht and Ω(τitp ) → ∞ as τitp → 1. Therefore, by the mean value theorem there
exists a unique τitp that satisfies (C.4) for all i ∈ {1, 2, ..., I}.
Lastly, we prove that in the free transfer scenario, all competitive equilibria that satisfy
static eﬀiciency have ineﬀiciency intangible capital investment. The competitive equilibrium
condition of intangible capital in this scenario, with static eﬀiciency, isijii
X ujc,t Gji,t Fz,tFl,t1 − τjtp
=1+
i
ii
Hl,t
Fz,t
ui Gi F ii (1 − τitp )j̸=i c,t i,t z,t(C.5)

It amounts to show that there does not exist a set of corporate income taxes {τitp }∀i thatsatisfiesijij
X ujc,t Gjî,t Fz,tX ujc,t Gji,t Fz,t1 − τjtp+ Σit , ∀i1+
p = 1+
i
i
ii
i
i
ii
u
G
F
(1
−
τ
)
u
G
F
c,t
z,t
c,t
z,t
i,t
it
i,t
j̸=i
j̸=i
which can be written as a system of I linear equations with I unknowns:−Ψ1 Υ21 ... ΥI1
1 − τ1tp Υ12 −Ψ2 ... ΥI2   1 − τ2tp 



 ...
  ...  = 0
...
...
...



p
Υ1I
Υ2I ... −ΨI
1 − τIt|{z}Ξwhere we define ΨI =

 j

P

ij
uc,t Gj Fz,t

j̸=i



î,t
ii
uic,t Gii,t Fz,tuj Gj F ij

+ Σit and Γij = uic,t Gii,t Fz,tii . It is straightforward toc,ti,t z,tshow that Ξ is a strictly diagonally dominant matrix as
ij
j
Fz,t
X ujc,t Gî,tij
X ujc,t Gji,t Fz,tX
|Ψi | =+
Σ>=|Υji |
itui Gi F ii
ui Gi F ii
j̸=i c,t i,t z,tj̸=i c,t i,t z,tj̸=i

By Gershgorin’s circle theorem, matrix Ξ is non-singular, then the above linear systemof equations has a unique solution:
τitp = 1, ∀i.

74

However, corporate income taxes are strictly smaller than 1. Hence, we cannot findcorporate income taxes that implements eﬀicient intangible capital allocation in the freetransfer scenario.
Proof of Proposition (2).
The budget constraint of the household is given by
∞
X





Qt (1 + τitc ) pit cit + pit (kit+1 − (1 − δ + rit ) kit ) − 1 − τith wit hit = Q−1 bi0 + 1 + rf fi0 +Vi0

t=0

where Vi0 =

(C.6)

P∞

t=0 Qt dit . Using the household’s first order conditions

β t uic,t − λQt (1 + τitc ) pit = 0
β t uih,t − λQt 1 − τith wit = 0
and the Euler equation−λQt pit + λQt+1 pit+1 (1 − δ + rit ) = 0
we can obtainui
Qt (1 + τitc ) pit cit = β t ict (1 + τi0c ) pi0 cit
uc0
ui(1 + τi0c ) pi0 hit
Qt 1 − τith wit hit = β t ht
uic0
and

Qt pit= 1 − δ + rit .
Qt+1 pit+1

Plugging these into equation (C.6), we then have
∞
X
t=0


β

t



uiht
uict
c
c
(1 + τi0 ) pi0 cit − i (1 + τi0 ) pi0 hit −[pi0 (1 − δ + rit )ki0 ] = Q−1 bi0 + 1 + rf fi0 +Vi0
i
uc0
uc0

with the transversality condition thatlim Qt kit+1 = 0.

t→∞

75

The constraint can be further simplified into∞X
β uict cit − uiht hit =
t

t=0



uic0
(1 + τi0c )bi0
fi0 Vi0
Ri0 + Q−1
+ (1 + rf )+pi0
pi0 pi0, ∀i = 1, ..., I

where Ri0 = (1 − δ + ri0 )ki0 .
We now show Vi0 = 0 under all three model scenarios. The dividend in the free transfer
scenario is:X
dit = πiit (zit ) − (1 − τitp ) wit litz +πijt (zit )
j̸=i

where
"
πiit (zit ) = (1 − τitp ) piit qiit +#x(1 − τijt)pijt qijt − wit liit − δpit kiit − rit kiitj̸=i"
= (1 − τitp )Xpiit qiit +X#x(1 − τijt)pijt qijt − wit liit − (rit + δ (1 − τitp ) pit ) kiit

j̸=i

and

πijt (zit ) = 1 − τjtp [p̂ijt q̂ijt − wjt lijt − δpjt kijt ] − rjt kijt 
= 1 − τjtp [p̂ijt q̂ijt − wjt lijt ] − rjt + δ 1 − τjtp pit kijt .
The sum of discounted dividends is
"
#
∞
∞
X
X
X
p
Qt dit =Qt πiit (zit ) − (1 − τit ) wit litz +πijt (zit )
t=0

t=0

=

∞
X

"

j̸=i

"

Qt (1 − τitp ) piit qiit +

t=0

X

#

x
(1 − τijt)pijt qijt − wit liit − (rit + δ (1 − τitp ) pit ) kiit

j̸=i

+

X

1 − τjtp

[p̂ijt q̂ijt − wjt lijt ] − rjt + δ

j̸=i

76

1 − τjtp

pjt kijt#− (1 − τitp ) wit litz.

The first-order conditions for the intermediate-good producers are:jiii
= p̂jit Fl,tliit : wit = piit Fl,t
X 1 − τjtp

ij
i
ii
i
Hl,t
+
liz : wit = piit Fz,tp̂ijt Fz,tHl,tp
(1 − τit )
j̸=i


ij
kijt : rjt = 1 − τjtp p̂ijt Fk,t
− pjt δxpijt .
qijt : piit = 1 − τijt

Plugging in these equations, we have∞X
t=0

Qt dit =

∞
X

"



ii
ii
ii z
i
Qt (1 − τitp ) piit F ii (liit , kiit , zit ) − Fl,tliit − Fk,tkiit − Fz,tlit Hl,t

t=0

+

X

1 − τjtp

p̂ijt F

ij

ij
ij
ij z
i
(lijt , kijt , zit ) − Fl,tlijt − Fk,tkijt − Fz,tlit Hl,tj̸=iWith constant return-to-scale production functions F and H, we have
ii
ii
ii z
i
F ii (liit , kiit , zit ) − Fl,tliit − Fk,tkiit − Fz,tlit Hl,t
=0

and
ij
ij
ij z
i
F ij (lijt , kijt , zit ) − Fl,tlijt − Fk,tkijt − Fz,tlit Hl,t= 0, ∀j ̸= i.

Thus, we have dit = 0 and
Vi0 =

∞
X

Qt dit = 0.

t=0The dividend in the transfer pricing scenario is:
dit = πiit (zit ) − (1 − τitp )wit litz +X
πijt (zit ) + (τjtp − τitp )ϑijt zit .

j̸=i

The first-order conditions for the intermediate-good producers are:jiii
= p̂jit Fl,tliit : wit = piit Fl,t
X
ij
ii
i
i
liz : wit = piit Fz,tHl,t+p̂ijt Fz,tHl,tkijt : rjt =1 − τjtp



j̸=i
ij
p̂ijt Fk,t
− pjt δxqijt : piit = 1 − τijt
pijt .
77





#

ij
Following the same procedure and assuming that ϑijt = p̂ijt Fz,t, the sum of discounted
dividends can be derived as
"
∞
∞
X
X


ii
ii
ii z
i
liit − Fk,tkiit − Fz,tlit Hl,t
Qt dit =Qt (1 − τitp ) piit F ii (liit , kiit , zit ) − Fl,t
t=0

t=0

+

X


p


i



#

ij
ij
ij z
lijt − Fk,tkijt − Fz,tlit Hl,t
1 − τjt p̂ijt F ij (lijt , kijt , zit ) − Fl,t

j̸=i

= 0.
The dividend in the profit shifting scenario is:"
dit = πiit (zit ) + (1 − τitp )+X−wit litz + (1 − λit )

X



#

ϑijt zit + λit (φ − C(λit ))νit zit − ϑiit zit

j̸=i
πijt (zit ) − (1 − τjtp )ϑijt zit + (1 − τTp Ht )λit (1 − φ)νit zit .

j̸=i

The first-order conditions for the intermediate-good producers are:ji
ii
liit : wit = piit Fl,t
= p̂jit Fl,t
h
X
ii
i
i
i
liz : wit = piit Fz,tHl,t+ (1 − λit ) Hl,tϑijt − λit Hl,tϑiiti
− Ci (λit ) Hl,t

X

j̸=i
i
ϑijt + φλit Hl,t

X

ϑijt

j
j

p
X 1 − τjt X
 (1 − τTp Ht )iiiiiλit (1 − φ) Hl,tp̂ijt Fz,t Hl,t − Hl,t ϑijt +
ϑijt
+
p
p
(1 − τit )(1 − τit )jj̸=i,T H

kijt : rjt = 1 − τjtp

ijp̂ijt Fk,t
− pjt δxqijt : piit = 1 − τijt
pijt .

ij
Following the same procedure and assuming that ϑijt = p̂ijt Fz,t, the sum of discounted78

dividends can be derived as
"
∞
∞
X
X


ii
ii
ii z
i
Qt dit =Qt (1 − τitp ) piit F ii (liit , kiit , zit ) − Fl,tliit − Fk,tkiit − Fz,tlit Hl,t
t=0

t=0

+

X


p


i



#

ij
ij
ij z
1 − τjt p̂ijt F ij (lijt , kijt , zit ) − Fl,tlijt − Fk,tkijt − Fz,tlit Hl,t

j̸=i

= 0.

Proof of Proposition (3). The proof that balance of payments is satisfied follows directlyfrom arguments in Chari et al. (2023). Toward showing that one can implement eﬀicient
allocation of intangible capital, first impose static eﬀiciency to get
ii
Fl,t
=
ii
i
Fz,t
Hi,t

1+

X

ij
ujc,t Gjî,t Fz,t

j̸=i

ii
uic,t Gii,t Fz,t!!
Ω (τitp )(C.7)

now we want to find taxes that implement the (52) using C.7. Thus equating the right handsides we get:

1+

X

ij
ujc,t Gjî,t Fz,t

j̸=i

ii
uic,t Gii,t Fz,t!!
Ω (τitp ) =

"
#
mi
nm
m
m
mm
m X
n
X Hi,tX vc,tGim̂,t Fz,t
Gm
vc,t
F
Hi,t
Gm
m,t z,tn̂,t Fz,t
1+
+
+ i
+
i
ii
ii
i
ii
v i Gi F ii
Hi,t
Gii,t Fz,tvc,t Gii,t Fz,tvi
Hi,tGii,t Fz,tm̸=im̸=i c,t i,t z,tm̸=i c,t n̸=i,mm m im
X vc,tGî,t Fz,tthus we have
#
!
"
i
mi
m
nm
m
m X
m
m
mm
n
X
X
G
F
G
F
H
v
G
F
v
H
z,t
z,t
i,t
c,t
m,t
z,t
c,t
i,t
n̂,t
m̂,t
+
+ i
+
Ω (τitp ) = 1 +
i
i
ii
i
i
ii
i
ii
i
ii
v
G
F
H
G
F
v
G
F
vi
Hi,t
Gii,t Fz,t
c,t
z,t
z,t
c,t
z,t
i,t
i,t
i,t
i,t
m̸=i
m̸=i
m̸=i c,t n̸=i,m
ij !!−1
X ujc,t Gjî,t Fz,t
× 1+
ii
uic,t Gii,t Fz,tj̸=i
m m im
X vc,tGî,t Fz,twhere we have from Lemma 2 thatlim
Ω (τitp ) = ∞p

τit →1lim Ω (τitp ) = 0pτit→τTp H79

then by mean value theorem there exists a unique τbitp > τTp H that implements Ramsey allocation.
Optimal conditions when t = 0
In period 0, we use replace equation (B.30) with (B.37) as the first-order condition withrespect to consumption We can first get the no intratemporal wedge condition as
−

i
i0
vc,0
+ φi ∂W
∂ci0
i
vh,0

=

1
ii
Gii,t Fl,0=1
jiGiĵ,t Fl,0∀i ∀j̸=i

The no intertemporal wedge condition is
i
i0
vc,0
+ φi ∂W
∂ci0
i
βvc,1


 
ji
ii
= (1 − δ) + Giĵ,1 Fk,1
= (1 − δ) + Gii,1 Fk,1

∀i ∀j̸=i

The static eﬀiciency condition is




n
n0
+ φn ∂W
Gnn,0 vc,0
Gin,0
∂cn0


=
∂Wm0
Gim,0
m
m
n
Gm,0 vc,0 + φ ∂cm0∀i ∀m,n̸=i

And the dynamic eﬀiciency condition is


Gij,0
Gij,1


ii
(1 − δ) + Gii,1 Fk,1
=

Gjj,0



∂W
c
vc,0
+ φj ∂cj0j0
jGjj,1 βvc,1




∀j

Finally, the intangible eﬀiciency isjijiiGjî,0 Fz,0X vc,0
Fl,0
=1+
ii
i
Fz,0
Hi,0
v i Gi F ii
j̸=i c,0 i,0 z,0
!
#
" j
ji
j
jj
k v j Gj F kj
X Hi,0
X Hi,0Giĵ,0 Fz,0vc,0Gjj,0 Fz,0c,0 k̂,0 z,0+ i i ii +
+
i
ii
i
c
ii
Hi,0
Gii,0 Fz,0
vc,0 Gi,0 Fz,0
Hi,0
vi,0
Gii,0 Fz,0k̸=i,jj̸=iD∀iQuantitative model details

Here we provide more details about the quantitative model, which builds closely on Dyrda
et al. (2022). Time is discrete and indexed by t = 1, 2, . . .. There are I regions indexed by iand j, each populated by a representative household, a measure of heterogeneous firms, and80

a government. Regions differ in population, total factor productivity, trade costs, FDI costs,
labor income taxes, and corporate income taxes. Households choose consumption, labor
supply, tangible investment, and bond holdings. Firms in each region decide the following:where to export and where to open foreign subsidiaries; how much labor to hire and tangible
capital to rent in the parent division and each subsidiary; and how much intangible capitalto produce in the parent division. As in McGrattan and Prescott (2009). intangible capital
is nonrival and is used simultaneously in all of a firm’s divisions, both foreign and domestic.
Multinational firms (firms with foreign aﬀiliates) use transfer pricing to allocate the costsof producing intangible capital across their foreign aﬀiliates in proportion to the scale at whichthese aﬀiliates use this capital. Aﬀiliates license the right to use intangible capital from thedivision that owns this capital, and MNEs can shift profits by selling their intangible capital
to aﬀiliates in lower-tax regions. We denote the region with the lowest corporate income
tax rate by LT . Additionally, there is an unproductive tax haven that is populated by arepresentative household and a government, labelled as T H, where no economic activity
takes place. MNEs based in high-tax regions can transfer their intangible capital rights toeither the low-tax region or the tax haven, provided that they have established aﬀiliates
there.
Throughout this appendix, we use capitals to denote aggregate variables and lower-cases
to denote microeconomic firm-level variables. We omit time subscripts where appropriate forbrevity,

D.1

Households

Each region i has a representative household with preferences over sequences of consumption,∞{Cit }∞
t=0 , and labor supply, {Lit }t=0 , given by
∞
X


β

t


log

t=0

Cit
Ni





Lit
+ ψi log 1 −
Ni


.

(D.1)

Households choose consumption, labor supply, tangible investment, {Xit }∞t=0 , and internationally∞
traded bonds, {Bit+1 }t=0 to maximize utility subject to a sequence of budget constraints,
Pit [(1 + τict )Cit + Xit ] + Pbt Bit+1 = (1 − τiℓt ) Wit Lit + (1 − τikt )Rit Kit + Bit + Dit ,(D.2)

a law of motion for tangible capital,
Kit+1 = (1 − δ)Kit + Xit ,
81

(D.3)

and initial conditions on capital and bonds, Ki0 and Bi0 . Households take the wage, Wit ,the labor income tax rate, τiℓt , the rental rate, Rit , the bond price, Pbt , and dividends, Dit ,as given. The first-order conditions of the household’s problem are
Cit /Ni(1 − τi )Wit=,
1 − Lit /Ni(1 + τcit )Pit
Cit Pt+1
Pbt = β,Cit+1 Pt
Cit
(1 − τikt+1 )Rit+1
1=β
1−δ+
.
Cit+1
Pit+1

(D.4)
(D.5)
(D.6)

We define rit as the ex-depreciation rental rate on tangible capital. Note that in a steadystate, the Euler equation for capital is(1 − τik )Ri
1=β 1−δ+
Pi

D.2





Pi
1−β
⇒ Ri =
δ .
(1 − τik )
β

(D.7)

Final goods

Each region has a representative final-good producer that combines domestic and foreign
products into a nontradable aggregate that is bought by households and the government for
consumption. The final good is a constant-elasticity-of-substitution aggregate of products
from different source countries.

Qit =

ρ
# ρ−1

" J Z
X
j=1

qjit (ω)

ρ−1
ρ

dω

,

(D.8)

Ωjit

where qjit (ω) is the quantity of variety ω from region j, Ωjit is the set of goods from j available
in i (determined by firms’ exporting and FDI decisions specified later), and ρ is the elasticity
of substitution between varieties. The demand curve for each variety can be written as1

pjit (ω) = Pit Qitρ qjit (ω)− ρ .
1

(D.9)

The aggregate price index is

Pit =

" J Z
X
j=1

1
# 1−ρ

pjit (ω)1−ρ dω
Ωjit

82

.

(D.10)

D.3

Intermediate firm’s problem

A firm’s objective is to maximize its dividend payout,dit (a) = πiit (a, z; JX ) +Xπijt (a, z) + 1{λT H >0} πi,T H,t (a, z; JF ),(D.11)

j∈JF

in each period by choosing the following objects: where to export and open foreign aﬀiliates,
JX and JF ; how much intangible capital to produce, z; how much local labor and tangible
capital to hire in each of its divisions, ℓ = (ℓij )j∈I and k = (kij )j∈I ; how much to sell to eachof its markets through exporting and/or FDI, q = (qij , q̂ij )j∈I ; and how much of its intangible
capital property rights to shift, λ = (λLT , λT H ).
The domestic parent corporation’s after-tax profits, πii (a, z; Jx ), are given by
πiit (a, z; JX ) = (1 − τit ) piit (qiit )qiit +Xpijt (qijt )qijt +

j∈JXt


−Wit ℓiit +X(1 − λLT t − λT Ht )ϑijt (z)z

j∈JF

X

X

X

F

z
+
κiX +
κiF + κi,T H 1{λT H >0}
Ai j∈Jj∈J(D.12)

− δPit kit
−Wit [Ci,T H (λT H ) + Ci,LT (λLT )] νit (z)z − (λT H + λLT )ϑiit (z)z− rit kit .

The first line contains revenues from sales and licensing the portion of intangible capitalthat is not transferred to the low-tax region or the tax haven. The second line contains
labor costs of domestic production workers, workers hired to set up export relationships andforeign aﬀiliates, and depreciation expenses. The last line contains labor costs of workers
hired to engage in profit shifting, licensing fees paid to the low-tax region and the tax haven,and capital expenditures net of depreciation. After-tax profits in foreign aﬀiliates in high-tax
regions are given by
πijt (a, z) = (1−τjt ) [pijt (q̂ijt )q̂ijt − Wjt ℓj − δPjt kjt − ϑijt (z)z]−rjt kjt , j ∈ JF \{LT }. (D.13)
The low-tax aﬀiliates’ after-tax profits are
πi,LT,t (a, z; JX ) = (1 − τLT t ) pi,LT,t (q̂i,LT,t )q̂i,LT,t +X
j∈JF ∪{i}\{LT }(D.14)

λLT,t ϑijt (z)z


−WLT,t ℓLT,t − δPLT,t kLT,t − (1 − λLT,t )ϑi,LT,t (z)z − rLT,t kLT,t .

83

The first line includes revenues from sales and licensing fees generated by the portion of
intangible capital that is transferred to this aﬀiliate. The second line includes labor andcapital costs, and licensing fees paid on the portion of intangible capital that is retainedby the parent. Finally, the after-tax profits of the tax-haven aﬀiliate, which only include
licensing fees, are given by(1 − τT H,t ) XλT H ϑijt (z)z  .

(D.15)

j∈JF t ∪{i}

We denote the firm’s policy functions by zit (a), JiXt (a), JiF t (a), ℓt (a), kt (a), qit (a),pit (a), λt (a), and ϑit (a). It is also useful to define πt (a) and π̃t (a) as the after-tax andtaxable profits, respectively, in each division implied by these policy functions.

D.4

Aggregation and accounting measures

We revert to expressing firms’ choices as functions of their varieties (ω) for notational brevity
in defining national accounting measures and other macroeconomic aggregates.
Gross domestic product. Nominal GDP in each region i is the total value of goods produced by domestic firms and local aﬀiliates of foreign MNEs:
GDPit =

I Z
X
j=1

pjit (ω)yjit (ω) dω.

(D.16)

ω∈Ωj ,i∈JF t (ω)

We compute real GDP by deflating by the consumer price index Pit defined in (D.10).
Goods trade. Aggregate exports and imports of goods are given by
EXitG =XZ
j̸=i

IMitG =

XZ
j̸=i

X
pX
ijt (ω) (1 + ξij ) qijt (ω) dω,(D.17)XpX
jit (ω) (1 + ξji ) qjit (ω) dω.

(D.18)ΩiΩjServices trade. Intangible capital licensing fees enter the national accounts as net exports84

of intellectual property services. High-tax regions’ services trade flows are given by
EXitS =IMitS =

XZ

[1 − λLT,t (ω) − λT H,t (ω)] ϑijt (ω)zit (ω) dω,

j̸=i

Ωi

j̸=i

Ωi

XZ

[λLT,t (ω) + λT H,t (ω)] ϑijt (ω)zit (ω) dω +(D.19)

XZ

ϑjit (ω)zit (ω) dω.

(D.20)

λLT,t ϑjit (ω)zit (ω) dω,(D.21)j̸=iΩjThe low-tax region’s services trade flows are
S=
EXLT,tS
IMLT,t=XZ[1 − λT H,t (ω)] ϑijt (ω)zit (ω) dω +

j̸=i

Ωi

j̸=i

Ωi

XZ

λT H,t (ω)ϑijt (ω)zit (ω) dω +

XZ

XZ

j̸=i

j̸=i

Ωj

Ωj

[1 − λLT,t (ω)]ϑjit (ω)zit (ω) dω. (D.22)

We can also write the tax haven’s services exports as
EXTSH,t =

I Z
X
j=1

λT H,t ϑjit (ω)zit (ω) dω.

(D.23)Ωj

Net factor receipts and payments. Net factor receipts from (payments to) are the sum
total of the dividends paid by foreign subsidiaries of domestic multinationals (domestic subsidiaries of foreign multinationals):
N F Rit =N F Pit =

XZ
j̸=i

Ωi

j̸=i

Ωj

XZ

(1 − τj )πijt (ω) dω,(D.24)(1 − τi )πjit (ω) dω.

(D.25)

Profit shifting. Following Dyrda et al. (2022), we define π̂ijt (ω) as the profits a firm would
have reported in region j if it did not shift profits, holding fixed all of its other policy
functions. Then, we can define the profits shifted out of region j by firm ω as
psijt (ω) = π̂ijt (ω) − πijt (ω).

(D.26)

When psijt (ω) > 0, this indicates that the firm would book more profits in region j in theabsence of profit shifting, i.e., the firm has shifted profits away from region j. Aggregating85

firm-level shifted profits yields the total profits shifted out of region j:
P Sjt =

I Z
X

D.5

psijt (ω) dω.

(D.27)Ωi

i=1Market clearing and equilibrium

In equilibrium, the government’s budget constraint must be satisfied, the markets for labor,
capital, and final goods must be satisfied, and the balance of payments must hold in eachproductive region.
Government budget constraint. Government spending, Gi , must equal revenue from labor income taxes and corporate taxes:
Pit Git = τiℓt Wit Lit + τi

I Z
X
j=1

π̃jit (ω) dω.

(D.28)Ωj

Government consumption, Gi , is an exogenous parameter. In our calibration we set it equal
to total tax revenues, but in our counterfactual experiments we hold it fixed and adjust thelabor income tax rate, τilt , to restore fiscal balance.
Labor market. Labor demand comes from four sources: production of intermediate goods;
production of intangible capital; fixed costs of exporting and setting up foreign aﬀiliates;
and the costs of transferring intangible capital. The labor market clearing condition can be
written as


Z
Z
I
X
X
X
 zit (ω) +Lit =ℓjit (ω) dω +κiX +κiF + 1{λT H,t (ω)>0} κi,T H  dω
A
i
Ω
Ω
j
i
j=1
j∈JXt (ω)
j∈JF t (ω)
Z+[Ci,T H (λT H,t (ω)) + Ci,LT (λLT,t (ω))] νit (ω)zit (ω) dω.
(D.29)Ωi

Capital market. The capital market clearing condition is
Kit =

I Z
X
j=1

kjit (ω) dω.

(D.30)ΩjFinal goods market. Final goods market clearing requires that production of final goods

86equals the sum of private consumption, public consumption, and investment in each region:
Qit = Cit + Gi + Xit .

(D.31)

Balance of payments. Each region’s balance of payments must hold:
EXitG + EXitS − IMitG − IMitS + N F Rit − N F Pit = Pbt Bit+1 − Bit .

(D.32)

Competitive equilibrium. Given a set of parameters, an equilibrium in our model is asequence of bond prices, {Qt }∞t=0 , a sequence aggregate prices and quantities for each region, {Wit , Pit , Cit , Lit }∞t=0 , and a sequence of firm-level policy functions for each region,{JiXt (ω), JiF t (ω), zit (ω), ℓit (ω), kit (ω)qit (ω), pit (ω), πit (ω), λit (ω)}∞
t=0 , that satisfy
1. the representative household’s utility maximization problem (D.1)–(D.10);
2. the firm’s profit-maximization problem (D.11);
3. the labor market clearing condition (D.29);
4. the capital market clearing condition (D.30);
5. the government’s budget constraint (D.28); and6. the balance of payments (D.32).
A stationary equilibrium is a competitive equilibrium in which the objects listed above areconstant over time. In this paper, we restrict attention to stationary equilibria in which all
regions have balanced current accounts, i.e., Bit+1 = 0 for all i.99

In this class of model there are many possible stationary equilibria, each associated with a different
vector of bond holdings (Kehoe et al., 2018). Given a set of initial conditions, the stationary equilibrium towhich the model will converge in the long run is unique, but solving for transition dynamics is an immense
computational undertaking. Our assumption that all countries have balanced current accounts is common in
the literatures on both trade and international macro.

87