Economic integration and the optimal corporate tax
structure with heterogeneous firms
Christian Bauer∗
Ronald B. Davies†
Andreas Haufler‡
Final version, December 2013
forthcoming in: Journal of Public Economics
Abstract
This paper links recent tax-rate-cut-cum-base-broadening reforms of corporate
taxation to the closer integration of international trade. We study the corporate
tax structure in a small open economy with heterogeneous firms, in a setting
where it is optimal to subsidize capital inputs by granting a tax allowance in excess of the true costs of capital. Economic integration reduces the optimal capital
subsidy and drives low-productivity firms from the small country’s home market,
replacing them with high-productivity exporters from abroad. This endogenous
policy response creates a selection effect that increases the average productivity of home firms when trade barriers fall, in addition to the well-known direct
effects.
Keywords: corporate tax reform, trade liberalization, firm heterogeneity
JEL Classification: H25, H87, F15
∗
University of Munich, 80799 Munich, Germany; e-mail: [email protected]
†
University College Dublin, Belfield, Dublin 4, Ireland; e-mail: [email protected]
‡
Corresponding author: Department of Economics, University of Munich, Akademiestr. 1/II, 80799
Munich, Germany. Phone: +49-89-2180-3858, fax: +49-89-2180-6296, e-mail: [email protected]
1
Introduction
Corporate tax reform has been a core issue on the agenda of most countries over the last
decades. Starting with the tax reforms in the United Kingdom and the United States
during the mid-1980s, a prominent type of tax reform among the OECD countries has
been to combine a reduction in the statutory corporate tax rate with a broadening
of corporate tax bases. On average, statutory tax rates in the OECD countries have
fallen from roughly 50% in 1980 to 30% in 2010, while depreciation allowances for
investment have simultaneously become less generous.1 Klemm and van Parys (2012,
Figure 1) report evidence of similar reforms in a sample of 40 developing countries in
Latin America, the Caribbean and Africa, where tax rate cuts have been combined
with lower investment allowances and shorter periods of tax holidays.
Despite the popularity of tax-rate-cut-cum-base-broadening reforms, their motivation
is still only imperfectly understood. The existing literature (referenced below) has
explained these reforms as the result of increased international mobility of capital and
firms, arguing that tax rate cuts help to attract highly mobile, multinational firms and
their profits to the country undertaking the reform. Most of these analyses, however,
either keep tax revenues or effective tax rates constant, and therefore do not offer
an independent explanation for the base-broadening element of existing corporate tax
reforms.2
In particular, an important stylized fact of many corporate tax systems in the early
1980s was the wide divergence of effective marginal tax rates (EMTRs) by sector, type
of investment, and source of finance. King and Fullerton (1984) stress the substantial
distortions caused by generous investment credits in conjunction with tax-deductible
debt financing, which in several cases resulted in negative EMTRs.3 The view is also
1
See Devereux et al. (2002) for a detailed analysis and Auerbach et al. (2010) for a recent survey.
Becker and Fuest (2011, Figure 1) list a total of 12 tax-rate-cut-cum-base broadening reforms in
selected OECD countries that have taken place during the period 1982-2003. Similarly, for the period
1980-2004 and a larger sample of 29 OECD countries, Kawano and Slemrod (2012) report 37 instances
where a tax rate cut was accompanied by a broadening of the tax base in the same year.
2
See, for example, the critical comment on this literature by Ottaviani (2002).
3
The EMTR is defined as EM T R = (coc − r)/coc, where coc is the after-tax cost of capital and r
is the competitive interest rate (Devereux et al., 2002, p. 461). As a marginal investment just covers
1
widespread that the tax-rate-cut-cum-base-broadening reforms enacted subsequently
led to a convergence of EMTRs that has improved both investment efficiency (Keen,
2002; p. 611) and tax equality across different sectors (Ottaviani, 2002). What has
not been explained, however, is whether the initial introduction of large depreciation
allowances, as well as their subsequent (partial) repeal, can be rationalized as an equilibrium response to a changing tax environment.
Against this background, the present paper offers a different approach to explain the
observed pattern of corporate tax reforms, which is based on the integration of international trade in a model with firm heterogeneity. Our argument relies on a two
sector economy with monopolistic competition in the differentiated goods sector with
heterogeneous firms, and a constant-returns-to-scale homogeneous goods sector in the
background. This setting implies an inefficiently low output in the differentiated goods
sector and offers a reason for governments to subsidize capital in this sector by means
of generous depreciation allowances. The importance of imperfect competition in trade
flows is well documented empirically, dating back to the seminal work of Grubel and
Lloyd (1975).4 Indeed, the current empirical trade literature takes the presence of imperfect competition as a given. In their recent review of this literature, Melitz and
Trefler (2012) show that intra-industry trade, which serves as an indicator of imperfect
competition, accounts for at least one third of world-wide trade flows (and nearly twice
that when using broader definitions of intra-industry trade).5
Our first main result is that, as economic integration proceeds, the optimal capital
subsidies are reduced for two different reasons. Firstly, economic integration implies
that the benefits to consumers which result from capital subsidies increasingly accrue
to foreigners. Secondly, cheaper imports from abroad mitigate the undersupply of goods
in the imperfectly competitive sector that motivates the subsidy. The last argument
is reinforced by firm heterogeneity, because foreign exporters have a higher average
its financing cost, the deduction for the interest cost of debt fully offsets the taxation of the return to
investment. Hence negative values for the EMTR result whenever investment are debt financed and
the depreciation allowances exceed true economic depreciation.
4
See Leamer and Levinsohn (1995) for a discussion of the early evolution of the empirical work on
imperfect competition in international trade.
5
An example of a more detailed analysis is Broda and Weinstein (2006), who document the increase
in the number of traded varieties for the example of the United States.
2
productivity than domestic producers. The resulting cut in optimal capital subsidies
can be achieved by a broadening of the corporate tax base, a reduction in the corporate
tax rate, or by a combination of both. Overall, we thus show that the observed pattern
of corporate tax reforms can be motivated in a setting with trade integration only, with
no need to rely on the mobility of capital or firms. This is important because in an era
of increasing trade liberalization, a failure to recognize the implication of trade flows in
and of themselves for tax policy has the potential for missing critical aspects of policy
formation.
By incorporating firm heterogeneity we are also able to analyze how changes in tax
policy affect firms with different productivities. Our second main result is that the
well-known productivity improvements brought about by falling trade barriers in the
presence of firm heterogeneity (Melitz, 2003) will be enlarged when tax policy is endogenous. In particular, we show that the endogenous policy response in our model
reinforces the selection effect arising from economic integration and thus strengthens
the reallocation of resources towards the most productive firms. As the effective capital subsidy on marginal investment is reduced, this forces low-productivity firms in the
home country to exit the market, adding to the effect of stronger foreign competition
resulting from a more integrated economy.
There is some further, suggestive evidence that our trade-based explanation of corporate tax reforms captures empirically relevant effects. This comes from the development
of additional, discrete investment incentives offered by 24 OECD countries during the
1980s and 1990s (see OECD, 1998). The OECD report stresses (p. 40) that out of 157
programmes classified as offering general investment incentives, only one is solely directed towards direct investment from abroad. This suggests that investment incentives
are primarily used to enhance domestic production and employment, rather than as a
means to attract FDI. Moreover, for the period 1989-1993, the OECD report shows a
noticeable decline of roughly 12% in the expenditures on investment subsidies (Table 1,
p. 27) and links this, among other factors, to the accelerating globalization of industrial
activities (Murphy and Pretschker, 1997). The patterns underlying the development of
these discrete investment incentives are thus very similar to the ones hypothesized here
for the general corporate tax system.
Our results also offer two distinct explanations for the puzzling fact that statutory
3
corporate tax rates have fallen significantly during the last decades while corporate tax revenues have simultaneously increased in many countries (see, for example,
Sørensen, 2007). The first argument from our analysis is that the tax-rate-cut-cumbase-broadening reforms have unambiguously reduced effective subsidy levels for all
investments facing negative EMTRs. This by itself increases corporate tax revenues.
The second effect working towards higher revenues is that economic integration and
the endogenous response of the tax structure both lead to a surge in the average profitability of firms, thus raising the base of the corporation tax.
Our analysis can be linked to several strands in the literature. A relatively small number
of papers on corporate taxation simultaneously analyzes optimal tax rate and tax base
policies in settings with capital and firm mobility. For example, Haufler and Schjelderup
(2000), Fuest and Hemmelgarn (2005), Devereux et al. (2008) consider different models
of income shifting within multinational firms and link this to the observed tax-rate-cutcum-base-broadening patterns of corporate tax reforms. Becker and Fuest (2011) focus
instead in the location choice of internationally mobile firms and show that the optimal
combination of tax rate and tax base policies depends critically on whether mobile
firms are more or less profitable than immobile firms. Egger and Raff (2011) analyze,
both theoretically and empirically, tax competition via tax rates and tax bases for
an internationally mobile monopolist. With the exception of Becker and Fuest (2011),
however, these models either hold corporate tax revenues or effective marginal tax rates
on capital constant. Moreover, in all these models it is FDI that links countries, and
the resulting tax changes come about from mobile multinational firms responding to
taxation.
A second strand of research has analyzed the effects of exogenous trade and tax policies in open economies with heterogeneous firms. Demidova and Rodriguez-Clare (2009)
compare the effects of import tariffs and export subsidies on aggregate productivity
and welfare in a small open economy. Chor (2009) analyzes the effects of a production
subsidy in an economy that competes for FDI, whereas Davies and Paz (2011) consider tariffs and value-added taxes in the presence of an informal sector. Closer to our
setting, Baldwin and Okubo (2009) study the effects of tax rate and tax base policies
on the location of internationally mobile firms. They show that a tax-rate-cut-cumbase-broadening reform that keeps the effective tax rate constant for the marginal firm
4
always increases tax revenues. Finally, Finke et al. (2013) perform a microsimulation
analysis to evaluate the impact of the German 2008 corporate tax reform, which followed a pattern of tax rate cut cum base broadening, on heterogeneous firms. They
show that firms with low productivity benefitted least from the reform, because they
were hit most by the reduction of depreciation allowances. These papers, however, do
not endogenize optimal government policies.
A recent, third set of papers derives optimal tax policies in open economy models
with heterogeneous firms. Pflüger and Südekum (2013) analyze optimal subsidies to
market entry in an open economy model of policy competition. Davies and Eckel (2010)
analyze tax rate competition for internationally mobile, heterogeneous firms, whereas
Krautheim and Schmidt-Eisenlohr (2011) derive Nash equilibrium tax rates when the
location of firms is fixed but profits can be shifted between countries. These papers
focus solely on tax rate competition, however, rather than on the optimal tax structure.
We are aware of only one other paper, Dharmapala et al. (2011), which analyzes the
optimal combination of tax instruments in the presence of firm heterogeneity. Their
setting, however, is very different from ours as they study the optimal taxation of firms
in a closed economy when there are administrative costs of tax collection.
The present paper is organized as follows. Section 2 describes the basic model employed
in our analysis. Section 3 derives the small country’s optimal tax structure. Section 4
analyzes the effects of economic integration on the government’s optimal policy response and on the entry and exit decisions of firms with different levels of productivity.
Section 5 discusses various extensions of our basic model. Section 6 concludes.
2
The model
We study a two-country model of a small open economy (the home country) and a large
rest of the world (the foreign country, whose variables are denoted by an asterisk).
The focus of our analysis lies on the tax policy in the small home country, whose
government chooses an optimal corporate tax structure taking as given the degree of
economic integration. The two countries produce and trade two goods, a homogeneous
numéraire good Y and a differentiated good X. Following Melitz (2003), firms in the
differentiated sector X are heterogeneous with respect to their unit production costs.
5
Consumers in the small home country hold a total endowment of K units of capital.
Capital is the only factor of production, and it is used in the production of both goods.
2.1
Consumers
Consumers in home are homogeneous and value the two private goods X and Y . The
direct utility function is quasi-linear and given by
Z
D
U ≡ µ ln X + Y ,
α
X≡
q (j) dj
α1
.
(1)
j∈Ω
In eq. (1), Y D is the quantity consumed of the numéraire good and X is the DixitStiglitz composite of all varieties in the monopolistically competitive sector that are
available to home consumers. The set of these varieties is given by Ω, elements of which
can include home- and foreign-produced varieties. Varieties are consumed in quantity
q (j), where j is the index for the firm producing the variety. Varieties are substitutes
and the elasticity of substitution between any two varieties is given by ε ≡ 1/(1−α) > 1,
where α ∈ (0, 1).
Utility maximization requires that the ratio of marginal utilities for the two private
goods equals their relative price. From the utility function (1) this implies µ/X = P
and thus fixes the expenditures for the differentiated good X at µ.6 This yields isoelastic
demand functions for each variety
P
q (j) =
p (j)
ε
µ
,
P
(2)
where the price index for good X is
Z
P =
−(ε−1)
p (j)
1
− ε−1
dj
.
(3)
j∈Ω
Finally, with µ spent on the differentiated good, the remainder of income I is spent
on good Y . Income in home is composed of three sources: the return to the fixed
6
This simplifying result of the quasi-linear preference structure has been exploited by several appli-
cations of heterogeneous goods to questions of policy; see Chor (2009), Pflüger and Südekum (2013),
and Cole and Davies (2011). Since changes in trade costs are reflected only in changes in the numéraire,
this setup allows us to avoid complications caused by changes in income driven by declining trade
costs (see Cole and Davies, 2011, for more discussion).
6
capital endowment K, the net profits of domestic firms, and tax revenues R, which are
redistributed to consumers as a lump sum. This implies:
Y D = I − µ.
2.2
(4)
Producers
In the numéraire sector Y , one unit of capital is used to produce one unit of output.
Capital is internationally mobile and can be traded against the numéraire good Y . This
fixes the return to capital at unity.7 In the differentiated X sector, each country has an
exogenous mass of internationally immobile potential entrants (‘entrepreneurs’), N e in
home and N e∗ in foreign, who are capable of producing a variety of the differentiated
good. We normalize N e ≡ 1. Since the home country is small, its policy changes do
not affect the mass of active firms in foreign (cf. Flam and Helpman, 1987).8 Each
entrepreneur receives the residual profit from the firm producing the variety, which
then enters into the income of the country where production takes place (in contrast
to the return to internationally mobile capital, which accrues in the country where the
capital owner resides). Entrepreneur j, and thus firm j in the X sector, is exogenously
assigned a unit capital requirement a (j). The distribution of these productivities across
entrepreneurs is given by G(.).9 Since firms differ only with respect to their unit costs
7
Note that the assumption of international capital mobility is made only for expositional conve-
nience. Alternatively, we could allow capital to be internationally immobile but fix the international
price of the numéraire to one. In this case, as long as each nation had sufficient capital supplies so that
both the heterogenous good and the numéraire are produced in equilibrium, the resulting equilibrium
would be the same. This is the reason why we have stated in the introduction that our analysis does
not have to rely on international capital mobility, in order to derive its results. Finally, note that even
if capital is mobile, as its movement does not imply the movement of technology or corporate control,
this is quite different from contemporary notions of foreign direct investment.
8
This definition of ‘small’ has been applied to the heterogenous firms literature by Demidova and
Rodriguez-Clare (2009), among others. In Section 5.3 we discuss the implications of relaxing the small
country assumption for home.
9
Some heterogeneous firm models, including Melitz (2003), assume instead that entrepreneurs
draw from this distribution of productivities at a cost. As discussed by Cole (2009) and Jørgensen
and Schröder (2008), this approach and ours yield generally comparable results. This is because, when
firms pay to learn their productivity, they continue to do so until the expected profit from doing so is
7
a(j), in our discussion we will often replace the firm index j with the firm-specific costs
a.
If a home firm decides to produce for the domestic market, it must pay a uniform
overhead cost of Fd . This can be interpreted as ‘entrepreneurial services’ and captures
the amount of capital needed for startup. In addition, if it chooses to service the
export market, it incurs a further fixed cost Fx ≥ Fd as well as per unit (iceberg)
transportation costs. These costs are such that in order for one unit of output to arrive
overseas, the firm must produce 1 + τ units. Combining the fixed costs with the unit
production requirements results in domestic (subscript d) and export (subscript x)
capital demands for a firm with input coefficient a of:
kd (a) ≡ aqd (a) + Fd ,
kx (a) ≡ (1 + τ )aqx (a) + Fx .
(5)
Similarly, revenues of a firm with variable unit costs a in the two markets are:
ρd (a) ≡ pd (a) qd (a) ,
ρx (a) ≡ px (a) qx (a) .
(6)
In order to write the profit equation, we must first describe the tax system. The government of home determines both the tax rate and the tax base for the profit-making,
differentiated sector X. Taxable profits are subject to the corporate tax rate t ∈ (0, 1).
The base of the corporation tax is given by the firm’s revenue less a tax-deductible
share δ of the total capital outlays that are incurred by each firm. Thus δ incorporates
the tax deductibility of the costs of financing the investment and of its real economic
depreciation. Recalling that the competitive return to capital is normalized to unity,
the after-tax profits of a firm with costs a in market i ∈ {d, x} are then given by
πi (a) = ρi (a) − ki (a) −t [ρi (a) − δki (a)] ≡ (1 − t)πig ,
{z
} |
{z
}
|
gross profits
(7)
taxable profit base
where
πig ≡ ρi (a) − ∆ (t, δ) ki (a)
∀ i ∈ {d, x},
(8)
zero, resulting in zero aggregate profits. In our case, since excess profits are spent on the numéraire,
production in the X sector is unaffected by aggregate profits. Thus, in each case, there is no impact
from changes in aggregate profits on the heterogenous goods sector. We further discuss relaxing this
assumption in Section 5.3 below.
8
are the ‘gross profits’ of a firm10 in market i and
∆ (t, δ) ≡ 1 +
t
(1 − δ)
1−t
(9)
is the tax factor with which the competitive rental rate of capital must be multiplied
for all X producers in the home country. Eq. (9) immediately shows that any given
level of ∆ can be obtained from an infinite number of combinations of the government’s
primary tax parameters t and δ. Moreover, since the rental rate is unity, ∆ equals the
(after-tax) cost of capital in our framework.11
Our formulation (7)–(8) allows for a simple representation of after-tax profits by regarding the corporate tax as a proportional levy on the difference between revenues
ρi and the total capital cost ∆ki . In the special case where capital costs can be fully
deducted from the corporate tax base (δ = 1) the cost of capital is ∆ = 1 from (9)
and the corporate tax is a tax on pure profits only. When the tax rate t is positive and
the tax deductibility of inputs is incomplete (δ < 1), then the cost of capital is ∆ > 1
and the corporate tax includes a partial taxation of capital inputs. Conversely, if t > 0
and δ > 1, then the corporation tax implies ∆ < 1 and capital inputs are effectively
subsidized. Note also that when ∆ < 1, an increase in ∆ can be achieved by a reduction
in the tax rate, a base-broadening decrease in δ, or both. In any case, capital market
equilibrium will ensure positive values for the cost of capital and thus ∆ > 0.
Using the definition of the effective marginal tax rate (see footnote 3), the EMTR in
our model can be expressed as
EM T R =
∆−1
.
∆
(10)
Hence the EMTR is positive when ∆ > 1, but negative for ∆ < 1.
Substituting (5) and (6) into (7) and optimizing yields profit-maximizing prices and
quantities for each firm in the domestic market:
pd (a) =
10
∆a
,
α
qd (a) =
h α iε
P ε−1 µ.
∆a
(11)
Strictly speaking, πig are the profits before deducting the corporate tax rate t, but incorporating
the tax-inclusive cost of capital ∆. For brevity, though somewhat loosely, we will refer to this term as
‘gross profits’ in the following.
11
Our analysis assumes that the tax treatment is the same for variable and for fixed capital costs.
The same qualitative conclusions would hold, but the analysis would become more complicated, if the
fixed costs were not subject to investment taxes or subsidies.
9
This shows that more productive firms (firms with a lower a) charge lower prices
and sell larger quantities. Also an increase in the cost of capital ∆ raises prices and
reduces quantities. For future discussion, note that this reduction in output is greater
for high-productivity firms (i.e. those with low values of a), making them relatively
more responsive to tax policy than low-productivity firms.
Similarly, firm-specific prices and quantities in the export market are given by
ε
α
(1 + τ ) ∆a
,
qx (a) =
(P ∗ )ε−1 µ∗ ,
px (a) =
α
∆ (1 + τ ) a
(12)
where P ∗ is the aggregate price index in foreign and µ∗ is foreign’s expenditure share for
good X. These parameters are fixed from the perspective of the small home country.12
Note also that, as a result of transport costs, export prices are higher and export
quantities are lower for any given level of a than in the domestic market.
These choices give maximized after-tax profits in the domestic market equal to
)
(
ε−1
αP
µ − ∆Fd ,
πd (a) = (1 − t) (1 − α)
∆a
(13)
indicating that more productive firms earn larger profits. Moreover, since ε > 1, profits
are unambiguously falling in the cost of capital parameter ∆.
Comparably, the additional after-tax profits for an exporting firm equal
(
)
ε−1
αP ∗
πx (a) = (1 − t) (1 − α)
µ∗ − ∆Fx .
(1 + τ ) ∆a
(14)
Finally, we assume that the foreign firms in the monopolistically competitive sector
face the same distribution of costs. Ignoring taxes in foreign (i.e. assuming t∗ = 0 and
∆∗ = 1), the maximized profits of a foreign firm that also chooses to export to the
home country are given by13
πx∗ (a)
αP
= (1 − α)
(1 + τ ∗ ) a
ε−1
µ − Fx∗ .
(15)
12
Cf. the discussion after eq. (18) below.
13
Note that the assumptions on t∗ and ∆∗ are not necessary, but are made simply to reduce notation.
10
2.3
Equilibrium
Market Entry Decisions: The home country’s firms will only be active in the
domestic market if πd (a) ≥ 0 holds. Setting πd (a) = 0 in eq. (13) determines a cutoff
productivity (or a maximum cost threshold) ad , given by
1
ε
µ ε−1
− ε−1
.
ad ≡ αP ∆
εFd
(16)
All firms with unit costs a ≤ ad will choose to be active in the domestic market.
An increase in the cost of capital ∆ reduces the cutoff value ad , implying that fewer
firms enter home’s market. This result holds even when we account for the general
equilibrium impact on P (see Appendix B).
With the exogenous mass of firms normalized to N e = 1, the number of domestic firms
operating in home’s market is then given by N = G(ad ) < 1, where G(ad ) is the value
of the cumulative distribution function at the cutoff level of costs.
Similarly, active home firms choose to export if πx (a) ≥ 0 in (14). This yields a
maximum cost threshold ax for exporting:
ε
αP ∗ − ε−1
ax ≡
∆
1+τ
µ∗
εFx
1
ε−1
.
(17)
All firms with unit costs a ≤ ax will choose to export. In the following we assume that
the parameters in (16) and (17) are such that ad > ax . A sufficient set of conditions
for this to hold is, for example, P = P ∗ , µ = µ∗ , τ > 0 and Fx > Fd . This will ensure
that some firms produce only for the domestic market, whereas other, more productive
firms will also export.
Finally, foreign firms choose to export if πx∗ (a) > 0 holds in (15). This yields a cutoff
cost level for foreign producers a∗x equal to
a∗x
αP
≡
1 + τ∗
µ∗
εFx∗
1
ε−1
.
(18)
The foreign export cutoff, and hence the number of foreign exporters M ∗ = G (a∗x ), are
only affected through the domestic price index P . The total number of active firms in
the rest of the world, N ∗ , is exogenous in our model by the small country assumption.
Note that, by the foreign equivalent of (16), this implies that P ∗ is fixed. Also, as this
implies that the home country takes the foreign price index as given, it thus mirrors
the traditional notion of a small country being a price-taking one.
11
Capital Market Clearing: For the purpose of aggregation, we denote the total
capital demand of a firm serving both the domestic and the export market by k (a) ≡
kd (a) + kx (a). Similarly, total sales revenue is ρ (a) ≡ ρd (a) + ρx (a) and total profits
are π (a) ≡ πd (a) + πx (a).
Capital market clearing is then derived as follows. In the home country, the total
demand for capital in the Y and X sectors is given by14
Z ad
S
k (a) dG(a),
KY = Y , KX ≡
0
where Y S stands for the production of good Y . Any discrepancy between home’s capital
endowment K and its capital demand KY +KX is met by international trade in capital.
Tax Revenues: Tax revenues R are determined as the difference between the firms’
gross value added and their net profits. Using πig from (8) gives
Z ad
Z ad
π g (a)dG(a)
[ρ(a) − k(a)]dG(a) − (1 − t)
R =
0
Z ad
Z ad0
=t
π g (a)dG(a) + (∆ − 1)
k(a)dG(a).
0
(19)
0
Intuitively, tax revenues can be decomposed into a profit tax on the base π g and a
tax on capital inputs levied at the rate (∆ − 1). The first is a lump-sum tax, whereas
changes in the cost of capital ∆ have allocative effects.
We assume that tax revenues are redistributed to consumers in a lump-sum fashion.
Note that this assumption is not restrictive in the present framework, and it yields the
same resource allocation as if we required a fixed (and feasible) level of revenues and
public good supply. This is because only the cost of capital (∆) matters for resource
allocation and the home country has two instruments in our model to achieve any
desired level of ∆. Hence, by raising the corporate tax rate (t) and simultaneously
increasing the tax deductibility of capital expenditures (δ) to keep ∆ constant, the
home country is able to increase the taxation of inframarginal profits and thus to raise
tax revenues in a lump-sum way. We will further discuss this property of our model
below.
14
Recall that ad > ax . Hence aggregating over the capital demands of all firms with cost levels up
to ad includes the capital demands for exports of all firms that serve both markets.
12
Income: Recall that income is the sum of tax revenues, the return on the fixed
endowment of capital K, and the after-tax profits of domestic firms. Hence we can
express income of the representative consumer as
Z ad
g
I ≡ K + (1 − t)
π (a)dG (a) + R,
0
or, by including the level of tax revenues and netting out the lump-sum tax component:
Z ad
Z ad
g
k(a)dG(a).
(20)
π (a)dG(a) + (∆ − 1)
I=K+
0
0
The Price Level: The cutoff levels in (16) and (18) include the domestic price level
P , which is endogenous. To derive a first expression for P , we define harmonic means
of the marginal costs of domestic producers and of foreign exporters:
1
1
− ε−1
Z ad
− ε−1
Z a∗x
a−(ε−1) dG (a)
ãd (ad ) ≡
a−(ε−1) dG (a)
,
ã∗x (a∗x ) ≡
.
0
0
We can then apply the mark-up pricing rules in (11) and (12) to these harmonic means
to obtain
P
3
−(ε−1)
=
Ƌd
α
−(ε−1)
(1 + τ ∗ ) ã∗x
+
α
−(ε−1)
.
(21)
The optimal tax structure
The home government chooses its capital tax structure so as to maximize the utility
of the representative consumer. Using (20) and (4) in (1) yields indirect utility
Z ad
Z ad
g
π (a)dG(a) + (∆ − 1)
k(a)dG(a) − µ ln P + C,
V =
(22)
0
0
where C ≡ K − µ + µ ln µ is a constant.
The cost of capital ∆, which incorporates the effective taxation of capital inputs, is the
government’s critical choice parameter in our model. Differentiating (22) with respect
to ∆ yields the first-order condition
Z ad
Z ad
Z ad
∂
∂
µ ∂P
g
π dG +
k (a) dG + (∆ − 1)
k (a) dG −
= 0.
∂∆ 0
∂∆ 0
P ∂∆
0
It is shown in Appendix A that this can be simplified to
"Z
#
ε−1
Z ad
ad
µ ∂P
qd (a)P ε
∂
α
dG − 1 + (∆ − 1)
k (a) dG = 0.
P ∂∆ 0
µ
∂∆ 0
13
(23)
Appendix B further shows that the first term in (23) is strictly negative. A rise in ∆
raises the input costs of all firms and this is passed on via mark-up pricing into the price
level. Note that, due to the markup arising from firms’ market power, the price level
rises by more than the cost of capital. Other things equal, this increase in the price level
raises producer profits. The positive effect on the income of the representative consumer
is insufficient, however, to compensate the consumer for the loss in purchasing power
resulting from the increase in P . Intuitively this is because a tax-induced increase in
the cost of capital aggravates the distortion arising from imperfect competition in the
differentiated sector and further contributes to the underconsumption of good X.
In an interior optimum for ∆, the second effect in (23) must therefore be positive. The
aggregate change in the demand for capital in this term is unambiguously negative.
Intuitively an increase in the cost of capital ∆ reduces the cutoff levels of costs ad
and ax at which firms can profitably enter the domestic and the foreign market [see
eqs. (16) and (17)]. Hence the mass of active firms falls both in the domestic market
and in the exporting market. At the same time, all firms that remain active reduce
their output, lowering their variable demand for capital [see eqs. (5) and (11)–(12)]. It
is shown in the appendix that these negative effects on the demand for capital cannot
be overcompensated by the rise in the price level (and thus firm profitability) that
accompanies the increase in ∆. It thus follows that (∆ − 1) must be unambiguously
negative in an interior tax optimum. This is summarized in:
Proposition 1 The optimal policy in the small open economy is to set ∆ < 1, implying
that capital inputs are effectively subsidized. For a positive corporate tax rate t > 0, the
tax allowance for capital inputs thus exceeds their true costs (δ > 1) and the tax base
is narrower than under a pure profit tax.
Proof: See Appendix B.
From Proposition 1 the optimal policy for the small country is to grant a capital subsidy
to all active domestic firms. Intuitively, the small country has an incentive to expand
output in the differentiated sector, in order to address the fundamental distortion
arising from imperfect competition. The capital subsidy achieves this by reducing the
price level of good X, thus increasing consumption. As in homogeneous firms models,
the capital subsidy increases the variable capital demand, and hence output, for all
14
active firms.15 Additional effects arise from firm heterogeneity in our model. Thus the
subsidization of capital for home producers leads to additional entry of domestic firms
into both the home and the foreign markets, whereas the number of foreign exporters
to the home market falls. These effects are discussed further in Section 4.2.
At this point it should be noted that a capital subsidy is equivalent to an output
subsidy in our model, since capital is the only factor of production. In a more general
model that includes labor as a second variable input, this equivalence no longer holds.
A capital subsidy, while addressing the output gap, will then simultaneously distort
the firms’ factor input mix. The first-best policy instrument would therefore be a direct
output subsidy, if no other distortions were present in the economy.
In practice, however, direct output subsidies are rarely observed in OECD countries,
even though imperfect competition is prevalent in many sectors. One possible reason
why capital subsidies are used instead of output subsidies is the presence of unemployment caused by union wage setting. As Fuest and Huber (2000) have shown, capital
subsidies dominate employment subsidies from a welfare perspective in such a setting,
as an employment subsidy would increase the union’s bargaining power and thus exacerbate the underlying distortion in the labor market. By the same argument, a capital
subsidy should also dominate an output subsidy, since the latter is equivalent to a simultaneous and equal subsidy to both capital and labor inputs. Therefore, even though
a more general model with labor inputs and collective wage bargaining is beyond the
scope of the present paper, our results could potentially carry over to such an extended
framework.
To provide a more detailed analysis of the first-order condition (23), we now introduce
15
It is important to note that this optimal policy is in response to underproduction, not under-entry.
As demonstrated by Arkolakis et al. (2012), in this environment the laissez-faire equilibrium generally
results in the socially preferred number of varieties because in equilibrium the benefit from a variety’s
existence (i.e. consumer willingness to pay) equals the cost of it entering into production. Thus, the
government’s objective is not to induce additional entry, but to expand output by those that do.
15
the assumption that the unit costs of firms follow the Pareto distribution16
G (a) =
a
a0
θ
0 < a ≤ a0 ,
,
θ > ε − 1.
(24)
The Pareto distribution is characterized by two parameters: the maximum cost level
a0 , which we assume to exceed ad in (16), and the exponent θ. The larger is θ the more
firms draw high cost levels approaching a0 , and hence the lower is the cost heterogeneity
between the different firms.
Using this specification, Appendix C derives reduced-form expressions for the four
principal endogenous variables in our analysis, the three cost thresholds ad , ax and a∗x
and the aggregate price level P [see eqs. (C.2)–(C.5)]. For later use, we summarize the
comparative static properties of these variables with respect to ∆:
∂P
> 0,
∂∆
∂a∗x
> 0,
∂∆
∂ad
< 0,
∂∆
∂ax
< 0.
∂∆
(25)
The next step is to obtain a compact first-order condition for ∆ based on the Pareto
distribution. The resulting optimality condition derived in Appendix C is:
εθ
θ − (ε − 1)
1 θ − (ε − 1)
+ α − ∆ Γ (∆) +
χ (∆) + χ (∆) =
.
∆
εθ
ε−1
θ(ε − 1)
(26)
In this equation,
Fx G(ax )
χ(∆) ≡
=
Fd G(ad )
1
1+τ
θ N
∗
θ
−(ε ε−1
−1)
∆
Fx
Fd∗
+
θ
ε−1
−1
θ
1
∗
1+τ
1−
ε−1
θ
Fd
Fx∗
θ
ε−1
−1
µ
εFd∗
≥0
(27)
gives the aggregated fixed costs incurred by home exporting firms, relative to the
aggregated fixed costs of all domestic firms, and
Γ (∆; τ
∗
, Fd /Fx∗ )
εθ
1 + ε−1
γ
≡
,
1+γ
γ≡∆
εθ
ε−1 −1
1
1 + τ∗
θ Fd
Fx∗
θ
ε−1
−1
>0
(28)
is a measure of the relative fixed costs incurred by domestic vis-à-vis foreign (exporting)
firms for serving the domestic market.
16
This distribution is frequently used in the literature on firm heterogeneity (e.g. Baldwin and
Okubo, 2009; Krautheim and Schmidt-Eisenlohr, 2011), as the Pareto distribution is analytically
convenient and it is also a good approximation of empirically observed cost distributions.
16
To interpret the first-order condition (26) more closely, we can state the following
properties of χ (∆) and Γ (∆) for special cases where either the import or the export
channel of trade in good X is shut down. In our setting this will arise when either
the fixed costs of foreign firms to export into the domestic market or the fixed costs
of domestic firms to export to the foreign market become prohibitively high, implying
Fx∗ → ∞ and Fx → ∞ respectively. In these cases we get:
(no exports of X) limFx →∞ χ(∆) = 0
(no imports of X) limFx∗ →∞ Γ(∆) = 1
(autarky in X)
limFx →∞ χ(∆) = 0 and
(29)
lim Γ(∆) = 1.
Fx∗ →∞
For the case of autarky, substituting χ = 0 and Γ = 1 in (26) yields, after some
manipulations,
∆aut = α ≡
ε−1
< 1.
ε
(30)
Equation (30) is a special case of the more general result in Proposition 1. It shows that,
for the case of autarky, the optimal capital subsidy is higher (i.e., ∆ is lower), when
there is weak competition between different varieties of the differentiated good (i.e.,
ε is low). Weak competition implies a high mark-up 1/α that each producer charges
on its marginal cost. As eq. (11) shows, the optimal capital subsidy in (30) leads to
output prices that equal before-tax input prices for each variety of good X. Hence the
capital subsidy exactly offsets the effects of monopolistic market power. For the case
of autarky, the rule (30) therefore guarantees a first-best allocation in our model.
4
4.1
Trade liberalization
Optimal policy response
In this section we analyze the optimal adjustment in the effective taxation of capital
as implied by ∆, when the home and foreign countries become more closely integrated.
As a first step in this analysis, we consider the discrete switch from a situation without
any trade in good X to the opening up of either imports or exports. The results for
this case are summarized in our next proposition.
17
Proposition 2 Consider a closed economy that opens up for imports, exports, or both.
Starting from the autarky solution ∆ = α, the optimal policy response to the opening
of trade is to reduce the effective subsidization of capital inputs and raise the cost of
capital to ∆ > α.
Proof: See Appendix D.
While the formal proof of Proposition 2 is relegated to the appendix, its essence can
easily be deduced from inspection of (26). The opening to trade raises the value of χ
above zero and the value of Γ above unity. Both of these effects tend to increase the lefthand side of (26), whereas the right-hand side of the equation stays constant. To offset
this change, ∆ must rise in equilibrium, thus reducing the value of the first bracketed
term on the left-hand side of (26) (which always remains positive in equilibrium).
To provide an intuitive understanding of Proposition 2, recall that the motivation
for the government to grant capital subsidies is to increase domestic consumption
of good X and thus counteract the distortion arising from imperfect competition in
that sector. The capital subsidy, however, cannot distinguish between firms nor the
destination of their output. When some domestic production is exported, the case for
subsidizing capital is weakened, other things being equal, as the production subsidy
now partially benefits foreign consumers. In addition, as is well known in heterogenous
firms models, there is also a selection effect brought about by heterogeneity. By offering
a capital subsidy, the home government encourages additional firms to export even
though their productivity is insufficient for that to be profitable in the absence of the
subsidy. Since the only gain to the home country from having these firms export is the
profits they earn, the net impact of these firms’ exporting on home income is negative.
In particular, because exporting benefits home only via the profits it generates, the
government would prefer that both the extensive and intensive exporting decisions
were based on a cost of capital equal to unity. This is because, since home is small and
the foreign price index is exogenous, the firm’s exporting decision at ∆ = 1 matches
that of the government.17 However, as it cannot discriminate between capital used for
domestic and foreign consumption, when the economy opens up for exports, the home
17
When home is large, it has an incentive to limit exports to exploit a terms of trade effect, as
discussed in Section 5.3.
18
government compromises by reducing its capital subsidy (i.e. it raising ∆) in order to
minimize this loss.
Turning to the import side, heterogeneity again plays a key role. First, recall that since
foreign firms do not benefit from home’s capital subsidy, the subsidy distorts international trade and drives out some foreign exporters by decreasing the domestic price
level [eq. (25)]. Since exporters are more productive than the average firm, this means
that low-cost imports are being replaced by high-cost domestic production. Thus, the
subsidy undermines the welfare improving selection effect discussed by Melitz (2003)
and others. As an alternative intuitive approach, recall that a firm’s responsiveness to
∆ is increasing in its productivity. When foreign competition drives out less-productive
home firms, this increases average responsiveness to tax policy. As such, a smaller capital subsidy is needed to achieve a comparable increase in output. Therefore, the switch
from autarky to trade causes the small country’s government to consider the adverse
consequences of capital subsidies for average productivity and leads to a reduction in
the optimal capital subsidy.
How does the optimal subsidy change with the degree of firm heterogeneity? To analyze
this question in the simplest possible way, we focus on the import side and assume
χ(∆) = 0. The first-order condition (26) then simplifies to
F ≡
1
1
Γρ
−
+ = 0,
∆
(ε − 1) θ
ρ≡
θ − (ε − 1)
+ α − ∆ > 0,
εθ
(31)
where ρ > 0 is implied from the first-order condition, since Γ, ∆ > 0 and θ > (ε − 1).
Differentiating with respect to θ and using (31) to substitute out for Γ/∆ gives
∂F
1
1
1 (ε − 1)
ρ ∂Γ
1
ρ ∂Γ
= 2
−
−ρ +
= 2 (∆ − α) +
.
∂θ
θ ρ
ε−1 θ
ε
∆ ∂θ
θ ρ
∆ ∂θ
(32)
The first term on the right-hand side of (32) is unambiguously positive since ∆ > α
from Proposition 2. The second term is ambiguous, in general.18 When θ is not much
larger than ε − 1, however, so that ρ in (31) is small, then the first effect in (32)
will dominate and ∂F/∂θ > 0. In this case, the implicit function theorem implies
18
Differentiating Γ in (28) gives
ε
γ
εθ
1
∂γ
∂Γ
=
+
−1
.
∂θ
ε − 1 (1 + γ)
ε−1
(1 + γ)2 ∂θ
The first term in this expression is positive, but the second term is negative since ∂γ/∂θ < 0 [see (28)].
19
∂∆/∂θ > 0 when the second-order condition for the optimal choice of ∆ is fulfilled.
Hence, an increase in firm heterogeneity (that is, a reduction in θ) reduces ∆ and thus
increases the optimal capital subsidy. Intuitively, a lower level of θ implies that there are
more productive domestic firms. This reduces the efficiency costs of the subsidy, which
arise from substituting less productive domestic firms for more productive exporters
from abroad.
So far, we have only dealt with a discrete switch from autarky to a situation with
trade in the differentiated good. Our next result shows that similar results apply for
continuous changes in economic integration, starting from an initial equilibrium with
trade in good X. There are two different measures of economic integration in our
model, the per-unit trade costs (τ , τ ∗ ), and the additional fixed costs of serving an
export market (Fx , Fx∗ ). The results in the following proposition hold for both of these
measures.
Proposition 3 (a) Consider a situation where the home country imports good X, but
does not export it. Then either a small reduction in the trade costs τ ∗ or a reduction in
the fixed exporting costs Fx∗ faced by foreign firms leads the home country’s government
to raise the cost of capital ∆.
(b) Consider a situation with bilateral trade in good X. Then either a small reduction
in the trade costs τ or a reduction in the fixed exporting costs Fx faced by domestic
exporters leads the home country’s government to raise ∆.
Proof: See Appendix E.
The fundamental effects behind Proposition 3 are the same as those discussed above. An
increase in either exports or imports of the differentiated good, caused by a reduction
in unit trade costs or in the fixed costs of serving an export market, will weaken the
link between domestic production and domestic consumption and therefore reduces
the incentive for the home government to subsidize domestic capital inputs. These
effects are strengthened by the selection effects caused by firm heterogeneity, which are
discussed in more detail below.
Proposition 3 implies that the subsidization of capital inputs should be continuously
reduced, and the cost of capital be accordingly increased, as economic integration
20
proceeds. In our model this increase in the cost of capital (or in the EMTR) can either
be brought about by a broadening of the tax base (i.e. a reduction in the deductibility
parameter δ), by reducing the corporate tax rate, or by a combination of both.19 This is
consistent with a tax-rate-cut-cum-base-broadening reform of the corporate tax system
when the initial value of the EMTR is negative [see eq. (10)].
4.2
Effects at the firm level
In this subsection, we delve deeper into the differing firm-level effects from falling
trade barriers with a particular eye towards how these effects interact with changes
in the optimal policy. As highlighted in Melitz (2003) and others, one of the primary
innovations of firm heterogeneity is that falling trade barriers lead to a selection effect
that shifts resources towards relatively more productive firms. Therefore it is important
to understand how endogenous tax policy also results in selection effects that encourage
production by low-cost firms relative to their higher-cost competitors.
In our model, the changes in productivity cutoffs in response to falling trade barriers occur both through direct effects as well as indirect ones which arise through the
endogenous policy response. While the direct effects have been widely discussed elsewhere, this is not the case for the indirect ones. As we will show, these indirect effects
can have important implications for the mass of firms in operation as well as average
productivity.
In the following we confine ourselves to changes in variable trade costs. It is straightforward, however, to establish that reductions in the fixed beachhead costs of exporters,
Fx and Fx∗ , have qualitatively very similar effects as the variable cost reductions studied
here. Moreover, our analysis conceptually separates between the variable trade costs
faced by domestic and foreign firms, respectively. Starting with the latter and turning
19
Note that, from eq. (9) the corporate tax rate acts as a multiplier for any given value of the
depreciation parameter in excess of true economic costs. Hence an isolated fall in tax rates will raise
the cost of capital ∆, other things being equal, while still maintaining a capital subsidy at the margin
(i.e. keeping ∆ below unity).
21
first to the productivity of the last operating domestic firm, we see that:
dad
∂ad ∂ad d∆
+
> 0,
=
dτ ∗ |{z}
∂τ ∗ |{z}
∂∆ |{z}
dτ ∗
(+)
(−)
(33)
(−)
where the first term is the direct effect and the second term is the indirect effect through
the induced change in the cost of capital ∆. As can be verified from eq. (C.3) in the
appendix, the direct effect is positive as a decline in trade costs leads to increased
competition by foreign firms. This reduces variable profits for the least productive
home firms below the fixed cost of production. In addition, however, it is necessary to
consider the indirect effect. By Proposition 3(a), falling trade costs result in an increase
in the cost of capital ∆, at least when there are no home exports. From eq. (25) this
increase in ∆ further reduces the cost cutoff for domestic firms. Therefore the direct
effect of trade liberalization is reinforced by the policy response, leading to a greater
reduction in ad than would occur if policy were exogenous. This in turn implies that
the increase in the average productivity of domestic firms to a reduction in trade costs
will be greater when the home country’s policy adjusts.
Turning to a decline in home trade costs τ , the effect on the cutoff cost level ad is:
∂ad ∂ad d∆
dad
+
> 0.
=
dτ
∂τ
∂∆ |{z}
dτ
|{z}
|{z}
(0)
(−)
(34)
(−)
In contrast to the change in the foreign trade cost, there is no direct effect on home
domestic activity from the home export cost since the cutoff ad is for a firm that does
not export.20 However, there remains an indirect effect. As before, the induced change
in tax policy increases the cost of capital, thereby driving low-productivity domestic
firms from the market.
In many situations economic integration will simultaneously reduce the trade costs for
home and foreign firms, thus combining the effects in eqs. (33) and (34).21 Clearly, both
effects work in the same direction so that the change in the cutoff level of domestic
20
This is a result of preferences being quasi-linear. If preferences are not quasi-linear, then there
can be income effects that would lead to potential changes in ad from a decline in τ (see footnote 6).
21
For example, trade in either direction would be eased by improvement of transport infrastructures,
easier international communication, or mutual trade concessions.
22
firms will be even larger when trade costs fall for domestic and foreign firms alike. Importantly, the indirect effects through the induced change in ∆ also mutually reinforce
each other, strengthening the selection effect that is caused by trade integration.
Similarly, we can analyze the effects of trade integration on the cutoff cost level of
foreign exporters, a∗x . Again beginning with τ ∗ , we see a comparable reinforcement of
the direct effect through the induced policy change:
da∗x
∂a∗x ∂a∗x d∆
+
< 0.
=
dτ ∗ |{z}
∂τ ∗ |{z}
∂∆ |{z}
dτ ∗
(−)
(+)
(35)
(−)
The direct effect of trade integration is now negative, as a decline in τ ∗ increases
profits for the marginal foreign exporter. This leads to additional entry by foreign
firms into the home market [see eq. (C.4) in the appendix]. As before, this direct effect
is reinforced by the tax policy response (Proposition 3a). The induced increase in the
cost of capital for domestic firms encourages exports by foreign firms [cf. eq. (25)].
Thus, in comparison to a setting in which the home country’s tax policy is static, the
increased penetration by foreign firms following a drop in trade barriers will be larger
when policy is endogenously chosen.
For the effect of domestic trade costs on the cutoff level a∗x we get
da∗x
∂a∗x ∂a∗x d∆
=
+
< 0.
dτ
∂τ
∂∆ |{z}
dτ
|{z}
|{z}
(0)
(+)
(36)
(−)
As in eq. (34), there is no direct impact on foreign exporters from a change in the trade
costs for home firms. However, there are still the indirect effects as the government
raises ∆ in response to the decline in trade barriers. As discussed above, this indirect
effect increases capital costs to home firms, thereby increasing a∗x . Combining (35)
and (36), it is again the case that simultaneous declines in bilateral trade barriers (i.e.
in both τ and τ ∗ ) work to reinforce one another. Hence trade integration unambiguously
increases the number of foreign exporters to the small home country, and the direct
effect of trade integration is reinforced by the effects of endogenous policy.
Finally, we consider the impact on the cutoff of home country’s exporters. The effects
of a decline in the variable trade costs faced by foreign firms are:
dax
∂ax ∂ax d∆
=
+
> 0.
∗
dτ
∂τ ∗ |{z}
∂∆ |{z}
dτ ∗
|{z}
(0)
(−)
23
(−)
(37)
The direct effect is zero, since the relevant trade costs for home exporters face are given
by τ , rather than τ ∗ [see eq. (C.5)]. With endogenous policy, however, there remains an
indirect effect. When the sufficient conditions described in Appendix E are fulfilled, the
induced increase in the cost of capital from trade liberalization reduces the profits from
exporting. As shown in (25), this increase in ∆ reduces ax and leads low-productivity
home exporters to quit the foreign market.
The effect of an increase in home’s trade costs on home’s exporting behavior is, however,
not clear cut:
dax
∂ax ∂ax d∆
=
+
≶ 0.
dτ
∂τ
∂∆ |{z}
dτ
|{z}
|{z}
(−)
(−)
(38)
(−)
The direct effect of a fall in trade costs is to increase the number of home exporters
since the reduction in trade costs makes exporting more profitable. The indirect effect,
however, works in the opposite direction as the decline in home trade costs raises the
cost of capital for home exporters. Hence the direct effect of the fall in foreign trade
costs is here mitigated by the indirect policy effect.
In general, therefore, the endogenous policy response may reinforce or counteract the
selection effects at the firm level that arise from trade integration. In the small country’s home market, however, the effects are unambiguous and summarized in our final
proposition.
Proposition 4 A reduction in τ ∗ under the conditions in Proposition 3(a), or in τ
under the conditions in Proposition 3(b), reduces the threshold level for domestic firms
ad and increases the threshold level for foreign exporters a∗x . Hence the number of domestic producers falls, whereas the number of foreign exporters rises. In both cases, the
selection effects caused by economic integration are unambiguously strengthened by the
endogenous policy adjustment in the domestic cost of capital.
We conclude this section by calculating the effects of economic integration on domestic
tax revenues, taking account of the induced change in the domestic capital cost ∆.
Since we know from Proposition 3 that various measures of economic integration have
similar effects, we restrict attention to the simplest scenario and study an isolated
reduction in the trade costs of foreign firms, τ ∗ . We further focus on the case without
24
home exports and assume that the change in the cost of capital is brought about solely
by a change in the depreciation parameter δ while leaving t unchanged.22 Appendix F
then derives:
#
"
Z ε−1
Z ad
dR
∂∆
αε µ ad P
Fd dG(a)
= (1 − t)
dG(a) +
∗
ε
dτ
∆ 0
a
∂τ ∗
0
Z ad α ε−1
∂∆
ε−2 ∂P
+ t(1 − α)µ(ε − 1)P
dG(a) ∗
∂∆ 0
∆a
∂τ
Z ad ε−1
ε
P
∆ ∂P
µα
∂∆
+ (∆ − 1) ε+1 −ε + (ε − 1)
dG(a) ∗
∆
P ∂∆ 0
a
∂τ
∂ad ∂ad ∂∆
+ (∆ − 1)kd (ad )
< 0.
+
∂τ ∗ ∂∆ ∂τ ∗
(39)
Equation (39) shows that a reduction in τ ∗ (as an indicator of economic integration)
raises domestic tax revenues. The first line gives the net revenue gain resulting from the
reduced capital subsidy (i.e., the rise in ∆) that is caused by the fall in τ . This effect
remain positive, even though it includes the reduction in the gross profits of domestic
firms, which lowers the tax base for the statutory corporate tax rate [see eq. (F.3) in
Appendix F]. The positive effect on domestic tax revenues is further strengthened by
the increase in the domestic price level, given in the second line. The third line in (39)
gives the gain in tax revenues that results from lower capital use by subsidized domestic
firms following the reduction in the investment subsidy.23 Finally, the fourth line gives
the corresponding tax savings that result from some of the subsidized firms leaving
the market when foreign competition becomes more intense, and when the subsidy is
reduced [see eq. (33)].
Appendix F also shows that after-tax profits in the X sector fall following the reduction
in τ ∗ , and this reduces private income (net of redistributed tax revenues). Therefore
the ratio of tax revenues over private income unambiguously rises following an increase
in economic integration, when the tax change is brought about solely by a change in
the corporate tax base. This is in line with the actual increase in tax revenues, as a
share of GDP, that has occurred in many countries (see the introduction). While it
22
This can be extended to the case with home exporters when the conditions in Appendix E for a
rise in ∆ from a decline in τ ∗ are met.
23
It is shown in Appendix F [eq. (F.4)] that the squared bracket in this line is negative, as the direct
effect of the reduced subsidy dominates the indirect effect resulting from the rise in the price index.
25
must be emphasized that the statutory tax rate has been held constant in our analysis,
the argument nevertheless shows that a rising share of tax revenues can be compatible
with a reduction in t under conditions of economic integration.
5
Extensions
In this section, we discuss the robustness of our baseline results, namely that the
government sets the cost of capital below unity but increases it with trade liberalization,
to alternative assumptions.
5.1
Introducing trade taxes
As noted above, the government’s preferred cost of capital rises with trade liberalization, in part because trade confers some of the benefits of subsidization to foreign
consumers. This effect is due to the government’s inability to discriminate between
output destined for foreign markets and that sold domestically. Introducing additional
policy instruments such as trade taxes, which by their nature distinguish between locations, makes it possible to achieve such discrimination. To understand the implications
that would then arise, consider a baseline case where full discrimination is possible. In
this setting, the government would prefer to implement a cost of capital equal to ∆dom
for capital used to produce goods intended for domestic consumption. This subsidy lies
between the one when discrimination is not possible and α, the subsidy it would choose
under autarky.24 At the same time, as noted above, the government would not subsidize the capital used for export production at all. Thus, with discrimination, the home
government would subsidize domestically-consumed output but not foreign-consumed
output.
As an alternative, consider the case without discrimination but with a tax, φ acting
like an additional iceberg trade cost.25 Here, the same result can be enacted by setting
this new cost equal to φ = (∆dom )−1 −1 > 0, thus ensuring that the total cost of capital
24
Note that ∆dom exceeds α due to the desire to avoid driving out low-cost foreign exporters in
favor of high-cost domestic firms.
25
See Cole (2011) for a discussion on how to map from an iceberg cost to an ad valorem trade tax.
26
of production for the foreign market equals (1 + φ)∆dom = 1. As exporting trade costs
change, this decoupling of domestic and export production means that there is no
longer a need to alter the capital subsidy when exports expand due to a fall in export
trade costs. However, as before, a fall in inbound trade costs will still result in a rise in
∆dom , implying a comparable offsetting change in the preferred export tax (although
one smaller than before since there is no longer a concern over subsidizing foreign
consumption). Combining these results shows that the baseline results hold, with the
additional implication that a capital subsidy will be met with a positive export tax.
When permitting an import tariff, the results are less clear-cut. This is because, in contrast to the preferred export tax, the optimal import tariff can be positive or negative.
As discussed by Cole and Davies (2011), with firm heterogeneity imports form a large
share of the consumption bundle because they are from highly-productive (and thus
low-cost) foreign firms. Just as with domestic production, there is a desire to subsidize
imports to offset the under-production by monopolistic firms, even though this crowds
out low-productivity home firms. This effect is, however, counteracted by the desire for
tariff revenue (which enters domestic welfare via the numéraire). As they show, which
effect dominates, and thus whether an import tariff or an import subsidy is optimal,
depends on parameter values. In any case, the use of such a policy does not eliminate
the desire to subsidize domestic production for domestic consumption. As a result, the
government will still subsidize capital. The extent to which it chooses to do so will,
however, depend on the size of the import tariff/subsidy, as that affects the set of home
firms in operation.
5.2
Introducing foreign direct investment
One of the goals of our model was to show that, in contrast to the presumption that
the observed tax changes are attributable solely to changes in the FDI landscape,
they can also arise from changes in trade in goods. Nevertheless, FDI is an undeniable
part of the current economic environment. With FDI, two opposing forces will enter
the government’s choice of the capital subsidy. As shown by Chor (2009), there is a
desire to subsidize inbound investment because of the under-production of these highly
productive firms (a rationale comparable to the import tariff subsidy above).
27
This, however, is countered by three effects. First, there is a desire to discriminate
between domestic and foreign firms because foreign firm profits do not enter home
income. This means that, all else equal, the government would prefer to offer foreign
firms a lower subsidy than it would domestic firms, placing upward pressure on ∆.
Second, the high relative productivity of inbound FDI predicted by models such as
Helpman et al. (2004) means that inbound FDI increases the average responsiveness
of firms producing in home, further increasing ∆. Third, as the capital subsidy would
apply only to production within home’s borders, it would distort outbound FDI by
encouraging home firms to produce at home rather than abroad. Comparable to the
discussion of an export tax, the optimal level of FDI would be achieved when the
decision is based on a cost of capital of ∆ = 1. Hence, this effect would put upward
pressure on ∆, working counter to the Chor (2009) subsidy. Combining the opposing
forces indicates that, although there is still a net benefit to setting ∆ < 1, it is unclear
how the resulting cost of capital would compare to that without FDI.
The implications of declining inbound trade barriers for changes in the tax structure,
however, likely depend on the motive for FDI. When, as in Helpman et al. (2004),
FDI is horizontal and substitutes for trade, declining trade costs would lead some
foreign multinationals to revert to exporting, reducing inbound FDI. Hence, if the net
effect of introducing FDI is to cause a higher cost of capital ∆, this reduction in FDI
might then remove some of the upward pressure on the cost of capital. In contrast,
with cost-driven vertical FDI (e.g. Bergstrand and Egger, 2007), falling trade costs can
instead spur FDI. In this case the upward pressure on the cost of capital would even
be reinforced by economic integration. However, because vertical FDI models typically
rely on endowment differences in a multiple factor model, moving into such a setting
would likely introduce additional optimal tax considerations not found in our setting.
5.3
Endogenizing the number of firms
In the above discussion, two assumptions were made about the number of firms. First,
we assumed that home was small and could not affect the number of active foreign
firms. Second, we assumed that the mass of potential home entrants was fixed. In this
subsection, we relax these assumptions in turn.
28
An important result stemming from the small home country assumption was that it
implies that the foreign price index is constant. Thus, as noted above, the government
has no incentive to intervene in the export behavior of firms and prefers that both the
extensive and intensive decisions are based on capital costs equal to unity. Alternatively, we can assume that the home country is large and recognizes its impact on the
foreign price index. As discussed by Helpman and Krugman (1989), with monopolistic competition an exporting government prefers to restrict exports via an export tax.
This is because although each exporter internalizes the impact of its exports on its own
price, it ignores the negative effect this has on other exporters’ profits operating via
the foreign price index. With heterogeneous firms, there is an additional selection effect
whereby restricting exports also shifts production to the more profitable exporters. In
contrast to the baseline model, where the home government would prefer that exporting
is based on ∆ = 1, the ability to manipulate the foreign price index (creating a terms
of trade effect) would lead it to prefer that exporting is based on ∆ > 1, i.e. where the
cost of capital exceeds unity and exports are restricted. On the importing side, however, since the marginal foreign firm does not export, the above analysis remains the
same. Similarly, there continues to be a rationale for subsidizing domestic production
for domestic consumption. If the terms of trade motivation for increasing the user cost
of capital outweighs the desire to subsidize domestic consumption, then it can be the
case that the preferred ∆ exceeds unity. This, however, requires a large asymmetry
across countries (such as having µ∗ be much greater than µ), which can result in the
empirically refuted possibility that all home firms export (i.e. that ax = ad ).
Moving to the effects of trade liberalization on ∆, two new effects emerge when exporting trade costs fall. First, there is an increase in home’s ability to manipulate the foreign
price index. This occurs both via an increase in the number of home firms selling in the
foreign market, and via a reduction in the number of active foreign firms. As a result,
home’s incentive to raise ∆ is increased. Second, as new home firms start exporting,
there is a decline in the average productivity of home exporters. As this reduces the
average sensitivity of home exporters to ∆, it puts additional upward pressure on the
cost of capital to achieve the desired terms of trade effect. Thus, endogenizing the number of foreign firms reinforces the tax movements in response to export liberalization.
On the other hand, as the least productive active foreign firm does not export and
29
the home exporting decision is independent of home’s import costs, there are no new
interactions between import liberalization and the cost of capital.
Turning to free entry into the mass of home firms, the standard framework assumes that
a firm must pay a fixed cost Fe to determine its productivity parameter. With free entry,
firms will continue to pay to take a draw from the productivity distribution until the
expected profits are exactly offset by Fe . As discussed in Pflüger and Südekum (2013),
although the subsidy does not alter the number of firms producing in equilibrium, it
does increase the average productivity of active firms. This creates a welfare-improving
selection effect. Since subsidizing capital in our model increases expected profits, we
anticipate that it would lead to a comparable increase in average productivity, giving
an additional motive for lowering the cost of capital below unity.26
The impact of trade liberalization, however, depends on whether we consider outbound
or inbound trade cost reductions. As outbound trade costs fall, expected profits rise,
encouraging additional firms to take a draw from the productivity distribution and
increasing average productivity. Compared to the baseline setting with an exogenous
mass of home firms, this increase in the average response to export liberalization would
therefore imply a larger increase in the optimal level of ∆. On the other hand, a fall
in import costs would lower expected profits, resulting in fewer home firms drawing
from the productivity distribution and a fall in average productivity. At the same time,
the productivity-improving selection effects arising from increased import competition
benefit consumers, reducing the underconsumption of good X. These counteracting
effects lead to potentially ambiguous changes in ∆, relative to the baseline case.
6
Conclusion
Over the past thirty years, many countries have enacted corporate tax reforms that
have combined significant reductions in corporate tax rates with some broadening of
corporate tax bases. While the previous literature has motivated these tax reforms by
the desire to attract foreign direct investment, this paper has linked the reforms to
26
When the home country is large, an additional terms of trade effect arises which, from our above
discussion, would tend to increase ∆. Hence in this case the effects of free entry of home firms on the
optimal choice of ∆ would be ambiguous.
30
the closer integration of international commodity trade, using a model of imperfectly
competitive, heterogenous firms.
We begin by showing that, as a result of imperfect competition, the government of a
small country has an incentive to offer capital subsidies in order to increase output and,
eventually, domestic consumption. When trade barriers fall, there are two reasons to
reduce the optimal capital allowance. First, economic integration decouples production
and consumption in the home country, reducing the effectiveness of capital subsidies
as a means of increasing domestic consumption. This effect would also arise in a model
of homogeneous firms. As a second effect, however, capital subsidies lead to additional
inefficiencies in the presence of heterogeneous firms, as they support low-productivity
domestic producers while barring more productive foreign firms from the home market.
These inefficiencies are exacerbated by economic integration and thus give a further
reason for the government to reduce the subsidies to domestic firms.
At the same time, this optimal policy reinforces the selection effects that are caused
by increasing trade openness in a heterogeneous firms framework. In our model trade
integration leads low-productivity home firms to exit the domestic market both as
a result of increased foreign competition and as a result of reduced capital subsidies.
Similarly, a larger number of foreign exporters enters the home country’s market due to
the direct effect of reduced trade barriers, but also because they are less discriminated
against after the induced policy change in the home country. These results imply that a
decline in trade barriers results in greater improvements to average productivity when
tax policy is endogenous than when it is not.
It is not, however, our contention that the observed policy changes are completely
unrelated to changes in FDI. Instead, by showing that these patterns can also result
from declines in barriers to trade alone, we hope to provide a more complete picture
of the interactions between the forces of globalization and tax policy. In particular,
our results suggest that even in industries or countries where FDI is rare, governments
may well need to be cognizant of the potential for welfare-enhancing policy shifts.
Thus, our results complement the discussion on FDI and taxation, hopefully providing
a framework for more successful policy choices.
31
Acknowledgements
We thank two anonymous referees and the editor, Jim Hines, for their detailed and constructive comments. This paper was presented at the Public Economic Theory (PET)
meeting in Bloomington, Indiana, at the Congress of the International Institute for
Public Finance in Ann Arbor, Michigan and at conferences in Frankfurt and Munich.
We thank conference participants, in particular Robert Cline, Michael Devereux, Dominika Langenmayr and Marco Runkel for helpful comments and discussions. Bauer
and Haufler gratefully acknowledge financial support from the German Research Foundation (Grant No. HA 3195/8). Davies produced this paper as part of the project
“Globalization, Investment and Services Trade (GIST) Marie Curie Initial Training
Network (ITN)” funded by the European Commission under its Seventh Framework
Programme - Contract No. FP7-PEOPLE-ITN-2008-211429.
32
Appendix
Appendix A: Derivation of equation (23)
We start from the first-order condition for ∆, which is repeated here for convenience:
Z ad
Z ad
Z ad
∂
µ ∂P
∂
g
k (a) dG + (∆ − 1)
π (a)dG +
k (a) dG −
= 0.
(A.1)
∂∆ 0
∂∆ 0
P ∂∆
0
Differentiating the maximized domestic profit function for a single firm with respect to
∆ and collecting terms gives
∂πdg
∂qd ∂qd ∂P
∂p
∂kd
∂p ∂P
=
+
qd + p − ∆
+
qd − kd .
∂∆
∂∆ ∂P ∂∆
∂q
∂qd
∂P ∂∆
(A.2)
The first term in (A.2) is zero from the optimal output choice of firms. Differentiating
the inverse demand function p(qd ) and using α = (ε − 1)/ε implies
∂p
=α
∂P
µ
qd P
1ε
.
(A.3)
Next, using the first-order condition for optimal quantities of exporters, the change in
exporting profits is equal to
∂πxg
= −kx (a).
∂∆
(A.4)
Integrating over all firms in (A.2) and (A.4), using πdg (ad ) = πxg (ax ) = 0 and combining
with (A.1) gives eq. (23).
Appendix B: Proof of Proposition 1
To show that the first term in (23) is negative, note first that ∂P/∂∆ > 0.27 Moreover,
Z
ad
α
0
27
qd (a)P
µ
ε−1
ε
α Z ad
α
P
PX
α
dG = α
qd dG (a) ≤ α
= α,
µ
µ
0
(B.1)
Assuming to the contrary that the price level is decreasing in ∆ implies a contradiction: if
∂P/∂∆ ≤ 0, raising ∆ would increase consumer prices and, from (16) and (18), decrease the mass
of domestic firms and foreign exporters. Taken together, this is not compatible with the presumed
reduction in P .
33
where the first equality follows from exchanging variables, the inequality in the middle
is strict when imports are positive, and the last equality follows from µ = P X. Since
α < 1 the squared bracket in the first term of (23) must thus be negative.
To prove that (∆ − 1) < 0 in the second term of (23), it remains to show that
Z ai
Z ai
∂ki (q (∆))
∂
∂ai
ki (ai ) g (ai ) +
dG < 0 ∀i ∈ {d, x}. (B.2)
ki (a) dG =
∂∆ 0
∂∆
∂∆
0
For i = x this must always be fulfilled since ∂ax /∂∆ < 0 from (17) and ∂kx /∂∆ < 0
from (12) and (5). For i = d the direct effects of ∆ are analogous, but there is a
counteracting effect from the increase in the price level (∂P/∂∆ > 0) on kd . To see
R ad
∂
that the net effect is negative, we need to show that ∂∆
(aqd + F )dG < 0. The
0
effect of ∆ on P is via the domestic producers (labeled N ) and via the mass of foreign
exporters (labeled M ∗ ), which depends on the foreign export cutoff. Given a∗x = a∗x (P )
from (18), we define an (inverse) measure of import prices
Z
M∗
p∗x
pM (P ) ≡
(j)
−(ε−1)
dj = M
∗
a∗x
Z
0
p∗x
(a)
−(ε−1)
0
g ∗ (a)
da =
G (a∗x )
Z
a∗x
p∗x (a)−(ε−1) dG (a) ,
0
with ∂pM /∂P > 0. With this notation,
P
−(ε−1)
Z
=
N
−(ε−1)
pd (j)
Z
dj +
0
M∗
p∗x
(j)
−(ε−1)
Z
ad
dj =
0
0
∆a
α
−(ε−1)
dG + pM (P ) .
(B.3)
Together with the optimal quantities in (11) this allows us to write
"Z −(ε−1) #
Z ad
ad
∆a
µα
µα aqd (a) dG = P ε−1
dG =
1 − P ε−1 pM .
∆
α
∆
0
0
Differentiating both sides of (B.4) with respect to ∆, it follows that
∂
∂∆
R ad
0
(B.4)
aqd (·) dG <
0, since the RHS is unambiguously falling in ∆. The latter follows since P ε−1 pM is
R ad
∂
increasing in P and ∂P/∂∆ > 0. We are thus left to show that ∂∆
F dG < 0. For
0
this it is sufficient to prove that an increase in ∆ lowers the domestic cut-off level of
costs, ∂ad /∂∆ < 0. Rearranging (B.3) gives
P
−(ε−1)
Z
− pM (P ) =
0
ad
∆a
α
−(ε−1)
dG.
(B.5)
Since the RHS of (B.5) is rising in ad for given ∆ and the LHS is strictly decreasing
in P , this implies ∂P/∂ad |∆ < 0 and, by an analogous argument, ∂P/∂∆|ad > 0. As a
34
final step, totally differentiating the definition of the cutoff πd (ad ; ∆) = 0 gives
∂πd /∂∆|ad
∂ad
=−
< 0.
∂∆
∂πd /∂ad |∆
(B.6)
The numerator in (B.6) must be negative from (13) as both fixed and variable capital
costs are rising in ∆. For given ad , the latter follows from the fact that
ε−1 Z ad a −(ε−1)
∆
=
dG + ∆ε−1 pM (P )
P
α
0
is increasing in ∆ so that the inverse expression (P/∆) in (13) rises in ∆. Finally, the
denominator in (B.6) is also negative, as its sign is given by the change in P/ad , which
is negative for given ∆ from (B.3). This completes the proof. Appendix C: Derivation of equation (26)
In a first step we derive the solution for the three cutoff variables ad , ax and a∗x and the
price index P under the Pareto distribution. Noting that the derivative of the Pareto
distribution function (24) (i.e., the density function) is
θa(θ−1)
,
aθ0
G0 (a) ≡ g(a) =
(C.1)
we obtain
1−
θ
(αP ) =
ad
a0
θ
1 ε ε−1 −1
∆
1
Fd
θ
ε−1
−1
θ
=
a∗x
a0
θ
ax
a0
θ
=
∆Fd 1 +
+
aθ0
θ
ε−1
−1
ε
µ
θ
1 ε ε−1 −1
∆∗
1 − ε−1
θ
θ
∆ ε ε−1 −1
∆∗
1−
µ
ε
θ
1
1+τ ∗
θ
1
1+τ ∗
ε−1
θ
µ
ε
1
1+τ
θ Fd∗
Fx
θ ε−1
∆∗
∆
1
Fx∗
θ
ε−1
−1
(C.2)
Fd
Fx∗
(C.3)
θ
ε−1
−1
θ
∗ ε−1
−1
∆∗ Fx∗ 1 + (1 + τ ∗ )θ FFxd
=
ε−1
θ
θ
ε ε−1
θ
∆∗ ε ε−1 −1
∆
N∗ .
(C.4)
(C.5)
This leads to the comparative static properties with respect to ∆:
∂P
> 0,
∂∆
∂a∗x
> 0,
∂∆
∂ad
< 0,
∂∆
35
∂ax
< 0,
∂∆
(C.6)
which follow from (C.2)–(C.5) and θ > ε − 1, ε > 1. These results are reproduced in
eq. (25) in the main text.
Next we rewrite the indirect utility function (22) in a more compact form. From the
definition of (8) the first and the second term in (22) can be combined, cancelling the
∆ terms. This gives
Z
ad
Z
0
ax
(ρx − kx ) dG − µ ln P,
(ρd − kd ) dG +
Ṽ =
(C.7)
0
where Ṽ ≡ V − C. We define
1
− ε−1
Z ai
−(ε−1) dG
,
āi ≡
a
G (ai )
0
i = {d, x}.
Under the Pareto distribution, ād and āx are linear in the respective cutoffs:
1
1
1
θ − (ε − 1) ε−1
ād =
≡ θ̄ ε−1 ad ,
āx = θ̄ ε−1 ax .
θ
(C.8)
Inserting firms’ price and quantity choices under the Pareto distribution gives
)
(
ε−1
Z ad
P
α
µ − Fd G (ad ) ,
(C.9)
(ρd − kd ) dG =
1−
∆ pd (ād )
0
(
)
ε−1
Z ax
α
P∗
(ρx − kx ) dG =
1−
µ − Fx G (ax ) .
(C.10)
∆ px (āx )
0
Using (C.2) and (C.3) together with (C.8) in (C.9) gives
Z ai
i
h
ε
(ρi − ki ) dG = (∆ − α) − 1 Fi G (ai ) ,
θ̄
0
i = {d, x}.
(C.11)
From (C.11) and (C.2) we obtain the indirect utility function
" εθ
#
θ
θ ε−1
−1
−1
h
i
ε
µ
1 ε−1
Fd
1
Ṽ1 = (∆ − α) − 1 [Fd G (ad ) + Fx G (ax )]+ ln
+
θ
∆
1 + τ∗
Fx∗
θ̄
(C.12)
#
"
1 εF
where Ṽ1 is a monotonous transformation of Ṽ , with Ṽ1 ≡ Ṽ + ln
a0 θ̄ θ
d
µ
α
1
1
ε−1 − θ
µ
.
The optimality condition derived from (C.12) is given by
h
i
ε
ε
∂ad
∂ad
[Fd G (ad ) + Fx G (ax )] + (∆ − α) − 1 Fd g (ad )
+ Fx g (ax )
∂∆
∂∆
θ̄
θ̄
θ
−ε ε−1
θ
µ
∆
=
ε
−1
.
(C.13)
θ
−1
θ
θ Fd ε−1
ε−1
θ 1 ε ε−1 −1
1
+ 1+τ ∗
∆
F∗
x
36
The RHS of this expression can be simplified using
1
∆
θ
ε ε−1
−1
+
1
1 + τ∗
θ Fd
Fx∗
θ
ε−1
−1
=
θ̄µ/εFd
θ
∆ε ε−1 G (ad )
.
(C.14)
Further we use the properties of the Pareto distribution
g (ad ) = θ
θ
aθ−1
d
= G (ad ) ,
θ
ad
a0
θ
G (ax ) .
ax
g (ax ) =
(C.15)
Using (C.14) and (C.15) in (C.13), and substituting θ̄ = [θ − (ε − 1)]/θ, the first-order
condition becomes
Fx G (ax )
θ̄
θ ∂ad Fx G (ax ) θ ∂ax
ε
1
1+
+ (∆ − α) −
+
=
− . (C.16)
Fd G (ad )
ε ad ∂∆ Fd G (ad ) ax ∂∆
ε−1 θ
Next we calculate the derivatives ∂ad /∂∆ and ∂ax /∂∆ in (C.16). Defining

θ̄µ
ε∆Fd

ad = 
1+∆
θ
−1
ε ε−1
1
1+τ ∗
 θ1
θ1
ζ

a0 ,
a0 ≡

θ
−1
θ Fd ε−1
ν
(C.17)
Fx∗
the derivative of ad with respect to ∆ is
∂ad
ad ∂ζ/∂∆ ∂ν/∂∆
=
−
,
∂∆
θ
ζ
ν
where
θ
1
∂ζ/∂∆
=− ,
ζ
∆
θ
ε ε−1 − 1
∂ν/∂∆
=
ν
∆
∆ε ε−1 −1
1+∆
Fd
Fx∗
θ
1
1+τ ∗
θ
1
1+τ ∗
θ
−1
ε ε−1
θ
ε−1
−1
Fd
Fx∗
.
θ
ε−1
−1
This yields, using the definition of Γ from (28) in the main text:
Γ (∆)
1 ∂ad
=−
.
ad ∂∆
∆θ
(C.18)
Differentiating ∂ax /∂∆ is straightforward. Using (C.5) gives
1 ∂ax
ε 1
=−
.
ax ∂∆
ε−1∆
(C.19)
Substituting (C.18)–(C.19) in (C.16) gives
Fx G (ax )
1
θ̄
θ Fx G (ax )
ε
1
1+
−
(∆ − α) −
Γ (∆) + ε
=
− . (C.20)
Fd G (ad )
∆
ε
ε − 1 Fd G (ad )
ε−1 θ
37
The final step is to incorporate G (ax ) and G (ad ) under the Pareto distribution:
G (ax ) =
G (ad ) =
ax
a0
a θ
d
ā
θ
≡
=
1
1+τ
θ
θ
−ε ε−1
∆
θ̄µ/ε
θ
∆Fd 1 + ∆ε ε−1 −1
Fd∗
Fx
θ
ε−1
N∗
(C.21)
θ
1
∗
1+τ
Fd
Fx∗
θ
ε−1
−1
(C.22)
Using this in (C.20) gives equation (26) in the main text. Moreover, using the definition
of ν in (C.17) we can rewrite G (ad ) as G (ad ) = (θ̄µ)/(ε∆Fd ν). Using this and the
definition of ν in (C.17) yields (27) in the main text.
Appendix D: Proof of Proposition 2
Consider first the opening up of imports. As long as the export channel remains closed
(Fx → ∞), χ = 0 holds from (29) and the optimality condition is:


1
1
1
1 ε − 1 1

−
.
 =1+
1 −
ε − 1 θ Γ (∆)
|ε {zθ } ∆
(D.1)
<1
In autarky, Γ = 1. Opening up for imports, Γ rises to a level Γ > 1 when Fx∗ < ∞.
Hence the RHS of (D.1) is now smaller. For the LHS to also fall, ∆ must increase. The
fact that ∆ = α under autarky completes the proof for this case.
Consider next the case where the small country exports good X, but there are no
imports. Hence Γ = 1 from (29). The first-order condition (26) then simplifies to:
θ
1
1ε−1
1
1ε−1
ε
1
1−ε
1−
1−
χ (∆) +
1−
=
− . (D.2)
ε−1
∆
ε θ
∆
ε θ
ε−1 θ
θ
The LHS of (D.2) is strictly increasing in χ, i.e. 1 − ε ε−1
1 − ∆1 1 − 1ε ε−1
> 0. To
θ
see this, suppose to the contrary that ∂LHS/∂χ ≤ 0. This implies
θ
1
1ε−1
1−ε
1−
1−
≤0
ε−1
∆
ε θ
(ε − 1) − εθ ≥ [(ε − 1) − εθ] ∆.
The term (ε − 1) − εθ must be negative since assuming to the contrary that (ε − 1) −
εθ ≥ 0 implies ε − 1 > [(ε − 1)/ε] ≥ θ, a contradiction. Thus, supposing ∂LHS/∂χ
38
≤ 0 implies 1 ≤ ∆, which is a contradiction to Proposition 1. Thus, the LHS is strictly
increasing in χ. In autarky, χ = 0 , and ∆ = α. Allowing firms to export, χ jumps up
to some χ > 0, so the LHS is now higher. Thus, as ∂LHS/∂∆ < 0, ∆ must increase
to restore optimality.
The case where the economy simultaneously opens up for imports and exports combines
the arguments made above. This completes the proof. Appendix E: Proof of Proposition 3
As a preliminary step, we group the different measures of economic integration into
two sets, l = {τ, Fx } and j = {τ ∗ , Fx∗ }. Differentiating χ in (27) and Γ in (28) with
respect to these different measures of economic integration gives
∂χ ∂χ
,
< 0,
∂l ∂j
∂Γ
= 0,
∂l
∂Γ
< 0.
∂j
(E.1)
Moreover, differentiating χ and Γ with respect to ∆ gives
∂Γ
> 0.
∂∆
∂χ
< 0,
∂∆
(E.2)
We start with part (a) of the Proposition, where there are no exports (χ = 0), but
there are imports (Γ > 1). The first-order-condition
1
1 α
1−
=1+
−
∆
θ
ε−1
in this case is
1
1
.
θ Γ (∆)
(E.3)
Next, we define
1 1
α
Ω̄ =
1−
−1 −
−
∆ | {z θ }
ε−1
|
{z
>0
>0
1
1
= 0.
θ Γ (∆)
}
(E.4)
Starting from an interior optimum ∆∗ in the initial equilibrium, it must be true that
∂ Ω̄/∂∆ < 0 for ∆ > ∆∗ . Moreover, for j = {τ ∗ , Fx∗ }
∂ Ω̄
∂ Ω̄ ∂Γ
=
< 0.
∂j
∂Γ ∂j
|{z}
|{z}
>0
<0
Thus, by the implicit function theorem:
∂∆
∂ Ω̄/∂j
(< 0)
=−
=−
< 0,
∂j
∂Ω/∂∆
(< 0)
39
(E.5)
which proves part (a) of Proposition 3.
Turning to part (b), the optimality condition without any restrictions on trade is:
1
θ
1
α̃ (∆) χ (∆) − α̃ (∆) Γ (∆) −
−
= 0,
(E.6)
Ω≡ 1−ε
ε−1
ε−1 θ
|
{z
}
>0
where we define
1
α̃ (∆) ≡ 1 −
∆
1ε−1
1−
,
ε θ
α̃ (∆)0 > 0.
(E.7)
Since the initial equilibrium represents a maximum, it must be true that ∂Ω/∂∆ < 0
for ∆ > ∆∗ . Hence raising ∆ is an optimal response to trade liberalization if and only
if ∂Ω/∂l < 0, where l = {τ, τ ∗ , Fx , Fx∗ }. Differentiating Ω gives
∂Ω
θ
∂χ
= 1−ε
α̃ (∆)
< 0,
∂l
ε−1
∂l
|{z}
|
{z
}
(E.8)
proving part (b) of Proposition 3. Note that with bilateral trade,
∂Ω
θ
∂χ
∂Γ
= 1−ε
α̃ (∆)
−α̃ (∆)
,
∂j
ε−1
∂l
∂j
|{z}
|{z}
|
{z
}
(E.9)
<0
>0
<0
>0
<0
the sign of which depends on α̃ (∆) and the factors feeding into these derivatives such
as N ∗ . A sufficient condition for ∂Ω/∂j < 0 is that α̃ (∆) < 0, which then implies a
rise in ∆ from a decline in inbound trade costs. Appendix F: Derivation of equation (39)
We start from the tax revenue expression (19). Substituting in the ‘gross profit’ expression πdg from (13) and the capital demand from (5) gives
#
#
ε−1
Z ad "
Z ad " ε ε−1
αP
P
α
R=t
(1 − α)µ
µ
− ∆Fd dG(a)+(∆−1)
+ Fd dG(a)
∆a
∆
a
0
0
(F.1)
where the integral in the first term represents aggregated ‘gross profits’ and thus the
variable part of domestic income in (20). The total change in tax revenues induced by
a change in the foreign trade cost τ ∗ is given by
dR
∂R
∂R ∂∆
= ∗+
.
∗
dτ
∂τ
∂∆ ∂τ ∗
40
Without home exports (χ = 0), the only direct effect of τ ∗ on tax revenues stems from
the change in ad . Indirect effects result from the induced change in ∆, which alters
gross profits, capital demands in the differentiated sector, and the price level.
Noting that πdg (ad ) = 0 yields in a first step
#
Z " ε−1
dR
P
∂∆
t(1 − α)µ(1 − ε)αε−1 ad
− Fd dG(a) ∗
=
∗
ε
dτ
∆
a
∂τ
0
"
#
ε−1
α ε Z ad
P
∂∆
+ Fd dG(a) ∗
µ
+
∆
a
∂τ
0
Z ad ∂P
α ε−1
∂∆
+ t(1 − α)µ(ε − 1)P ε−2
dG(a) ∗
∂∆ 0
∆a
∂τ
Z ad ε−1
ε
µα
P
∂∆
∆ ∂P
+ (∆ − 1) ε+1 −ε + (ε − 1)
dG(a) ∗
∆
P ∂∆ 0
a
∂τ
∂ad ∂ad ∂∆
+ (∆ − 1)kd (ad )
+
.
∂τ ∗ ∂∆ ∂τ ∗
(F.2)
Note first that ∂∆/∂τ ∗ < 0 from Proposition 3(a). Hence the first term in (F.2), which
incorporates the change in the gross profits of domestic firms, is positive from 1−ε < 0.
This implies that a reduction in τ ∗ reduces gross profits through the induced change
in ∆. The second and the third terms are negative, the latter because ∂P/∂∆ > 0
from (25). The sign of the fourth term depends on the expression in squared brackets.
The fifth term is again negative as ∆ < 1 and dad /dτ ∗ > 0 from (33).
To sign (F.2), we combine the positive first with the negative second effect. This gives
#
"
Z ε−1
Z ad
κµαε−1 ad P
∂∆
dG(a) + (1 − t)
Fd dG(a)
< 0,
(F.3)
ε
∆
a
∂τ ∗
0
0
where κ ≡ t(1 − α)(1 − ε) + α = α(1 − t) from ε = 1/(1 − α). Hence the sum of the
first two terms is unambiguously negative.
To sign the squared bracket in the fourth term, we use (C.2) in Appendix C. This gives
∆ ∂P
Ψ[εθ − (ε − 1)]
=
P ∂∆
θ(ε − 1)
where
θ
0<Ψ≡
αθ
θ
1 ε ε−1 −1
∆
θ
1 ε ε−1 −1
∆
α
θ
ε−1
−1
1
Fd
+
41
1
Fd
θ
ε−1
−1
θ
1 ε ε−1 −1
∗
∆
θ
1
∗
1+τ
1
Fx∗
< 1.
θ
ε−1
−1
Hence
−ε + (ε − 1)
∆ ∂P
Ψ(ε − 1)
= −ε(1 − Ψ) −
< 0.
P ∂∆
θ
(F.4)
Substituting (F.3) and (F.4) in (F.2) unambiguously signs dR/dτ ∗ < 0. This is given
in eq. (39) in the main text.
42
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