optimal tax policy and foreign direct investment under ambiguity

Journal of Macroeconomics 32 (2010) 185–200

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Journal of Macroeconomics

journal homepage: www.elsevier .com/locate / jmacro

Optimal tax policy and foreign direct investment under ambiguity

Takao AsanoFaculty of Economics, Okayama University, 3-1-1 Tsushimanaka, Kita-ku, Okayama 700-8530, Japan

a r t i c l e i n f o

Article history:Received 27 February 2007Accepted 22 December 2009Available online 4 January 2010

JEL classification:D81E62G31H21

Keywords:Optimal taxForeign direct investmentAmbiguity

0164-0704/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.jmacro.2009.12.006

E-mail address: [email protected] As in Aizenman and Marion (2004, p.126), vertica

each stage of production in the country in which it cproduct or service in multiple countries.

a b s t r a c t

We analyze the optimal timing of an irreversible foreign direct investment by a foreignfirm and the optimal tax policy by a host country under ambiguity. We derive the optimalGDP level at which the foreign firm switches from exporting to a foreign direct investment.Furthermore, we derive the optimal tax policy by the host country, and analyze the effect ofan increase in ambiguity on the optimal tax policy. We show that the host country shouldreduce the optimal corporate tax rate from the host government’s perspective in responseto an increase in ambiguity. Our result is different from the one obtained by Pennings(2005) that shows that an increase in risk induces an increase in the optimal corporatetax rate.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Suppose a firm that considers whether to export her product to a host country or to enter the market by undertaking aforeign direct investment (henceforth FDI). Compared with domestic markets, it can be considered that foreign firms thataim to enter foreign markets are faced with more uncertainty about the prospect of foreign markets than that about the pros-pect of their own domestic markets since economic and political stability in foreign countries cannot be easily predicted.Since uncertainty about the prospect of foreign markets directly leads to uncertainty about the prospect of profits earnedin the host country, it can be considered that uncertainty has a negative impact on FDI. As pointed out by Aizenman andMarion (2004), when we analyze FDI, focusing on uncertainty is important since (1) in general, host countries of FDI aredeveloping countries, and (2) business in developing countries is considered to be more uncertain than that in developedcountries. Thus, if greater uncertainty has a negative impact on FDI from developed countries, then it might deter economicdevelopment in developing countries, which implies that developing countries should adopt policies to encourage FDI.

Aizenman and Marion (2004) analyze the impact of risk in vertical and horizontal FDI.1 Based on data on US multinationalfirms (US Bureau of Economic Analysis), Aizenman and Marion (2004) find empirical evidence that (1) risk has a negative impacton FDI, (2) risk has a greater negative impact on vertical FDI than horizontal FDI, and (3) risk has a negative impact on corporatetaxes, which implies that a lower corporate tax significantly increases FDI. On the other hand, Pennings (2005) analyzes effectsof increases in risk on host country’s corporate tax rate, and shows that an increase in risk increases the corporate tax rate. Thisresult neither conforms to Aizenman and Marion (2004)’s empirical result nor our intuition that when firms are more uncertainabout the prospect of foreign markets, they are reluctant to undertake FDI, which makes host governments reduce their

. All rights reserved.

l FDI is adopted when multinational firms fragment their production processes internationally and locatean be produced at the least cost. Horizontal FDI is adopted when multinational firms produce the same

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186 T. Asano / Journal of Macroeconomics 32 (2010) 185–200

corporate taxes to encourage FDI.2 The purpose of this paper is to generalize the notion of uncertainty to the notion of ambiguity,which is distinguished by risk, and to show that the host government should reduce the optimal corporate tax rate from the hostgovernment’s perspective in response to an increase in ambiguity.3

The importance of the distinction between risk and ambiguity is pointed out by Ellsberg (1961), which provides someevidence that people tend to prefer to act on known rather than unknown or vague probabilities. Uncertainty that is capturedby a set of probability measures is called Knightian uncertainty or ambiguity. On the other hand, uncertainty that is capturedby a unique probability measure is called risk. Ambiguity can be analyzed within the framework of the Maxmin ExpectedUtility (henceforth MMEU). MMEU axiomatized by Gilboa and Schmeidler (1989) in order to overcome the Ellsberg paradoxstates that if a certain set of axioms is satisfied, then decision maker’s beliefs are captured by a set of probability measuresand her preferences are represented by the minimum of expected utilities over the set of probability measures. MMEU hasdeepened our understanding of decision maker’s behaviors under ambiguity. If foreign companies are assumed to be lessconfident about the prospect of economic and political stability in host countries and they are assumed to make decisionsvery cautiously, then their decisions can be well analyzed within the framework of MMEU. Therefore, this paper adopts acontinuous-time model of ambiguity proposed by Chen and Epstein (2002) in order to provide an economic foundationfor tax policies adopted by host governments that aim to encourage FDI.4

We consider a situation in which a foreign firm must decide whether to export her product to a host country or to enter themarket by undertaking an irreversible FDI under ambiguity.5 In this paper, FDI is irrversible in the sense that once the foreign firmdecides to shut down her plant at the host country, the cost of the plant cannot be recovered when she switches to exporting. Forthe evolution of the GDP level in the host country, we assume that the evolution follows a geometric Brownian motion. Contrary tothe standard model in which the evolution of the GDP level is characterized by a geometric Brownian motion based on a singleprobability measure, we assume that the firm is not perfectly confident about the evolution of the GDP level. In other words, thefirm is faced with ambiguity in which her beliefs are characterized by a set of probability measures, not by a single probabilitymeasure and her preferences are represented by the minimum of expected utilities over the set of probability measures.

We provide four specific models in Sections 3 and 4. In Sections 3.1 and 3.3, we consider Nash bargaining situations underrisk and under ambiguity where the foreign firm and the host government jointly maximize their present values of benefits,and we derive the critical GDP level, the optimal corporate tax rate and the optimal subsidy to the cost of investment. Theseoutcomes correspond to efficient solutions to a social planner. In Sections 3.2 and 3.4, we consider non-cooperative situa-tions under risk and under ambiguity in which the foreign firm and the host government maximize their own present valuesof benefits, and we derive the optimal GDP level, and the optimal corporate tax rate. In the non-cooperative situations, tax-ation by the host government leads to a double marginalization problem. This is because the firm requires a mark-up overthe cost of FDI and the host government charges a mark-up over the required after-tax payoff to the firm.

We show that an increase in ambiguity induces a decrease in the value of undertaking FDI, and that an increase in ambi-guity induces a decrease in the optimal corporate tax rate from the host government’s perspective. The first claim impliesthat the negative impact on the value of undertaking FDI makes the firm become more cautious about FDI than beforeand makes the firm postpone FDI. The second claim states that the host government should reduce the optimal corporatetax rate in response to an increase in ambiguity in order to encourage FDI by the foreign company that is reluctant to under-take FDI and is less confident about the evolution of the GDP level. Furthermore, the second claim also implies that the moreambiguity the foreign company has about the prospect of the GDP level, the greater is the incentive for the host governmentto reduce the optimal corporate tax rate than to increase. Our results are different from the ones obtained by Pennings (2005)that shows that an increase in risk does not affect the value of undertaking FDI, and that an increase in risk induces an in-crease in the optimal corporate tax rate from the host government’s perspective.6

2 In the literature on real option, risk is captured by volatility, which implies that volatility has both negative and positive effects on profits. Therefore, itcould be considered that Aizenman and Marion (2004)’s empirical results capture this negative effect on profits. This implies that an increase in risk induces adecrease in the corporate tax rate.

3 Mackie-Mason (1990), Hasset and Metcalf (1999) and Pennings (2005) analyze effects on investment of uncertainty about tax policy. Mackie-Mason (1990)and Hasset and Metcalf (1999) consider exogenous taxation, whereas Pennings (2005) and this paper consider endogenous taxation.

4 In static frameworks, Ghirardato et al. (2004) and Klibanoff et al. (2005) provide more general models than MMEU in which MMEU is derived as a specialcase. Klibanoff et al. (2009) generalize Klibanoff et al. (2005)’s model into a dynamic but discrete-time infinite horizon framework. In this paper, we adopt acontinuous-time model of ambiguity proposed by Chen and Epstein (2002) in order to analyze behaviors under ambiguity within a continuous-timeframework. For applications of Chen and Epstein (2002), Epstein and Miao (2003) analyze the home bias puzzle with the presence of ambiguity in generalequilibrium settings, and show that the puzzle can be resolved by incorporating ambiguity. Nishimura and Ozaki (2007) first introduce the concept ofambiguity into real option and analyze an optimal investment problem under ambiguity. Based on Nishimura and Ozaki (2007), Asano (2007) analyzes optimalenvironmental policies under ambiguity.

5 In this paper, we assume that a foreign firm determines the timing of switching from exporting to FDI when the host country’s GDP level reaches a certaincritical level. Within our model, the GDP level and the firm’s profitability are positively related. Thus, the firm’s problem can be formulated as follows:the firm’sproblem is to determine the timing of switching from exporting to FDI when the profit of undertaking FDI reaches a certain critical value. Contrary to our model,Pennings (2005) assumes that the foreign firm’s decision depends on the growth rate in households. See footnote 9.

6 In the literature on real option, it has been widely accepted that an increase in risk has a negative impact on critical values of optimal investments. Sarkar(2000) and Wong (2007) analyze the effect of risk on investment timing in a canonical real option model within the framework of the single-factorintertemporal capital asset pricing model. Wong (2007) shows that an increase in risk induces an increase in the critical value of the optimal investment forrelatively high levels of risk. Wong (2007) also shows that an increase in risk induces a decrease in the critical value of the optimal investment for relatively lowlevels of risk. On the other hand, Nishimura and Ozaki (2007) analyze the effect of ambiguity on investment timing within the framework of MMEU, and showthat an increase in ambiguity induces a decrease in the critical value of optimal investments.

T. Asano / Journal of Macroeconomics 32 (2010) 185–200 187

The organization of this paper is as follows. Section 2 provides continuous-time models under risk and ambiguity, andderives the value of FDI and the value of undertaking FDI. Section 3 describes a cooperative game and a non-cooperativegame framework, respectively. First, we analyze a Nash bargaining situation in which the foreign firm and the host govern-ment jointly maximize their surpluses. Second, we analyze a non-cooperative game situation where in the first stage thegovernment fixes the corporate tax rate and in the second stage the firm decides when to enter the market. Furthermore,Section 3 derives the critical level of the GDP in a cooperative game framework and in a non-cooperative game framework,respectively. Section 4 provides sensitivity analyses, and shows that an increase in ambiguity induces a decrease in the valueof undertaking FDI, and that an increase in ambiguity induces a decrease in the optimal corporate tax rate from the host gov-ernment’s perspective.

2. The value of foreign direct investment under ambiguity in continuous time

In this section, we provide continuous-time models in risk and ambiguity. Before we present a continuous-time modelunder ambiguity proposed by Chen and Epstein (2002), we review a continuous-time model under risk analyzed by Pennings(2005).

2.1. A general continuous-time model under risk

We consider a situation in which a firm is faced with the problem whether she exports her product from her home coun-try to a foreign country or she builds a plant by undertaking FDI and produces in the foreign country.7 We assume that thecost of building her plant by undertaking FDI is completely irreversible.8 Moreover, we consider situations in which there existonly two countries, and that the cost of exporting from a third country is extremely high, which implies that it is only profitablefor the firm to export her product from her country or to produce in the host country by undertaking FDI. We assume that thefirm’s decision depends on the evolution of the GDP level in the foreign country, which directly affects her profit. Before explain-ing the firm’s problem and the host government’s problem in details, some setups are in order. In this paper, the source of ambi-guity about FDI is captured by the growth of the GDP level in the host country.9

Let ðX;FT ; PÞ be a probability space, let ðBtÞ06t6T be a standard Brownian motion with respect to P, and let ðFtÞ06t6T be thestandard filtration for ðBtÞ06t6T . Let Yt be the evolution of the GDP level in the host country at time t. We assume that theevolution of the GDP level ðYtÞ06t6T follows a geometric Brownian motion:

7 In tshe wan

8 Asspecificsold to

9 Othin futurand theBuckleythe gropaper, aauthor

10 Wit

dYt ¼ lYtdt þ rYtdBt;

where l and r are some constants.10 Since the geometric Brownian motion is unbounded above in the long run, the assump-tion that the GDP level follows this geometric Brownian motion is made as an approximation for the evolution of the GDP levelin the short and medium run.

As in Pennings (2005), we consider the industry in which the firm does business is so small compared to the entire econ-omy that price changes in the industry do not affect the prices of all other industries. Thus, partial equilibrium analysis is inorder, and a representative household is assumed to maximize the following utility function U : R� R! R,

Uðx;mÞ ¼ uðxÞ þm;

where u : R! R is her felicity function, x is her consumption of the good, and m denotes the total expenditure on all othergoods. As in standard models, her felicity function u is assumed to be twice differentiable, u0ðxÞ > 0 and u00ðxÞ 6 0 for allx P 0. Then, the inverse demand function of the representative household is u0ðxÞ ¼ q, where q is the consumer price ofthe good.

The consumer price of the good can be written as q ¼ qE ¼ pE þx if the firm exports, and q ¼ qF ¼ pF if the firm under-takes FDI, where pE and pF denote the producer prices in the export and FDI, respectively, and x P 0 denotes the trade coststhat consist of transportation costs, x1 P 0, and a specific tariff, x2 P 0 such that x ¼ x1 þx2. Thus, x ¼ 0 if the firmundertakes FDI and produces in the host country. Let cE and cF be the constant marginal costs of production at the plantin the home country and the plant in the foreign country, respectively. We assume that the production of one unit of the

his paper, we assume that there exist no capacity constraints, that is, after undertaking FDI, the foreign firm can produce in the host country as much asts in order to maximize her profit.

Dixit and Pindyck (1994, p. 8) point out, investment expenditures are considered to be sunk costs and thus irreversible if they are firm or industry. If an investment expenditure is firm specific, then it cannot be sold to another firm. On the other hand, if it is industry specific, then it cannot be alsoanother firm. If a firm considers that her plant is useless and she wants to sell the plant, then it is also useless for all the firms in the same industry.er factors, for example, changes in producer prices or changes in investment costs can influence future profits of FDI. Pennings (2005) assumes that riske profits lies in the randomness of the evolution of the market size. As pointed out in Pennings (2005), his assumption fits with the agglomeration effectconcentration effect on FDI that are empirically found. For example, see Brainard (1997) and Head et al. (1995). Moreover, his approach generalizesand Casson (1981) that derive a critical market size at which a foreign firm switches from exporting to FDI within a deterministic framework. However,

wth rate in households seems to be the least significant source of ambiguity, because it evolves slowly and can be well predicted. Therefore, in thismbiguity about future profits from investment projects is assumed to lie in the randomness of the evolution of the GDP level in the host country. The

thanks one of the anonymous referees for pointing out this approach.hout loss of generality, let r > 0. For r < 0, let ð�BtÞ instead of ðBtÞ in the following analyses. We exclude r ¼ 0 that corresponds to a deterministic case.


good at the home country and the production at the foreign country entail constant marginal costs, cE and cF , respectively. Asin Blanchard (1981) that assumes that real profit is an increasing function of output, this paper assumes that profits from theinvestment project are modelled as a constant fraction of the total GDP level. That is, profits from the investment project arerepresented by kYt for some 0 < k < 1.11 Then, the firm’s maximized pre-tax and pre-tariff profit per unit household ispE ¼ ðpE � cEÞxðqEÞ if the firm exports, and is pF ¼ ðpF � cFÞxðqFÞ if the firm undertakes FDI and produces in the host country.Note that pE and pF denote the producer prices in the export and FDI that maximize producer’s pre-tax and pre-tariff profitsper unit household, respectively, and that qE ¼ pE þx and qF ¼ pF . In the sequel, xðqEÞ and xðqFÞwill be sometimes written xE

and xF for short, respectively.When the firm undertakes FDI, she has to bear the cost of setting up a plant in the host country, which is denoted by

I P 0. Furthermore, the firm is levied a corporate tax 0 6 s1 6 1 over her profit earned from FDI and a lump-sum tax (or sub-sidy) s2 P 0 (or �I 6 s2 6 0) by the host country. On the other hand, the firm is levied a corporate tax 0 6 s0 6 1 over herprofit from exporting by the home country.

In order to analyze firm’s decision problem in an infinite-time horizon, we assume that if the firm builds a plantin the host country by undertaking FDI, then the plant will never depreciate completely. Then, the value at t of undertakingFDI is

11 Sin

Wt �WðYtÞ ¼ EPZ 1

te�rðs�tÞYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds

� ��Ft

�¼ A

Yt

r � l; ð1Þ

where r is the discount factor, EP½�jFt � is the expectation with respect to P conditioned on Ft , and A is defined by

A � ð1� s1ÞpF � ð1� s0ÞpE� �

: ð2Þ

The firm’s problem determines the timing of switching from exporting to FDI. The optimal time is the solution to the opti-mal stopping problem of finding an (Ft)-stopping time, t0, that maximizes the value at time 0 of FDI under risk:

Fr0 � max

t02½0;1�EPZ 1

t0e�rsYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds� e�rt0 ðI þ s2Þ

� ��F0

�: ð3Þ

In order to ensure the existence of the expected present value of Yt , it is standard to assume that r > l. Now that the value attime 0 of FDI is defined by (3), the value at time t of FDI under risk, denoted Fr

t , is as follows:

Frt � max

t02½t;1�EPZ 1

t0e�rðs�tÞYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds� e�rðt0�tÞðI þ s2Þ

� ��Ft

�:

Note that Frt can be rewritten as follows:

Frt � FrðYtÞ ¼max

t0EP EP e�rðt0�tÞ

Z 1

t0AYse�rðs�t0 Þds� ðI þ s2Þ

� �� Ft0

��Ft

� �

¼maxt0

EP e�rðt0�tÞ EPZ 1

t0AYse�rðs�t0 Þds

� ��Ft0

� �� ðI þ s2Þ

� ��Ft

�¼max

t0EP e�rðt0�tÞ AYt0

r � l� ðI þ s2Þ

� �� Ft

�;

where the first equality follows from the law of iterated integrals, and the third equality follows from (1). Following the stan-dard argument in the literature on real option, FrðYtÞ can be obtained as follows:

FrðYtÞ ¼AYtr�l� ðI þ s2Þ if Yt P Y�r

Iþs2b�1

YtY�r

bif Yt < Y�r ;

8<: ð4Þ

where the optimal GDP level under risk Y�r and b are provided by

Y�r ¼r � l

Ab

b� 1ðI þ s2Þ and

b ¼�ðl� 1

2 r2Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl� 1

2 r2Þ2 þ 2rr2q

r2 ; ð5Þ

respectively. Note that b is the positive root of the characteristic equation ð1=2Þr2xðx� 1Þ þ lx� r ¼ 0, and that b > 1. (Forexample, see Dixit and Pindyck (1994).)

ce k does not affect analyses of our models, without loss of generality, we assume that k ¼ 1.


Next, we consider the government’s problem. The government decides a corporate tax rate, a lump-sum tax rate, and atariff in order to maximize the net domestic gain as follows:

12 Let

is equivfunctio

EPZ t0

tuðxEÞ � qExE� �

Yse�rðs�tÞds

" ��Ft

#þ EP

Z 1

t0uðxFÞ � qF xF� �

Yse�rðs�tÞds� ��Ft

�þ s1

pFY�rr � l

þ s2

� �EP e�rðt0�tÞ� ��Ft

þ EPZ t0

tx2xEYse�rðs�tÞds

" ��Ft

#;

where the first term is the present value of expected consumer surplus from export, the second is the present value of ex-pected consumer surplus from FDI, the third is the present value of expected tax income, the fourth is the present value ofexpected tariff income, and t0 is the first passing time defined by t0 � infft P 0 jYt R ð0;Y�r Þg. Pennings (2005) shows that thegovernment’s expected net domestic gain at time t under risk can be obtained as follows:

GrðYtÞ ¼YtðcsE þx2xEÞ

r � l þ Y�r ðcsF þ s1pF � csE �x2xEÞ þ s2ðr � lÞr � l

Yt

Y�r

� �b

;

where csE � uðxEÞ � qExE and csF � uðxFÞ � qF xF .

2.2. Ambiguity in continuous time

Let H be a set of density generators. (For the definition of density generators, see Appendix A.) For such a set H, define theset of probability measures, PH on ðX;FTÞ, generated by H, by12

PH ¼ Q hn ��h 2 H

o: ð6Þ

In order to analyze behaviors under ambiguity within the framework of MMEU, decision maker’s beliefs are assumed to becaptured by the set of probability measures equivalent to a probability measure P. If the set of probability measures PH be-comes large through the set of density generators H, then this implies that the decision maker considers a lot of situationsincluding the worst and the best ones.

Let us introduce the following class of density generators first proposed by Chen and Epstein (2002). The notion of i.i.d.ambiguity is introduced in order to solve dynamic optimization problems analytically. Ambiguity characterized by HK iscalled i.i.d. ambiguity if there exists a compact subset K of R such that 0 2 K and

HK ¼ ðhtÞ 2L2jhtðxÞ 2 Kðm� PÞ-a:s:� �

; ð7Þ

where m is the Lebesgue measure restricted on Bð½0; T�Þ, and Bð½0; T�Þ denotes the smallest Borel r-algebra containing ½0; T�.Note that the set HK is independent of state and time. In the case of j-ignorance, which is a special case of i.i.d. ambiguity HK ,we can parameterize the degree of ambiguity, that is, the set K in the case of j-ignorance is specified as K ¼ ½�j;j� for allj > 0 and the positive real number j is considered to represent the degree of ambiguity because the larger j, the larger theset of probability measures. By considering the case of j-ignorance, we can perform comparative static analyses on the ef-fects of ambiguity.

Finally, we derive the value at t of undertaking FDI under ambiguity. We assume that the firm is ambiguity averse, that is,her beliefs are captured by the set of probability measures, and she maximizes the infimum of her expected profit over PH.Furthermore, we assume that ambiguity is characterized by i.i.d. ambiguity, the planning horizon is infinite, and the plantwill never depreciate completely. These three assumptions enable us to derive the value at t of undertaking FDI under ambi-guity that does not depend on time t directly. This stationarity is crucial to solve the Hamilton–Jacobi–Bellman equation(henceforth HJB equation) under ambiguity (see Section 2.3). Thus, the value at t of undertaking FDI under ambiguity isas follows:

Vt � VðYtÞ ¼ infQ2PH

EQZ 1

te�rðs�tÞYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds

� ��Ft

�:

The following proposition can be proved similar to Proposition 1 in Nishimura and Ozaki (2007). Thus, the proof is omitted.

Proposition 1. Suppose that the firm is ambiguity averse, and her beliefs are characterized by HK , where HK is defined by (7) forsome ðKÞ. Then, the value at t of undertaking FDI under ambiguity is provided by

h be a density generator. Let ðzht Þ06t6T be a stochastic process defined by (31) in Appendix A. Then, a probability measure Qh on ðX;FT Þ defined by

ð8A 2FT Þ Q hðAÞ ¼Z

Xzh

T ðxÞvAðxÞdPðxÞ ¼ EP ½vAzhT �:

alent to P. Conversely, any probability measures equivalent to P can be obtained by a density generator in this way. Note that v denotes the indicatorn.

13 For14 In o

assump


VðYtÞ ¼AYt

ðr � ðl� rh�ÞÞ ; ð8Þ

where A is defined by (2), and h� is defined by

h� � argmax rxjx 2 Kf g ¼max K: ð9Þ

In the case of no-ambiguity (h� ¼ 0) analyzed by Pennings (2005), the right-hand-side of (8) turns out to VðYtÞ ¼AYt=ðr � lÞ. Then, risk captured by r does not affect the value of undertaking FDI.

2.3. The value of FDI under i.i.d. ambiguity

In this subsection, as in Section 2.2, we assume that ambiguity is characterized by i.i.d. ambiguity, the planning horizon isinfinite, and the plant will never depreciate completely. These assumptions enable us to provide further characterization ofthe value of FDI and analytically solve the HJB equation under ambiguity.

The firm’s problem is to determine the timing of switching from exporting to FDI under ambiguity contrary to Section 2.1in which the firm is faced with risk. Thus, the firm is supposed to find an ðFtÞ-stopping time, t0, that maximizes the value attime 0 of FDI:

minQ2PH

EQZ 1

t0e�rsYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds� e�rt0 ðI þ s2Þ

� ��F0

�:

Therefore, the maximum value at time t of FDI under ambiguity, denoted by Fat , is as follows:

Fat ¼ max

t02½t;1�minQ2PH

EQZ 1

t0e�rðs�tÞYs ð1� s1ÞpF � ð1� s0ÞpE

� �ds� e�rðt0�tÞðI þ s2Þ

� ��Ft

�: ð10Þ

Under the above three assumptions, the maximum value at t of FDI under ambiguity, defined by (10), turns out to be sta-tionary, which implies that we can denote Fa

t as Fat ¼ FaðVtÞ, where Fa : Rþ ! R is some real-valued function. Accordingly, Fa

can be obtained as the unique solution to the following HJB equation13:

FaðVtÞ ¼ max Vt � ðI þ s2Þ; minh2H

EQh

½dFat jFt � þ FaðVtÞ � rFaðVtÞdt

� �: ð11Þ

Therefore, in the continuation region, it follows that

minh2H

EQh

½dFat jFt � ¼ rFaðVtÞdt:

This equation states that the minimum expected capital gain of holding FDI opportunity is equal to its opportunity cost mea-sured in terms of the firm’s discount rate. By applying Ito’s lemma to (8) together with (34) in Appendix A, it follows that:

dVt ¼ ðl� rhÞVtdt þ rVtdBht ; ð12Þ

where V0 ¼ AN0=ðr � lþ rh�Þ and Bht is defined by (34) in Appendix A. Moreover, by applying Ito’s lemma to (10) together

with (12), it follows that:

dFat ¼

dFa

dVtðl� rhÞVtdt þ rVtdBh

t

þ 1

2r2V2

td2Fa

dV2t

dt:

In the continuation region, it can be shown that14

minh2H

EQh

½dFat jFt � ¼

dFa

dVtðl� rh�ÞVtdt þ 1

2r2V2

td2Fa

dV2t

dt: ð13Þ

Thus, in the continuation region, it follows that:

12r2V2

td2Fa

dV2t

þ ðl� rh�ÞVtdFa

dVt� rFaðVtÞ ¼ 0: ð14Þ

the derivation of (11), see Nishimura and Ozaki (2007).rder to derive (13), we assume that @Fa=@Vt is positive, and Fa is twice differentiable in the continuation region. We can show that these two

tions actually hold. For the derivation of (13), see Nishimura and Ozaki (2007).


We solve this differential equation under the following boundary conditions:

15 Conconjugato (14).B ¼ ððI þ

16 Whthat tooptimal

17 Sincmaximi

Fað0Þ ¼ 0; ð15ÞFaðV�Þ ¼ V� � ðI þ s2Þ; and ð16ÞðFaÞ0ðV�Þ ¼ 1; ð17Þ

where V� is the critical value of V at or above which the firm should enter the market immediately. Condition (15) reflectsthe fact that if Vt is always zero, then the value of FDI is zero. Thus, the value of FDI will remain to be zero. Condition (16) isthe value matching condition; when Vt ¼ V� and the firm decides to enter the market, it incurs a sunk cost I and obtains thenet payoff V� � ðI þ s2Þ. Condition (17) is the smooth pasting condition; if entering the market at V� is critical, then the deriv-ative of the value function must be continuous at V�. For the value matching condition and the smooth pasting condition, seeDixit and Pindyck (1994). By solving the differential equation with the three boundary conditions, we obtain the following15:

FaðVtÞ ¼Iþs2a�1

� �1�aa�aðVtÞa if Vt < V�

Vt � ðI þ s2Þ if Vt P V�;

(ð18Þ

where

V� ¼ aa� 1

ðI þ s2Þ; ð19Þ

a ¼� ðl� rh�Þ � 1

2 r2

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl� rh�Þ � 1

2 r2� �2 þ 2rr2

qr2 : ð20Þ

The uniqueness of the reservation value V�, the twice-differentiability of Fa, and the positivity of ðFaÞ0 follow from this opti-mal value of FDI. Note that the optimal GDP level Y�a under ambiguity can be obtained from (8) and (19) as follows16:

Y�a ¼r � ðl� rh�Þ

AV� ¼ r � ðl� rh�Þ

Aa

a� 1ðI þ s2Þ; ð21Þ

where A is defined by (2).

2.4. The value of the government’s expected gain

In this subsection, we consider the government’s problem. At first, we define the government’s objective function andderive the government’s expected net domestic gain under ambiguity. We assume that the government faces no ambiguityitself, but calculates her expected payoff given the firm faces ambiguity and evaluates the FDI under the firm’s worst casescenario. Then, the government decides a corporate tax rate, a lump-sum tax rate, and a tariff in order to maximize thenet domestic gain that depends on the firm’s timing of entry whose beliefs are captured by a set of probability measurescontrary to Section 2.1. When the firm is faced with ambiguity, the government’s payoff is as follows17:

minQ2PH

EQZ t0

tuðxEÞ � qExE� �

Yse�rðs�tÞds

" ��Ft

#þ min

Q2PHEQ

Z 1

t0uðxFÞ � qF xF� �

Yse�rðs�tÞds� ��Ft

�

þ minQ2PH

EQ s1pFY�a

r � ðl� rh�Þ þ s2

� �e�rðt0�tÞ

� ��Ft

�þ min

Q2PHEQ

Z t0

tx2xEYse�rðs�tÞds

" ��Ft

#: ð22Þ

Thus, the government objective is to maximize (22) with respect to s1, s2, and x2. Note that the first term is the present valueof expected consumer surplus from export, the second is the present value of expected consumer surplus from FDI, the thirdis the present value of expected tax income, the fourth is the present value of expected tariff income, and t0 is the first passingtime defined by t0 ¼ infft P 0 jYt R ð0;Y�aÞg. The following three lemmata that are proved in Appendix B enable us to derivethe government’s expected net domestic gain at time t under ambiguity. Recall that h�, a, and Y�a are defined by (9), (20) and(21), respectively.

Lemma 1. Suppose that the firm is ambiguity averse, and her beliefs are characterized by (7), the planning horizon is infinite, andthe plant will never depreciate completely. Then,

sider the characteristic equation for (14): ð1=2Þr2xðx� 1Þ þ ðl� rh�Þx� r ¼ 0. The solutions to this characteristic equation are provided by (20) and itste c. It can be shown that any solution to (14) can be uniquely expressed by FaðVtÞ ¼ BVa

t þ CVct for some B; C in R. Conversely, this is indeed a solution

In Appendix B, we show that a > 1 and c < 0. On the other hand, the negativity of c and (15) imply C ¼ 0. Then, C ¼ 0, (16) and (17) implys2Þ=ða� 1ÞÞ1�aa�a .

en the GDP level Yt is strictly less than Y�a , then the value at t of undertaking FDI under ambiguity, VðYtÞ (see (8)), is strictly less than V� , which implieswait is the optimal strategy. On the other hand, when Yt is greater than Y�a , then VðYtÞ is greater than V� , which implies that to stop right now is thestrategy. Therefore, Y�a can be considered to be the reservation value.e we assume that the government faces no ambiguity itself, but calculates her expected payoff given the firm faces ambiguity and chooses t0 in order toze (10), each of the terms in (22) is evaluated by min operators over the firm’s beliefs Ph .

18 Anbargainbecause

19 As iwr

F andambigu


minQ2PH

EQ e�rðt0�tÞ� ��Ft ¼ Yt

Y�a

� �a

:

From (18) and (21) and Lemma 1, FaðVtÞ can be rewritten as a function of Nt:

FaðYtÞ ¼I þ s2

a� 1Yt

Y�a

� �a

¼ ð1� s1ÞpF � ð1� s0ÞpE

r � ðl� rh�Þ Y�a � ðI þ s2Þ� �

minQ2PH

EQ e�rðt0�tÞ� ��Ft : ð23Þ


minQ2PH

EQZ t0

tYse�rðs�tÞds

" ��Ft

#¼ Yt

r � ðl� rh�Þ �Y�a

r � ðl� rh�ÞYt

Y�a

� �a

:


minQ2PH

EQZ 1

t0Yse�rðs�tÞds

� ��Ft

�¼ Y�a


Y�a

� �a

:

Equation (22) and Lemmas 1–3 enable us to obtain the government’s expected net domestic gain at time t under ambi-guity as follows:

GaðYtÞ ¼YtðcsE þx2xEÞr � ðl� rh�Þ þ

Y�aðcsF þ s1pF � csE �x2xEÞ þ s2ðr � ðl� rh�ÞÞr � ðl� rh�Þ

Yt

Y�a

� �a

; ð24Þ

where csE ¼ uðxEÞ � qExE and csF ¼ uðxFÞ � qF xF .

3. Cooperative game and non-cooperative game

In this section, we provide cooperative and non-cooperative solutions under ambiguity. The former is the situation inwhich the host country and the foreign company jointly maximize their profits. The latter is the situation where the hostcountry and the foreign company maximize their profits in a two-stage game in which the host country fixes her corporatetax rate in the first stage and the foreign company determines the optimal entry timing in the second stage. At first, we re-view the results obtained by Pennings (2005) under risk.

3.1. Cooperative solution under risk

In this subsection, we consider the situation where the host government levies the corporate tax, s1, and the lump-sumtax (or subsidy), s2, on the foreign firm under risk in a cooperative bargaining game.18

Let Frc : Rþ ! R and Grc : Rþ ! R be the firm and the government payoffs at time t under risk in a cooperative bargaininggame, respectively, and let N�rc be the critical value of the GDP level under risk in this cooperative bargaining framework, at orabove which the firm immediately switches from exporting to FDI, and below which she continues exporting her product tothe host country. Let wr

F and wrG be the market power of the firm and the government under risk, respectively, with

wrF þ wr

G ¼ 1.19 Then, the following proposition is obtained.

Proposition 2 (Pennings (2005)). Suppose that in a cooperative bargaining situation the firm and the governmentsimultaneously determine the timing of undertaking FDI and maximize their joint profits with respect to s1 and s2. Then, thecritical GDP level Y�rc is provided by

Y�rc ¼r � l

pF � ð1� s0ÞpE þ csF � csE �x2xE

bb� 1

I;

and sr1 and sr

2 satisfy the following:

FrcðYtÞ ¼ wrFðF

rcðYtÞ þ GrcðYtÞÞ and GrcðYtÞ ¼ wrGðF

rcðYtÞ þ GrcðYtÞÞ;

where sr1 and sr

2 denote optimal tax rates maximizing their joint profits. Furthermore, the total surplus is provided by

active role for the home government could be considered since by changing the corporate tax s0, the home government could affect the solution of theing process between the firm and the host government. We leave analyses of this topic not only under risk but also under ambiguity for future research

it is beyond the scope of this paper.n Rubinstein (1982), this paper assumes that the market power of the firm and the government wr

F and wrG is exogenously given. To endogenously derive

wrG is a very interesting task. But this task is beyond the scope of this paper. We also assume that the market power of the firm and the government under

ity is exogenously given. See Section 3.3.

20 In Panalyse


FrcðYtÞ þ GrcðYtÞ ¼I

b� 1Yt

Y�rc

� �b

þ YtðcsE þxxEÞr � l

;

where b is defined by (5).20

In this cooperative game situation, the firm undertakes FDI if the GDP level Yt reaches the critical GDP level Y�rc at whichthe total benefits are jointly maximized, and the total surplus is divided based on their bargaining power. Moreover, the out-come in the cooperative game situation reflects the outcome of a social planner that is socially desirable one. When the dif-ference between csF and csE increases, the critical GDP level Y�rc decreases, which makes the total surplus Frc þ Grc increase.

3.2. Non-cooperative solution under risk

In this subsection, we study the optimal taxation in a non-cooperative situation. The host government and the foreignfirm are assumed to maximize their own benfits in a two-stage game. The host government sets her corporate tax rate leviedon the foreign firm in the first stage, and the foreign firm determines the optimal entry timing in the second stage.

Let �s1 be the corporate tax rate such that

ð1� �s1ÞpF � ð1� s0ÞpE ¼ 0: ð25Þ

When s1 ¼ �s1, all profits exceeding benefits from exporting are taxed away by the host government. Let Frnc : Rþ ! R andGrnc : Rþ ! R be the firm and the government profits at time t under risk in this non-cooperative game, respectively, andlet Y�rnc be the critical GDP level in this non-cooperative game under risk. Then, the following proposition is obtained.

Proposition 3 (Pennings (2005)). Let s2 ¼ 0 (no subsidy). Then, the optimal corporate tax rate is provided by

sr1 ¼

�s1

b� b� 1

bcsF � csE �x2xE

pF: ð26Þ

In this case, the critical GDP level is

Y�rnc ¼b

b� 1Y�rc: ð27Þ

The firm’s profit at time t is given by

FrncðYtÞ ¼I

b� 1Yt

Y�rnc

� �b

;

and the government’s profit at time t is given by

GrncðYtÞ ¼bI

ðb� 1Þ2Yt

Y�rnc

� �b

þ YtðcsE þxxEÞr � l

;

where b is defined by (5).

Some comments are in order. First, we can show that FrcðYtÞ þ GrcðYtÞ > FrncðYtÞ þ GrncðYtÞ, which implies that taxation bythe host government leads to a double marginalization problem. This is because the firm requires a mark-up over the cost ofFDI and the host government charges a mark-up over the required after-tax payoff to the firm. Secondly, it follows from (27)that Y�rnc > Y�rc since b > 1, which implies that taxation delays the optimal entry timing compared with the case in which thehost government and the firm jointly maximize the total surplus. Moreover, the increase of the difference between consumersurplus under FDI and consumer surplus under exporting induces the decrease of the optimal tax rate and an incentive forthe firm to hasten FDI. Note that if the tax rate s1 is equal to �s1, then all benefits exceeding the benefits from exporting aretaxed away by the host government.

3.3. Cooperative solution under ambiguity

In this subsection, we consider the situation where the government levies the corporate tax, s1, and the lump-sum tax (orsubsidy), s2, on the firm under ambiguity in a cooperative bargaining game. Let Fac : Rþ ! R and Gac : Rþ ! R be the firm andthe government payoffs at time t under ambiguity in a cooperative bargaining game, respectively, and let Y�ac be the criticalvalue of the GDP level under ambiguity in this cooperative bargaining framework, at or above which the firm immediatelyswitches from exporting to FDI, and below which she continues exporting her product to the host country. Let wa

F and waG be

the market power of the firm and the government under ambiguity, respectively, with waF þ wa

G ¼ 1. Then, the following prop-osition is obtained.

ennings (2005), the last term YtðcsE þxxEÞ=ðr � lÞ is omitted. See Proof of Proposition 4 of this paper. But, it does not affect the comparative statics under risk.


Proposition 4. Suppose that in a cooperative bargaining situation the firm and the government simultaneously determine thetiming of undertaking FDI and maximize their joint profits with respect to s1 and s2. Then, the critical GDP level Y�ac is provided by

Y�ac ¼r � ðl� rh�Þ

pF � ð1� s0ÞpE þ csF � csE �x2xE

aa� 1

I;

and sa1 and sa

2 satisfy the following:

FacðYtÞ ¼ waFðF

acðYtÞ þ GacðYtÞÞ and GacðYtÞ ¼ waGðF

acðYtÞ þ GacðYtÞÞ;

where sa1 and sa

2 denote optimal tax rates maximizing their joint profits. Furthermore, the total surplus is provided by

FacðYtÞ þ GacðYtÞ ¼I

a� 1Yt

Y�ac

� �a

þ YtðcsE þxxEÞr � ðl� rh�Þ ;

where a is defined by (20).

Proof. See Appendix B. h

In this cooperative game situation, as in the case of no-ambiguity, the firm undertakes FDI if the GDP level Yt reaches thecritical GDP level Y�ac at which the total benefits are jointly maximized, and the total surplus is divided based on their bar-gaining power. Moreover, the outcome in the cooperative game situation reflects the outcome of a social planner that is so-cially desirable one. When the difference between csF and csE increases, the critical GDP level Y�ac decreases, which makes thetotal surplus Fac þ Gac increase.

3.4. Non-cooperative solution under ambiguity

In this subsection, we study the optimal taxation under ambiguity in a non-cooperative situation. The host governmentand the foreign firm are assumed to maximize their own benfits in a two-stage game. The host government levies her cor-porate tax rate on the foreign firm in the first stage, and the foreign firm determines the optimal entry timing in the secondstage.

Let Fanc : Rþ ! R and Ganc : Rþ ! R be the firm and the government profits at time t under ambiguity in this non-coop-erative game, respectively, and let Y�anc be the critical GDP level under ambiguity in this non-cooperative game. Then, thefollowing proposition is obtained.

Proposition 5. Let s2 ¼ 0 (no subsidy). Then, the optimal corporate tax rate is provided by

sa1 ¼

�s1

a� a� 1

acsF � csE �x2xE

pF; ð28Þ

where �s1 is defined by (25). In this case, the critical GDP level is

Y�anc ¼a

a� 1Y�ac:

The firm’s profit at time t is given by

FancðYtÞ ¼I

a� 1Yt

Y�anc

� �a

;

and the government’s profit at time t is given by

GancðYtÞ ¼aI

ða� 1Þ2Yt

Y�anc

� �a

þ YtðcsE þxxEÞr � ðl� rh�Þ ; ð29Þ

where a is defined by (20).


Some comments are in order. First, we can show that FacðYtÞ þ GacðYtÞ > FancðYtÞ þ GancðYtÞ, which implies that taxation bythe host government leads to a double marginalization problem. This is because the firm requires a mark-up over the cost ofFDI and the host government charges a mark-up over the required after-tax payoff to the firm. Secondly, since a > 1,Y�anc > Y�ac , which implies that taxation delays the optimal entry timing compared with the case in which the host govern-ment and the firm jointly maximize the total surplus. Furthermore, the increase of the difference between consumer surplusunder FDI and consumer surplus under exporting induces the decrease of the optimal tax and an incentive for the firm tohasten FDI.


4. Comparative statics

In this section, we analyze the effects of increases in risk and ambiguity on the value of undertaking FDI, the value of FDI,and the optimal corporate tax rate, respectively. We show that the host government should reduce the optimal corporate taxrate in response to an increase in ambiguity. This is because the firm’s less confidence in the prospect of the GDP level makesthe firm postpone FDI.

4.1. An increase in risk

We consider the case of no-ambiguity in order to analyze the effects of an increase in risk on the value of undertaking FDI,the value of FDI, and the optimal corporate tax rate. Since we consider the case of no-ambiguity, the value of undertaking FDI,the value of FDI, and the optimal tax rate are provided by (1), (4), (26), respectively.

Proposition 6 (Pennings (2005)). In the case of no-ambiguity, an incease in risk induces no change in the value of undertakingFDI, Wt, an increase in risk induces an incease in the value of FDI, Fr

t , in the continuation region. Furthermore, an increase in riskinduces an increase in the optimal corporate tax rate sr

1.

The first claim in this proposition states that an increase in risk does not have any effect on the value of undertaking FDI.The second claim states that an increase in risk leads to an increase in the value of FDI. Since an increase in the volatilityabout the GDP level leads to an increase in the volatility about the firm’s profit earned in the host country, the host countryraises the rate of the optimal corporate tax in order to gain the tax revenues as soon as possible. The optimal tax policy inresponse to an increase in ambiguity is different from the one in response to an increase in risk. In Section 4.2, we analyze theoptimal tax policy in response to an increase in ambiguity.

4.2. An increase in ambiguity

We assume that the firm is faced with j-ignorance, which is a special case of i.i.d. ambiguity, in order to analyze the effectsof an increase in ambiguity on the value of undertaking FDI, the value of FDI, and the optimal corporate tax rate.

Under the assumption of j-ignorance, h� defined by (9) is further characterized as follows:

h� � argmaxfrxjx 2 ½�j;j�g ¼ j:

Since h� is equal to j, the value of undertaking FDI, (8) and (20) turn out to be

VðYtÞ ¼AYt

r � ðl� rjÞ and ð30Þ

a ¼� ðl� rjÞ � 1

2 r2

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl� rjÞ � 1

2 r2� �2 þ 2rr2

qr2 ;

respectively. Recall that A is defined by (2). Since we consider the case of ambiguity, the value of undertaking FDI, the valueof FDI, and the optimal corporate tax rate are provided by (30), (18) and (28), respectively.

Proposition 7. Assume that the planning horizon is infinite, and the plant will never depreciate completely. Moreover, we assumethat the foreign firm is faced with j-ignorance. Then, an increase in ambiguity induces a decrease in the value of undertaking FDI,Vt, and an increase in ambiguity induces a decrease in the value of FDI, Fa

t , in the continuation region. Furthermore, an increase inambiguity induces a decrease in the optimal corporate tax rate sa

1.


The first claim in this proposition states that an increase in ambiguity has a negative effect on the value of undertakingFDI. The negative effect on the value of undertaking FDI implies that the firm becomes more cautious about FDI than beforeand it makes the firm postpone FDI. This result is one of the starkest contrasts to Pennings (2005) in which an increase in riskdoes not affect the value of undertaking FDI. The second one states that an increase in ambiguity leads to a decrease in thevalue of FDI. The last claim states that an increase in ambiguity has a negative effect on the optimal corporate tax rate. Thisresult implies that the less confidence the foreign firm has in the prospect of the GDP level, the greater is the incentive for thehost government to reduce the optimal corporate tax rate than to increase the optimal tax rate. Some comments are in order.First, since an increase in ambiguity has a negative impact on the value of undertaking FDI and makes the firm postpone FDI,it informs the government of increases in firm’s ambiguity about the evolution of the GDP level. Second, the difference ineffects of increases in risk and ambiguity on the optimal corporate tax rate stems from the foreign firm’s less confidencein the prospect of the GDP level. In the case of ambiguity, an increase in ambiguity implies the firm’s less confidence inthe prospect of the GDP level, which encourages the host country to reduce the optimal corporate tax rate for fear thatthe foreign firm is going to postpone FDI.


5. Conclusion

In this paper, we analyze effects of uncertainty on host country’s optimal tax policy when a foreign firm is uncertain aboutthe host country’s evolution of the GDP level. Focusing on uncertainty is important since (1) in general, host countries of FDIare developing countries, and (2) business in developing countries is considered to be more uncertain than that in developedcountries. Thus, if greater uncertainty has a negative impact on FDI from developed countries, then it might deter economicdevelopment in developing countries, which implies that developing countries should adopt policies to encourage FDI. Bydifferenciating ambiguity from risk, we analyze effects of increases in ambiguity on the optimal corporate tax rate and showthat the host country should reduce the optimal corporate tax rate levied on a foreign firm in response to an increase inambiguity. Our result conforms to Aizenman and Marion (2004)’s empirical result and our intuition that when firms aremore uncertain about the prospect of foreign markets, they are reluctant to undertake FDI, which makes host governmentsreduce their corporate taxes to encourage FDI.

Our result is different from Pennings (2005) that shows that an increase in risk induces an increase in the optimalcorporate tax rate. The difference in effects of increases in risk and ambiguity on the optimal corporate tax rate stemsfrom the foreign firm’s less confidence in the prospect of the GDP level. In the case of ambiguity, an increase in ambi-guity implies the firm’s less confidence in the prospect of the GDP level, which makes the host country reduce the opti-mal corporate tax rate in order to encourage FDI by the foreign firm. From a theoretical point of view, this paper isconsidered to provide some policy implications for developing countries that aim to encourage FDI from developedcountries.

Acknowledgements

I would like to appreciate a co-editor and two anonymous referees for their comments and suggestions that haveimproved this paper substantially. I am also grateful to Takanori Adachi, Kozo Kiyota, Noriaki Matsushima, Hiroyuki Ozaki,Akihisa Shibata, and Hajime Sugeta for their comments and discussions on this work. Financial support from MEXT (Grant-in-Aid for Scientific Research), Toyo University (Special Research), and the Zengin Foundation for Studies on Economics andFinance is greatly acknowledged. Needless to say, I am responsible for any remaining errors.

Appendix A

Let ðX;FT ; PÞ be a probability space, and let ðBtÞ06t6T be a standard Brownian motion with respect to P. Let ðFtÞ06t6T be thestandard filtration for ðBtÞ06t6T . Let L be the set of real-valued, measurable, and ðFtÞ-adapted stochastic processes onðX;FT ; PÞ with an index set ½0; T�, and let L2 be a subset of L that is defined by

L2 ¼ ðhtÞ06t6T 2L

Z T

0h2

t dt < þ1P-a:s:��

� �:

Given h ¼ ðhtÞ 2L2, define a stochastic process ðzht Þ06t6T by

ð8t 2 ½0; T�Þ zht ¼ exp �1

2

Z t

0h2

s ds�Z t

0hsdBs

� �: ð31Þ

By Ito’s lemma, we can define ðzht Þ06t6T as a unique solution to the stochastic differential equation, dzh

t ¼ �zht htdBt with zh

0 ¼ 1.A stochastic process ðhtÞ 2L2 is called a density generator if ðzh

t Þ is ðFtÞ-martingale. Novikov’s condition is one of the suf-ficient conditions for ðzh

t Þ to be ðFtÞ-martingale. For example, see Karatzas and Shreve (1991).

A set of density generators HKt ðxÞ is rectangular if there exists a set-valued stochastic process ðKtÞ06t6T such that

HKt ðxÞ ¼ ðhtÞ 2L2jhtðxÞ 2 KtðxÞ ðm� PÞ � a:s:� �

; ð32Þ

and, there exists a nonempty compact subset K of R such that for each t, Kt : X�K is compact-valued and convex-valued,the correspondence ðt;xÞ�KtðxÞ, when restricted to ½0; s� �X, is Bð½0; s�Þ �Fs-measurable for any 0 < s 6 T , and0 2 KtðxÞ ðm� PÞ-a.s., where m is the Lebesgue measure restricted on Bð½0; T�Þ. Note that i.i.d. ambiguity defined by (7) isa special case of rectangularity (32). Nishimura and Ozaki (2007) proves the following lemma.

Lemma 4. Let 0 6 s 6 t 6 T and let X be an FT -measurable function. If a set of density generators H satisfies rectangularity, thenthe following recursive structure holds:
� � � �
minh2H

EQh

XjFs½ � ¼minh2H

EQh

EQh

XjFt½ �h ��Fs ¼min

h2HEQh

minh02H

EQh0

XjFt½ ��Fs ; ð33Þ

as long as the minima exist.


Let h be a density generator, and let ðBtÞ06t6T be a standard Brownian motion with respect to P. Define a stochastic processðBh

t Þ06t6T by21

21 Not22 Thi

ð8t 2 ½0; T�Þ Bht ¼ Bt þ

Z t

0hsds: ð34Þ

Since ðzht Þ is ðFtÞ-martingale, it follows from Girsanov (1960)’s theorem that ðBh

t Þ06t6T is a standard Brownian motion withrespect to Q h. By Girsanov’s theorem, the stochastic differential equation capturing ambiguity in the development of theGDP level turns out to be

dYt ¼ ðl� rhtÞYtdt þ rYtdBht ð35Þ

for any h 2 H. By (35), and by applying Ito’s lemma to ln Yt considering Q h as the true probability measure, it follows that:

ð8h 2 HÞð8t 2 ½0; T�Þ Yt ¼ Y0 exp l� 12r2

� �t � r

Z t

0hsdsþ rBh

t

� �:

Appendix B

We provide proofs of Lemma 1–3, and proofs of Proposition 4, 5, 6 and 7.

Proof (Proof of Lemma 1). Let ht ¼ hðYtÞ �minQ2PH EQ ½e�rðt0�tÞ jFt �, where h : Rþ ! R is some real-valued function. Then, itfollows that

ht ¼ minQ2PH

EQ�

e�rðt0�tÞjFt

�¼ e�rdt min

h2HEQh�

EQhhe�rðt0�t�dtÞjFtþdt

i��Ft

�¼ e�rdt min

h2HEQh

minh02H

EQh0�

e�rðt0�t�dtÞjFtþdt

� ��Ft

�

¼ e�rdt minh2H

EQh

htþdt jFt½ � ¼ ð1� rdtÞ minh2H

EQh

½dhtjFt � þ ht

� �¼ min

h2HEQh

½dhtjFt � þ ht � rhtdt;

where the first equality holds by the definition of ht , the second holds by the definition of PH, (6), and the law of iteratedexpectations, the third follows from rectangularity, (33), the fourth follows from the definition of ht , the fifth follows fromapproximating e�rdt by ð1� rdtÞ and replacing dhtþdt with dht þ ht , and the last equality holds by eliminating higher termsthan dt. Thus, minh2HEQh

½dht jFt� ¼ rhtdt. By Ito’s lemma, it follows that:

dht ¼ h0ðYtÞðl� rhtÞYtdt þ h0ðYtÞrYtdBht þ

12

h00ðYtÞr2Y2t dt:

Moreover, by i.i.d. ambiguity, it follows that:

minh2H

EQh

dhtjFt½ � ¼ h0ðYtÞðl� rh�ÞYtdt þ 12

h00ðYtÞr2Y2t dt:

Then,

12

h00ðYtÞr2Y2t þ h0ðYtÞðl� rh�ÞYt � rhðYtÞ ¼ 0: ð36Þ

The ordinal differential equation (36) together with boundary conditions hðY�aÞ ¼ 1 and hð0Þ ¼ 0 has the general solution,hðYtÞ ¼ A1Ya1

t þ A2Ya2t , where a1 > 1 and a2 < 0 are solutions to the following characteristic equation ð1=2Þr2aða� 1Þþ

ðl� rh�Þa� r ¼ 0. By the boundary conditions, A2 ¼ 0 and A1ðY�aÞa1 ¼ 1, which implies that A1 ¼ ð1=Y�aÞ

a1 . Thus,hðYtÞ ¼ ðYt=Y�aÞ

a1 . h

Proof (Proof of Lemma 2). Let lt ¼ lðYtÞ �minQ2PH EQ ½R t0

t Yse�rðs�tÞdsjFt �, where l : Rþ ! R is some real-valued function. Sup-pose that l0ð�ÞP 0.22 Then, it follows that:

lt ¼ minQ2PH

EQZ t0

tYse�rðs�tÞds

" ��Ft

#¼ Ytdt þ e�rdt min

h2HEQh

EQhZ t0

tþdtYse�rðs�t�dtÞds

��Ftþdt

" #" ��Ft

#

¼ Ytdt þ e�rdt minh2H

EQh

minh02H

EQh0Z t0

tþdtYse�rðs�t�dtÞds

" ��Ftþdt

" #��Ft

#

¼ Ytdt þ e�rdt minh2H

EQh

½dltþdt jFt � ¼ Ytdt þ ð1� rdtÞ minh2H

EQh

½dlt jFt� þ lt

� �¼ Ytdt þmin

h2HEQh

½dltjFt � þ lt � rltdt;

where the first equality follows from the definition of lt , the second follows from the definition of PH, (6), and the law ofiterated expectation, the third holds by rectangularity, (33), the fourth holds by the definition of lt , the fifth follows from

e that under the assumption of i.i.d. ambiguity, ht in (34) is independent of time and state. Therefore, ht is replaced with some constant h.s assumption is indeed satisfied for Yt 6 Y�a .


approximating e�rdt by ð1� rdtÞ and replacing dltþdt with dlt þ lt , and the last equality follows from eliminating higher termsthan dt. Thus, Ytdt þminh2HEQh

½dlt jFt � ¼ rltdt. By Ito’s lemma, it follows that:

dlt ¼ l0ðYtÞðl� rhtÞr2Y2t dt þ l0ðYtÞrYtdBh

t þ12

l00ðYtÞrY2t dt:

Thus, as in Proof of Lemma 1, i.i.d. ambiguity implies the following ordinary differential equation:

12

l00ðYtÞr2Y2t þ ðl� rh�Þl0ðYtÞYt � rlðYtÞ þ Yt ¼ 0: ð37Þ

This ordinary differential equation (37) together with boundary conditions, lðY�aÞ ¼ 0 and lð0Þ ¼ 0, has the general solution,lðYtÞ ¼ B1Ya1

t þ B2Ya2t þ Yt=ðr � ðl� rh�ÞÞ, where a1 > 1 and a2 < 0 are the solutions to the following characteristic equation,

ð1=2Þr2aða� 1Þ þ ðl� rh�Þa� r ¼ 0. By the boundary conditions, B2 ¼ 0 and B1 ¼ �ðY�aÞ1�a1=ðr � ðl� rh�ÞÞ, which implies

that

lðYtÞ ¼Yt

r � ðl� rh�Þ �Y�a


Y�a

� �a1

: �

Proof (Proof of Lemma 3). Let ft ¼ f ðYtÞ �minQ2PH EQ ½R1

t0 Yse�rðs�tÞdsjFt �, where f : Rþ ! R is some real-valued function.Then, it follows that:

ft ¼ minQ2PH

EQZ 1

t0Yse�rðs�tÞds

� ��Ft

�¼ e�rdt min

h2HEQh

EQhZ 1

t0Yse�rðs�t�dtÞds

� ��Ftþdt

� ��Ft

�

¼ e�rdt minh2H

EQh

minh02H

EQh0Z 1

t0Yse�rðs�t�dtÞds

� ��Ftþdt

� �jFt

�

¼ e�rdt minh2H

EQh

ftþdtjFt½ � ¼ ð1� rdtÞ minh2H

EQh

½dft jFt� þ ft

� �¼min

h2HEQh

½dftjFt � þ ft � rftdt;

where the first equality follows from the definition of ft , the second follows from the definition of PH, (6), and the law ofiterated expectations, the third follows from rectangurality, (33), the fourth holds by the definition of ft , the fifth follows fromapproximating e�rdt by ð1� rdtÞ and replacing dftþdt with dft þ ft , and the last equality holds by eliminating higher terms thandt. Thus, minh2HEQh

½dft jFt � ¼ rftdt. By Ito’s lemma, it follows that:

dft ¼ f 0ðYtÞðl� rhtÞYtdt þ f 0ðYtÞrYtdBht þ

12

f 00ðYtÞr2Y2t dt:

Then, as in Proof of Lemma 1, i.i.d. ambiguity implies the following differential equation:

12

f 00ðYtÞrY2t þ f 0ðYtÞðl� rh�ÞYt � rf ðYtÞ ¼ 0: ð38Þ

This ordinal differential equation (38) with boundary conditions f ðY�aÞ ¼ Y�a=ðr � ðl� rh�ÞÞ and f ð0Þ ¼ 0 has the general solu-tion, f ðYtÞ ¼ C1Ya1

t þ C2Ya2t , where a1 > 0 and a2 < 0 are solutions to the following characteristic equation,

ð1=2Þr2aða� 1Þ þ ðl� rh�Þa� r ¼ 0. By the boundary conditions, C2 ¼ 0 and f ðY�aÞ ¼ C1ðY�aÞa1 ¼ Y�a=ðr � ðl� rh�ÞÞ, which

implies that C1 ¼ ðY�aÞ1�a1=ðr � ðl� rh�ÞÞ. Thus,

f ðYtÞ ¼Y�a


Y�a

� �a1

�

Proof (Proof of Proposition 4). Let a1 ¼ pF=ðr � ðl� rh�ÞÞ, a2 ¼ ðcsF � csE �xxEÞ=ðr � ðl� rh�ÞÞ, a3 ¼ pEð1� s0Þ=ðr � ðl�rh�ÞÞ, and a4 ¼ ðx2xE þ csEÞYt=ðr � ðl� rh�ÞÞ. Note that the value at time t of FDI can be rewritten as follows:

FaðYtÞ ¼ð1� s1ÞpF � ð1� s0ÞpE

r � ðl� rh�Þ Y�a � ðI þ s2Þ� �

Yt

Y�a

� �a

:

Hence

FacðYtÞ ¼ ð�a1s1 þ a1 � a3ÞY�ac � ðI þ s2Þ� � Yt

Y�ac

� �a

;

where Fac : Rþ ! R and Y�ac denote the firm’s profit at time t and the critical value of the GDP level under ambiguity in a coop-erative bargaining game, respectively. The net domestic gain can be rewritten as follows:

GacðYtÞ ¼pF

r � ðl� rh�Þ s1 þcsF � csE �xxE

r � ðl� rh�Þ

� �Y�ac þ s2

� �Yt

Y�ac

� �a

þ csE þx2xE

r � ðl� rh�Þ

� �Yt :


Therefore,

GacðYtÞ ¼ ða1s1 þ a2ÞY�ac þ s2� � Yt

Y�ac

� �a

þ a4;

where Gac : Rþ ! R denotes the government’s profit at time t under ambiguity in a cooperative bargaining framework.In the framework of a cooperative bargaining, the firm and the government maximize the following objective function

with respect to s1 and s2:

ðFacðYtÞÞwaF ðGacðYtÞÞw

aG :

The first order condition with respect to s1 and s2 provides

waFGacðYtÞ � wa

GFacðYtÞ ¼ 0: ð39Þ

Thus, Fac ¼ waFðF

ac þ GacÞ and Gac ¼ waGðF

ac þ GacÞ. The first order condition with respect to Y�ac implies that

waFGac @Fac

@Y�acþ wa

GFac @Gac

@Y�ac¼ 0: ð40Þ

Eqs. (39) and (40) imply that @Fac=@Y�ac ¼ �@Gac=@Y�ac. Thus,

@ðFac þ GacÞ@Y�ac

¼ 0:

Therefore,

Y�ac ¼a

a� 1r � ðl� rh�Þ

pF � ð1� s0ÞpE þ csF � csE �xxEI: ð41Þ

By inserting (41) into (23) and (24), we obtain the total surplus as follows:

FacðYtÞ þ GacðYtÞ ¼I

a� 1Yt

Y�ac

� �a

þ YtðcsE þxxEÞr � ðl� rh�Þ : �

Proof (Proof of Proposition 5). Define a1; a2; a3, and a4 as in Proof of Proposition 4. By substituting s2 ¼ 0 into (21), (23) and(24), and replacing Y�a with Y�anc in (23) and (24), it follows that:

Y�anc ¼a

a� 1I

ð1� s1Þa1 � a3and ð42Þ

FancðYtÞ ¼I

a� 1Yt

Y�anc

� �a

;

GancðYtÞ ¼ ða1s1 þ a2ÞY�ancYt

Y�anc

� �a

þ a4; ð43Þ

where Y�anc , Fanc : Rþ ! R, and Ganc : Rþ ! R denote the critical value of the GDP level, the firm’s and the government’s profitsat time t under ambiguity in a non-cooperative game, respectively. The first order condition with respect to s1 provides

sa1 ¼

a1 � a3 � ða� 1Þa2

a1a¼

�s1

a� a� 1

acsF � csE �x2xE

pF:

Substituting s1 ¼ sa1 into (42) and (43) imply that Y�anc ¼ a=ða� 1ÞY�ac and (29), respectively. h

Proof (Proof of Proposition 6). It can be easily shown that @Wr=@r ¼ 0, which shows the first claim. Recall that b is providedby (5). It can be also shown that @b=@r < 0 and @FrðYtÞ=@b < 0. Thus, @FrðYtÞ=@r ¼ @FrðYtÞ=@b � @b=@r > 0, which shows thesecond claim. For the last claim, since @b=@r < 0, it follows that:

@sr1

@r¼ �

�s1

b2

@b@r� 1

b2

csF � csE �x2xE

pF

@b@r

> 0: �

Proof (Proof of Proposition 7). Differentiating (30) with respect to j shows that @Wa=@j < 0, which implies the first claim.Recall that a is provided by (20). As in Nishimura and Ozaki (2007), it can be shown that @a=@j > 0 and @FaðYtÞ=@a < 0. Thus,@FaðYtÞ=@j ¼ @FaðYtÞ=@a � @a=@j < 0, which shows the second claim. For the last claim, since @a=@j > 0, it follows that:

@sa1

@j¼ @a@j

��s1

a2 �1a2

csF � csE �x2xE

pF

� �< 0: �


Proof (Proofs of a > 1 and c < 0.). Since K is a compact subset of R and 0 2 K , h� ¼ argmaxfrxjx 2 Kg ¼max K P 0. Thus,r > l� rh� since r > l, r > 0, and h� P 0, which implies that

a >� ðl� rh�Þ � 1

2 r2

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl� rh�Þ � 1

2 r2� �2 þ 2ðl� rh�Þr2

qr2 ¼

�ðl� rh� � 12 r

2Þ þ jl� rh� þ 12 r

2jr2 :

The claim a > 1 follows since the right-hand side of this inequality is equal to 1 if l� rh� þ ð1=2Þr2 P 0, and it is more than1 if l� rh� þ ð1=2Þr2 < 0.

For c, note that

c ¼� ðl� rh�Þ � 1

2 r2

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðl� rh�Þ � 1

2 r2� �2 þ 2rr2

qr2 <

�ðl� rh� � 12 r

2Þ � jl� rh� � 12 r

2jr2 ;

where the strict inequality follows since r > 0. The claim c < 0 follows since the right-hand side of this inequality is less than0 if l� rh� � ð1=2Þr2 P 0, and it is equal to 0 if l� rh� � ð1=2Þr2 < 0. h

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optimal tax policy and foreign direct investment under ambiguity

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