Congestion Externalities of Tourism, Dutch Disease and Optimal Taxation:
Macroeconomic Implications
Lee-jung Lu
Department of Economics, Feng Chia University, Taiwan
Juin-jen Chang
Institue of Economics, Academia Sinica
Department of Economics, National Central University, Taiwan
Shin-wen Hu
Department of Economics, Feng Chia University, Taiwan
April 2008
Abstract
This paper develops a dynamic optimizing, two-sector macro model of a small open economy sheds light on
two tourism stylized facts, namely, (i) the congestion externalities caused by tourism expansion and (ii) the
wealth effect generated by the revenues from oversea tourism taxation.
Based on the two salient
characteristics, our positive analysis indicates that while a tourist boom results in de-industrialization, this
phenomenon of the Dutch disease is less pronounced in the presence of congestion externalities of tourism.
We also find a perhaps surprising result: If wealth effect is positive and substantial, a tax on tourism may
result in an expansion, rather than a contraction, in the scale of the tourism sector.
In a normative analysis,
we show that to correct this distortion caused by the negative externality of tourism, a government should
levy a general tax on tourism consumption.
To generate the revenues from oversea tourism taxation, the
government should discriminate between domestic and overseas tourism consumptions, such that a positive
tax surcharge is imposed on foreign tourists.
Besides, the factors that governing the optimal rates of
general tax and tax surcharge are also examined in this paper.
Keywords: Congestion externalities of tourism; Dutch disease; optimal taxation
JEL classification: E60; H40
Address for Correspondence: Juin-jen Chang, Institute of Economics, Academia Sinica, Nankang,
Taipei 115, Taiwan, R.O.C.; Fax: 886-2-27853946; e-mail: [email protected]
I. Introduction
Tourism is a growing industry in many economies.
According to the 2007 annual report of the
World Tourism Organization (WTO), tourism is a major source for earning foreign exchange.
The
number of international tourist arrivals was 69 million in 1960 and it reached 842 million in 2006,
while tourism revenue was US$6 billion in 1960 and jumped to $735 billion in 2006.
The recent
“Tourism: 2020 Vision” issued by WTO forecasts that 1.6 billion tourists will visit foreign countries
annually by the year of 2020, with revenue of US$2 trillion per year.
In addition, a 2007 report from
the World Travel & Tourism Council (WTTC) indicates that for almost half of 176 countries, tourism is
expected to contribute more than 10.4% of GDP and 8.3% of total employment.
Tourism-based
economies have been shown to have displayed faster growth on average than other countries (Brau et
al., 2003, and Sequeira and Campos, 2005).
Although international tourism is today one of the most important economic sector, its
importance in an economy has been greatly neglected in the literature on economic development and
macroeconomic-related issues.
It is surprising how little we definitely know about that the rapid
growth of the tourism sector has generated many important macroeconomic and policy-related
questions.
In the economics literature, there is a lack of comprehensive theoretical analysis of
tourism, and, in particular, the relevant public regulation of tourism and social welfare analysis has not
been formally modeled.
This paper makes a first step in this direction by developing an optimizing
macro model of a small open economy to conduct both the relevant positive and normative analyses.
The economic benefits of tourism are clear (more employment and the high tax revenue potential),
but it also generates substantial attendant externalities, such as congestion and environmental
degradation (Gooroochurn and Sinclair, 2003, 2005 and Gago et al., 2006).
This distinguishing
feature makes public regulation of tourist activity a necessity.
Among the public policies affecting the
tourism sector, taxation plays an especially important role.
This is due, on the one hand, to the
magnitude of the potential tax revenue and, on the other hand, to the corrective function that tourism
taxes can be given.
WTO (1998) has identified around forty different taxes levied on tourism; among
fifty destinations surveyed, 73% increased taxes on tourism in the past five years.1
WTO (1998) also
has estimated that tax receipts generated by tourism commonly represent about 10-25% of the tax
1
Because of the lookout for new sources of taxation revenue, the rapid growth in tourism around the world has caught the
attention of governments. Tourism is seen as an easy sector to tax – the taxation points are readily identified, taxes can be
collected by the industry itself, and those who are taxed. (WTO, 1998)
1
revenue collected by some developed countries.
In some small specialized countries, such as
Maldives accounted for 40% of government revenue and the Bahamas, over 50% of government
revenue is generated from tourism.
According to the report of McAleer, et al. (2005), this figure can
approach 100% in certain small tourist economies.
In spite of incremental taxes on tourism, the
question arises as to whether tourism is taxed appropriately?
Whether different tourism taxes act as
part of integrated strategy responding to accepted economic principle?
consequences of tourism taxation?
What are the macroeconomic
How does an economy reallocate the production resources
between sectors in response to a tourist boom?
How is the so-called Dutch disease caused by a tourist
boom related to the externalities of tourism?
Our study is a theoretical attempt to explore these
related questions.
To the end, we build a dynamic optimizing two-sector macro model of a small open economy
sheds light on two tourism stylized facts, namely, (i) the congestion externalities caused by tourism
expansion and (ii) the wealth effect generated by the revenues from oversea tourism taxation.
It is
well-known that the environment is of key important to tourism and environmental resources have
been subject to market failure owing to externalities and their public good nature, resulting in
non-optimal utilization and limited returns to production of tourism good.
Given that the price of
natural resources cannot be fully internalized, they are subject to congestion and environmental
degradation through over-use.
Besides, tourism expansion also generates congestion on
tourism-related infrastructures, such as on roads, highways, airports, etc. (See Sinclair, 1998, Tisdell,
2000, and Sinclair and Stabler, 2002, for detailed discussions)
These external costs, including
environmental costs and congestion, are not only a by-product of the tourism sector but also an input of
its production function.2
To be concrete, in our model the negative externalities enter the production
function and are unfavorable to the production of tourism good.
With regard to characteristic (ii), we
emphasize that unlike other levies, tourism taxes can increase domestic welfare if the revenues from
oversea tourism are distributed to residents.3
Due to that, different from a standard macro model, the
revenue from aboard enters the economy-wide resource constraint and gives rise to the welfare effect
in terms of affecting agents’ behaviors.
2
3
As we will see later, the welfare effect is central in both
Dwyer and Forsyth (1993) argue that the concentration of large numbers of visitors can lead to overcrowding of beaches,
national parks, recreation areas, mountain resorts etc and place excessive demands on existing infrastructure such as roads,
public transport systems, water and electricity supply. As a result, congestion leads to loss of amenity, loss of leisure time,
increased accidents, increased fuel consumption and greater pollution.
As stressed by Gooroochurn and Sinclair (2003), if tax incidence falls mainly on the oversea tourists, tourism taxes may
increase domestic welfare due to “tax exportability.”
2
positive and normative analyses.
Our main findings are summarized as follows.
In a positive analysis, we complement the recent
studies (Copeland, 1991, Chao, et al., 2006, and Capó et al., 2007) that highlight the phenomenon of
the so-called “Dutch disease” regarding de-industrialization could be induced by a tourist boom.4
Of
importance, while a tourist boom draws the production resources away from the industrial sector to the
tourism sector, the Dutch disease is less pronounced when the economy is in the presence of
congestion externalities of tourism.
Moreover, Based on the wealth effect, tourism taxation can have
a mixed effect on the resource allocation between the manufacturing and tourism sectors.
Perhaps
surprisingly, if wealth effect is positive and substantial, a tax on tourism will result in an expansion,
rather than a contraction, in the scale of the tourism sector.
This potentially suggests that the Dutch
disease cannot be cured by tourism taxation.
In a normative analysis, since congestion and environmental degradation caused by tourism
expansion essentially are externalities that are not taken into account by individual tourism firm,
“unaccounted-for-externalities” drives a wedge of resource allocation between the decentralized
economy and the centralized economy.
Given that the externality is negative, tourism expansion is
more desirable to an individual firm than it is to society, so leading to excessive output (labor) in the
tourism sector and too little output (labor) in the manufacturing sector to the social optimum.
To
correct this distortion, in the social optimum the government should tax tourism consumption so as to
draw the resources away from the industrial sector to the tourism sector.
In particular, in face of
higher degree of externality or more demand from aboard, the distortion becomes more significant.
Thus, a higher rate of tax is necessary to fully internalize the tourism externality.
In addition to levy a general tourism tax, to generate the revenues from oversea tourism taxation
the welfare-maximizing government also should discriminate between domestic and overseas tourism
consumptions, such that a positive tax surcharge is imposed on foreign tourists.
This result provides a
convincing explanation to differential taxation implemented in some countries.
In India, for example,
there are separate entry tickets for local residents and foreign tourists to view the Taj Mahal.
In
Kenya foreign tourists pay higher charges for admission to wildlife parks than local residents.
4
A boom caused by a favorable market situation draws production away from industrial sectors to the economy’s emergent
raw material and service sectors, worsening conditions for the manufacturing sector and sending it into a decline. In the
economic literature these circumstances have been termed “Dutch Disease”. The literature can be traced to work by
Corden and Neary (1982) and Corden (1984).
3
Besides, more foreign demand for tourism allows the welfare-maximizing government to conduct tax
discrimination, imposing a higher surcharge rate on foreign tourists.
By contrast, a higher degree of
the tourism externality gives rise to a negative effect on tourism output, therefore, the government has
less monopoly power to discriminate between domestic residents and foreign tourists.
Related literature
Empirical studies have developed a useful body of work on tourism, while they are
predominantly based on partial equilibrium analysis of specific sectors of an economy.
By contrast,
researchers pay relatively little attention to provide theoretical underpinning for practical phenomenon.
Nevertheless, so far, trade economists have devoted little attention the economic issues facing the
tourism industry.
Copeland (1991) argues that, in a static trade model, an increase in tourism may
increase welfare, provided that distortions are absent from the economy, even though it may lead to
de-industrialization.
Subsequent studies, as summarized by Chao, et al. (2006), extend the
tourism-related analysis in two distinctive directions.
with distortions.
One direction is to examine static economies
For example, Nowak, et al. (2003) introduce increasing returns to scale into the
economy and show that increased tourism may result in a fall in manufacturing output and welfare.
The other direction of research is to undertaken the analysis of tourism in dynamic models.
For
example, Hazari and Sgro (1995) examine the effects of an increase in tourism on domestic capital
accumulation and welfare.
Recently, in two-sector dynamic models, Chao, et al. (2005) and Chao, et
al. (2006) explore the effects of an expansion in tourism on capital accumulation, employment, welfare,
and resource allocation between the manufacturing and tourism sectors.
Our analysis differs from the existing literature cited above in two aspects.
In the aspect of
model setting, we develop an optimizing dynamic macro model, instead of a trade model, with a
particular emphasis on two distinguishing features of tourism, noted previously.
By solving the
optimization problems, this allows us to address the exact responses of domestic agents to tourism
taxation and a tourist boom.
Besides, going beyond their studies, our analysis mainly focuses on the
normative respect, deriving the optimal taxation of tourism.
To investigate the questions of how to
tax tourism, by how much, and who will pay, is especially urgent, given its growing importance in both
developed and developing countries. This paper can be viewed as one of the first attempt within a
dynamic optimizing macro model to address the optimal tourism taxation.
4
2. Analytical Framework
We consider an open economy with two final goods: manufacturing good X (which is tradable)
and tourism good Y (which in non-tradable).
Although tourist consumption in the receiving
country is predominantly of non-traded goods (and services), it contributes to foreign currency
earnings when consumers are foreign tourists.
In the model economy there are three types of decision
makers: firms, households and a government.
Firms maximize their profits by producing goods from
labor and capital through Cobb-Douglas technology. Domestic residents (households) consume both
goods X and Y , while foreign tourists consume only the non-traded tourism good Y .
households, subject to their budget constraint, seek to maximize the lifetime utility.
Domestic
The government
balances its budget each period; it provides lump-sum transfers to domestic residents by leaving taxes
on income and consumption.
In particular, we assume that the government can discriminate between
domestic and overseas tourism consumptions, such that the latter may be taxed at a higher rate.
2.1. Firms
In the paper the setting of production environment greatly follows Copeland (1991), Hazari and
Sgro (2004), and Chao, et al. (2006).
There are two sectors: the X sector of manufacturing good
and the Y sector of tourism good.
For simplification, we assume that both sectors are perfectly
competitive.
The X good is viewed as the numeraire and, accordingly, the relative price of tourism
good is denoted as P .
Manufacturing good sector X
In the manufacturing good sector each producer employs labor ( L X ) and capital ( K ) to produce
good X by using a symmetric technology as follows:
X = AK α ( L X )1−α ; 0 < α < 1 .
where A is a constant technology factor.
(1)
The term α (1 − α ) is the shares of capital (labor),
implying that the production function exhibits homogeneity of degree one in all input factors.
Given the production function (1), the optimization problem of the traded good producer is to
choose labor ( L X ) and capital ( K ) so as to maximize profits, π X , i.e.:
5
max π X = X − rK K − wL X ,
(2)
where w is the wage rate and rK is the rental rate of physical capital.
Solving the optimization
problem (2) leads to the following first-order conditions:
rK =
αX
K
and w =
(1 − α ) X
.
LX
(3)
Equation (3) refers to the common MC = MR conditions.
Tourism good sector Y
In accordance with the realistic observations and the common specification in the tourism
literature, we assume that in the tourism good sector firms use labor ( LY ) and land ( V ) to produce the
tourism good Y.
Land acquisition apparently is important for tourism construction, while tourism
expansion may deplete the country’s natural resource base (Sinclair, 1998). In this model, V can
broadly refer to a physical plant: a site, natural resource, or facility such as a waterfall, wildlife, resort,
or hotel (see, Copeland, 1991, Sinclain, 1998, and Chao, et al., 2006, for the similar specification).
Due to that the tourism resource is limited, we assume that it is supplied in-elastically and normalized
to V = 1 , for simplification.
Besides, as addressed by Smith (1994), the physical plant requires the
input of “labor services” to make it useful for tourists.
Labor services LY refer to the performance
of specific tasks required to meet the needs of tourists.
For example, a hotel needs management, front
desk operation, housekeeping, maintenance, and food and beverage provisions to function as a hotel.5
Thus, the production function of tourism good producer is given by:
Y = BV ω ( LY )1−ω Y − β
where B is a constant technology factor.
(4)
Comparing with the manufacturing good sector, the
tourism good sector needs more labor services and is relatively labor-intensive.
Thus, it is ignored
without loss of significant generality.
As emphasized in the Introduction, environmental resources, in reality, are subject to market
failure owing to their public good nature and congestion externalities.
To incorporate this reality into
our model, Y − β (where Y denotes the aggregate output of tourism good) enters the tourism
production function and captures negative externalities caused by tourism expansion, namely
congestion and environmental degradation. For example, when over-use in natural resources and
5
Fitzsimmons and Sullivan (1982) and Normann (2001) provide an overview of a wide range of service management issues.
6
tourism-related infrastructures create congestion on roads, highways, airports, etc.
For an individual
tourism firm, this congestion gives rise to an externally negative effect in the production of tourism
good.
Besides, the negative externalities also occur as hotel construction creates additional
environmental damages; such environmental degradation is unfavorable to the production of tourism
good. (See Sinclair and Stabler, 2002, for detailed discussions)
Given V = 1 and (4), we can express the optimization problem of the tourism producer as:
max π Y = PY Y − rV − wLY ,
(5)
where P is the (relative) price of tourism good and rV is rent of land (the unit cost of using
environmental resources).
The corresponding first-order conditions are:
w=
(1 − ω ) PY Y
and rV = ωPY Y .
LY
(6)
By defining L being the labor endowment of
Population growth is abstracted from our model.
the economy, for analytical convenience we assume that the level of employment of the X sector is
L X = uL (where 0 < u < 1 is the shares of labor allocated to the manufacturing good sector),
meaning equivalently that LY = (1 − u ) L .
Suppose that labor is perfectly mobile between the X
sector and the Y sector (but is not mobile internationally). Thus, the workforce will move around
until wage levels in all two are the same due to “the factor price equalization,”, i.e.,
w = (1 − α ) AK α (uL) −α = (1 − ω ) PB((1 − u ) L) −ω Y − β .
(7)
Given that K and V are specific factors, we can rewrite the optimal conditions concerning capital
and land as:
rK =
αX
= αAK α −1 (uL)1−α ,
K
rV = ωPY .
(8)
(9)
Our analysis is confined to a symmetric equilibrium under which Y = Y and, as a result, the
aggregate production function is:
{
1−ω
Y = B[(1 − u ) L]
1
1+ β
}
.
(4’)
Finally, since both sectors X and Y are perfect competition, free entry and exit guarantee zero profits
for each producer, i.e. π X = π Y = 0 .
7
2.2. Households
The economy is populated by a unit measure of identical and infinitely lived households. The
representative household derives utility from consuming both manufacturing and tourism goods,
denoted by C X and CY , respectively.
Given that the instantaneous utility is denoted by U , the
household facing its budget constraint chooses C X , CY , and K so as to maximize the discounted sum
of future instantaneous utilities:
∞
∫
max W = U (C X , CY )e
− ρt
∞
dt =
0
s.t.
∫
0
(C φX CY1−φ )1−σ − 1 − ρt
e dt ; 1 > φ > 0 ,
1−σ
K& = (1 − t )(rK K + rV + wL ) − C X − (1 + τ ) PCY + R .
(10)
(11)
where an overdot denotes the rate of change with respect to time, ρ is the time preference rate, and
σ is the inverse of the elasticity of intertemporal substitution which are exogenously given and
constant over time.
and τ , respectively.
Moreover, the household incomes and tourism consumption are taxed at the rate t
The term R stands for the lump-sum transfers provided by the government.
To focus our point, consumption tax on manufacturing good is ignored.
This assumption does not
alter our results.
The optimal conditions necessary for this optimization problem are as follows:
φ
CX
(C φX CY1−φ )1−σ = λ1 ,
1 − φ φ 1−φ 1−σ
(C X CY )
= λ1 (1 + τ ) P ,
CY
λ1 (1 − t )rK = −λ&1 + λ1 ρ ,
together with (11) and the transversality condition lim λ1 Ke − ρt = 0 .
t →∞
(12a)
(12b)
(12c)
The term λ1 is the co-state
variable, which can be interpreted as the shadow value of the capital stock, measured in utility terms.
The optimal conditions (12a) and (12b) indicate that the marginal utilities of consumption C X and
CY should be equal to their marginal costs.
Equation (12c) shows that the change in the shadow
value of capital depends upon the difference between the rate of time preference and the after-tax
return rate of capital.
The transversality condition reveals that the value of the household’s per capita
assets must approach to zero as time approaches to infinity.
8
As mentioned previously, there are two types of consumers in the economy.
Domestic residents
consume both manufacturing ( C X ) and tourism goods ( CY ), while foreign tourists only demand
tourism good (their consumption is denoted by DY ).
By following Chao, et al. (2005), the demand
of foreign tourists takes the following functional form without loss of generality:
DY = DY ((1 + τ (1 + η )) P, T ) = −(1 + τ (1 + η )) P + θT .
(13)
where T represents a shift parameter for capturing the tourist activity and it is assumed that
∂D Y
D YT =
> 0 . In addition, DY is assumed to be decreasing in the after-tax rate of price
∂T
(1 + τ (1 + η )) P in order to satisfy the rule of demand and, accordingly, we have:
∂D
∂D
∂DY
DYP =
< 0 , DYτ = Y < 0 , and DYη = Y < 0 . Notice that since the government can
∂P
∂τ
∂η
discriminate between domestic and overseas tourism consumptions, we specify that there is a tax
surcharge rate η , leading to a higher tax rate imposed on foreign tourists.
The market-clearing condition for tourism good requires:
CY + DY = Y ,
(14)
indicating that the quantity of demand of both domestic and foreign tourists equals the quantity of
supply of the tourism good producers.
2.3. The Government
The government levies taxes on households’ incomes (rK K + rV + wL ) , domestic and foreign
tourism consumptions P (CY + DY ) and redistributes these tax revenues to domestic residents
(households) as a transfer payment R in a lump-sum manner.
Given foreign tourists are taxed at a
higher rate with surcharge η , the government’s budget constraint can be expressed as:
R = t (rK K + rV + wL ) + τP (CY + (1 + η ) DY ) = t ( X + PY ) + τP (CY + (1 + η ) DY ) .
Due to the zero-profit condition π X = π Y = 0 , we immediately have rK K + rV + wL = X + PY .
(15)
To
isolate the effects of taxation, we further assume that the government balances its budget in any period
by adjusting the lump-sum transfer R .
By combining the household’s budget constraint (11), the definitions of both manufacturing and
tourism firms (2) and (5), and the government’s budget constraint (15), the economy-wide resource
9
constraint is given by:
K& = X + PY − C X − PCY + τ (1 + η ) PDY .
(16)
It is worth noting that departing from the common aggregate resource constraint of a closed economy,
the revenue from oversea tourism taxation τ (1 + η ) PDY enters the aggregate resource constraint and
plays an important role in terms of affecting the economy’s performance. As we will see in the
positive analysis below, since the government redistributes its tax revenues to domestic residents, the
revenue from foreign tourism taxation will give rise to an additional wealth effects in terms of
governing the government’s policy effectiveness.
Meanwhile, in Section 4 concerning the normative
analysis, we further show that the wealth effect stemming from oversea tourism revenues jointed with
tourism congestion externalities will give the social planner an incentive to set the optimal tourism tax
policies in order to achieve the Pareto optimum.
3. General Equilibrium
3.1. Definition
An equilibrium in this model economy is defined by a sequence of prices {P, rK , rV , w}t =0 , real
∞
allocations {C X , CY , K , u , X , Y }t =0 , and policy variables {t ,τ , η , R}t =0 , that satisfy:
∞
∞
(i) the optimization problems of households, i.e., (12a)-(12c);
(ii) the optimization problems of manufacturing good firms (3) and tourism good firms (6);
(iii) the budget constraints of households (11) and the government (15);
(iv) the market-clearing condition for the labor market with perfect mobility (7);
(v) the market-clearing condition for the tourism good (14);
(vi) the aggregate resource constraint (16).
At the steady-state equilibrium, the economy is characterized by λ&1 = K& = 0 .
Accordingly, six
~
equations (8), (12a)-(12c), (14) and (16) allow us to solve the following six endogenous variables: C X ,
~
~
~
~
CY , u~ , P , K , and λ , where a tilde (~) over variables denotes their steady-state level.
10
~
Furthermore, substituting u~ and K into the production functions of manufacturing and tourism
~
~
goods (1) and (4) yields the steady-state X and Y .
3.2. Comparative statics and Dutch disease
We are now already to conduct the comparative statics analysis.
Based on the definition of
competitive equilibrium above, we thus establish the following proposition.
Proposition1. (The effects of an increase in the demand of foreign tourism) At the steady-state
~
~
~
~
~
~
dC X
dK
dP
∂X
∂Y
du~
dCY >
<0,
equilibrium, we have:
>0,
>0,
<0,
< 0 , and
> 0.
0,
dT
dT
dT
dT
∂T
∂T
dT <
Proof: See the Appendix A. ■
The results of Proposition 1 are intuitive.
First of all, an expansion in T will increase the demand for
~
tourism, therefore the relative price of tourism good P rises as a response. As the tourism good
becomes more expensive, the substitution effect will lead domestic residents to consume more
non-tourism good and consume less tourism good. On the other hand, an expansion in T increases the
revenue from oversea tourism taxation which is redistributed to domestic residences.
Once domestic
residents’ income accordingly, the wealth effect leads households to increase their consumption in both
tourism and non-tourism goods. It is clear from the substitution effect and the wealth effect that an
~
increase in the foreign tourism demand will increase C X unambiguously, while it has a mixed impact
~
on CY .
In addition, in response to an increase in the foreign demand for tourism good, labor will move
away from the manufacturing good sector toward the tourism good sector.
employment of the tourism sector increases, i.e., u~ decreases.
As a result, the portion of
At the same time, due to that labor
and capital are technical complements for each other, the specific factor of the manufacturing good
~
sector – capital ( K ) – also decreases. Since a tourist boom draws the production resources away
~
from the industrial sector to the tourism sector, the output of tourism good Y rises and the output of
~
non-tourism good X falls. The decline of the manufacturing sector is construed as an optimum
response to the increased wealth generated from oversea tourism.
These results are similar to those of
Chao et al. (2006) and Capó et al. (2007).
Proposition 1 potentially points out that a tourist boom tends to raise the demand for tourism
11
good that expands the tourism production at the cost of the shrink of the manufacturing sector.
literature, this circumstance of “de-industrialization” is termed “Dutch disease.” 6
In the
Based on
Proposition 1, our analysis further shows that the magnitude of Dutch disease is crucially related to the
congestion externality of tourism.
Corollary 1. (Tourism congestion and Dutch Disease) A tourist boom results in “Dutch Disease”
~ ~
∂ (Y − X )
> 0 ). However, the problem of Dutch disease is alleviated when the tourism congestion
(
∂T
~ ~
∂ ∂ (Y − X )
]< 0 .
[
effect is greater, i.e.,
∂β
∂T
Proof: See the Appendix A. ■
Corollary 1 provides us an important implication: In the presence of congestion externalities of tourism
the problem of Dutch Disease could be alleviated. When negative congestion externalities caused by
tourism expansion are considered as more serious ( β is large), a tourist boom will have a limited
effect in terms of drawing the production resources away from the industrial sector to the tourism
sector.
Given that the effect on the allocation of resource is restricted, the magnitude of
de-industrialization then is reduced, and as a result, Dutch disease becomes less significant.
It is also interesting to investigate the macro effects of tourism taxation.
Since the revenue from
foreign tourism taxation will give rise to the wealth effect in terms of governing the macroeconomic
effects of taxation, we then have the following lemma:
Lemma 1. By defining FR = τ (1 + η ) PDY , we have:
∂FR
= (1 + η ) PDY (1 − ε ) <> 0 if 1 <> ε ,
∂τ
where ε = −
∂ ( PDY ) τ
> 0 is the price elasticity with respect to the expenditure of foreign
∂τ
( PDY )
tourism.
Lemma 1 in essence is the so-called Laffer curve. Based on the 124 studies, Dwyer and Forsyth
(2006) report that the price elasticities vary across countries, with the range from 0.15 to 7.01.
6
Since
The term Dutch disease was first used by The Economist on 26 November 1977 (Corden, 1984) to describe the sharp rise in
wealth in Holland during the 1960s due to the discovery of large reserves of gas. This finding led to the restructuring of
national production and the strong appreciation of the currency, the guilder, with particularly adverse effects on the
manufacturing sector.
12
the price elasticity can be larger than one, the WTO cites a lot of examples where governments have
ended up with less revenue after increasing a tax rate.
Proposition 2. (The Macroeconomic Effects of Tourism Tax Surcharge) Under Lemma 1, we have:
(i) if ε > 1 , increasing tourism tax will decrease domestic manufacturing consumption
~
~
~
∂C X
∂C
∂P
< 0 ), tourism consumption ( Y < 0 ),the relative price of tourism good (
< 0 ),
(
∂τ
∂τ
∂τ
~
∂Y
∂X
< 0 ), but it will increase the manufacturing output (
> 0 ),
and the tourism output (
∂τ
∂τ
~
∂K
∂u~
> 0 ), the ratio of labor of the manufacturing sector (
physical capital (
> 0 ).
∂τ
∂τ
~
∂X
(ii) if ε < 1 , increasing tourism tax will decrease the manufacturing output (
< 0 ), physical
∂τ
~
∂u~
∂K
capital (
< 0 ), the ratio of labor of the manufacturing sector (
< 0 ), but it will
∂τ
∂τ
~
~
∂C X
∂C
increase domestic manufacturing consumption (
> 0 ), tourism consumption ( Y > 0 ),
∂τ
∂τ
~
dY
∂P
> 0 ).
> 0 ), and the tourism output (
the relative price of tourism good (
dτ
∂τ
Proof: See the Appendix B. ■
Proposition 2 clearly points out that the welfare effect plays a decisive role in terms of governing the
macroeconomic effects of tourism taxation.
We first discuss the case where ε > 1 .
Intuitively,
when tourism good becomes expensive in response to an increase in tourism tax τ , both domestic
~
consumption CY and foreign consumption DY decrease. Meanwhile, manufacturing consumption
~
C X will increase, due to the substitution effect. However, by following the well-documented feature
of a Cobb-Douglas utility function, increasing the after-tax price will decrease domestic households’
real income; the income effect gives rise to an opposing effect which offsets the substitution effect (see,
for example, Pindyck and Rubinfeld, 2005).7 It turns out that at this stage tourism taxation has a
~
~
negative impact on CY , but has no impact on C X , due to there is no cross effect of a change in price.
In the 1 < ε case, increasing τ will decrease the revenue from foreign tourism taxation and,
consequently, the domestic residents’ wealth. The wealth effect, on the one hand, decreases
~
∂C X
< 0 ) and, on the other hand, further decreases tourism
manufacturing consumption (
∂τ
7
Given a Cobb-Douglas utility function, the cross-price elasticity of demand is 0.
13
~
∂ (CY + DY )
<0.
consumptions from both domestic residents and oversea tourists
∂τ
Due to the
decreased demand for tourism good, the equilibrium relative price of tourism good falls as a response
~
∂P
< 0 ). A fall in the relative price of tourism good harms to the marginal revenue of the
(
∂τ
production on tourism good, so drawing production resources away from the tourism sector to the
~
~
~
dY
∂u~
∂K
∂X
>
0
,
>0,
> 0 , and
< 0 ).
manufacturing sector (
∂τ
dτ
∂τ
∂τ
In contrast, in the case where ε < 1 , the welfare effect will give rise to a positive effect on both
~
∂C X
manufacturing consumption (
> 0 ) and tourism consumption. Since the wealth effect dominates
∂τ
the price effect stemming from increasing tourism tax, a rise in τ eventually increase tourism
~
∂C
consumption ( Y > 0 ). Moreover, under the condition ε < 1 , increasing τ has a positive effect
∂τ
~
∂ (CY + DY )
> 0 ,8 therefore, the relative price of
on the total amount of tourism consumption, i.e.,
∂τ
~
∂P
tourism good also rises (
> 0 ). Given the fact that the marginal revenue of the production on
∂τ
tourism good increases, production resources are drawn away from the manufacturing sector to the
~
~
~
∂u~
dY
∂K
∂X
<0,
< 0,
< 0 , and
> 0 ).
tourism sector (
∂τ
dτ
∂τ
∂τ
It is worthwhile to highlight some interesting results yielded from Proposition 2. First, if wealth
effect is considered and becomes positive significantly (i.e., in the case where ε < 1 ), while a tax on
tourism decrease unambiguously the oversea demand for tourism, it may increase, rather than decrease,
domestically tourism consumption.
Under such a situation, perhaps surprisingly, a tax on tourism
may result in an expansion, rather than a contraction, in the scale of the tourism sector. Besides, since
the effect of the tourism tax surcharge on oversea tourists η is similar to that of the general tourism
tax τ , we will abstract it from this paper.
Nevertheless, a detailed mathematical deduction is available
upon request.
4. Optimal Taxation
In this model there are two economic rationales for the government to intervene in the markets by
8
~
⎡ ∂ (C + D ) ⎤
We have: sgn ⎢ Y∂τ Y ⎥ = sgn(1 − ε ) . See Appendix B for mathematical proof.
⎣
⎦
14
setting the socially-optimal tax policy: remedying the congestion externalities caused by tourism
expansion and increasing domestic wealth generated by the revenues from oversea tourism taxation.
Due to that in the decentralized economy tourism firms and households do not account for the
congestion effect stemming from tourism expansion and the wealth effect stemming from oversea
tourism tax revenue, the social planner will internalize these two externalities in order to achieve the
Pareto optimum.
In this section, we derive the optimal tax policy by comparing the decentralized
system with the centralized system. To the end, we will first examine the social planner optimization
problem in a centralized economy.
4.1. Centralized economy
The optimization problem of the social planner can be expressed by:
∞
max W =
∫
0
s.t.
(C φX CY1−φ )1−σ − 1 − ρt
e dt ,
1−σ
K& = X + PY − C X − PCY + τ (1 + η ) PDY ,
CY + DY = Y (or equivalently, P =
CY + θT − Y
),
(1 + τ (1 + η ))
X = AK α (uL)1−α ,
Y = [ B ((1 − u ) L)
1
1−ω 1+ β
]
(10)
(16)
(14)
(1)
.
(4’)
Unlike the individual in the decentralized economy, the social planner make his decision, subject to the
market equilibrium conditions (14) and (16) (rather than the individual’s budget constraint (11)) and
taking account of the production technology in the aggregate level (1) and (4’) (rather than taking the
factor prices as given).
The constraints (16) and (4’) clearly indicate that the social planner
internalize the congestion effect stemming from tourism expansion and the wealth effect stemming
from oversea tourism tax revenue as he makes a decision.
By substituting (1), (4’), and (14) into (16) and defining a current Hamilton function with a
multiplier λ2 , we can solve the optimization problem for C X , CY , K, λ2 , and u.
The
corresponding optimal conditions are:
φ
CX
(C φX CY1−φ )1−σ = λ 2 ,
1 − φ φ 1−φ 1−σ
(C X CY )
= λ 2 (θT − 2 DY ) ,
CY
15
(17a)
(17b)
λ 2 {αAK α −1 (uL)1−α } = −λ&2 + λ 2 ρ ,
(17c)
K& = X + PY − C X − PCY + τ (1 + η ) PDY ,
(16)
1
α
(1 − α ) AK (uL)
−α
−1
(1 − ω )
1+ β
=
[ B((1 − u ) L)1−ω ]
B((1 − u ) L) −ω (θT − 2 DY ) ,
1+ β
and the transversality condition lim λ 2 Ke − ρt = 0 .
(17d)
To ensure an interior solution for the optimization
t →∞
problem, we assume that θT − 2 DY > 0 in (17b) and (17d).
4.2. Optimal taxation
By comparing the systems of the decentralized and centralized economies, we derive the optimal
tax rates and reported in the following proposition.
Proposition 3. (Optimal Taxation) To correcting the congestion externalities caused by tourism
expansion and increasing domestic wealth generated by the revenues from oversea tourism taxation,
the optimal income tax rate t * , the tourism tax rate τ * , and the tax surcharge rate η * on oversea
tourism are given by, respectively:
t* = 0 , τ * =
β (θT − DY ) + DY
(1 + β )(θT − DY ) DY
> 0 , and η * =
>0.
(θT − 2 DY )
(θT − 2 DY )[ β (θT − DY ) + DY ]
Proof: By comparing (12a) with (17a), we have λ1 = λ 2 . Substituting this condition into (12c) and
(17c), together with the expression for the equilibrium rental rate of capital (8), yields: t * = 0 .
Comparing (17d) with (7) associated with the aggregate consistency condition Y = Y , we derive the
optimal consumption tax on tourism as follows:
β (θT − DY ) + DY
τ* =
>0.
(θT − 2 DY )
(18)
This implies that by levying τ * , the congestion distortion caused by tourism expansion can be
removed, leading the allocation of labor u in the competitive equilibrium to achieve the optimal level.
Moreover, given the relationship λ1 = λ 2 and the market-clearing condition for the tourism
C + θT − Y
, (12b) and (17b) allow us to derive:
sector P = Y
(1 + τ (1 + η ))
η* =
(1 + β )(θT − DY ) DY
>0.
(θT − 2 DY )[ β (θT − DY ) + DY ]
(19)■
First of all, Proposition 3 clearly indicates that the distortionary income tax is not able to serve as
instruments in remedying these both tourism externalities mentioned above, i.e., t * = 0 .
16
The main
reason is that consumption tax and the tourism tax surcharge can more effectively correct these
distortions and that both goods and factor markets are perfect competition.
Secondly, congestion and environmental degradation caused by tourism expansion essentially are
negative externalities that are not taken into account by tourism firms in the decentralized economy.
Therefore, tourism expansion will be more desirable to an individual firm than it is to society. By
comparing (7) with (17d), we can see that the competitive equilibrium leads to excessive output (labor)
in the tourism sector (and, accordingly, results in too little output (labor) in the manufacturing sector)
to the social optimum.
To correct this distortion, a tax should be imposed on tourism consumption,
i.e., τ > 0 , drawing the production resources away from the industrial sector to the tourism sector,
*
i.e., increasing the portion u of labor in the manufacturing sector.
Thirdly and more interestingly, Proposition 3 points out that to achieve social optimum there is
tax surcharge, i.e., η * > 0 , leading to a higher tourism tax rate on foreign tourists.
As stressed
previously, the economic rationale for taxing tourism is to (i) correct congestion externalities caused by
tourism expansion and (ii) increase domestic wealth generated by the revenues from oversea tourism
taxation.
A unified consumption tax τ * can reach the first purpose, while it is unable to increase
domestic wealth.
To increase domestic wealth, the social planner should implement tax
discrimination between domestic residents and foreign tourists and collect the revenues from oversea
tourism taxation.
Therefore, a positive tax surcharge ( η * > 0 ) is set in order to achieve social
optimum.
Corollary 2. (Optimal Taxes, Tourism Congestion, Foreign Tourism Demand) To achieve social
optimum,
(i)
the optimal tourism tax rate is increasing in both the congestion externality of tourism
and the oversea demand for tourism.
(ii) the optimal tax surcharge rate on foreign tourist is decreasing in the congestion
externality of tourism, but increasing in the oversea demand for tourism.
Proof: Differentiating (18) and (19) with respect to β and DY immediately yield:
∂τ *
∂τ *
(θT − DY )
(1 + β )θT
=
> 0,
=
>0,
∂β (θT − 2 DY )
∂DY (θT − 2 DY ) 2
17
∂η *
(1 + β )θT [ β (θT − DY ) 2 + DY2 ]
(θT − DY ) DY (θT − 2 DY )
∂η *
=
> 0 .■
=−
<0,
∂β
∂DY {(θT − 2 DY )[ β (θT − DY ) + DY ]}2
(θT − 2 DY )[ β (θT − DY ) + DY ] 2
As argued in Proposition 3, the negative externalities caused by tourism expansion result in
excessive output (labor) in the tourism sector and too little output (labor) in the manufacturing sector.
The magnitude of misallocation resource becomes more serious if the externality parameter β is
greater.
To fully internalize this externality, the optimal tax rate of tourism consumption should
increase with the externality parameter β .
Similarly, in face of an increase in the foreign demand
for tourism DY , the tourism firm will be more likely to expand its scale as a response and, as a result,
the degree of the tourism externality turns out to be larger. Thus, the social planner should tax a
higher rate of tourism consumption to respond to the increase in the degree of the tourism externality.
The intuition of ∂η * / ∂DY > 0 reported in Corollary (ii) is straightforward.
More foreign
demand for domestic tourism good will allow the welfare-maximizing government to conduct tax
discrimination, imposing a higher surcharge rate on foreign tourists.
In addition, as addressed in
Corollary 1, a higher degree of the tourism externality will give rise to a negative effect on tourism
~
∂ ∂Y
output, i.e.,
( ) < 0 . Under such a situation, the government has less monopoly power to
∂β ∂T
discriminate between domestic residents and foreign tourists.
Therefore, the tax surcharge rate levied
on oversea tourists should be reduced in the social optimum.
5. Concluding Remarks
This paper has developed a dynamic optimizing macro model with two sectors and used it to
explore macroeconomic implications of tourism. Two silent novel characteristics concerning tourism
are highlighted: (i) the congestion externalities caused by tourism expansion and (ii) the wealth effect
generated by the revenues from oversea tourism taxation. Based on the two salient characteristics, we
conduct both positive and normative analyses with particular emphasis on the latter one.
In the positive analysis, we have complemented recent studies that highlight the phenomenon of
the Dutch disease regarding de-industrialization could be induced by a tourist boom.
Our analysis has
pointed out that while a tourist boom draws the production resources away from the industrial sector to
the tourism sector, the problem of Dutch Disease is less pronounced when the economy is in the
presence of congestion externalities of tourism.
Besides, given that the revenues from oversea
tourism taxation gives rise to a beneficial wealth effect on the economy, tourism taxation in general has
18
a mixed effect on the resource allocation between the manufacturing and tourism sectors.
If wealth
effect is positive and substantial, a tax on tourism may result in an expansion, rather than a contraction,
in the scale of the tourism sector.
In the normative analysis, we have derived the socially optimal taxation on tourism.
Since the
negative externality caused by tourism expansion is not internalized in tourism firms’ costs, tourism
expansion is more desirable to an individual firm than it is to society, so leading to excessive output
(labor) in the tourism sector and too little output (labor) in the manufacturing sector to the social
optimum.
To correct this distortion, a tax should be imposed on tourism consumption so as to draw
the resources away from the industrial sector to the tourism sector.
In particular, in face of higher
degree of externality or more demand from aboard, the distortion becomes more significant.
Thus,
the social planner should tax a higher rate of tourism consumption to fully internalize the tourism
externality. In addition to levy a general tourism tax, in the social optimum the government also
should discriminate between domestic and overseas tourism consumptions, such that a positive tax
surcharge is imposed on foreign tourists in order to generate the revenues from oversea tourism
taxation.
It also been proved that the optimal rate of tax surcharge is decreasing in the congestion
externality of tourism, but increasing in the oversea demand for tourism.
19
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2
Appendix A (The proof of Proposition 1 and Corollary 1)
We restrict the analysis to a symmetric equilibrium under which Y = Y and, hence the
aggregate production of tourism good is Y = [ B ((1 − u ) L)
dynamic system is characterized by λ&1 = K& = 0 .
1
1−ω 1+ β
]
.
At the steady-state equilibrium, the
Thus, we rewrite (7), (12a)-(12c), (14), and (16) as
follows:
1
~
~
(1 − α ) AK α (u~L) −α = (1 − ω ) P B 1+ β ((1 − u~ ) L)
− (ω + β )
1+ β
,
φ ~ φ ~ 1−φ 1−σ ~
= λ1 ,
~ (C X CY )
CX
(A1)
(A2)
1 − φ ~ φ ~ 1−φ 1−σ ~
~
~ (C X CY ) = λ1 (1 + τ ) P ,
CY
(A3)
~
(1 − t )αAK α −1 (u~L)1−α − ρ = 0 ,
(A4)
1−ω
1
~
~
CY + DY = B 1+ β ((1 − u~ ) L) 1+ β ,
(A5)
1
~
~
~
~
~~
[ AK α (u~L)1−α − C X ] + P{[ B ((1 − u ) L)1−ω ] 1+ β − CY } + τ (1 + η ) P DY = 0 ,
(A6)
~
~
~
~ ~
These six equations above allows us to solve for six variables: P , u~ , K , C X , CY , and λ1 .
~
When these six variables are determined, (1) and (4) allow us to further solve recursively solve for X
~
and Y from
~
X = AK α (uL)1−α ,
(A7)
1
Y = [ B ((1 − u~ ) L)1−ω ] 1+ β .
(A8)
From (A1)-(A8) and by the Cramer’s rule, it is easy to derive the comparative statics
concerning T as follows:
~
∂P 1
= ΩΦ > 0
∂T Δ
(A7)
∂u~
1
X
= − (1 − t ) MPKK
MPLY Φ < 0
∂T
Δ
(A8)
~
∂K 1
X
= (1 − t ) MPKL
MPLY Φ < 0
∂T Δ
(A9)
3
~
~
∂X
dK
du~
= MPKX
+ MPLX L
<0
∂T
dT
dT
~
∂Y
du~
1
=−
MPLY L
>0
∂T
dT
1+ β
(A10)
(A11)
~
∂C X 1
~
~
~~
= {(−(1 + τ ) P U XY + U YY )(ΩW + MPLY E ) + Ωλ1 (1 + τ ) P DYT (1 + τ (1 + η ))} > 0 (A12)
∂T
Δ
>
~
∂CY
~
1
~
~
Y
= − {(−(1 + τ ) P U XX + U YX )(ΩW + MPL E ) + λ1 (1 + τ ) DYT Ω} = 0
∂T
Δ
<
(A13)
where
~
X
Δ = −(1 − t ) MPLY MPKL
MPKX (−(1 + τ ) P U XX + U YX )
~
~
+ Ω[(−(1 + τ ) P U XX + U YX ) S + λ1 (1 + τ )]
~
~
~
X
+ [ DYP Ω − (1 − t ) MPLY MPLY MPKK
][(−(1 + τ ) P U XY + U YY ) − P (−(1 + τ ) P U XX + U YX )]
~
~
X
+ (1 − t ) MPLY MPKK
( MPLX + P MPLY )(−(1 + τ ) P U XX + U YX )
~
Where MPLX = (1 − α ) AK α (u~L) −α > 0
since
1
1+ β
− (ω + β )
= (1 − ω ) B ((1 − u~ ) L) 1+ β > 0 ,
~
X
MPLK
= α (1 − α ) AK α −1 (u~L) −α > 0 ,
~
MPKX = αAK α −1 (u~L)1−α > 0 ,
~
X
MPKK
= α (α − 1) AK α − 2 (u~L)1−α < 0 ,
~
~
U XX = φ[φ (1 − σ ) − 1]C Xφ (1−σ ) − 2 CY(1−φ )(1−σ ) < 0 ,
~
~
U YX = C = (1 − φ )φ (1 − σ )C Xφ (1−σ ) −1CY(1−φ )(1−σ ) −1 < 0 ,
~
~
U XY = φ (1 − φ )(1 − σ )C Xφ (1−σ )−1CY(1−φ )(1−σ )−1 < 0 ,
~
~
U YY = D = (1 − φ )[(1 − φ )(1 − σ ) − 1]C Xφ (1−σ ) CY(1−φ )(1−σ ) − 2 < 0 ,
~
~
~
− (1 + τ ) P U XX + U YX = (1 − φ )C Xφ (1−σ ) −1CY(1−φ )(1−σ ) −1 > 0 ,
~
~
~~
− (1 + τ ) P U XY + U YY = −φ (1 + τ CY ) P C Xφ (1−σ ) −1CY(1−φ )(1−σ ) −1 < 0 ,
MPLY
1 ~ ~
~~
S = − ~ ( X − C X ) + τ (1 + η ) P DYP~ < 0
P
~
Ω = −ω ((1 − u~ ) L) −1 L(1 − τ )α (1 − α ) 2 A 2 K 2α − 2 (u~L)1− 2α < 0
~
~
~
and J = −(1 + τ ) P U XY + U YY + P[(1 + τ ) P U XX + U YX ] < 0 , hence Δ < 0 。
~~
~
~
~
~~
~
Φ = [τ (1 + η ) P DYT (−(1 + τ ) P U XX + U YX ) − DYT (−(1 + τ ) P U XY + U YY ) + P DYT (−(1 + τ ) P U XX + U YX )]} > 0 then,
~
we have ∂P = 1 ΩΦ > 0 (see (A7)). That is an increase in the tourism transforming the tourism goods
∂T Δ
to become exports will lead to an improvement in the price of tourism goods. As a result, equations
4
~
~
(A8)-(A11) indicate that the an expansion in tourism will lead to decrease u~ , K , and X , and an
~
increase in Y . Moreover, we have:
1
~
1+ β
W = − DYT [ B ((1 − u~ ) L)
E=
consumption
~
(−(1 + τ ) P U XX
~
~
− CY + τ CY DY ] < 0 ,
τ (1 − α ) − (α + β )
~
~
(1 − τ )α (1 − α ) A 2 K 2α − 2 (u~L)1− 2α LDYT > 0 ,
1+ β
~
∂C X
>0
then
∂T
if τ (1 − α ) > (α + β ) ,
(1−ω )
1+ β
equation
(13),
Since
>
~
∂CY
~
~
+ U YX )(ΩW + MPLY E ) > 0 and λ1 (1 + τ ) DYT Ω < 0 , hence
= 0 , an expansion of
∂T
<
of
good
X
by
(see (A12)), it shows an expansion of tourism increase the
domestic
resident.
See
tourism, the CY may be increase, decrease or unchanged.
Combining Equation (A10) with (A11) has:
~ ~
~
~
~
du~
∂ (Y − X ) ∂Y ∂X
1
Y
X
X dK
MPL + MPL ) L
=
−
= −(
− MPK
>0
dT
∂T
dT
∂T ∂T
1+ β
(A14)
~ ~
~
~
∂ ∂ (Y − X )
∂ ∂Y
∂ ∂X
−
[
]=
∂β
∂T
∂β ∂T ∂β ∂T
where
(A15)
−β
~
1−ω
∂ ∂Y
−1
∂Z du~
, since
)[ B((1 − u~ ) L)1−ω ]1+ β +
}L
=−
B ((1 − u~ ) L) −ω {(
1+ β
1+ β
∂β ∂T
∂β dT
−β
[ B ((1 − u ) L)1−ω ]1+ β and
−β
~
∂Z
−1
∂ ∂Y
1−ω
= [ B((1 − u ) L)1−ω ]1+ β (
)
ln[
B
((
1
−
u
)
L
)
]
<
0
,
hence
< 0.
∂β
∂β ∂T
(1 + β ) 2
1
− (ω + β )
~
~
1 − ω 1+ β
∂ ∂X
∂ du~
∂ dK
~
~
~
B ((1 − u ) L) 1+ β + (1 − α ) AK α (u~L) −α }L
={
+ αAK α −1 (u~L)1−α
1+ β
∂β ∂T
∂β dT
∂β dT
(A16)
From equations (A8) and (A9):
∂ du~
1
X
= − 2 (1 − t ) MPKK
Φ( F + Q + H )
∂β dT
Δ
(A17)
~
∂ dK
1
X
= 2 (1 − t ) MPKL
Φ( M + N + O)
∂β dT Δ
(A18)
where
5
(1 − ω )
∂Z
u~
~
X
Ω−
(1 − α ) AK α (u~L) −α −1 LMPLY (1 − t ) MPKK
]
~
2
∂β
(1 + β ) (1 − u )
~
~
{−[−(1 + τ ) P U XX + U YX ]S + λ1 (1 + τ )}
(1 − ω )
∂Z
u~
~
X
Q = [(1 − ω ) B ((1 − u ) L) −ω
Ω−
(1 − α ) AK α (u~L) −α −1 LMPLY (1 − t ) MPKK
]DYP
~
2
∂β
(1 + β ) (1 − u )
~
~~
~
~
[−φ (1 + τ CY ) P C Xφ (1−σ ) −1CY(1−φ )(1−σ ) −1 − P (−(1 + τ ) P U XX + U YX )]
F = [(1 − ω ) B ((1 − u ) L) −ω
H = ( MPLY ) 2
−1
~
~
X
(1 − α ) AK α (u~L) −α L(1 − t ) MPKK
(−(1 + τ ) P U XX + U YX ) > 0
2
(1 + β )
M = (1 − ω ) B((1 − u ) L) −ω
N ={
∂Z
∂G
X
Ω( I + DYP J ) − MPLY DYP
(1 − t ) MPKK
J
∂β
∂β
∂R
∂G
~
X
MPLY +
S}MPLY (1 − t ) MPKK
( −(1 + τ ) P U XX + U YX )
∂β
∂β
O=−
∂G
X ~
MPLY (1 − t ) MPKK
λ1 (1 + τ )
∂β
since
~
~
~ ~
~
~~
I = {−[−(1 + τ ) P U XX + U YX ][(Y − CY ) + τ (1 + η ) DY + τ (1 + η ) P DYP~ ] + λ1 (1 + τ )} > 0
u~
(1 − ω )
∂G
~
(1 − α ) AK α (u~L) −α −1 L > 0 ；
=
∂β (1 + β ) 2 (1 − u~ )
∂R
−1
~
(1 − α ) AK α (u~L) −α L < 0 ；
=
∂β (1 + β ) 2
hence F > 0 , Q > 0 , M > 0 , N > 0 , O > 0 , substituting these result into equations (A16), (A17),
~
~ ~
~
~
∂ ∂ (Y − X )
∂ ∂X
∂ dK
[
> 0 and
]<0.
and (A18), we obtain: ∂ du > 0 ,
> 0,
∂β
∂T
∂β ∂T
∂β dT
∂β dT
Appendix B (The proof of Proposition 2)
We can next discuss the effect of an increase in the consumption tax rate ( τ ) on the
economy’s steady-state. From equations (A1)-(A6), applying Cramer’s rule, we derive that:
~
∂C X
1
= {ab + cd − e}
∂τ
Δ
(A19)
~
∂CY
1
~
~
~
~
= {[ −(1 + τ ) P U XX + U YX ]DYτ b + fd − λ1 (1 + τ )ΩDYτ }
∂τ
Δ
(A20)
~
∂P 1
= Ω{− f + g}
∂τ Δ
(A21)
X
(1 − t ) MPKK
MPLY
∂u~
=−
{− f + g}
∂τ
Δ
6
(A22)
~
X
∂K (1 − t ) MPLY MPKL
=
{− f + g}
∂τ
Δ
(A23)
~
~
∂X
∂u~
X ∂K
= MPK
+ MPLX L
∂τ
∂τ
∂τ
(A24)
~
∂Y
1
∂u~
MPLY L
=−
∂τ
1+ β
∂τ
(A25)
where
~~
~
~
a = − DYτ ( − (1 + τ ) P U XY + U YY ) − λ1 P < 0
~
X
X
b = MPLY [(1 − t ) MPKL
MPKX − (1 − t ) MPKK
( MPLX + P MPLY )] + ΩS > 0
~
~
~
~ ~~
c = ( − (1 + τ ) P U XY + U YY )( DY + τDYτ )(1 + η ) P − λ1 P 2 < 0
X
d = ΩDYP − (1 − t ) MPKK
MPLY MPLY > 0
~
~
~
~
e = −λ1 (1 + τ )ΩP{(1 + η ) DY + [1 + τ (1 + η )]DYτ } > 0
~ ~
f = −(1 + η − φη ) P{DY −
<
1
~
~
~
~
[φY − (1 − φ )(1 + η )τDYτ ]}C Xφ (1−σ ) −1C Y(1−φ )(1−σ ) −1 0
>
(1 + η − φη )
1
~ >
~
~
if DY
[φY − (1 − φ )(1 + η )τDYτ ]
< (1 + η − φη )
~
~
~
~
g = DYτ [(−(1 + τ ) P 2U XX + P U YX ) − (−(1 + τ ) P U XY + U YY )]} < 0
~
∂C X
substituting these result into equations (A19)-(A25), can be derived:
> 0,
∂τ
~
~
~
∂K <
∂X <
∂Y >
∂u~ <
0,
0,
0。
0,
∂τ >
∂τ >
∂τ >
∂τ <
~
~
∂CY >
∂P >
0,
0,
∂τ <
∂τ <
Next, we compute the effect of an increase in the consumption tax rate ( τ ) on the “Foreign
Tax Relief” ( R F ):
1− ω
~
⎛
⎞
X
(1 − t ) MPLY MPKK
PL((1 − u ) L) −1 Y [ B −1 − 1]}⎟
⎜ − f {( DY + PDYP )Ω +
1
β
+
⎜
⎟
⎟
1⎜
∂R F
1−ω
~~
Y
X
−1
= (1 + η ) P DY + τ (1 + η ) ⎜ + g{DY Ω +
PY [ B ((1 − u ) L)] L(1 − t ) MPL MPKK }
⎟
∂τ
Δ⎜
1+ β
⎟
~
~
⎜
⎟
P
[
(
1
τ
)
P
U
U
]
D
b
λ
(
1
τ
)
PD
−
−
+
+
+
+
Ω
XX
YX
Yτ
1
Yτ
⎜
⎟
⎝
⎠
(A26)
~
The above equation, if DY is relatively huge, lead to the first item’s value [ (1 + η ) PDY ] is larger than
7
second item. That is, an increase in the consumption tax rate ( τ ) will lead to an improvement in the
∂R F
~
terms of R F . Contrary, if DY is relatively small,
can be either positive or negative.
∂τ
Finally, we discuss the effect of an increase in the consumption tax rate ( τ ) on the tourism
demand. In the previous text, an increase in τ , effect (I) reduces the DY , effect (II) reduces or
~
~
increase the CY . If DY is relatively high, increases in the τ will cause the CY raise. If DY is
~
∂CY
relatively small, increases in the τ ,
will be uncertain. Hence, compare the two effects:
∂τ
~
~
∂CY ∂DY
~
1 ~~
~
X
+
= {P C Xφ (1−σ ) −1CY(1−φ )(1−σ ) −1 ( Λ − DY )[ΩDYP − (1 − t ) MPLY MPLY MPKK
]}
∂τ
∂τ
Δ
(A27)
1
~
φY − DYτ [φ (1 + τ ) + τ (1 + η )(1 − φ ) + (1 − φ )] > 0 . If DY is relatively high, lead
(1 + η − φη )
~
~
dCY ∂DY
X
to
+
> 0 , (since Δ < 0 and [ΩDYP − (1 − t ) MPLY MPLY MPKK
] > 0 . With τ coming up,
dτ
∂τ
~
~
dCY ∂DY
+
< 0 , that is,
tourism demand can be rise. If DY is relatively small, then it follows that
∂τ
dτ
where Λ =
increases the τ will lead to tourism demand reduces.
8