Tourism Demand Def

Development strategies for tourism destinations: tourism

sophistication vs. resource investments∗

Rainer Andergassen†, Guido Candela

November 9, 2012

Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126, Bologna, Italy

Abstract

This paper investigates the effectiveness of development strategies for tourism destinations.

We argue that resource investments unambiguously increase tourism revenues and that increas-

ing the degree of tourism sophistication, that is increasing the variety of tourism related goods

and services, increases tourism activity and decreases the perceived quality of the destination’s

resource endowment, leading to an ambiguous effect on tourism revenues. We disentangle these

two effects and characterize situations where increasing the degree of tourism sophistication is a

viable development strategy and where it is impracticable and describe the optimal policy mix.

Keywords: Tourism destination; tourism sophistication; resource investments; tourism de-

mand; development strategy.

JEL: L83; O1; D11.

∗We would like to thank two anonymous referees for their comments and suggestions.†Corresponding author. Tel. +390512098666. E-mail addresses: [email protected],

[email protected]

1

1 Introduction

Considering the worldwide distribution of tourism activity we observe regions with highly developed

destinations and regions where tourism is still absent. Policy makers in these latter areas, which

include many developing countries, view the promotion of tourism activity, with its inherently strong

forward and backward linkages, as a leading growth and development strategy (see, for example,

UNCTAD, 2007; Lee and Chang, 2008; Sequeira and Nunes, 2008). This raises the policy issue of

identifying features that allow for a successful tourism take-off and of finding instruments apt to foster

the transformation of a region into a flourishing tourism destination. Given the region’s cultural and

natural resource endowments and given the variety of goods and services offered to actual or potential

tourists, the policy maker’s problem is how to promote tourism activity.

To address this question we investigate the characteristics of tourism demand and revenues in

a destination, identify two policy instruments and analyze the policy makers’ optimal development

strategy. In our model tourists are attracted by the presence of natural and/or cultural resources (see,

for example, Melian-Gonzalez and Garcia-Falcon, 2003; Papatheodorou, 2003; Correani and Garofalo,

2010) and exhibit love of variety preferences for tourism related goods and services, such as restaurants,

recreational facilities and so on. It is this novel feature that allows us to show that overnight stays,

which are a proxy for tourism activity, depend positively on the degree of differentiation of tourism

related goods and on the destination’s resource endowment. In other words, the destination’s capability

to attract tourists depends positively on the variety of tourism related goods and services and on the

cultural and/or natural resource endowment. This result is in keeping with the conceptual model

of destination competitiveness developed in Ritchie and Crouch (2003), in which it is argued that

among the most important factor are physiography, culture, mix of tourist activities, special events and

entertainment. From the policy viewpoint this opens the possibility of furthering tourism development,

measured in terms of tourism revenues1, either by increasing the variety of tourism related goods and

services, that is, by increasing the degree of sophistication of the tourism product or by enriching

the destination’s resource endowment. Examples of this latter policy option are investments aimed at

reclaiming or creating artificial beaches, the production of artificial snow or the creation of new ski1Candela and Figini (forthcoming, ch. 3.4) argue that the policy maker’s aim is to maximize tourism expenditures

instead of profits because “... the destination does not have the same decision power of an individual firm ...” and “...even if the destination had such power, the computation skills needed to gather and elaborate data on the productioncosts of hundred of individual ’productive units’ would simply not be available in most cases.”

2

slopes or ski runs2, or increasing the opening hours of museums3. We investigate the trade-off between

these two policy options and argue that while resource investments unambiguously increase tourism

revenues, increasing the degree of tourism sophistication has potentially an ambiguous effect on tourism

revenues. On the one hand tourism sophistication affects revenues positively by increasing tourism

activity (that is, overnight stays); on the other hand it may affect revenues negatively by decreasing

tourism quality because of resource congestion issues. In particular, an increase in tourism activity

may lead to an over-utilization of available cultural or natural resources, leading to a lower perceived

quality and to lower tourism expenditures. We disentangle these two effects and describe situations

where the former effect dominates the latter one, so that increasing the degree of sophistication is

a viable development strategy. In particular, we show that if the perceived tourism quality strongly

reacts to changes in tourism activity, then furthering the degree of sophistication may reduce tourism

revenues4.

We characterize in a static model the policy trade-off between sophistication and resource invest-

ments and describe the optimal policy mix, that is, the optimal investment strategy for a policy maker

who aims at developing a thriving tourism destination. In particular, by introducing costs to the

policy maker to enrich the destination’s resource endowment and costs of increasing the degree of

tourism product sophistication we describe the optimal policy intervention. We show that in general

the optimal strategy consists of a mix of both policies, where the solution to the trade-off depends

on the size of marginal costs and marginal benefits (that is, marginal increase in tourism revenues).

The analysis also shows that if the perceived tourism quality strongly reacts to increases in tourism

activity, then the policy maker should invest relatively more in resources; in the extreme case where

this reaction is very strong, a corner solution may be optimal where increasing the degree of sophisti-

cation of the tourism product is not desirable. Finally we highlight situations where tourism take-off

may be unfeasible. This discussion is closely aligned with recent contributions on sustainable tourism2The tourism industry is becoming more and more technological, since today’s technologies allow destinations to

expand and enrich their natural resource endowments (see, for example, Bruinsma et. al, 2010; Candela and Figini,forthcoming).

3In a similar vein, Stabler et. al (2010) base their analysis on the distinction between natural attractions and buildones, that is, human made.

4For instance, consider the case of a policy maker trying to promote tourism activity in a destination whose maintourism attraction is a beautiful small beach that is greatly appreciated by the few tourists (i.e. high perceived tourismquality). Suppose that by increasing the variety of tourism products and services offered, for example, by creating newattractions, such as a shopping mall, the policy maker succeeds in increasing tourism activity. This may lead to anovercrowding of the primary tourism attraction, the small beautiful beach, which now may no longer be perceived bytourists as beautiful. In other words, the sophistication strategy reduces the perceived quality of the tourism product.If this reduction is sufficiently strong, then it leads to a reduction in overall tourism expenditures and the destination’stourism revenues, while if this reduction is sufficiently weak, then it leads to an increase in tourism revenues. Thus, whatmatters is how strongly the perceived tourism quality decreases as tourism activity increases.

3

and sustainable tourism development, in which growth and development of destinations are related

to the long-term survival of tourism (see for example Sharpley and Telfer, 2002; Butler, 2005; and

for a survey see Stabler et. al, 2010). Our contribution formalizes the relationship between tourism

take-off, or long-term survival of the destination, and resource investments, that is, the sustainability

of the destination, by showing that if development through tourism sophistication is not viable, then

resource endowments pose a binding constraint on tourism take-off and hence resource investments are

necessary.

The remaining part of the paper is organized as follows. In Section 2 we present the formal model

and the main results. Section 3 contains some concluding remarks and all proofs are listed in the

Appendix.

2 The model

We consider a continuum of measure one of identical individuals, each endowed with a constant elastic-

ity of substitution (CES) utility function exhibiting Dixit and Stiglitz (1977) love of variety preferences

for differentiated tourism related goods. The utility function of the representative consumer j is

U (y (j) , h (j) , x1 (j) , ..., xi (j) , ..., xn (j)) =

y (j)

β+ zβ

[h (j)

γ+(∑n

i=1xi (j)

α) γα

] βγ

1β

(1)

where y is a composite non-touristic good5, h are overnight stays and xi, for i = 1, ..., n, represent

differentiated tourism related goods. We call T the tourism product, consisting of overnight stays (h)

and differentiated tourism related products (xini=1), i.e. T = (h, xini=1). z indicates the perceived

quality of the destination’s resource endowment, such as beaches, mountains, museums or more in

general heritages, on which tourism is based. In particular, the greater z is, the greater the relative

importance of the tourism product in utility terms 6. We assume that at least one variety has to be

offered such that tourism is viable, i.e. n ≥ 1; in other words, for n = 0, total overnight stays are nil.

n is the degree of tourism product diversification and we consider it to be a proxy for the degree of

tourism sophistication7. We neglect for simplicity the index j wherever this does not lead to confusion.5y could also include tourism consumption related to other destinations.6Let us rewrite the utility function as follows U

(y, T

)=(yβ + zβ Tβ

) 1β , where T is the sub-utility of the tourism

product, then the marginal rate of substitution between y and T is dy

dT= − Uy

UT

= − 1zβ

(Ty

)1−βand thus the greater z

is, the lower∣∣∣ dydT

∣∣∣, that is, the greater the relative importance of the tourism product in utility terms.7See also Andergassen and Candela (2009).

4

Throughout the paper we assume the following.

Assumption 1 (i) 0 < β < 1, (ii) −∞ < γ < 0, (iii) 0 < α < 1.

Assumption 1 (i) implies that the non-touristic good y and the tourism product T are gross sub-

stitutes; for β → 1 they are perfect substitutes. Assumption 1 (ii) implies that overnight stays and

tourism related products are gross complements, where for γ → −∞ they are perfect complements.

Assumption 1 (iii) implies that goods/services xi, i = 1,...,n, are gross substitutes.

The representative consumer faces the budget constraint

y + phh+∑n

i=1pixi = I (2)

where I is his income, ph is the price of a single overnight stay and pi the price of xi; the price of the

non-touristic good y has been normalized to 1. Note that tourism implies by definition a movement

of persons in space and time and thus the tourists’ income (I) in the present partial equilibrium

framework is generated in regions other than the destination considered. As a consequence, we treat I

as an exogenous variable. Moreover, this definition implies that tourism expenditures and the residents’

income from tourism activity do not lead to further tourism activity in the destination 8.

We define λβ ≡ β1−β ∈ (0,∞), λγ ≡ γ

1−γ ∈ (−1, 0), λα ≡ 1−αα ∈ (0,∞). Let H =

´ 1

0h (j) dj be

aggregate overnight stays, Xi =´ 1

0xi (j) dj aggregate consumption of each differentiated good/service

and Y =´ 1

0y (j) dj aggregate consumption of non-touristic goods.9

We assume that tourism is based on natural and/or cultural resources. Let R be a measure of the

destination’s natural and/or cultural resource endowment, such as beaches or mountains, museums or

shrines. We treat R as an exogenous variable, but assume that tourism activity affects the perceived

quality of these resources. In particular, we assume that for a given resource endowment, an increase

in tourism activity reduces the perceived quality of the destination’s resource endowment (z) because

of congestion problems (i.e. common pool resources). On the other hand, we assume that the policy

maker can undertake investments that enrich the destination’s resource endowment. Therefore, we

conjecture that z depends negatively on total overnight stays (H), which is a proxy for the size of the

tourism activity, and positively on the destination’s resource endowment (R). Let εzH = zHz H < 0

be the elasticity of z with respect to H, a measure for the degree of quality depreciation as tourism8For a discussion of the definition and its implication see Candela and Figini (forthcoming, ch 1).9Note that the assumption of a unit measure of individuals is without loss of generality since a change in the measure

of individuals can always be interpreted as a change in the income of individuals.

5

activity increases.

Assumption 2 z = z (H,R), where (i) zH < 0, zHH < 0; (ii) zR > 0 and zRR ≤ 0; (iii) zHR ≥ 0.

zH is the degree of quality depreciation as a consequence of tourism activity and (i) states that the

depreciation becomes stronger the stronger tourism activity is. zR is the degree of quality appreciation

as a consequence of an increase in the destination’s resource endowment where we assume that there are

decreasing returns to scale to investments aimed at increasing R; (iii) implies that resource investments

mitigate congestion issues.

Consider next the supply side. We assume that the degree of sophistication of the tourism product

is centrally devised, that is, the policy maker decides the type and how many tourism related goods

and services are offered in the destination, that the unit production costs of differentiated tourism

goods and services xi is c for each i = 1, ..., n and that they are supplied in a perfectly competitive

way10. Consequently, at the equilibrium, we have that pi = p = c for each i = 1, ..., n. We assume

that the aggregate supply function for overnight stays is ph = S (H), with∞ > SH ≥ 0, where SH > 0

represents the case in which there are decreasing returns to scale due to, for example, environmental

constraints, while SH = 0 represents the case in which such constraints are not binding and overnight

stays are in perfectly elastic supply 11. Let εSH = SHS H be the elasticity of S with respect to H and

we assume that εSH is non-increasing in H.

We characterize total tourism revenues as n and R vary.

The destination’s tourism revenue Ω is defined as

Ω (n,R) = phH + p

n∑i=1

Xi

where ph = S (H) and p = c. Since the representative consumer’s income I is constant and because of

the aggregate budget constraint, characterizing Ω implies characterizing I − Y .

Before analyzing the effect of n and R on Ω we show that the aggregate demand of overnight stays

H? is an increasing function of resource endowment (R) and the degree of tourism sophistication (n).10One could assume that there exists a product/firm set-up cost, and determine n through an entry and exit decision of

firms, by assuming that firms have some market power (e.g. monopolistic competition). The policy variable in this casewould be the size of these entry costs (e.g. the policy maker can subsidize the entry cost). This would greatly complicatethe exposition and the solution of the model, but without affecting the qualitative results. Thus, for simplicity’s sakewe consider n to be the policy variable. One way to interpret our assumption is that there exist product/firm set-upcosts, but these are fully subsidized by the policy maker, and that competition among firms to serve the market drivesthe price down to marginal costs. If the policy maker wants to increase n he simply subsidizes an additional market; ifhe wants to reduce n, he increases the entry costs such that no firm is interested in serving the market.

11One possible microfoundation of this supply function is to assume that firm costs for overnight stays H are κ (H),with κ′ (H) > 0, κ′′ (H) ≥ 0, and where κ′′ (H) > 0 represents the case of increasing marginal costs due to environmentalconstraints, and that firms behave in a perfectly competitive way.

6

Lemma 1 (i) H? is increasing in R; (ii) H? is increasing in n; (iii) the larger |εzH | or SH is, the

lower ∂H?

∂R and ∂H?

∂n .

An increase in the destination’s resource endowments increases the perceived tourism quality (z)

and hence increases overnight stays. The stronger the negative feedbacks through price increases (if

the supply of overnight stays is not perfectly elastic) and congestions issues (a decrease in z due

to the increase in overnight stays) are, the smaller the increase in overnight stays. Because of the

complementarity assumption between overnight stays and tourism related goods, an increase in n

increases overnight stays. The intuition for this result is that the more tourism related goods and

services are available, the more time tourists spend in a given destination. The greater |εzH | or SH is,

the lower the positive impact of tourism sophistication and resource investments on tourism activity.

Proposition 1 Ω is increasing in R.

Increasing the destination’s resource endowment increases consumer demand and expenditure for

overnight stays and for tourism related goods and services, thereby increasing tourism revenues.

We define M ≡ εzHλγ − εSH (1 + λγ).

Lemma 2 Ω is decreasing in the degree of tourism sophistication (n) if M > 1, while they are in-

creasing if M < 1.

Tourism revenues are decreasing or increasing in the degree of product diversification, depending

on the degree of complementarity between overnight stays and tourism goods/services (λγ), on the

degree of quality depreciation as tourism activity increases (εzH) and on the elasticity of the supply

function (εSH). To better understand this result, consider first the case in which the supply of overnight

stays is perfectly elastic (that is, SH = 0). In this case, result is driven by the interplay between two

opposing forces: a love of variety effect which positively affects tourism expenditures and a quality

depreciation effect, which negatively affects tourism revenues. If the complementarity between tourism

goods and overnight stays is weak, that is if n weakly affects H?, or if z does not strongly react to

changes in H, then the relationship between n and z is weak. Consequently, the love of variety effect

is relatively stronger and thus n positively affects tourism revenues. In other words, because of the

increased variety of tourism related goods and services available, tourists stay longer in the destination

and overall spend more, thereby leading to greater tourism revenues. On the other hand, if they are

strong complements, that is, if an increase in n leads to a strong increase in H?, or if z strongly reacts

7

to changes in H, then an increase in n leads to a strong reduction of the perceived quality of tourism.

Consequently, the quality depreciation effect dominates, and thus the reduction in expenses on tourism

related goods and services is stronger than the increased expenditure on overnight stays and thus total

tourism revenues decrease as n increase12. In other words, tourists stay longer in the destination and

thus spend more for overnight stays, but because of the perceived poorer quality they spend less on

tourism related goods and services, which leads to lower total tourism revenues. Note that this effect

is stronger, the stronger the degree of complementarity (λγ) and/or the larger |zH |. If the supply

of overnight stays is not perfectly elastic (SH > 0), then the resulting price increase mitigates the

resource congestion effect produced by an increase in n. In other words, the greater εSH is, the weaker

the influence of n on H? and thus on z.

Summing up, if quality depreciation is strong enough, consumers increase their expenditure on non-

touristic goods and reduce their expenses on tourism; a sufficiently strong price increase eliminates the

resource congestion effect, leading to a positive relationship between Ω and n.

Let us define M ≡ limn→∞M. The following result holds.

Proposition 2 (i) If M > 1, then there exists a n? where Ωn ≷ 0 for each n ≶ n?, where ∂n?

∂R ≷ 0 if

MR ≶ 0; (ii) if M < 1 then Ωn > 0 for each n.

If we consider n as a policy instrument for the development of a tourism destination, then n?

is an upper limit for the usefulness of the instrument. The existence of n? depends on the degree

of resource depreciation; as stated in Proposition 2, a sufficient condition for the existence of n? is

that limn→∞M > 1. Tourism sophistication leads to an increase in tourism activity (Lemma 1) and

therefore to a perceived quality depreciation of the destination’s resource endowment. Since the quality

depreciation gets the stronger the greater the degree of tourism sophistication is (Assumption 2 (i)), it

may happen that, as the process of tourism sophistication proceeds, the degree of quality depreciation

becomes sufficiently strong such that a further increase in n leads to a reduction in tourism revenues.

In this case there exists a degree of tourism sophistication (n?) that, for a given resource endowment,

maximizes tourism revenues.

An increase in R has a twofold effect on n?. On the one hand, for given overnight stays, it increases

the perceived quality of the destination’s resource endowments and thus increases n?, on the other

hand it increases overnight stays thereby increasing congestion issues and thus decreasing tourism

quality, which decreases n?. For MR > 0 the latter effect dominates and thus resource investments12Note also that since H? is always increasing in n, it is

∑ni=1 X

?i that is decreasing in n for M > 1.

8

instead of alleviating worsen congestion issues, while for MR < 0 the former effect dominates and thus

resource investments weaken the constraints posed by congestion issues.13

If the degree of quality depreciation remains small as tourism activity increases, then tourism

sophistication always increases tourism revenues.

We treat the two cases in a unified way considering n? <∞ if M > 1 and n∗ =∞ if M < 1. If we

consider R and n as two policy instruments (Proposition 1 and Proposition 2, respectively), then the

following Corollary describes the policy trade-off.

Corollary 1 LetMRSn,R ≡ −ΩRΩn

denote the marginal rate of substitution between n and R. MRSn,R

is negative for each n < n? and is positive for each n > n?.

As long as MRSn,R is negative, the policy maker can use both instruments to promote tourism

development and the optimal policy mix depends on the relative costs and benefits of doing so. But

once n > n? tourism development via sophistication is no longer viable. Note that if n? = ∞, then

sophistication is always a viable development strategy.

Let us assume that the policy maker can affect the degree of tourism product sophistication and

enrich the destination’s resource endowment at some costs. In particular, letKR (R−R°)+Kn (n− n°)

be the social opportunity cost of increasing resource endowments by ∆R = R − R° and the degree of

tourism product sophistication by ∆n = n − n°, where R° and n° are the initial levels of R and n,

respectively. KR > 0 is the constant marginal social (opportunity) cost of enriching the destination’s

resource endowment and that Kn > 0 is the constant social marginal (opportunity) cost of increasing

the degree of tourism sophistication. Assume also that the policy maker faces the budget constraint

KR (R−R°)+Kn (n− n°) ≤ K. LetMRTSn,R ≡ −KRKn be the marginal rate of technical substitution,

that is, the relative social marginal cost of increasing R and n 14. We characterize the optimal static

policy mix if the aim is to maximize overall tourism revenues minus the cost of the policy intervention,

13Using the implicit function theorem one obtains that dn∗

dR= −MR

Mn, where Mn > 0, and

MR =(zHR

H?

z− zHzRH

?

z2

)λγ

+

(zHH

H?

z− z2

HH?

z2 + zH1z

)λγ − (1 + λγ)

∂εSH∂H

H?R

where the first term is negative while the second one is positive, since εSH is by assumption non-increasing in H and H?R

is positive.14Social costs of increasing n could, for example, be the social costs incurred by the policy maker in subsidizing firms’

entry costs, while those of increasing R could be due to resources spent in reclaiming and creating artificial beaches, orcreating new ski slopes.

9

subject to the budget constraint:

maxR,n Ω (n,R)−K (R−R°, n− n°)

s.t. KR (R−R°) +Kn (n− n°) ≤ K

We simplify the analysis and assume that the supply of overnight stays is perfectly elastic and that

resource investments alleviate congestion issues15.

Proposition 3 If εSH = 0 and MR < 0, then the optimal policy mix solves the problem |MRSn,R| =

|MRTSn,R|.

Proposition 3 describes the optimal policy mix. Since |MRSn,R| and |MRTSn,R| are relative

marginal gains and relative marginal costs of resource investments, respectively, the policy maker has to

choose ∆n and ∆R such that the condition |MRSn,R| = |MRTSn,R| is satisfied. Thus, in general it is

optimal to increase both n and R. Note that the largerM is, that is, the stronger the perceived quality

depreciation as tourism activity increases, the lower marginal gains from a sophistication strategy and

hence the more the policy maker should invest in resources. Moreover, if M is sufficiently large

(in particular, if M > 1), then a corner solution may be optimal and thus the enrichment of the

destination’s resource endowment is the unique optimal development strategy.

A final issue concerns the viability of a tourism industry. For a given resource endowment, Ω ≡

Ω |n=n? is the maximum of revenues achievable through tourism sophistication. If Ω is too low to

guarantee tourism take-off, then resource endowments pose a binding constraint to the development

process and hence resource investments are necessary 16.

3 Conclusion

The main problem for policy makers of destinations is how to foster or how to kick-off the develop-

ment of a tourism industry. We investigated the effectiveness of tourism sophistication and resource

investments as development strategies. We showed that the success of fostering tourism development

through tourism sophistication may be constrained by the destination’s resource endowment. Tourism

sophistication increases tourism activity, thereby affecting positively tourism revenues, but aggravating15Using a continuity argument, the result holds also for small values of εSH . On the other hand, for εSH sufficiently

large, monotonicity of∣∣MRSn,R

∣∣, and consequently the uniqueness of the optimal policy mix, is no longer guaranteed.16Suppose that n? <∞ in which case Ω <∞, then taking the total derivate of Ω with respect to R we obtain, using

the envelope theorem, dΩdR

= ∂Ω∂R

> 0.

10

resource congestion issues. In particular, we argued that if the perceived quality depreciation of the

destination’s resource endowment as a consequence of tourism activity is strong enough, then engaging

in a sophistication strategy may well reduce tourism revenues, obstructing the kick-off of the tourism

industry. To overcome this hurdle, our analysis suggests that policy makers should engage in invest-

ments aimed at enriching the destination’s natural and/or cultural resource endowment since they

positively affect tourism demand and revenues. Those regions where these investments are not feasible

or too costly cannot become tourism destinations since the perceived quality of such destinations is

too low to generate sustained revenues.

A limitation of the present model is the partial equilibrium framework. It would be interesting

to extend the model to a general equilibrium framework, where two or more economic regions are

considered and where tourists’ income is endogenously determined. In this framework one could also

analyze competition between tourism destinations and how it affects regional development strategies.

Tourism activity in a given destination, by increasing the income of its residents, would lead in this

case to tourism activity in the other destination, leading to interesting feedbacks that policy makers

should take into account. We leave these open issues for a possible future research.

Appendix

Proof of Lemma 1. We first calculate individual demand functions, and then aggregate over individ-

uals. Since there is a continuum of consumers, each one has a negligible effect on the perceived tourism

quality z. Using Lagrange for solving the problem of maximizing (1) under the budget constraint (2),

the first order conditions for the representative consumer read:

yβ + zβ

[hγ +

(∑n

i=1xαi

) γα

] βγ

1β−1

yβ−1 = λ (3)

yβ + zβ

[hγ +

(∑n

i=1xαi

) γα

] βγ

1β−1

zβ[hγ +

(∑n

i=1xαi

) γα

] βγ−1

hγ−1 = λph (4)

yβ + zβ

[hγ +

(∑n

i=1xαi

) γα

] βγ

1β−1

zβ[hγ +

(∑n

i=1xαi

) γα

] βγ−1 (∑n

i=1xαi

) γα−1

xα−1i = λpi,

(5)

11

for i = 1, ..., n, where λ is the Lagrange multiplier. Using the assumption that all firms producing

tourism related goods are symmetric we have pi = p and hence obtain xi = x, for each i = 1, ..., n.

From (4) and (5) we obtain

x = h

(p

phn1− γα

) 1γ−1

(6)

while from (4) and (3) we obtain ph =zβ(hγ+n

γα xγ

) βγ

−1hγ−1

yβ−1 which, using (6), reads as

y = h (ph)1

1−β zββ−1

[1 + n

γ1−γ

1−αα

(p

ph

) γγ−1

]( βγ−1) 1β−1

(7)

Finally, we calculate h substituting (6) and (7) into the budget constraint (2) and obtain

h (n, z) =I

ph

[1 + nλγλα

(pph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ(8)

where hn > 0. Substituting (8) back into (6) and (7) one obtains

x (n, z) =I

p

[n−λγλα

(pph

)λγ+ 1

] 1

n+ pλβh z−λβ

[n− λγλβ + n

λγλα−λγλβ

(pph

)−λγ]−λβλγ (9)

and

y (n, z) = I1

1 + p−λβh zλβ

[1 + nλγλα

(pph

)−λγ]λβλγ . (10)

Since all individuals are identical, h (j) = h and x (j) = x, and consequently H = h and since

pi = p = c it follows that Xi = X = x. Next we calculate the aggregate demand function H (n,R),

where the consumers’ choice H feeds back into the perceived tourism quality z. Using (8), we have to

solve the following fixed point problem:

H = f (ph, n, z (H,R)) ≡ I

ph

[1 + nλγλα

(pph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ(11)

which yields the solution H = H (n,R). In view of Assumption 2, fH(ph, n, z

(H, R

))≤ 0, with

12

f (ph, n, z (0, R)) > 0 and a unique H? solving f(ph, n, z

(H, R

))= H exists, and thus the aggregate

demand function for overnight stays can be written as ph = D (H,n,R). Using the implicit function

theorem one obtains that ∂ph∂H = 1−fH

fph, ∂ph∂n = − fn

fphand ∂ph

∂R = − fRfph

, where fph < 0, fH ≤ 0, fn > 0

and fR > 0 and thus DH < 0, DR > 0 and Dn > 0.

Equating demand and supply, equilibrium overnight stays H? are implicitly defined by S (H) =

D (H,n,R). Applying the implicit function theorem one obtains that ∂H?

∂n = DnSH−DH > 0 and ∂H?

∂R =

DRSH−DH > 0, or equivalently ∂H?

∂n = −fnSHfph−(1−fH) > 0 and ∂H?

∂R = −fRSHfph−(1−fH) > 0, respectively.

After rearranging terms one obtains

∂H?

∂n=

−H?2 phI λγλαn

λγλα−1(pph

)−λγ 1 + pλβh z−λβ

(1− λβ

λγ

)[1 + nλγλα

(pph

)−λγ]−λβλγ−SHfph + 1−H? ph

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]1−λβλγ

εzH

> 0

(12)

and

∂H?

∂R=

H?2 phI

[1 + nλγλα

(pph

)−λγ]pλβh λβz

−λβ zRz

[1 + nλγλα

(pph

)−λγ]−λβλγ−SHfph + 1−H? ph

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]1−λβλγ

εzH

(13)

where

fph = −H?2

I

[

1 + λβ + (1 + λγ)nλγλα(p

ph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(p

ph

)−λγ]−λβλγ− λβ

(14)

Using (9) and (10), one obtains that X (n,R) = x (n, z (H?, R)) and Y (n,R) = y (n, z (H?, R)).

Comparative statics results directly follow from (12).

Proof of Proposition 1. From the aggregate budget constraint one obtains that Ω = I − Y .

Consequently, substituting (10) into the expression Ω = I − Y , finding the common denominator and

dividing the numerator and the denominator by p−λβh zλβ[1 + nλγλα

(pph

)−λγ]λβλγone obtains

Ω (n,R) =I

1 + pλβh z (H?, R)

−λβ[1 + nλγλα

(pph

)−λγ]−λβλγ (15)

13

The derivative of Ω with respect to R reads

ΩR = −Ω2

Iλβp

λβh z−λβ

[1 + nλγλα

(p

ph

)−λγ]−λβλγ −11

ph

∂ph∂R−(zRz

+zHzH?R

)[1 + nλγλα

(p

ph

)−λγ]

Since ∂ph∂R = SH

∂H?

∂R and using 13 we can rewrite this expression as follows

ΩR = −Ω2

I λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]−λβ

λγ

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

zRz ×SHH?2 1

I pλβh λβz

−λβ[1 + nλγλα

(pph

)−λγ]−λβλγ− (1− SHfph)

Using (14) and rearranging terms we obtain

ΩR =Ω2

I λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]−λβ

λγ

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

zRz ×SH H?2

I

[1 + (1 + λγ)nλγλα

(pph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ+ 1

Using the definition of εSH , the definition of Ω and taking into account that ph = S (H) and that

phH?

Ω=

1[1 + nλγλα

(pph

)−λγ] (16)

we obtain

ΩR =

Ω2

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ−SHfph + 1−H? ph

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]1−λβλγ

εzH

zRz

εSH1 + (1 + λγ)nλγλα

(pph

)−λγ1 + nλγλα

(pph

)−λγ + 1

(17)

which is always positive.

14

Proof of Lemma 2. The derivative of the denominator of Ω with respect to n is

−pλβh λβz−λβ zH

z H?n

[1 + nλγλα

(pph

)−λγ]−λβλγ+

+pλβh z−λβ

(−λβλγ

)[1 + nλγλα

(pph

)−λγ]−λβλγ −1

λγλαnλγλα−1

(pph

)−λγ+λβp

λβh

1ph

∂ph∂n z

−λβ[1 + nλγλα

(pph

)−λγ]−λβλγ −1

where ∂ph∂n = SH

∂H?

∂n . After substituting the expression for ∂H?

∂n in the first line of the derivative and

rearranging terms we obtain

−λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ −1

λαnλγλα−1

(pph

)−λγ×

−H? phI λγεzH

[1+nλγλα

(pph

)−λγ]1+p

λβh z−λβ

[1+nλγλα

(pph

)−λγ]−λβ

λγ

−SHfph+1

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

+

+λβpλβh

1phSH

∂H?

∂n z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ −1

Using the equilibrium expression for H? (11) the derivative of Ω with respect to n can be written as

Ωn (R,n) = −Ω2

Iλβp

λβh z−λβ

[1 + nλγλα

(p

ph

)−λγ]−λβλγ −1

×

λαn

λγλα−1(pph

)−λγ−SHfph + 1−H? ph

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]1−λβλγ

εzH

(εzHλγ − 1 + SHfph) +1

phSHHn

Substituting the expression for ∂H?

∂n we obtain

Ωn (R,n) = −Ω2

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ −1λαn

λγλα−1(pph

)−λγ

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

×

(εzHλγ − 1 + SHfph) + 1phSHH

? (−λγ)1+p

λβh z−λβ

(1−

λβλγ

)[1+nλγλα

(pph

)−λγ]−λβ

λγ

[1+nλγλα

(pph

)−λγ]1+p

λβh z−λβ

[1+nλγλα

(pph

)−λγ]−λβ

λγ

15

Using the equilibrium expression for H? (11) can be written as

Ωn (R,n) = −Ω2

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ −1λαn

λγλα−1(pph

)−λγ

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

×

(εzHλγ − 1 + SHfph) + SH (−λγ) H?2

I

1 + pλβh z−λβ

(1− λβ

λγ

)[1 + nλγλα

(pph

)−λγ]−λβλγ

We next focus on the second line of this expression, which, after substituting the expression for fph

(14), reads

εzHλγ − 1− SH H?2

I

[1 + λβ + (1 + λγ)nλγλα

(pph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ− λβ

+SHH?2

I

−λγ + (λβ − λγ) pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγSimplifying this expression one obtains

εzHλγ − 1− SH H?2

I

[1 + (1 + λγ)nλγλα

(pph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ−λγSH H?2

I

1 + pλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγor

εzHλγ − 1− SHH?2

I(1 + λγ)

[1 + nλγλα

(p

ph

)−λγ]1 + pλβh z−λβ

[1 + nλγλα

(p

ph

)−λγ]−λβλγUsing the expression for Ω and εSH we can write and taking into account (16) one obtain the final

expression for Ωn

Ωn = −Ω2

I λβpλβh z−λβ

[1 + nλγλα

(pph

)−λγ]−λβλγ −1λαn

λγλα−1(pph

)−λγ

−SHfph+1−H? phI λβpλβh z−λβ

[1+nλγλα

(pph

)−λγ]1−

λβλγεzH

×

εzHλγ − 1− εSH (1 + λγ)

(18)

and hence the result in the lemma follows.

16

Proof of Proposition 2. Taking the derivative of M (n) with respect to n we obtain

Mn = H?n

(zHH

H?

z− z2

H

H?

z2+ zH

1

z

)λγ − (1 + λγ)

(SHH − S2

H

H?

S2+ SH

1

S

)

which, under Assumption 2 and the assumption that εSH is non-increasing in H is always positive.

Moreover, since for n = 0, H? = 0, if follows that εzH = εSH = 0 and thus M (0) = 0 < 1. Therefore, if

M > 1, then there exists a n? such that M (n) < 1, or Ωn > 0, for n < n?, and M (n) > 1, or Ωn < 0,

for n > n?. On the other hand, if M < 1, then Ωn > 0 for all values of n.

Proof of Corollary 1. The corollary is a direct consequence of Lemma 2, Proposition 1 and

Proposition 2.

Proof of Proposition 3. From the first order conditions of the maximization problem one

obtains the the optimal (n,R) is given by the following condition MRSn,R = MRTSn,R. Using (17)

and (18), MRSn,R one obtains

MRSn,R = −zRzn

εSH

[1 + (1 + λγ)nλγλα

(pph

)−λγ]+ 1 + nλγλα

(pph

)−λγλαnλγλα

(pph

)−λγ 1 + εSH (1 + λγ)− εzHλγ

and for εSH = 0 the expression becomes

MRSn,R = −zRzn

1 + nλγλα(pph

)−λγλαnλγλα

(pph

)−λγ(1− εzHλγ)

If MR < 0, then |MRSn,R| is decreasing in R and increasing in n for each n < n? and decreasing in n

for each n?, while |MRTSn,R| is increasing in R and decreasing in n and consequently the result in

the proposition follows.

References

[1] Andergassen, R. - Candela G. (2009), Less Developed Countries, Tourism Investments

and Local Economic Development, WP n. 676, Department of Economics, University of

Bologna, 2009, http://www2.dse.unibo.it/wp/676.pdf .

17

[2] Bruinsma, F. - Kourtit K. - Nijkamp P. (2010), An Agent-Based Decision Support Model

for the Development of E-Services in the Tourist Sector, Review of Economic Analysis,

2, 232 - 255.

[3] Butler, R.W. (2005), Developing the Destinations: Difficulties in Achieving Sustainability,

In Alejziak W. andWiniarski R. (eds), Tourism in Scientific Research, Krakow, University

of Information Technology and Management, Rzeszow, 33-46.

[4] Candela, G. - Figini P. (Forthcoming). Economics of Tourism Destinations. Springer

Verlag, Berlin.

[5] Correani, L. - Garofalo G. (2010), Chaotic Paths in the Tourism Industry, Economia

Politica, Journal of Analytical and Institutional Economics, 27, 103 - 146.

[6] Dixit, A. K. - Stiglitz J. E. (1977), Monopolistic Competition and Optimum Product

Diversity, American Economic Review, 67, 297-308.

[7] Lee, C.C. - Change C.P. (2008), Tourism Development and Economic Growth: A Closer

Look at Panels, Tourism Management, 29, 180 - 192.

[8] Melian-Gonzalez, A. - Garcia-Falcon, J. M. (2003), Competitive Potential of Tourism in

Destinations, Annals of Tourism Research, 30, 720-740.

[9] Papatheodorou, A. (2003), Modelling tourism development: a synthetic approach,

Tourism Economics, 9 , 407 - 430.

[10] Ritchie, J.R.B. - Crouch G.I. (2003), The Competitive Destination: A Sustainable Tourism

Perspective, Wallingford, CABI.

[11] Sequeira, T. N. - Nunes P.M. (2008), Does Tourism Influence Economic Growth? A

Dynamic Panel Data Approach, Applied Economics, 40, 2431 - 2441.

[12] Sharpley, R. - Telfer D.J. (2002), Tourism and Development, Clevedon, Channel View.

[13] Stabler, M.J. - Papatheodorou A. - Sinclair M.T. (2010), The Economics of Tourism,

Second Edition, Routledge, London - New York.

[14] UNCTAD (2007), FDI in Tourism: The Development Dimension, United Nations, New

York and Geneva.

18