Investment in Tourism Market: A Dynamic Model of Differentiated Oligopoly

Abstract

We present a theoretical dynamic model in tourism economics, assuming that the market for tourism is an oligopoly with differentiated products. Destinations can invest in order to improve their stock of physical, natural or cultural resources. Tourism flows yield current revenues, but they are usually detrimental for the stock of resources. We find the solution of the dynamic model, and in particular we find the open-loop Nash equilibrium of the game among destinations, under alternative settings, depending on whether the degree of differentiation among destinations is exogenous or endogenous. In particular, under the latter case, an increase of the number of destinations leads to a higher degree of product differentiation in steady state.

Appendix A

First order condition (14) in Text establishes the equality between the marginal revenue and the marginal cost (revenue and cost include investment in enhancing the stock of resources, along with the effect of tourists’ presence). Different cases must be considered.

$${\rm{If\ } [2B+D(n-1)]\geq\frac{1-z(\rho+\delta)}{\delta z(\rho +\delta)},}$$

(\rm A)

the marginal revenue is decreasing and non-positive for any positive value of x. Consequently, the corner solution x=0 is the optimum, i.e., the allocation associated to the maximum profit.

$$\hbox{If } [2B+D(n-1)] < \frac{1-z(\rho+\delta)}{\delta z(\rho+\delta)}$$

(\rm B)

, the marginal revenue is positive (so that both the marginal revenue and the marginal cost are positively sloped), and the intersection between the marginal revenue curve and the marginal cost curve represents the maxim profit point if and only if the marginal cost curve intersect the marginal revenue curve from below. Hence, we distinguish three sub-cases:

$$\hbox{If }\frac{1-z(\rho+\delta)}{\delta z(\rho+\delta)} - [2B+D(n-1)]> c^{\prime\prime}(x) + \frac{1}{(\rho+\delta )},$$

(\rm B.1)

the optimum is $x\to +\infty$ (This is due to the fact that the marginal revenue increases at a speeder pace than the marginal cost, as x increases. Unless some capacity constraint on the tourism flows is operative, like $x \leqslant x^{\wedge}$, there is no finite solution for x);

$$\hbox{If }\frac{1-z(\rho+\delta)}{\delta z(\rho+\delta)}-[2B+D(n-1)] = c^{\prime\prime}(x) + \frac{1}{(\rho+\delta)}$$

(\rm B.2)

the optimum is indeterminate (marginal cost and marginal revenue coincide);

$$\frac{1-z(\rho+\delta)}{\delta z(\rho+\delta)} - [2B+D(n-1)] < c^{\prime\prime}(x) + \frac{1}{(\rho+\delta)}$$

(\rm B.3)

the internal critical point denotes the maximum profit.

In sum, the dynamic problem can lead to a steady state with a positive and finite value for x, only under condition $[2B+D(n-1)] < <$> <$>[1-z(\rho+\delta)]/[\delta z(\rho+\delta)]$ in case (B.3). The system of these two disequations condition coincide with the condition given by (15) in Text.