Closed‐form solutions of an economic growth model of tourism

Abstract

The economic growth model for tourism aims at optimizing the quality of life of the residents. The model is studied under the limitations of crowding conditions, environmental, and economic resources. The current value Hamiltonian function is effectively used to determine the necessary first conditions for the optimal values of consumption and number of tourists in the destination. When the necessary conditions are sufficient for optimality, the Arrow sufficient theorem provides the special case. The partial Hamiltonian approach is applied to construct the first integrals of the economic growth model for two cases. These first integrals were utilized to compute the closed‐form solution of the model under consideration. As it is well‐known, different Ramsey‐type models are solved with restrictions on the parameters. The system dynamics were studied mainly by describing the trajectories of the variables and the equilibrium, which is not enough to understand the evolution of any economy. This study is to examine the existence of closed‐form solutions for the growth model of tourism. We apply this newly developed technique to analyze the dynamics of the total income for different initial stock ratios. The effects of natural amenities on the control and state variables were also part of this study. It is also shown that crowding aversion saves the stock of natural resources and services. We offer that for the long‐term growth of the tourism, the optimal number of the tourist at the destination plays an important role.