Dynamic Model of Economic Growth in a Small Tourism Driven Economy

Abstract

International tourism is one of the fastest growing industries, accounting for more than 10% of total international trade and almost half of total trade in services, and can be considered as one of the world’s largest export earners. In many countries, foreign currency receipts from tourism exceeds currency receipts from all other sectors together. Thus, tourism, which is an alternative form of exports, contributes to the balance of payments through foreign exchange earnings and proceeds generated from tourism expansion.

Appendix

1.1 Derivation of $\dot{b}(t)$

Time differentiating $b \equiv B/{Y^\sigma }$ gives

$$ \dot{b} = \frac{{\dot{B}}}{{{Y^\sigma }}} - \sigma \frac{B}{{{Y^\sigma }}}\frac{{\dot{Y}}}{Y} = \frac{{\dot{B}}}{{{Y^\sigma }}} - \sigma nb. $$

(10.18)

Dividing (10.1a) by ${Y^\sigma }$, noting that from (10.3b) we can write $\Phi (I,K) = \frac{{{q^2} - 1}}{{2h}}K$, we get

$$ \frac{{\dot{B}}}{{{Y^\sigma }}} = pA\frac{K}{{{Y^\sigma }}} - \frac{C}{{{Y^\sigma }}} - \frac{{{q^2} - 1}}{{2h}}\frac{K}{{{Y^\sigma }}} + r\frac{B}{{{Y^\sigma }}}. $$

(10.19)

Inserting (10.19) into (10.18), applying the definitions

$$ k \equiv \frac{K}{{{Y^\sigma }}},\quad c \equiv \frac{C}{{{Y^\sigma }}},\quad b \equiv \frac{B}{{{Y^\sigma }}} $$

and rearranging terms results in

$$ \dot{b} = (r - \sigma n)b + \left( {pA - \frac{{{q^2} - 1}}{{2h}}} \right)k - c, $$

(10.20)

which is (10.13) in the text.

1.2 Solution of $b(t)$

Linearizing (10.13) around a hypothetical steady-state, noting that ${b^{\hskip-2pt\dot\tilde}} = 0$, we get

$$ \begin{array}{ll}\dot{b} - (r - \sigma n)(b - \tilde{b})= & A\tilde{k}(p - \tilde{p}) - \left( {\tilde{p}A - \frac{{{{\tilde{q}}^2} - 1}}{{2h}}} \right)(k - \tilde{k}) \\&- \frac{{\tilde{q}\tilde{k}}}{h}(q - \tilde{q}) - (c - \tilde{c}).\end{array} $$

(10.21)

Using $c(t) = c(0){ \exp }[({\psi_C} - \sigma n)t]$, the stable solutions for $k$, (10.12) and $q$, (10.9b), and the definition of the steady-state of $b$,

$$ - (r - \sigma n)\tilde{b} = \left( {\tilde{p}A - \frac{{{{\tilde{q}}^2} - 1}}{{2h}}} \right)\tilde{k} - \tilde{c}, $$

Equation (10.21) can be written as

$$ \dot{b} - (r - \sigma n)b = L{e^{{\mu_1}t}} - c(0){e^{({\psi_C} - \sigma n)t}} + M. $$

(10.22)

$L$ and $M$ are defined as

$$ L \equiv \left[ {A\tilde{k} + \left( {\tilde{p}A - \frac{{{{\tilde{q}}^2} - 1}}{{2h}} + {\mu_1}\tilde{q}} \right)\frac{{\varepsilon \tilde{k}}}{{\tilde{p}}}} \,\right]\left( {{p_0} - \tilde{p}} \right) $$

(10.23)

$$ M \equiv \left[ {\tilde{p}A - \frac{{{{\tilde{q}}^2} - 1}}{{2h}}} \right]\tilde{k} $$

(10.24)

where for notational convenience we have made use of the fact that from the system’s eigenvectors it follows that

$$ - \frac{{{\mu_1}\varepsilon h}}{{\tilde{p}}} \,= \,\frac{A}{{r - \sigma n - {\mu_1}}} \,> \, 0. $$

$L$ denotes the difference between output and investment costs along the stable saddle path. $M$ measures the difference between steady-state production and steady-state investment costs.

Multiplying (10.22) by the integrating factor ${e^{ - (r - \sigma n)t}}$, and performing the integration yields

$$ \begin{array}{ll} {b(t) = } & {\left[ {{b_0} - \displaystyle\frac{L}{{{\mu_1} - r + \sigma n}} + \frac{{c(0)}}{{{\psi_C} - r}} + \frac{M}{{r - \sigma n}}} \right]{e^{(r - \sigma n)t}}} \hfill \\{} \hfill & { + \displaystyle\frac{L}{{{\mu_1} - r + \sigma n}}{e^{{\mu_1}t}} - \frac{{c(0)}}{{{\psi_C} - r}}{e^{({\psi_C} - \sigma n)t}} - \frac{M}{{r - \sigma n}}} \hfill \\\end{array} $$

(10.25)

The transversality condition for $B$, $\mathop {{ \lim }}\nolimits_{t \to \infty } \lambda B{e^{ - \beta t}} = 0$, can be rewritten as $Y_0^\sigma \lambda (0)\mathop {{ \lim }}\nolimits_{t \to \infty } b(t){e^{(\sigma n - r)t}} = 0$. Inserting (10.25), this is met if

$$ {b_0} - \frac{L}{{{\mu_1} - r + \sigma n}} + \frac{{c(0)}}{{{\psi_C} - r}} + \frac{M}{{r - \sigma n}} = 0 $$

(10.26)

$$ {\psi_C} < r. $$

(10.27)

Equation (10.26) is the economy’s intertemporal budget constraint and determines $c(0)$ and thus $\lambda (0)$. Equation (10.27) introduces an upper bound on ${\psi_C} \equiv \displaystyle\frac{{r - \beta }}{{1 - \gamma }}$ and can be rewritten as $\gamma < \frac{\beta }{r}$, and defines thus an upper bound on the intertemporal elasticity of substitution $1/(1 - \gamma )$. Hence, the solution of $b$ consistent with long-run solvency becomes

$$ b(t) = \frac{L}{{{\mu_1} - r + \sigma n}}{e^{{\mu_1}t}} - \frac{{c(0)}}{{{\psi_C} - r}}{e^{({\psi_C} - \sigma n)t}} - \frac{M}{{r - \sigma n}}. $$

(10.28)