Article Contents
Article Contents

# Population dynamics and economic development

• * Corresponding author: Luca Gori
• This research develops a continuous-time optimal growth model that accounts for population dynamics resembling the historical pattern of the demographic transition. The Ramsey model then becomes able to generate multiple determinate or indeterminate stationary equilibria and explain the process of the transition from a state with high fertility and low income per capita to a state with low fertility and high income per capita. The article also investigates the emergence of damped or persistent cyclical dynamics.

Mathematics Subject Classification: C61; C62; J1, J22; O41.

 Citation:

• Figure 1.  A portion of the stable manifold on which the trajectories converging to the steady state $P^{\ast }$ occur. Parameter set: $\alpha = 0.18$, $\beta = 0.84$, $\delta = 0.13$, $\varepsilon = 0.4$, $\rho = 0.3$, $\sigma = 1.25$, $a = 40.5$, $b = 0.5$. Stationary state equilibrium: $P^{\ast }: = (k^{\ast },c^{\ast },L^{\ast }) = (0.257,0.749,82.315)$

Figure 2.  Existence of multiple equilibria depending on $\beta$

Figure 3.  Global indeterminacy. Starting from the same initial conditions $k_{0} = 0.455$ and $L_{0} = 104.843$, an infinity of initial choices on $c_{0}$ lead to the indeterminate stationary state equilibrium $P_{2}^{\ast }$, while there exists a unique choice ($c_{0}^{1} = 0.833$) leading to the saddle $P_{1}^{\ast }$. Parameter set: $\alpha = 0.18$, $\beta = 0.84$, $\delta = 0.13$, $\varepsilon = 0.4$, $\rho = 0.3$, $\sigma = 1.25$, $a = 40.5$, $b = 0.5$ and $x = 0.7$. Stationary state equilibria: $P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.594,0.833,80.677)$ and $P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (4.372,2.925,79.676)$

Figure 4.  Local indeterminacy scenario in terms of capital (Panel A) and population size (Panel B)

Figure 5.  Three-dimensional phase portrait in the space $(k,c,L)$ showing: (i) the trajectory converging to $P_{1}^{\ast }$ starting from the initial condition $(k_{0},c_{0}^{1},L_{0}) = (0.346,0.565,3.178)$, and (ii) a trajectory converging to a limit cycle $\Gamma$ around $P_{2}^{\ast }$ starting from the initial condition $(k_{0},c_{0}^{2},L_{0}) = (0.346,0.452,3.178)$. Economic and demographic variables permanently fluctuate around $P_{2}^{\ast }$. Parameter set: $\alpha = 0.54$, $\beta = 1$, $\gamma = 1.5672$, $\delta = 0.7$, $\varepsilon = 0.4$, $\rho = 0.3$, $\sigma = 7.04$, $a = 0.37$, $b = 0.2$ and $x = 1.1$. Stationary state equilibria: $P_{1}^{\ast }: = (k_{1}^{\ast },c_{1}^{\ast },L_{1}^{\ast }) = (0.221,0.288,3.305)$ and $P_{2}^{\ast }: = (k_{2}^{\ast },c_{2}^{\ast },L_{2}^{\ast }) = (0.673,0.336,3.03)$

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