## Section: New Results

### Economic growth models

Participants : Jacques Lévy Véhel, Lining Liu.

*In collaboration with D. La Torre, University of Milan.*

We study certain economic growth models where we add a source of randomness to make the evolution equations more realistic. We have studied two particular models:

An augmented Uzawa-Lucas growth model where technological progress is modelled as the solution of a stochastic differential equation driven by a Lévy or an additive process. This allows for a more faithful description of reality by taking into account discontinuities in the evolution of the level of technology. In details, we consider a closed economy in which there is single good which is produced by combining physical capital $K\left(t\right)$ and human capital $H\left(t\right)$. The laws of motions of $K\left(t\right)$ and $H\left(t\right)$ are:

$\dot{K}\left(t\right)=A{\left(t\right)}^{\gamma}{\left[u\left(t\right)H\left(t\right)\right]}^{\xi}K{\left(t\right)}^{1-\xi -\gamma}-{\beta}_{K}K\left(t\right)-C\left(t\right),$ (17) $K\left(0\right)={K}_{0};$$H\left(0\right)={H}_{0},$where $A\left(t\right)$ is the level of technology, $H\left(t\right)$ is the total stock of human capital, $u\left(t\right)$ is the proportion to the production of good, $\gamma \in (0,1)$, $\xi \in (0,1)$ and $1-\xi -\gamma \in (0,1)$ are the shares of income accruing to $A\left(t\right)$, $u\left(t\right)H\left(t\right)$ and $K\left(t\right)$, respectively, ${\beta}_{K}\in [0,1]$ is the constant rate of depreciation of physical capital, ${\beta}_{H}\in [0,1]$ is the rate of depreciation of human capital and $\eta \ge 0$ is the productivity of human capital.

We assume that the level of technology evolves according to the following stochastic differential equation:

$dA\left(t\right)=\mu A\left(t\right)dt+\sigma A\left(t\right)dW\left(t\right)+\delta \int A\left({t}^{-}\right)z(\tilde{N}(dt,dz)-\nu (dt,dz)),$ (19) where $\mu \in \mathbb{R}$ is the drift rate, $\sigma >0$ is the volatility, $0\le \delta \le 1$, $W$ is a standard Brownian motion and $\tilde{N}$ is Poisson random measure with intensity measure $\nu $ which satisfies

$\underset{s\to {t}^{+}}{lim}\frac{1}{s-t}{\int}_{t}^{s}{\int}_{-1}^{1}{z}^{2}\nu (dz,dx)+\underset{s\to {t}^{+}}{lim}\frac{1}{s-t}{\int}_{t}^{s}{\int}_{1}^{\infty}z\nu (dz,dx)<\infty ,$${\int}_{0}^{t}{\int}_{-1}^{1}{z}^{2}\nu (dz,dx)+{\int}_{0}^{t}{\int}_{1}^{\infty}z\nu (dz,dx)<\infty ,$With a CIES utility function, the optional inter-temporal decision problem can be formulated as

$\underset{[C,u]}{max}\mathbb{E}\left[{\int}_{0}^{\infty}\frac{C{\left(t\right)}^{1-\phi}-1}{1-\phi}{e}^{-\rho t}dt\right],$ (20) where $\rho >0$ is the rate of time preference and $\phi >0$. We denote $V(H,K,A)$ the maximum value function associated with the stochastic optimisation problem. For given $t$, the maximum expect utility up to time $t$ obtained when applying the stochastic control $\left[C\right(t),u(t\left)\right]$ is defined by

$V(H\left(t\right),K\left(t\right),A\left(t\right))=\underset{[C,u]}{max}\mathbb{E}\left[{\int}_{0}^{t}\frac{C{\left(x\right)}^{1-\phi}-1}{1-\phi}{e}^{-\rho x}dx\right].$ (21) We have been able to solve this program under some simplifying assumptions. Numerical simulations allow one to assess precisely the effect of (tempered) multistable noise on the model.

A stochastic demographic jump shocks in a multi-sector growth model with physical and human capital accumulation. This models allows one to take into account sudden changes in population size, due for instance to wars or natural catastrophes. The laws of motions of physical capital $K\left(t\right)$ and human capital $H\left(t\right)$ are:

$\dot{K}\left(t\right)=AM{\left(t\right)}^{1-\xi -\beta}{\left[u\left(t\right)H\left(t\right)\right]}^{\beta}K{\left(t\right)}^{\xi}-{\eta}_{K}K\left(t\right)-c\left(t\right)M\left(t\right),$ (22) with initial conditions $K\left(0\right)={K}_{0}$ and $H\left(0\right)={H}_{0}$, where $M\left(t\right)$ is the population size, $H\left(t\right)$ is the human capital, $u\left(t\right)$ is the share of human capital employed in production, $\beta \in (0,1)$, $\xi \in (0,1)$ and $1-\xi -\beta \in (0,1)$ are the shares accruing to $M\left(t\right)$, $u\left(t\right)H\left(t\right)$ and $K\left(t\right)$, respectively, ${\eta}_{K}\in [0,1]$ is the constant rate of depreciation of physical capital, ${\eta}_{H}\in [0,1]$ is the rate of depreciation of human capital and $A\ge 0$, $B\ge 0$ are the productivities of physical capital and human capital.

We assume that the population size evolves according to the following stochastic differential equation:

$dM\left(t\right)=\mu M\left(t\right)dt+\sigma M\left(t\right)dW\left(t\right)+\delta \int M\left({t}^{-}\right)z(\tilde{N}(dt,dz)-\nu (dt,dz))$with initial condition $M\left(0\right)={M}_{0}$, where $\mu \in \mathbb{R}$ is the drift rate, $\sigma >0$ is the volatility, $0\le \delta \le 1$, $W$ is a standard Brownian motion and $\tilde{N}$ is Poisson random measure with intensity measure $\nu (dt,dz)$.

Here again, we are able to solve an optimisation program under some simplifying assumptions. This sheds light on the effect of demographic shocks on macroeconomic growth.