Growth Model

Macro

Growth model is deigned to be model of capital accumulation process.

A feasible allocation for the Growth Model is a list of sequence $\{c_t,h_t,k_t{\}}_{t=0}^{\infty }$ such that:

$$c_t+k_{t+1}=F(k_t,h_t)+(1-\delta )k_t$$

where,

$$0\le h_t\le 1,c_0\ge 0,k_t\ge 0,k_0=\widehat{k_0}$$

$F(k,h)$ is the Aggregate Production Function, if still hard to remember, you could see it as $Y$. $k$ is the capital input, $h$ is the labor input, $\delta$ is the depreciation rate.

In Continuous Time

In continuous time setting, we no longer use $\sum$, instead, we use $\int$.

For a representative household, its utility function would be:

$${\int }_0^{\infty }{e}^{-\rho t}u(c(t))dt$$

In a continuous case, the Pareto optimal allocation is the solution to the following problem:

$$V(\widehat{k_0})=\underset{\{c(t){\}}_{t=0}^{\infty }}{\max }{\int }_0^{\infty }{e}^{-\rho t}u(c(t))dt$$

subject to: $\dot{k}(t)=f(k(t))-\delta k(t)-c(t),k(0)=\widehat{k_0}$

This is also Law of Motion

$V(\widehat{k_0})$ is a value function, it represents the maximum lifetime utility a household could achieve under a given initial value of capital stock $\widehat{k_0}$.

In a simple optimization problem, suppose ${\max }_{x}F(x)$, we can just take a derivative and set it to zero, i.e. $F^{\prime }(x)=0$.

However, in a dynamic optimization problem, we are not maximizing over a single variable $x$, instead, we are maximizing $c(t)$, which is a function of time $t$. So we need to use the Hamiltonian method.

In short, Loci are curves that things stop changing.