---
tags: macro, memo,public
---

# Optimal Growth Model

## The Question

Maximize the lifetime utility with utility function $u$, production function $f$, and depreciation rate $\delta$. Given $k_0$, the problem is,

\begin{gather*}
\max_{\{c_t, k_t\}_{t=0}^\infty} \sum_{t=0}^\infty \beta^t u(c_t)\\
\text{s.t. }c_t +k_{t+1} = f(k_{t}) + (1 - \delta) k_{t}
\end{gather*}

## The Solution
### Euler equation

Substitute the budget constraint into the objective function, we have

\begin{gather*}
\max_{\{k_t \}_{t=1}^\infty} \sum_{t=0}^\infty \beta^t u \left (f(k_{t}) + (1 - \delta) k_{t} - k_{t+1} \right )\\
\end{gather*}

The first-order condition of $k_t$ is

$$
\beta^t u' \left (f(k_{t}) + (1 - \delta) k_{t} - k_{t+1} \right ) \left ( f'(k_t) + 1 -\delta \right) = \beta^{t-1} u' \left (f(k_{t-1}) + (1 - \delta) k_{t-1} - k_{t} \right )
$$

Simplify this Euler equation. We get a familiar form,

$$
\beta u' \left (c_{t} \right ) \left ( f'(k_t) + 1 -\delta \right) = u' \left (c_{t-1} \right ).
$$

### Steady state

Under steady state, $c_t = c_{t+1}=c$, $k_t = k_{t+1}=k$, and $u'(c_t) =u'(c_{t+1})$.

$$
\beta \left ( f'(k) + 1 -\delta \right) = 1.
$$

We still have the resource constraint,

$$
c = f(k) - \delta k.
$$

### Phase Diagram

We can plot the above two equations on the plane of $k, u'(c)$.

For the Euler equation, assuming $f'(k)>0, f''(k)<0$, there is an unique $k$ given any pair of $(\beta, \delta)$. However, this equation is irrelevant to $c$, so this equation corresponds to a vertical line in the plot.

For the resource constraint, assuming $u'(c)>0$, the line should be a negative slope when $c$ increase, that is, $f'(k) - \delta >0$. With the assumption of the production function, $f'(k) - \delta <0$ when $k$ is large enough. However, the rational agent will not accumulate that amount of capital.

Hence, the resource constraint corresponds to a negative line with a small $k$ in the graph.

In the below diagram, the black curve corresponds to the resource constraint, and the two blue lines correspond to the Euler equation with different values of $\beta$.

Since $f''(k)<0$, the blue line will move rightward when $\beta$ increases.


#### Directions

Given any pair $(k_t, u'(c_t))$, by the Euler equation,

\begin{align}
u'(c_{t+1}) > u'(c_t) &\iff \beta u'(c_{t+1}) > \beta u'(c_t) \\
&\iff \frac{u'(c_t)}{( f'(k_{t+1}) + 1 -\delta )} > \beta u'(c_t) \\
& \iff \frac{1}{( f'(k_{t+1}) + 1 -\delta )} > \beta \\
& \iff 1 > \beta ( f'(k_{t+1}) + 1 -\delta ). \\
\end{align}

Let $k^{SS}$ be the capital at steady-state. We know
$$
k > k^{SS} \iff f'(k) < f'(k^{SS}) \iff 1 > \beta ( f'(k) + 1 -\delta ).
$$

Hence, when $k > k^{SS}$, $u'(c)$ will increase, vice versa.

On the other hand, $k_{t+1}$ will increase if $c_t < f(k_{t}) - \delta k_t$. Since $u''<0$, for the pair $(k_t, u'(c_t))$ above the ARC, $k_{t+1}$ will increse, vice versa.