Combining {eq}`constraint1` and {eq}`constraint2` gives

$$

u'\left(C_t\right)\left[(1-\delta)+f'\left(K_t\right)\right]-u'\left(C_{t-1}\right)=0

\beta u'\left(C_t\right)\left[(1-\delta)+f'\left(K_t\right)\right]-u'\left(C_{t-1}\right)=0

\quad \text{ for all } t=1,2,\dots, T+1

$$

```{math}

:label: l12

u'\left(C_{t+1}\right)\left[(1-\delta)+f'\left(K_{t+1}\right)\right]=

\beta u'\left(C_{t+1}\right)\left[(1-\delta)+f'\left(K_{t+1}\right)\right]=

u'\left(C_{t}\right) \quad \text{ for all } t=0,1,\dots, T

```

$$

\begin{aligned} C_{t+1} =\left(\beta C_t^{\gamma}[f'(K_{t+1}) +

(1-\delta)]\right)^{1/\gamma} \notag\\= C_t\left(\beta [f'(K_{t+1}) +

(1-\delta)]\right)^{1/\gamma} \end{aligned}

(1-\delta)]\right)^{1/\gamma}

%\notag\\= C_t\left(\beta [f'(K_{t+1}) +

%(1-\delta)]\right)^{1/\gamma}

\end{aligned}

$$

which we can combine with the feasibility constraint {eq}`allocation` to get

$$

\begin{aligned}

C_{t+1} & = C_t\left(\beta [f'(F(K_t,1)+ (1-\delta) K_t - C_t) +

(1-\delta)]\right)^{1/\gamma} \\

K_{t+1} & = F(K_t,1)+ (1-\delta) K_t - C_t .

\end{aligned}

$$

This is a non-linear first-order difference equation that an optimal sequence $\vec C$ must satisfy.

This is a pair of non-linear first-order difference equations that map $C_t, K_t$ into $C_{t+1}, K_{t+1}$ and that an optimal sequence $\vec C , \vec K$ must satisfy.

It must also satisfy the initial condition that $K_0$ is given and $K_{T+1} = 0$.

Below we define a `jitclass` that stores parameters and functions

that define our economy.