import numpy as np

```

## The Model

## The model

We consider an infinite time horizon $t=0, 1, 2, 3..$

* $c_t$ is called the **control variable** or the **action**

* $\beta$ and $\gamma$ are **parameters**

### Trade-Off

### Trade-off

The key trade-off in the cake-eating problem is this:

Let's see if this is true.

## The Value Function

## The value function

The first step of our dynamic programming treatment is to obtain the Bellman

equation.

The next step is to use it to calculate the solution.

### The Bellman Equation

### The Bellman equation

To this end, we let $v(x)$ be maximum lifetime utility attainable from

the current time when $x$ units of cake are left.

Hence, $v(x)$ equals the right hand side of {eq}`bellman-cep`, as claimed.

### An Analytical Solution

### An analytical solution

It has been shown that, with $u$ as the CRRA utility function in

{eq}`crra_utility`, the function

plt.show()

```

## The Optimal Policy

## The optimal policy

Now that we have the value function, it is straightforward to calculate the

optimal action at each state.

plt.show()

```

## The Euler Equation

## The Euler equation

In the discussion above we have provided a complete solution to the cake

eating problem in the case of CRRA utility.

This is because, for more difficult problems, this equation

provides key insights that are hard to obtain by other methods.

### Statement and Implications

### Statement and implications

The Euler equation for the present problem can be stated as

The following arguments focus on necessity, explaining why an optimal path or

policy should satisfy the Euler equation.

### Derivation I: A Perturbation Approach

### Derivation I: a perturbation approach

Let's write $c$ as a shorthand for consumption path $\{c_t\}_{t=0}^\infty$.

This is just the Euler equation.

### Derivation II: Using the Bellman Equation

### Derivation II: using the Bellman equation

Another way to derive the Euler equation is to use the Bellman equation {eq}`bellman-cep`.