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[cake_eating_problem] Correct section heading capitalizations per QuantEcon style guide #570

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[cake_eating_problem] Correct section heading capitalizations per QuantEcon style guide #570

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20 changes: 10 additions & 10 deletions lectures/cake_eating_problem.md
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Expand Up @@ -45,7 +45,7 @@ plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
import numpy as np
```

## The Model
## The model

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

Expand Down Expand Up @@ -115,7 +115,7 @@ In this problem, the following terminology is standard:
* $c_t$ 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:

Expand Down Expand Up @@ -145,14 +145,14 @@ parameters*.

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.
Expand Down Expand Up @@ -199,7 +199,7 @@ If $c$ is chosen optimally using this trade off strategy, then we obtain maximal

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
Expand Down Expand Up @@ -249,7 +249,7 @@ ax.legend(fontsize=12)
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.
Expand Down Expand Up @@ -309,7 +309,7 @@ ax.legend()
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.
Expand All @@ -323,7 +323,7 @@ Euler equation.
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

Expand Down Expand Up @@ -376,7 +376,7 @@ see proposition 2.2 of {cite}`ma2020income`.
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 $\{c_t\}_{t=0}^\infty$.

Expand Down Expand Up @@ -444,7 +444,7 @@ $$

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`.

Expand Down
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