@@ -41,11 +41,10 @@ In what follows, we require the following imports:
4141
4242``` {code-cell} ipython
4343import matplotlib.pyplot as plt
44- plt.rcParams["figure.figsize"] = (11, 5) #set default figure size
4544import numpy as np
4645```
4746
48- ## The Model
47+ ## The model
4948
5049We consider an infinite time horizon $t=0, 1, 2, 3..$
5150
@@ -115,7 +114,7 @@ In this problem, the following terminology is standard:
115114* $c_t$ is called the ** control variable** or the ** action**
116115* $\beta$ and $\gamma$ are ** parameters**
117116
118- ### Trade-Off
117+ ### Trade-off
119118
120119The key trade-off in the cake-eating problem is this:
121120
@@ -145,14 +144,14 @@ parameters*.
145144
146145Let's see if this is true.
147146
148- ## The Value Function
147+ ## The value function
149148
150149The first step of our dynamic programming treatment is to obtain the Bellman
151150equation.
152151
153152The next step is to use it to calculate the solution.
154153
155- ### The Bellman Equation
154+ ### The Bellman equation
156155
157156To this end, we let $v(x)$ be maximum lifetime utility attainable from
158157the current time when $x$ units of cake are left.
@@ -199,7 +198,7 @@ If $c$ is chosen optimally using this trade off strategy, then we obtain maximal
199198
200199Hence, $v(x)$ equals the right hand side of {eq}` bellman-cep ` , as claimed.
201200
202- ### An Analytical Solution
201+ ### An analytical solution
203202
204203It has been shown that, with $u$ as the CRRA utility function in
205204{eq}` crra_utility ` , the function
@@ -249,7 +248,7 @@ ax.legend(fontsize=12)
249248plt.show()
250249```
251250
252- ## The Optimal Policy
251+ ## The optimal policy
253252
254253Now that we have the value function, it is straightforward to calculate the
255254optimal action at each state.
@@ -309,7 +308,7 @@ ax.legend()
309308plt.show()
310309```
311310
312- ## The Euler Equation
311+ ## The Euler equation
313312
314313In the discussion above we have provided a complete solution to the cake
315314eating problem in the case of CRRA utility.
@@ -323,7 +322,7 @@ Euler equation.
323322This is because, for more difficult problems, this equation
324323provides key insights that are hard to obtain by other methods.
325324
326- ### Statement and Implications
325+ ### Statement and implications
327326
328327The Euler equation for the present problem can be stated as
329328
@@ -376,7 +375,7 @@ see proposition 2.2 of {cite}`ma2020income`.
376375The following arguments focus on necessity, explaining why an optimal path or
377376policy should satisfy the Euler equation.
378377
379- ### Derivation I: A Perturbation Approach
378+ ### Derivation I: a perturbation approach
380379
381380Let's write $c$ as a shorthand for consumption path $\{ c_t\} _ {t=0}^\infty$.
382381
444443
445444This is just the Euler equation.
446445
447- ### Derivation II: Using the Bellman Equation
446+ ### Derivation II: using the Bellman equation
448447
449448Another way to derive the Euler equation is to use the Bellman equation {eq}` bellman-cep ` .
450449
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