<p>Suppose we wish to solve a simple infinite horizon cake eating problem,
such that:
<span class="math-container">$$
\max_{\left\{ c_{t}\right\} _{t=0}^{\infty}}\sum_{t}^{\infty}\beta^{t}u\left(c_{t}\right)
$$</span>
subject to:
<span class="math-container">$$
c_{t}+a_{t+1}=a_{t}
$$</span>
where <span class="math-container">$a_{t}$</span> is the total amount of the cake left over for tomorrow.
We can use Lagrangian techniques, obtain the Euler equations, and
pin down the optimal levels of <span class="math-container">$c^{*}.$</span> Alternatively, we can also
write this as a Bellman equation as:
<span class="math-container">$$
V\left(a\right)=\max_{a'}\left\{ u\left(a-a'\right)+\beta V\left(a'\right)\right\} 
$$</span>
and then either solve it numerically (or even obtain an analytical
solution in this case, by guess and verify and matching coefficients).
My question is this: what are the advantages of using the Bellman
approach? The textbook answer is: it helps breaks down an infinite
period problem into sequences of two period problems. But, it is still
not obvious to me how this is in fact helpful, preferable over simple
Lagrangian based optimality conditions. What am I missing?</p>

CC BY-SA 4.0
2022-08-26T14:04:45.480Z
Hi @Kwame. I also found myself unsatisfied with the textbook answer when I was studying these issues. A more complete story accounts for the fact that breaking the model down into stages allows it so be solved computationally. An infinite-horizon Lagrangian by comparison does not share the same benefits.
EB3112
CC BY-SA 4.0
2022-08-26T19:52:52.053Z
Does this answer your question? [Dynamic optimisation](https://economics.stackexchange.com/questions/14275/dynamic-optimisation)
Alecos Papadopoulos
<dynamic-programming>
Advantages of using Bellman equations
Kwame Brown
# Advantages of using Bellman equations

(Asked by: username_0 on 2022-08-26)

Suppose we wish to solve a simple infinite horizon cake eating problem,
such that:
$$
\max\_{\left\{ c\_{t}\right\} \_{t=0}^{\infty}}\sum\_{t}^{\infty}\beta^{t}u\left(c\_{t}\right)
$$
subject to:
$$
c\_{t}+a\_{t+1}=a\_{t}
$$
where $a\_{t}$ is the total amount of the cake left over for tomorrow.
We can use Lagrangian techniques, obtain the Euler equations, and
pin down the optimal levels of $c^{\*}.$ Alternatively, we can also
write this as a Bellman equation as:
$$
V\left(a\right)=\max\_{a'}\left\{ u\left(a-a'\right)+\beta V\left(a'\right)\right\}
$$
and then either solve it numerically (or even obtain an analytical
solution in this case, by guess and verify and matching coefficients).
My question is this: what are the advantages of using the Bellman
approach? The textbook answer is: it helps breaks down an infinite
period problem into sequences of two period problems. But, it is still
not obvious to me how this is in fact helpful, preferable over simple
Lagrangian based optimality conditions. What am I missing?

**Tags:** dynamic-programming

> **Comments:**
> **username_1**: Hi @username_0. I also found myself unsatisfied with the textbook answer when I was studying these issues. A more complete story accounts for the fact that breaking the model down into stages allows it so be solved computationally. An infinite-horizon Lagrangian by comparison does not share the same benefits.  
> **username_2**: Does this answer your question? [Dynamic optimisation](https://economics.stackexchange.com/questions/14275/dynamic-optimisation)  

---

## 0 answers

