<p>Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:</p>
<p><strong>Assumption 1:</strong> <span class="math-container">$\Gamma(x)$</span> (our set of feasible values) is non-empty for all <span class="math-container">$x\in X$</span>.</p>
<p><strong>Assumption 2:</strong><span class="math-container">$\lim_{t\rightarrow\infty}\sum_{t=0}^\infty \beta^t F(x_t,x_{t+1}) $</span> exists for all <span class="math-container">$\tilde{x}\in \Pi(x_0)$</span> (<span class="math-container">$\Pi(x_0)$</span> being the correspondence of <span class="math-container">$x$</span> considering an initial <span class="math-container">$x_0$</span>)</p>
<p><strong>Assumption 3:</strong><span class="math-container">$|V^*(x)&lt;\infty|$</span> for all <span class="math-container">$x \in X$</span>.</p>
<p><strong>Assumption 4:</strong> For any <span class="math-container">$x_0$</span>, there exists a plan <span class="math-container">$\tilde{x}\in \Pi(x_0)$</span> such that <span class="math-container">$u(\tilde{x})=V^*(x_0)$</span>.</p>
<p>In your story here since <span class="math-container">$\beta&lt;0$</span> we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian:
<span class="math-container">$$\mathcal{L}=\sum_{t=0}^\infty\{\beta^tu(c_t)+\sum_i\lambda_{t,i}(F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}\}$$</span></p>
<p>Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.</p>
<p>Hope this helps</p>
<p>Edit: After looking at your addendum im guessing the proceedure would be the same</p>
<p><span class="math-container">$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]\}+\max_{c_{t+1},...,c_{T}}\{\sum_{t+1} \ [\beta^{t+1}u(c_{t+1})+\sum_i\lambda_{it}[F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]]\} $$</span></p>
<p>or
<span class="math-container">$$V_t(k_t)=max_{c_t}\{u(c_t)+\lambda [F(K_{t+1},E_t-E_{i,t+1})-E_{j,t+1}]+V(k_{t+1})\}$$</span></p>

CC BY-SA 4.0
2020-11-05T19:06:09.650Z
Hi John, really appreciate your comments above. Really appreciate your time. I published an edited version of my question to show what I mean in terms of the problem with an implicit constraint. Would love to see your feedback.
EB3112
CC BY-SA 4.0
2020-11-06T19:14:12.673Z
@AndrewCoyle Check my edit. Some of the notation may be off but the idea is still there.
EconJohn
CC BY-SA 4.0
2020-11-07T11:06:56.483Z
Thanks John, I appreciate your time again. Helps a lot.  In terms of notation Would the subscripts under the max operator be reduced to consumption alone, as opposed to the other variables?
EB3112
CC BY-SA 4.0
2020-11-07T23:37:49.787Z
@AndrewCoyle it would probably make more sense to have your choice variables be $E_{t+1}$ and $K_{t+1}$ just based on the problems I've seen.
EconJohn
CC BY-SA 4.0
2020-11-10T23:54:13.307Z
@EB3112 if the excellent answer here happened to answer your Q consider accepting it
1muflon1
CC BY-SA 4.0
2020-11-05T18:33:40.753Z
2020-11-06T19:12:40.100Z
2020-11-06T19:12:40.100Z
EconJohn
<p>Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality?</p>
<ol>
<li><span class="math-container">$$\ max \Sigma\beta^s U(C_t)$$</span></li>
</ol>
<p>Subject to the following resource constraint:</p>
<ol start="2">
<li><span class="math-container">$$C_t + K_{t+1}= F(K_F, E_F,S_t)$$</span></li>
</ol>
<p>Where:
<span class="math-container">$$E_t = F_E(K_E,E_2,R_t, R_{t+1})$$</span></p>
<p><span class="math-container">$$S_t = \Sigma(1-d)E_t$$</span></p>
<p><span class="math-container">$$R_{t+1}=R_t -E_t$$</span></p>
<p>And the Bellman equation is given:</p>
<ol start="3">
<li><span class="math-container">$$V_t (K_t,R_t,E_t)= max_{C_1,K_2,E_1,E_t} {U(C_t)+\beta V_{t+1}(K_{t+1},R_{t+1},E_{t+1}) + \lambda_2 (F_2 (K_2,E_t-E_1)-E_t)}$$</span></li>
</ol>
<p>This problem is taken from eq. (27) and eq. (39) of the following textbook: <a href="https://are.berkeley.edu/%7Etraeger/pdf/KarpTraegerDraft.pdf" rel="nofollow noreferrer">https://are.berkeley.edu/~traeger/pdf/KarpTraegerDraft.pdf</a>.</p>
<p>The specified parameters in the model are:</p>
<ul>
<li>Consumption <span class="math-container">$C_t$</span></li>
<li>Capital <span class="math-container">$K_t$</span></li>
<li>The energy sector <span class="math-container">$E_t$</span></li>
<li>The carbon stock <span class="math-container">$S_t$</span></li>
<li>The finite resource <span class="math-container">$R_t$</span></li>
</ul>
<p>Would genuinely appreciate being able to see how this is conducted on this type of model.</p>

<mathematical-economics><dynamic-programming>
The Principle of Optimality and the Bellman Equation
EB3112
# The Principle of Optimality and the Bellman Equation

(Asked by: username_0 on 2020-11-05)

Given (1) and (2), is it possible to show the existence of a Bellman equation (3), using Bellman's Principle of Optimality?

1. $$\ max \Sigma\beta^s U(C\_t)$$

Subject to the following resource constraint:

2. $$C\_t + K\_{t+1}= F(K\_F, E\_F,S\_t)$$

Where:
$$E\_t = F\_E(K\_E,E\_2,R\_t, R\_{t+1})$$

$$S\_t = \Sigma(1-d)E\_t$$

$$R\_{t+1}=R\_t -E\_t$$

And the Bellman equation is given:

3. $$V\_t (K\_t,R\_t,E\_t)= max\_{C\_1,K\_2,E\_1,E\_t} {U(C\_t)+\beta V\_{t+1}(K\_{t+1},R\_{t+1},E\_{t+1}) + \lambda\_2 (F\_2 (K\_2,E\_t-E\_1)-E\_t)}$$

This problem is taken from eq. (27) and eq. (39) of the following textbook: [https://are.berkeley.edu/~traeger/pdf/KarpTraegerDraft.pdf](https://are.berkeley.edu/%7Etraeger/pdf/KarpTraegerDraft.pdf).

The specified parameters in the model are:

* Consumption $C\_t$
* Capital $K\_t$
* The energy sector $E\_t$
* The carbon stock $S\_t$
* The finite resource $R\_t$

Would genuinely appreciate being able to see how this is conducted on this type of model.

**Tags:** mathematical-economics, dynamic-programming


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## 1 answers

### Answer by username_1 on 2020-11-05. Score: 4

Recall that the Principle of Optimality states that the solution to Our Bellman Functional Equation is the same as the solution to the sequential problem if:

**Assumption 1:** $\Gamma(x)$ (our set of feasible values) is non-empty for all $x\in X$.

**Assumption 2:**$\lim\_{t\rightarrow\infty}\sum\_{t=0}^\infty \beta^t F(x\_t,x\_{t+1}) $ exists for all $\tilde{x}\in \Pi(x\_0)$ ($\Pi(x\_0)$ being the correspondence of $x$ considering an initial $x\_0$)

**Assumption 3:**$|V^\*(x)<\infty|$ for all $x \in X$.

**Assumption 4:** For any $x\_0$, there exists a plan $\tilde{x}\in \Pi(x\_0)$ such that $u(\tilde{x})=V^\*(x\_0)$.

In your story here since $\beta<0$ we know that this is bounded all we are doing is adding an additional constraint. This is equivalent to writing the legrangian:
$$\mathcal{L}=\sum\_{t=0}^\infty\{\beta^tu(c\_t)+\sum\_i\lambda\_{t,i}(F(K\_{t+1},E\_t-E\_{i,t+1})-E\_{j,t+1}\}$$

Again I may not be iterating the whole story because there are two sectors here but this is how I'd go about thinking about if the principle of optimality is satisfied.

Hope this helps

Edit: After looking at your addendum im guessing the proceedure would be the same

$$V\_t(k\_t)=max\_{c\_t}\{u(c\_t)+\lambda [F(K\_{t+1},E\_t-E\_{i,t+1})-E\_{j,t+1}]\}+\max\_{c\_{t+1},...,c\_{T}}\{\sum\_{t+1} \ [\beta^{t+1}u(c\_{t+1})+\sum\_i\lambda\_{it}[F(K\_{t+1},E\_t-E\_{i,t+1})-E\_{j,t+1}]]\} $$

or
$$V\_t(k\_t)=max\_{c\_t}\{u(c\_t)+\lambda [F(K\_{t+1},E\_t-E\_{i,t+1})-E\_{j,t+1}]+V(k\_{t+1})\}$$

> **Comments:**
> **username_0**: Hi John, really appreciate your comments above. Really appreciate your time. I published an edited version of my question to show what I mean in terms of the problem with an implicit constraint. Would love to see your feedback.  
> **username_1**: @AndrewCoyle Check my edit. Some of the notation may be off but the idea is still there.  
> **username_0**: Thanks John, I appreciate your time again. Helps a lot.  In terms of notation Would the subscripts under the max operator be reduced to consumption alone, as opposed to the other variables?  
> **username_1**: @AndrewCoyle it would probably make more sense to have your choice variables be $E_{t+1}$ and $K_{t+1}$ just based on the problems I've seen.  
> **username_2**: @username_0 if the excellent answer here happened to answer your Q consider accepting it  

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