<blockquote>
  <p>Consider the Gale cake problem with $u(c) = log(c)$, so that the problem becomes:$$\max_{x}\sum_{t=0}^{\infty} \beta^{t}log(x_t-x_{t+1})\: sub\: 0&lt;x_{t+1}\leq x_t,\: x_0&gt;0\ given$$ </p>
</blockquote>

<p>It is pretty easy to show that the function $v:(0,x_0]\rightarrow\mathbb{R}$ given by $v(x)=A+B\:log(x)$, where $A=\frac{\beta}{(1-\beta)^2}log\beta+\frac{1}{1-\beta}log(1-\beta)$ and $B=\frac{1}{1-\beta}$ is a solution of the Bellman equation. I also know that it is the value function of the cake problem but the classic verification principle, i.e. $\lim_{t \to +\infty} \beta^tv(x_t) = 0$ is not met. </p>

<p>How should i proceed to actually show that $v(x)$ is the value function of the problem?</p>

CC BY-SA 3.0
2016-01-07T20:19:37.673Z
Perhaps you did not complete the expression for the verification principle? To what should the limit evaluate? And I guess $0<\beta <1$?
Alecos Papadopoulos
CC BY-SA 3.0
2016-01-09T06:18:22.037Z
Sorry for the delay. The esxpression above is one of the verification principle, which does not work in this case. I know there are others. $\beta \in (0,1)$, you get it.
PostDocing
CC BY-SA 3.0
2016-01-09T15:38:25.147Z
I cannot understand the expression, because  it just writes down a limit. It does not write what the value of the limit should be. As is, it is not a verification principle, it is only an incomplete mathematical expression.
Alecos Papadopoulos
CC BY-SA 3.0
2016-01-09T23:15:16.197Z
The value of the limit is not significant. It is enough to know that it is not equal to zero, i.e. the verification principle I wrote is not satisfied. Are there other ways to show that $v(x)$ is the value function of the RF problem above?
PostDocing
CC BY-SA 3.0
2016-01-09T23:25:12.923Z
So it appears that you should write in your question "$\lim_{t \to +\infty} \beta^tv(x_t) =0$ is not met". Why do you presume that readers will know that the verification principle implies that this limit should be zero?
Alecos Papadopoulos
CC BY-SA 3.0
2016-01-10T09:19:51.400Z
I thought it was pretty clear that verification principle worked like this, sorry
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<optimization><dynamic-programming>
Dynamic programming: verification principle
PostDocing
# Dynamic programming: verification principle

(Asked by: username_0 on 2016-01-07)

> Consider the Gale cake problem with $u(c) = log(c)$, so that the problem becomes:$$\max\_{x}\sum\_{t=0}^{\infty} \beta^{t}log(x\_t-x\_{t+1})\: sub\: 0<x\_{t+1}\leq x\_t,\: x\_0>0\ given$$

It is pretty easy to show that the function $v:(0,x\_0]\rightarrow\mathbb{R}$ given by $v(x)=A+B\:log(x)$, where $A=\frac{\beta}{(1-\beta)^2}log\beta+\frac{1}{1-\beta}log(1-\beta)$ and $B=\frac{1}{1-\beta}$ is a solution of the Bellman equation. I also know that it is the value function of the cake problem but the classic verification principle, i.e. $\lim\_{t \to +\infty} \beta^tv(x\_t) = 0$ is not met.

How should i proceed to actually show that $v(x)$ is the value function of the problem?

**Tags:** optimization, dynamic-programming

> **Comments:**
> **username_1**: Perhaps you did not complete the expression for the verification principle? To what should the limit evaluate? And I guess $0<\beta <1$?  
> **username_0**: Sorry for the delay. The esxpression above is one of the verification principle, which does not work in this case. I know there are others. $\beta \in (0,1)$, you get it.  
> **username_1**: I cannot understand the expression, because  it just writes down a limit. It does not write what the value of the limit should be. As is, it is not a verification principle, it is only an incomplete mathematical expression.  
> **username_0**: The value of the limit is not significant. It is enough to know that it is not equal to zero, i.e. the verification principle I wrote is not satisfied. Are there other ways to show that $v(x)$ is the value function of the RF problem above?  
> **username_1**: So it appears that you should write in your question "$\lim_{t \to +\infty} \beta^tv(x_t) =0$ is not met". Why do you presume that readers will know that the verification principle implies that this limit should be zero?  
> **username_0**: I thought it was pretty clear that verification principle worked like this, sorry  

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## 0 answers

