<p>I've been playing around with a lot of cake eating problems and have been messing with how uncertainty could enter the model. One thing that I'm thinking about is whether we can solve a cake eating problem with uncertain time preferences. In this case we have uncertainty about how our individual values the future each period.</p>

<p>The advantage of this framework is that it allows for considerations for behavioral interpretation.</p>

<p>Lets define a cake eating problem sequentially as:</p>

<p><span class="math-container">$$\max_{c_t} \  U(c_t)=\mathbb{E}_0 \left[\sum_{t=0}^\infty\theta^t_t\ln(c_t)\right] $$</span></p>

<p>Subject to:</p>

<p>1.<span class="math-container">$ \ \  f(k_t)=c_t+x_t$</span> (resource constraint <span class="math-container">$c_t$</span> is consumption, <span class="math-container">$x_t$</span> is investment).</p>

<p>2.<span class="math-container">$ \ \  f(k_t)=k_t$</span> (Goods defined as dependent on cake size/capital at time <span class="math-container">$t$</span> as denoted by <span class="math-container">$k_t$</span>).</p>

<p>3.<span class="math-container">$x_t=k_{t+1}$</span> (law of motion). </p>

<p>4.<span class="math-container">$k_0&gt;0$</span> (Initial capital stock).</p>

<p>5.<span class="math-container">$\theta_t\in(0,1)$</span> and is a Random variable realized in period <span class="math-container">$t$</span> which doesn't follow an iid process.</p>

<p>Im guessing that the bellman that follows from this sequential definition is:</p>

<p><span class="math-container">$$v(k_t,\theta_t)=\max_{k_{t+1}} \left\{\ln(k_t-k_{t+1})+\mathbb{E}[\theta]\mathbb{E}[ v(k_{t+1},\theta_{t+1}] \right\}$$</span></p>

<p>I know that this is definitely a contraction mapping based on the bounds placed on <span class="math-container">$\theta$</span>.</p>

<p>However in terms of setting up this problem I'm not sure.</p>

<p>Any tips?</p>

CC BY-SA 4.0
2020-04-20T02:26:43.390Z
Just a question: have you tried simulating this model using a package like GAMS or R?
Brennan
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2020-04-20T02:30:38.053Z
@Brennan I have not. I should though my mathematical programming skills are pretty weak.
EconJohn
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2020-04-20T02:33:19.600Z
the policy functions would be the same as the regular cake eating problem in this setup. Except $\beta$ would be replaced by $\mathbb{E}[\theta]$
EconJohn
CC BY-SA 4.0
2020-04-20T10:05:17.280Z
The Bellman equation you've stated is strange. You treat $\theta_t$ as a state variable, suggesting it should be known at time $t$, but then you take expectations over it on the RHS. It's also not clear that the expectation factors that way. That is, I suspect you might have that $E[\theta v(k_{t+1},\theta_{t+1})] \neq E[\theta] E[ v(k_{t+1},\theta_{t+1})]$.
Theoretical Economist
CC BY-SA 4.0
2020-04-20T10:08:18.560Z
Also, if $\theta$ is not iid then surely your objective function should say $\theta_t^t$ instead of just $\theta^t$.
Theoretical Economist
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2020-04-20T18:36:13.087Z
This does not directly answer your question but Martin Weitzman has a whole series of papers on uncertainty in discount rates. He basically shows that uncertainty results in a declining discount rate.
Maarten Punt
CC BY-SA 4.0
2020-04-20T18:59:39.643Z
@MaartenPunt any to reccomend? Sounds like what im looking for
EconJohn
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2020-04-20T19:00:14.927Z
@TheoreticalEconomist ah would make it a mess. Though i think you're right.
EconJohn
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2020-04-20T20:00:28.650Z
@TheoreticalEconomist under your specification I think it would still be a contraction mapping. what do you think?
EconJohn
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2020-04-20T20:19:53.760Z
@EconJohn it's been a while since I've last read those but the gamma discounting paper in AER is very readable. Knowing Weitzmann he will probably use the maximum principle though instead of Bellman. Still you may learn something.
Maarten Punt
CC BY-SA 4.0
2020-04-20T23:09:19.980Z
@EconJohn That sounds right to me. What I don’t understand is why you’ve jumped straight into the infinite horizon problem. Why don’t you try the two-period version first? That’ll give you a better sense of the problem.
Theoretical Economist
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2020-04-20T23:15:30.527Z
@TheoreticalEconomist computationally less rigorous i guess. Im working on a handout of sorts on this material.
EconJohn
CC BY-SA 4.0
2020-04-22T14:15:32.937Z
@EconJohn Actually, having thought about it some more, it's not clear you have a contraction mapping. It's not even clear you can write the Bellman equation that way. However, if $\{\theta_t\}$ was a Markov process, and each $\theta_t$ were supported on $(0,1-\varepsilon)$ for some $\varepsilon > 0$, then you should get a simple Bellman equation where the Bellman operator is a contraction.
Theoretical Economist
CC BY-SA 4.0
2020-04-22T22:16:24.120Z
@TheoreticalEconomist Well the value function is a maximum value so we can treat it as a constant so really we can pass the operator through it. Im saying this because if we guess $v(k_t,\theta_t)=C_1+C_2\ln(k_t)$ then we can just pass it through. This is a valid guess as it considers the structure of our preferences along with our "technology" (here its just the cake size). so our last term should be $\mathbb{E}[\theta_{t+1}][C_1+C_2 \ln (k_{t+1})]$.
EconJohn
CC BY-SA 4.0
2020-04-22T22:35:16.837Z
@EconJohn I think you’re mistaken, unless the $\theta$’s are IID. In general the value function will depend on the entire path of the $theta$’s up to the current period. Assuming the $\theta$’s are Markov will let you drop the history aside from the most recent one. The value function may still take that form but you don’t know that your $C_1$ and $C_2$ don’t actually depend on the realisations of $\theta$.
Theoretical Economist
<bellman-equations><recursive-macroeconomics>
Eating a Cake with Uncertain Preferences
EconJohn
# Eating a Cake with Uncertain Preferences

(Asked by: username_0 on 2020-04-20)

I've been playing around with a lot of cake eating problems and have been messing with how uncertainty could enter the model. One thing that I'm thinking about is whether we can solve a cake eating problem with uncertain time preferences. In this case we have uncertainty about how our individual values the future each period.

The advantage of this framework is that it allows for considerations for behavioral interpretation.

Lets define a cake eating problem sequentially as:

$$\max\_{c\_t} \ U(c\_t)=\mathbb{E}\_0 \left[\sum\_{t=0}^\infty\theta^t\_t\ln(c\_t)\right] $$

Subject to:

1.$ \ \ f(k\_t)=c\_t+x\_t$ (resource constraint $c\_t$ is consumption, $x\_t$ is investment).

2.$ \ \ f(k\_t)=k\_t$ (Goods defined as dependent on cake size/capital at time $t$ as denoted by $k\_t$).

3.$x\_t=k\_{t+1}$ (law of motion).

4.$k\_0>0$ (Initial capital stock).

5.$\theta\_t\in(0,1)$ and is a Random variable realized in period $t$ which doesn't follow an iid process.

Im guessing that the bellman that follows from this sequential definition is:

$$v(k\_t,\theta\_t)=\max\_{k\_{t+1}} \left\{\ln(k\_t-k\_{t+1})+\mathbb{E}[\theta]\mathbb{E}[ v(k\_{t+1},\theta\_{t+1}] \right\}$$

I know that this is definitely a contraction mapping based on the bounds placed on $\theta$.

However in terms of setting up this problem I'm not sure.

Any tips?

**Tags:** bellman-equations, recursive-macroeconomics
> **Comments:**
> **username_1**: Just a question: have you tried simulating this model using a package like GAMS or R?  
> **username_0**: @Brennan I have not. I should though my mathematical programming skills are pretty weak.  
> **username_0**: the policy functions would be the same as the regular cake eating problem in this setup. Except $\beta$ would be replaced by $\mathbb{E}[\theta]$  
> **username_2**: The Bellman equation you've stated is strange. You treat $\theta_t$ as a state variable, suggesting it should be known at time $t$, but then you take expectations over it on the RHS. It's also not clear that the expectation factors that way. That is, I suspect you might have that $E[\theta v(k_{t+1},\theta_{t+1})] \neq E[\theta] E[ v(k_{t+1},\theta_{t+1})]$.  
> **username_2**: Also, if $\theta$ is not iid then surely your objective function should say $\theta_t^t$ instead of just $\theta^t$.  
> **username_3**: This does not directly answer your question but Martin Weitzman has a whole series of papers on uncertainty in discount rates. He basically shows that uncertainty results in a declining discount rate.  
> **username_0**: @MaartenPunt any to reccomend? Sounds like what im looking for  
> **username_0**: @TheoreticalEconomist ah would make it a mess. Though i think you're right.  
> **username_0**: @TheoreticalEconomist under your specification I think it would still be a contraction mapping. what do you think?  
> **username_3**: @EconJohn it's been a while since I've last read those but the gamma discounting paper in AER is very readable. Knowing Weitzmann he will probably use the maximum principle though instead of Bellman. Still you may learn something.  
> **username_2**: @EconJohn That sounds right to me. What I don’t understand is why you’ve jumped straight into the infinite horizon problem. Why don’t you try the two-period version first? That’ll give you a better sense of the problem.  
> **username_0**: @TheoreticalEconomist computationally less rigorous i guess. Im working on a handout of sorts on this material.  
> **username_2**: @EconJohn Actually, having thought about it some more, it's not clear you have a contraction mapping. It's not even clear you can write the Bellman equation that way. However, if $\{\theta_t\}$ was a Markov process, and each $\theta_t$ were supported on $(0,1-\varepsilon)$ for some $\varepsilon > 0$, then you should get a simple Bellman equation where the Bellman operator is a contraction.  
> **username_0**: @TheoreticalEconomist Well the value function is a maximum value so we can treat it as a constant so really we can pass the operator through it. Im saying this because if we guess $v(k_t,\theta_t)=C_1+C_2\ln(k_t)$ then we can just pass it through. This is a valid guess as it considers the structure of our preferences along with our "technology" (here its just the cake size). so our last term should be $\mathbb{E}[\theta_{t+1}][C_1+C_2 \ln (k_{t+1})]$.  
> **username_2**: @EconJohn I think you’re mistaken, unless the $\theta$’s are IID. In general the value function will depend on the entire path of the $theta$’s up to the current period. Assuming the $\theta$’s are Markov will let you drop the history aside from the most recent one. The value function may still take that form but you don’t know that your $C_1$ and $C_2$ don’t actually depend on the realisations of $\theta$.  

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## 0 answers

