# Eating a Cake with Uncertain Preferences

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?

• Just a question: have you tried simulating this model using a package like GAMS or R? – Brennan Apr 20 at 2:26
• @Brennan I have not. I should though my mathematical programming skills are pretty weak. – EconJohn Apr 20 at 2:30
• 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 Apr 20 at 2:33
• 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 Apr 20 at 10:05
• Also, if $\theta$ is not iid then surely your objective function should say $\theta_t^t$ instead of just $\theta^t$. – Theoretical Economist Apr 20 at 10:08