$\begingroup$

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?