The Cake Eating Problem with Depreciation (Modelling difficulties)

Question

How does one go about modelling the cake eating problem with depreciation? (i.e The cake goes bad over time)

The problem I have is the following.

Lets define a cake eating problem sequentially as:

$$\max_{c_t} \ U(c_t)=\sum_{t=0}^\infty\beta^t\ln(c_t) $$

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.$k_{t+1}=(1-\delta)k_t+x_t$ (law of motion).

4.$k_0>0$ (Initial capital stock).

when dealing with the case where $\delta=1$ the problem is fairly straight forward to solve recursively with the bellman equation of: $$v(k_t)=\max_{k_{t+1}} \left\{\ln(k_t-k_{t+1})+\beta v(k_{t+1}) \right\}$$

However If we were to consider the case of where "the cake goes bad" over time (meaning there is a cost to saving) it seems that modifying the standard framework would be necessary.

This is because if we allow for $\delta\neq0$ we end up with a result of "re-eating" of previously consumed cake. How do we go about addressing this problem?

Answer 1

It would seem that the way you've formulated your production function/law of motion has introduced double counting into the problem. Note that substituting 1 and 2 into 3 gives: $$k_{t+1}=(1-\delta)(c_t+x_t)+x_t$$ Where investment in period t is counted twice.

The correct law of motion is simply:

$$ 3. \: k_{t+1}=(1-\delta)x_t $$

And the general form of the Bellman equation would be: $$v(k_t)=\max_{k_{t+1}}\left\{\ln\left(k_t-\frac{k_{t+1}}{1-\delta}\right)+\beta v\left(\frac{k_{t+1}}{1-\delta}\right)\right\}$$

Hope this helps!