# The Cake Eating Problem with Depreciation (Modelling difficulties)

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?

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.
$$3. \: k_{t+1}=(1-\delta)x_t$$
And the general form of the Bellman equation would be: $$v(k_t)=\max_{k_{t+1}}\left\{\ln\left(k_t-\frac{k_{t+1}}{1-\delta}\right)+\beta v\left(\frac{k_{t+1}}{1-\delta}\right)\right\}$$
• After examining the topic of dynamic programming more in depth, I'm convinced that the argument of the second part of the Bellman equation should be $k_{t+1}$, as this is the amount of cake that the agent has to consume/save in the following time period. Dividing by $1-\delta$ assumes that depreciation is not a factor. Jan 9 '20 at 19:33