<p>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:
<span class="math-container">$$k_{t+1}=(1-\delta)(c_t+x_t)+x_t$$</span>
Where investment in period t is counted twice.</p>

<p>The correct law of motion is simply:</p>

<p><span class="math-container">$$ 3. \: k_{t+1}=(1-\delta)x_t $$</span></p>

<p>And the general form of the Bellman equation would be:
<span class="math-container">$$v(k_t)=\max_{k_{t+1}}\left\{\ln\left(k_t-\frac{k_{t+1}}{1-\delta}\right)+\beta v\left(\frac{k_{t+1}}{1-\delta}\right)\right\}$$</span></p>

<p>Hope this helps!</p>

CC BY-SA 4.0
2020-01-09T19:33:41.387Z
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.
H Rogers
CC BY-SA 4.0
2020-01-06T17:41:23.407Z
2020-01-08T00:44:10.127Z
2020-01-08T00:44:10.127Z
H Rogers
<p>How does one go about modelling the cake eating problem with depreciation? (i.e The cake goes bad over time)</p>

<p>The problem I have is the following. </p>

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

<p><span class="math-container">$$\max_{c_t} \  U(c_t)=\sum_{t=0}^\infty\beta^t\ln(c_t) $$</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">$k_{t+1}=(1-\delta)k_t+x_t$</span> (law of motion). </p>

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

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

<p>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.</p>

<p>This is because if we allow for <span class="math-container">$\delta\neq0$</span> we end up with a result of "re-eating" of previously consumed cake. How do we go about addressing this problem?</p>

<dynamic-programming><recursive-macroeconomics>
The Cake Eating Problem with Depreciation (Modelling difficulties)
EconJohn
# The Cake Eating Problem with Depreciation (Modelling difficulties)

(Asked by: username_0 on 2020-01-06)

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?

**Tags:** dynamic-programming, recursive-macroeconomics

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## 1 answers

### Answer by username_1 on 2020-01-06. Score: 3 | Accepted answer

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!

> **Comments:**
> **username_1**: 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.  

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