<p>Consider an infinitely lived agent born in time zero, endowed with a cake of size <span class="math-container">$x_0$</span>. The cake is storable (without depreciation) and infinitely divisible. The agent derives contemporary utility from (cake) consumption through <span class="math-container">$u(c_t)$</span>, and as usual, maximizes his infinitely discounted sum of utility streams. The discount rate is, as usual, given by β.</p>
<p>My guess is that the sequence problem to this is: <span class="math-container">$\sum_{i=0}^{\infty} \beta^tu(c_t)$</span> s.t <span class="math-container">$c_t + x_{t+1} ≤ x_t$</span>;</p>
<p><span class="math-container">$x_{t+1}≥0$</span> is the amount of cake left at the end of the current period ;</p>
<p><span class="math-container">$c_t≥0$</span></p>
<p>But I feel like I'm missing something in the constraints.</p>

CC BY-SA 4.0
2020-12-06T21:45:19.010Z
The constraints: (1) $c_t \geq 0$, (2) $c_t + x_{t+1} = x_t$ and (3) $x_0$ given, are all you need.
Walrasian Auctioneer
CC BY-SA 4.0
2020-12-06T21:49:16.223Z
@WalrasianAuctioneer I see, thank you for the clarification.
Maybeline Lee
<macroeconomics><dynamic-programming>
Cake eating problem
Maybeline Lee
# Cake eating problem

(Asked by: username_0 on 2020-12-06)

Consider an infinitely lived agent born in time zero, endowed with a cake of size $x\_0$. The cake is storable (without depreciation) and infinitely divisible. The agent derives contemporary utility from (cake) consumption through $u(c\_t)$, and as usual, maximizes his infinitely discounted sum of utility streams. The discount rate is, as usual, given by β.

My guess is that the sequence problem to this is: $\sum\_{i=0}^{\infty} \beta^tu(c\_t)$ s.t $c\_t + x\_{t+1} ≤ x\_t$;

$x\_{t+1}≥0$ is the amount of cake left at the end of the current period ;

$c\_t≥0$

But I feel like I'm missing something in the constraints.

**Tags:** macroeconomics, dynamic-programming
> **Comments:**
> **username_1**: The constraints: (1) $c_t \geq 0$, (2) $c_t + x_{t+1} = x_t$ and (3) $x_0$ given, are all you need.  
> **username_0**: @WalrasianAuctioneer I see, thank you for the clarification.  

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