<p>I'd say let <span class="math-container">$K_0=K$</span> and for all <span class="math-container">$t\ge 0$</span> define iteratively <span class="math-container">$K_{t+1}=K_{t}\texttt{\\}\{C_{it}\}$</span>, where <span class="math-container">$C_{it}\in\arg\max_{C\in K_t} u_i(C)$</span> and <span class="math-container">$u_i$</span> represents <span class="math-container">$≽_i$</span>.</p>

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2022-05-19T13:22:54.363Z
2022-05-19T13:22:54.363Z
VARulle
<p>As the title says, I would like to mathematically denote that a consumer behaves according to their preference structure at every time <span class="math-container">$t$</span>,  <span class="math-container">$t+1$</span>,  <span class="math-container">$t+2$</span> and so on in a <strong>finite</strong> consumption set <span class="math-container">$X$</span>, by consuming their most preferred good in each time period. Assume that it takes 1 unit of time for a consumer to consume 1 good in <span class="math-container">$X$</span>.</p>
<p><strong>This is what I have - may be wrong/incomplete:</strong></p>
<p>Let <span class="math-container">$C_{it} (K)$</span> denote the type of commodity consumer <span class="math-container">$i$</span> consumes at time <span class="math-container">$t$</span>, where <span class="math-container">$K$</span> is equal to the set of <span class="math-container">$n$</span> commodity types <span class="math-container">$[1,…,n]$</span>. Consumer <span class="math-container">$i's$</span> consumption behaviour will be such that for any pairs <span class="math-container">$j,k\in K$</span>, <span class="math-container">$C_{it} (K)=j$</span>, <span class="math-container">$C_{i,t+h} (K)=k$</span> if and only if <span class="math-container">$j≽_i k$</span>.</p>

<mathematical-economics><consumer-theory><preferences>
How to mathematically denote that a consumer behaves according to their preference structure at every point in time?
Eli J
# How to mathematically denote that a consumer behaves according to their preference structure at every point in time?

(Asked by: username_0 on 2022-05-19)

As the title says, I would like to mathematically denote that a consumer behaves according to their preference structure at every time $t$, $t+1$, $t+2$ and so on in a **finite** consumption set $X$, by consuming their most preferred good in each time period. Assume that it takes 1 unit of time for a consumer to consume 1 good in $X$.

**This is what I have - may be wrong/incomplete:**

Let $C\_{it} (K)$ denote the type of commodity consumer $i$ consumes at time $t$, where $K$ is equal to the set of $n$ commodity types $[1,…,n]$. Consumer $i's$ consumption behaviour will be such that for any pairs $j,k\in K$, $C\_{it} (K)=j$, $C\_{i,t+h} (K)=k$ if and only if $j≽\_i k$.

**Tags:** mathematical-economics, consumer-theory, preferences

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## 1 answers

### Answer by username_1 on 2022-05-19. Score: 4 | Accepted answer

I'd say let $K\_0=K$ and for all $t\ge 0$ define iteratively $K\_{t+1}=K\_{t}\texttt{\\}\{C\_{it}\}$, where $C\_{it}\in\arg\max\_{C\in K\_t} u\_i(C)$ and $u\_i$ represents $≽\_i$.


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