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3 votes
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How to mathematically denote that a consumer behaves according to their preference structure at every point in time?

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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6 votes
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Does Debreu's representation theorem of ordinal utility require Hausdorff topology?

No. However, the problem can be reduced to representing preferences on a Hausdorff space. Instead of trying to represent a complete preorder on a set, one can try to represent linear orders on the ...
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2 votes
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Preference notation by agent $i$

For all pairs $x,x' \in X$ we have $$ x \preceq_1 x' \Longleftrightarrow x \preceq_2 x' $$ and $$ x \preceq_1 x' \Longleftrightarrow x \succeq_3 x'. $$
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2 votes

Convex Preference but Convex Utility

Example given by Herr K. is perfect. Let me give another example of a dis-continuous utility function which is quasi-concave, but not concave. Consider $u:\mathbb{R}^2_+ \rightarrow \mathbb{R}$ ...
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2 votes

Deriving indifference curves

Another way to represent the preference is: \begin{eqnarray*} u(x_1, x_2, x_3) & = & x_1 + x_2 + x_3 - \min(x_1, x_2, x_3) \\ & = & \max(x_2+x_3, x_1+x_3, x_1+x_2)\end{eqnarray*} You ...
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0 votes

Does continuous preference imply upper-hemi continuous demand correspondence?

I think I find the solution: this short paper includes a generalized version of Berge's Theorem https://www.jstor.org/stable/2526431?seq=1 The binary relation is used instead of utility. The binary ...
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2 votes

Demand correspondence is both upper and lower hemi-continuous; is the preference continuous?

If the commodity space is $\mathbb{R}^2_+$ and the preference is Lexicographic, then with the standard budget sets $B=B(p_X, p_Y, M) = \{(x, y) \in \mathbb{R}^2 | p_Xx + p_Yy \leq M\}$ where $(p_X, ...
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