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8 votes
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(Preference Relation/Set) Continuous $\succsim$ imply closedness of upper and lower contour sets

Looking more closely at your question, I think things should not be overly complicated. From Mas-Colell et.al. Definition 3.C.1: The preference relation $\succsim$ on X is continuous if it is ...
Kitsune Cavalry's user avatar
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7 votes
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Can the Certainty Equivalent be negative?

If you start out with €0, then the certainty equivalent of losing €2.5 with probability 1 is -€2.5. Your exercise basically asks you to calculate what difference winning the lottery with a small ...
Giskard's user avatar
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6 votes

a risk lover agent preferences and the preference of risk natural agent may be the same

Don't commit the cardinal mistake of equating preferences with choices. In the context of Expected Utility Theory, the fact that a risk-averse agent ($RA$) would choose $N$ over $M$ implies that $...
Alecos Papadopoulos's user avatar
6 votes
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Utility representation of single peaked preferences

No. Basically, you can encode a form of lexicographic preferences, probably the most familiar example of non-representable preferences, as single-peaked preferences on $\mathbb{R}$. Define $\succeq$ ...
Michael Greinecker's user avatar
5 votes
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Continuous rational and monotone preference relation implies $x\succsim0$?

If we take the definition of monotonicity to be if $x\geqq y$ then $x \succeq y$, you can simplify the proof (though it looks right). Note $\mathbf{0}\leq x$ for all $x\in \mathbb{R}_+^l$. So by ...
Pburg's user avatar
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5 votes

What is the point of the indirect utility function?

Recall that if $x$ and $y$ are consumption bundles, $u(x)$ is the consumer's utility function, and $u(x)>u(y)$ means the consumer strictly prefers bundle $x$ to bundle $y$. The indirect utility ...
manofbear's user avatar
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5 votes
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What is one dimensional, ordered type?

Judging from the reference you provide, this refers to whether the set $\Theta$ is ordered or not. For example, natural numbers or the alphabet are ordered sets. In the context of moral hazard ...
luchonacho's user avatar
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5 votes
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Envelope theorem for discrete choice sets?

There is an envelope theorem for the setting you describe. Have a look at “Envelope Theorems for Arbitrary Choice Sets” by Milgrom and Segal (2002).
Theoretical Economist's user avatar
5 votes

Understanding the Choice Rule in MWG

I am having trouble understanding what $C(\{x, y, z\}) = \{x, y\}$ means. MWG already explain this on p.10: When [$C(B)$ contains more than one element], the elements of $C(B)$ are the alternatives ...
Herr K.'s user avatar
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5 votes

Market with changing number of goods and services

I think the best candidate would be monopolistic competition as introduced by Dixit and Stiglitz (1977) Monopolistic Competition and Optimum Product Diversity, in which two models are introduced. ...
Jesper Hybel's user avatar
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5 votes
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Weak preferences and negative transitivity

Probably it can be done easier if you do both steps separately ($\implies$ and $\impliedby$), but here is a proof that does both at the same time: \begin{align*} &x\succ y \vee x\sim y\\ \iff\;&...
LudwigNagasena's user avatar
5 votes
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Hick's and Slutsky's approaches lead to different income effects. Why?

EMP being a dual of UMP is unrelated to income effects. If $(c_0, b_0)$ denotes the initial demand, and $(c_1, b_1)$ denotes the demand after the price of coffee has changed, then To find the ...
Amit's user avatar
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5 votes
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Choice Function and Empty Set

The choice correspondence is typically assumed to be nonempty-valued. As these notes by John Nachbar explain: [The requirement of nonemptiness of the choice correspondence] eliminates the possibility ...
Herr K.'s user avatar
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4 votes
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Boots' Theory by Pratchett

Rampini elaborates on your idea (and also the Pratchett quote) in his AER article "Financing Durable Assets". See the abstract: This paper studies how the durability of assets affects ...
Bayesian's user avatar
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4 votes

Why does Figure 2.F.1(b) (MWG page 30) satisfy the WARP (Definition 2.F.1)?

Intuitively, this just says that if the bundle you choose was possible under wealth $w$ and price $p$ but it is not $x(p, w)$, then it must be that you cannot obtain it when price is $p'$ and wealth ...
Art's user avatar
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4 votes
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Difference between social choice functions and social decision functions?

Let $X$ be the set of alternatives. A social decision function maps profiles of preference orderings to relations on $X$ such that every nonempty subset of $X$ has at least one maximum under this ...
Michael Greinecker's user avatar
4 votes

Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function

If the preference is continuous, reflexive, transitive and complete, then there exists a continuous utility representation $g$, and since $U$ also represents the same preference, so there must be a ...
Amit's user avatar
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3 votes
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a risk lover agent preferences and the preference of risk natural agent may be the same

Another way of looking at this problem is to consider the means and variances of the lotteries. A risk averse agent (RA) likes high mean and low variance A risk neutral agent (RN) likes high mean ...
Herr K.'s user avatar
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3 votes
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Vocabulary/Name for Utility of a Set of Choices

What you seem to be interested is the subfield of decision theory that goes by the name of "menu choice." The starting point of this literature is the paper Kreps, David M. "A representation theorem ...
Michael Greinecker's user avatar
3 votes
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Relation between linear utility function and U=max{x,y}

The optimal choice set for a max function and a perfect substitutes function with equal relative prices share some solutions [i.e, boundary solutions], but in general, the indifference curves, and ...
thewhitetie's user avatar
3 votes
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Can there be sensible choice behavior that violates the Weak Axiom of Revealed Preference?

I think that your example relies on a misunderstanding of the meaning of the choice function. $C(\{x,y,z\})$ does not contain the elements that the decision-maker would choose simultaneously if he ...
Oliv's user avatar
  • 3,242
3 votes
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Exact definition of one-player Bayesian Correlated Equilibrium

The concept of the BCE from their 2016 paper is similar to what you have. I think Bergemann and Morris' intuitive explanation is valuable so I'll paraphrase it here. Each player in the game has a ...
corran_horn's user avatar
3 votes

Convex rationalization when the budget sets are segments?

Take a dataset $D = (B^t, x^t)_{t \in T}$ such that for all $t$, $x^t \in B^t$. I'll say that $D$ is rationalisable by the utility function $u$ if for all $t$ and all $x \in B$: $u(x^t) \ge u(x)$. If ...
tdm's user avatar
  • 12.3k
3 votes
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Convex rationalization when the budget sets are segments?

The idea is to consider the hyperplane tangent at the indifference curve through $x^t$ as a "linear budget". These linear budgets have to include the set $\overline{y^t, z^t}$. Then making ...
tdm's user avatar
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3 votes
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Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function

I took the question as asking for a representation with $f:\mathbb{R}\to\mathbb{R}$. If we only require $f$ to be defined on a subset of $\mathbb{R}$, Amit's answer solves the problem. Here is a proof ...
Michael Greinecker's user avatar
3 votes

Convexity of preferences (dissimilar definitions)

For Q 1: Let me give you a preference relation on $\mathbb{R}^2_+$ $(x_1, y_1) \succsim (x_2, y_2)$ if and only if $(x_1^2 + y_1^2 > x_2^2 + y_2^2)$ or $(x_1^2 + y_1^2 = x_2^2 + y_2^2 \ \wedge x_1 \...
Amit's user avatar
  • 8,966
3 votes
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Proof of the tangency condition in UMP

The proof below is not very rigorous - I do not discuss discontinuities - but I think it captures the logic well. From Wikipedia: Suppose $f$ is a function of one real variable defined on an ...
Giskard's user avatar
  • 29.3k
3 votes
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State dependent preferences vs state independent preferences in utility theory

Starting from the textbook, I would highly recommend any textbook for stochastic dynamic optimization. Then I would recommend you to get acquainted with markov chains, because it is relatively good ...
Athaeneus's user avatar
  • 834
3 votes

Minimal assumption for a “certainty equivalence” exists

I take it that $u: \mathbb{R} \to \mathbb{R}$ and not $u: \mathbb{R}^N \to \mathbb{R}$ (as in the question). Otherwise $u(c)$ for $c \in \mathbb{R}$ does not make sense. tldr: if $u$ is continuous, a ...
tdm's user avatar
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