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Questions tagged [preferences]

Binary relations that reflect which states of the world an agent considers to be most desirable. Preferences are a fundamental ingredient in the axiomatic study of consumer choice decision theory.

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0 answers
25 views

Reconciling Continuity of Binary/Preference Relations with Continuity of Functions/Correspondences

There are various ways to express the concept of continuity of a binary relation, but one I've come across seems to imply the closed-graph property is sufficient. That is to say: For a nonempty set $X$...
2 votes
1 answer
105 views

Conflicting Definitions of Weak Monotnocity (preferences)

Strong Montonicity my sources seem to agree on Strong monotonicity, i state equivalent definitions below. But weak montonicity i keep finding what appear to be conflicting definitions. In the ...
2 votes
1 answer
56 views

Equivalence of two definitions of monotone preference

In MWG, the definition of weak preference is for all $x,y \in X$, $y>>x$ implies $y\succ x$ . But I have read some other articles that define weak preference as $y\geq x\implies y\succeq x$. ...
1 vote
1 answer
79 views

What are the most common axioms to define a strict preference relation?

I am wondering which are the standard axioms with which indifference and (particularly) strict preference are usually defined. Normally, a weak preference relation for agent $i$ on a set $X$, denoted $...
3 votes
1 answer
71 views

Proof: Let $\epsilon>0$ and $x'\in\mathbb{R}^L_+$ be such that $\|x'-x\|\geq\epsilon$. Then $\alpha(x')$ belongs to some $[\alpha_0,\alpha_1]$

See Proposition 3.C.1 from MWG Continue from this post, the book (MWG) then started the proof that $\alpha(x)$ is a continuous function: We now argue that $\alpha(x)$ is a continuous function at all $...
6 votes
1 answer
63 views

Question About Proving $\alpha(\cdot)$ is Continuous in the Proof of Proposition 3.C.1 from MWG

See Proposition 3.C.1 from MWG Continue from this post, Microeconomic Theory by Mas-Colell et al. book said the following: What remains is to establish that all convergent subsequences of $\{\alpha(x^...
5 votes
1 answer
72 views

Question About the Step Proving $\alpha(x)$ Represents Preferences in the Proof of Proposition 3.C.1 from MWG

Continue from this question, the book Microeconomic Theory by Mas-Colell et al. said We now take $\alpha(x)$ as our utility function; that is, we assign a utility value $u(x)=\alpha(x)$ to every $x$. ...
2 votes
0 answers
47 views

Question About the Step Involving the Connectedness of $\mathbb{R}_+$ in the Proof of Proposition 3.C.1 from MWG

This is a small issue, but it confuses me. In the proof of Proposition 3.C.1 of Microeconomic Theory by Mas-Colell et al, the book said ... $\mathbb{R}_+\subset (A^+\bigcup A^-)$. The nonemptiness ...
3 votes
1 answer
84 views

Question About Proof of Proposition 3.C.1 in MWG - Step 1

I have difficulties understanding the first step of the proof of Proposition 3.C.1 in MWG. Proposition 3.C.1$\quad$ Suppose that the rational preference relation $\succsim$ on $X$ is continuous. Then ...
1 vote
1 answer
32 views

Revealed preferences - Commuter preferences

I'm working on a paper about commuter preferences and their level of satisfaction in a South American country. Do they prefer the metro or informal buses? That sort of thing. The data tells me that ...
2 votes
0 answers
82 views

Prove that any lexicographic preference $(u_1,u_2)$ must be complete and transitive

Let $\succsim$ be a lexicographic preference represented with $(u_1,u_2)$. $x\succsim y$ if $u_1(x)>u_1(y)$ OR $u_1(x)=u_1(y)$ and $u_2(x)\geq u_2(y)$. Is it obvious that $\succsim$ must be both ...
1 vote
1 answer
63 views

Prove a preference preserved under limits if and only if its upper and lower contour is closed

I'm concerned with the reverse direction, that upper and lower contour is closed implies the preference is continuous, that is for any sequence $x_n$ and $y_n$, $x_n\succcurlyeq y_n$ for all $n$, ...
2 votes
0 answers
33 views

Positive monotonic transformation

I want to draw an indifference curve for this preference relation on $ℝ^2$ $$x ≿ y \leftrightarrow (x_1+x_2)^2 ≥ (y_1+y_2)^2$$ Can I just take the square root(and then drop the absolute value sign ...
1 vote
0 answers
39 views

Indifference curves given preference relation

Let's say I have a preference relation on $ℝ^2$ given by $$x≥y \leftrightarrow x_1+x_2 ≥ y_1+y_2$$ So, from my understanding, the bundle x is preferred at least as much as the bundle y if and only if ...
6 votes
2 answers
1k views

Continuous rational and monotone preference relation implies $x\succsim0$?

I updated my proof to a general version as follows: please share your thoughts & 2cent. Thanks Show a monotone continuous complete preorder on $\mathbb{R}^L_+$ has $y\geq x\rightarrow y\succsim x$....
7 votes
1 answer
6k views

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

[ADDED/MODIFIED] : I have put my proof where the commodity space is simply $\mathbb{R_+}$(e.g. nonnegative reals) for simplicity below. Please share your 2 cent. I have put words to aid my own ...
2 votes
2 answers
89 views

labor leisure model, quasilinear preferences

There is a quasilinear utility function $u= (1-t)wl - p(l)$, where $l$ is labor supply. I don't quite understand what happens, if the budget changes (due to $w$ or $t$) since it is quasilinear. Does ...
0 votes
0 answers
12 views

Equivalent assumptions in Revealed Preference and Classical Preference Theories

I have read that there are dualities between Revealed Preference Theory and Classical Preference Theory. Suppose, I have $X$ as the set of all conceivable choices, and $\mathscr{B} \subseteq 2^X$ the ...
3 votes
1 answer
73 views

Understanding the definition of monotone

In Microeconomic Theory by Mas-Colell, Whinston, and Green, the definition of monotone preference relations is given as follows: Definition 3.B.2$\quad$ The preference relation $\succsim$ on $X$ is ...
2 votes
0 answers
20 views

Preference aggregation rules that satisfy the property that the most frequent best alternative is the collective best alternative?

Let $X$ be the set of alternatives, and $(R_i)_{i\in N}$ a profile of partial orders for $n= \#N$ individuals. I make the assumption that those partial orders are so that they all give rise to one ...
1 vote
1 answer
61 views

Can the following statement be rationalized if it yields a choice function?

A person choose an alternative to maximize another person's suffering. I thought we could define a sort of relation where the person suffers more from x than y. And if we can always do this, we can ...
1 vote
0 answers
41 views

What is a utility function rationalizes these preferences?

For a consumer deciding between goods A and B, with a budget of w: If A and B are the same price (or if A is cheaper), the consumer will spend their entire budget on A. As the relative price of A ...
3 votes
1 answer
361 views

How to solve a general equilibrium problem with lexicographic preferences?

I have been unable to find a good example of this type of GE problem in our textbooks, and our professor has indicated that something like this may appear on our exam. So, here is a hypothetical ...
1 vote
1 answer
66 views

How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?

Let $i$ be an agent, and let $A=\{x,y,z\}$ be a set of three alternatives. Then, suppose that player $i$’s linear order (i.e., complete, transitive, antisymmetric and reflexive binary relation) on $A$,...
0 votes
0 answers
27 views

Varian Analysis Non satiation question

Consider preferences defined over the nonnegative orthant by (xl,x2)> (yl,y2) if X1 + X2 < y1+ y2. Do these preferences exhibit local nonsatiation? If these are the only two consumption goods ...
1 vote
1 answer
40 views

Convex preference and convex utility

What are the differences between convex preferences and covex utility function? Why are convexity preferences usually represented by the quasi-concave function and not the convex function?
1 vote
1 answer
108 views

Cobb-Douglas utility function

Why in Cobb-Douglas utility function, the exponents have to sum to one? Can they not be equal to 1 and why?
5 votes
1 answer
90 views

Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)

I am doing exercises in Chapter 2 of MWG. I feel I got completely lost in exercise 2.F.3 parts (e) and (f). $\textbf{Exercise}$ Here is the question: I have solved parts (a) to (d). In particular, I ...
1 vote
1 answer
61 views

Stock market existence and saving money

I have a generic question that perhaps has more to do with psychology than with anything else. The question is, how effective is the stock market's existence in being effective in making people save ...
3 votes
1 answer
143 views

Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation

I am solving the following problem (from Exercise 2.F.3 (b) in MWG) and I got confused by the weak axiom of revealed preference and the definition of the revealed preference relation. Here is the ...
1 vote
0 answers
38 views

Considering there are four different goods {w,x,y,z}, how many different relations can be defined using all goods

In this problem, you can either prefer a good to another, disprefer a good to another or be indifferent between the two goods. While I have been able to do it mechanically, I am very confused as to ...
0 votes
0 answers
23 views

What does GARP in revealed preferences mean and how to regocnize it graphically?

I have a froblem understanding GARP in revealed preferences. I know what WARP means, and that SARP adds transivity such that it allows for indirect preferences, but what about GARP? How can you ...
1 vote
1 answer
216 views

Does strongly monotone preference imply local non-satiation?

How to prove this? I understand monotonicity implies local non-satiation but does strongly monotone also imply it? How to prove it like this - https://felixmunozgarcia.files.wordpress.com/2017/08/...
0 votes
0 answers
30 views

How do I prove 2 indifference curves have the same properties?

I understand questions a) but I'm completely stumped at c). What do they mean by U'=1 having same properties as U=10?? And w,for questions b and c, what are they asking by "general expression for ...
1 vote
0 answers
98 views

Proof for convex preference relation

⪰ is a strictly convex preference relation on a set X which is the set of all N-tuples of nonnegative real numbers. Further are x,y,z element of this set X and i have the preferences x ≻ y ≻ z given. ...
4 votes
1 answer
83 views

In revealed preference (RP), is any two points $x,y$ related by the indirect revealed preference relation?

Let $X$ be the closed compact convex set of alternative and $B$ be a closed compact convex subset of $X$. $C$ is defined on all closed compact convex set $B\subseteq X$. $X$ is ordered by a strictly ...
1 vote
1 answer
123 views

Determine Whether A Preference Relation Satisfies The Continuity Axiom - from Exercise 1.1 in Game Theory: Analysis of Conflict by Roger Myerson

I am self-studying game theory using Game Theory: Analysis of Conflict by Roger Myerson. Here is an exercise from the textbook. I tried it myself, but I am not sure if it is correct. I would really ...
1 vote
1 answer
140 views

Why is monotonic preference and monotonic utility function non-decreasing?

Obviously a monotonic function can be either nondecreasing or nonincreasing: However, in Economics, a quick google search gives: I am interested in the history or the motivation behind the econ ...
1 vote
0 answers
9 views

What was the study that shows peoples' tax preferences shift when they get more information about their income?

I am trying to find a study I read a few years ago, but I forgot the names of the authors and the title of the study. I have already tried to search for the study on Google Scholar without much luck. ...
7 votes
1 answer
965 views

Lexicographic Preference Relation on the QxR

I would like to ask for your help. I recently learned that the Lexicographic Preference relation can be represented by a utility function $u:X\to\mathbb{R}$ on $\mathbb{Q}\times\mathbb{R}$ (but not $\...
0 votes
0 answers
39 views

Example to Demonstrate that with Uncountably Infinite Outcomes We do not Have A Representing Utility Function [duplicate]

Suppose we define $\Omega$ = $\mathbb R_+$. The preference relation $\succeq $ is defined as $$(x_1, x_2)\succeq(y_1,y_2)\iff x_1>y_1 \text{ or } [x_1=y_1 \text{ and }x_2\geq y_2]$$ where $x_1, x_2,...
1 vote
1 answer
377 views

Archimedean but not mixture continuous

In the context of preferences on a set of lotteries on a finite set $X$, what is an example of a preference that is independent, Archimedean but not mixture continuous? I know the mixture continuous ...
0 votes
1 answer
214 views

Does Preference have a Hierarchy? A Silly Question

I have what is probably a very silly question, but I have gone down the rabbit hole and can’t get back out..... Is there is a hierarchy of preference, and within each level of choice do we reset the ...
1 vote
1 answer
92 views

y is weakly preferred over x if and only if x+y ≤ 4 defines a preference relation on {0,1,2,3} why is this incomplete?

y is weakly preferred over x if and only if x+y ≤ 4 defines a preference relation on {0,1,2,3}. True or False?. I can see why it's not transitive but I was told it was incomplete if we take 2 and 3. ...
2 votes
1 answer
107 views

Transitivity of Preferences paper

I am going through some of my old grad school notes, and in my microeconomics notes on transitive preferences, the teacher made a note of a behavioral economics result where when presented with two ...
1 vote
1 answer
47 views

Give bundles $x,y\in \mathbb R^n$, there must exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?

For a problem in revealed preference. Give bundles $x,y\in \mathbb R^n$, must there exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$? Intuitively, this mean that we have two bundles, and ...
2 votes
2 answers
370 views

Lexicographic preference relation not continuous

I’m having trouble understanding why lexicographic preference relations aren’t continuous. I need inspiration on how this proof would work.
0 votes
0 answers
45 views

Properties of Consumer Preferences - Monotonicity

Was reviewing topics and I came across this question. I am confused because there is no reference to strict or weak monotonicity in this case. I first thought that monotonicity is violated b/c an ...
3 votes
1 answer
155 views

Continuity of preference

"A preference is continuous if for any $a,b\in X$ with $a\succsim b$ there are some neighborhoods $N_{\varepsilon}(a)$, $N_{\delta}(b)$ around $a$ and $b$ such that for every $x \in N_{\...
0 votes
1 answer
744 views

Quasiconvex and quasiconcave utility function

I saw that the model of the quasi-convex utility function is similar to the concave utility function and also the quasi-concave utility function is similar to the convex utility function. How can ...

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