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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Is it possible to have mean-variance preferences with different states of nature? Mean-Variance and Expected utility together?

I have to maximize mean-variance preferences like this (where Pi is a profit function): \begin{align} \label{eq:9} \max\limits_{Q_{F}^{\{x\}}}Z_{w}= E[\pi{\{x\}}]-\frac{A}{2}Var[\pi{\{x\}}] \nonumber\\...
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Convexity of preferences (dissimilar definitions)

Varian's Intermediate Microeconomics describes convexity as $$\text{Given } x, y \in X: x \sim y \implies \forall t \in [0,1], tx + (1-t)y \succeq x,y$$ The other definition I read everywhere is: $$\...
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Are these preferences locally non satiated? U(x1,x2)=(x1-1)/(2-x2)^2

I got this utility function representing certain preferences. Are these preferences locally non satiated? Can somebody please explain me with the exact definition of local non satiation why these ...
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Does Concavity or quasi-concavity imply local non satiation?

I got a utility function with convex Indifferences curves and therefore convex preferences. Convexity of preferences implies Quasi-concavity. I would like to know if there is a relation between ...
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Representation theorem for $\succsim\supset>\cup\sim$

On $\mathbb R^2$, define $x=(x_1,x_2)>(y_1,y_2)=y$ if $x_i\geq y_i$ for all $i$ and $x_j>y_j$ for some $j$. Let $\sim $ be an equivalence relation that $x\sim y$ implies $x\not> y$. Define ...
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Proof that weak Monotonicity and local non satiation imply monotonic preference

Weak monotonicity in my case is defined as follows: If x is weakly larger than y, then x must be weakly preferred over y. Monotonicity is defined as follows: If x is strictly larger than y, then x ...
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Say a preference is "constant“, by analogy with a constant function

Let $f$ be a function with range of $\{-1,1\}$ and $f(x,y)=-f(y,x)$. Let the preference $\succ\subset X\times X$ where $x\succ y \iff f(x,y)=1$. In math we can define $f$ to be a constant function on ...
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Is this a function or vector space?

I want to make a function that states consumer $i's$ consumption $C_i$ in a shared bundle $X^n$ $\subset$ $\mathbb R^n$ depends on their preferences. That is, something like $C_i(\succeq_i)$. Since $\...
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How to mathematically denote that a consumer behaves according to their preference structure at every point in time?

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

By Debreu's theorem of ordinal utility, any continuous weak order on $X$ is represented with a continuous utility function, if $X$ is a second countable or connected separable topological space. My ...
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Preference notation by agent $i$

I want to say that the preferences of agent 1 on a consumption set $X$ are the same as agent 2's and the inverse of agent 3's (hence 2's is also the inverse of 3's). How would I do this using the ≽ ...
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Does continuous preference imply upper-hemi continuous demand correspondence?

Let alternative $x,y,z\in R^N$. $\succsim$ is convex, rational, monotonic, and continuous. Let $B=[y,z]$ be a budget segment. Let demand correspondence be $D[y,z]=\{x\in B||x\succsim B\}$ $D[y,z]$ is ...
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Indifference curves representable by real-valued functions on $\mathbb{R}$ of a continuous preference relation

Let $X = \mathbb{R}^2$. Suppose $\succeq$ denotes a continuous preference relation. If every indifference curve can be represented by functions from $\mathbb{R}$ to $\mathbb{R}$, will it mean the ICs ...
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Discontinuous function $U$ with continuous preferences can be written as a composition of discontinuous & monotone function and a continuous function

Conjecture: Every discontinuous utility function $U$ representing continuous preferences can be written as $U = f \circ g$ for some continuous $g$ and discontinuous strictly monotone $f$. The goal is ...
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What would you call a preference relation that is intransitive yet complete?

I am trying to make sense of the following terminology and putting it into a table helps me keep concepts straight: If I am understanding correctly, irrational behavior is describe as a preference ...
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Lagrangian when ICs are tangent to the budget line

Suppose the graph below shows three Indifference Curves such that $t > s > r$, and the budget line $p_1x + p_2y = I$. I was wondering if we set the Lagrangian as $\mathcal{L}= U(x,y) - \lambda (...
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Why do some game theory textbooks explicitly require preference relations to be reflexive?

A binary relation on a set of outcomes is called a preference relation if it is complete and transitive. Completeness of course implies reflexivity. But the authors of some game theory textbooks add ...
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Risk Premium for Prospect Theory-like value function

I am curious how to calculate the following risk premium for a utility function that is not linear in $w$. What i'm asking is the following: Consider an agent with utility function $u$, initial wealth ...
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Proving that Every preference relation can be represented by utility function

Im looking the prove the following: every preference relation ⪰ over finite/countable set X can be represented as a utility function u such that ...
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Preference terminologies

I am reading one paper by [Maskin et al. 1979] and cannot figure out some notations. Specially, they defined some states of nature $A$, and each player in the player set $I$ has some preferences over ...
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Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?

Epstein-Zin (EZ) utility is the solution to: DEU is relatively simple: $\sum_t \delta ^t\mathbb E[u(c_t)]$. Is DEU a special case of EZ? How are those two models compared? Since EZ is a solution of a ...
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Conditions to use the Lagrangian method

I have seen that the prices and $\text{MU}_{i}$ are assumed to be positive (or, the preferences monotonic). This is always mentioned when a utility maximization problem is being solved with the ...
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Price Offer Curve for a U(q1,q2)=max{q1,q2} Utility Function

Can someone help me understand how to draw out the price offer curve, or price consumption curve (PCC), for a U(q1,q2)=max{q1,q2} function? There's no information about this in my textbook and it's ...
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Perfect substitutes mathematical definitions not equivalent

Statement: Consider goods $X$ and $Y$ (and we denote the quantities of by the same notation) such that they are perfect substitutes with the substitution ratio $1:n$. Assume the basic axioms ...
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Mathematical definition of perfect substitutes

If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$, how do we get the mathematical equation from it? I know the equation is of the form $ax+by$ (and $U =...
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Marginal utility meaning and properties

Consider goods $X$ and $Y$ such that the marginal utility of a unit of good $X$ is always that of $n$ units of good $Y$. $X$ and $Y$ are perfect substitutes. Question 1: What does the above mean ...
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How to measure utility functions in the stock market

i am looking for some articles on how to identify and estimate utility functions in the stock market. My own search results yielded some papers by Blackburn and Ukhov https://www.researchgate.net/...
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Most utility functions under risk and uncertainty generalizes expected utility. What is deadly wrong if a model does not include EU as special case?

Why do people generalize EU instead of making an entirely new model, or create a model that is neither a special case nor an extension of EU? To my knowledge, most utility functions under risk and ...
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Homothetic preferences for international trade

In international trade class, we assume homothetic preferences for every country, and each country has an endowment. Why do we assume homothetic preferences? Is this because we see (from data) that ...
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Is Varian's definition of continuity of preference equivalent to standard definitions?

Here are two definitions of continuity of preferences. Denote the (weak) preference relation by ≽. We assume completeness, reflexivity and transitivity. Assume non-satiation or strict monotonicity ...
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Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function

my current thinking is i have to dis/prove two things cardinality continuity but im not sure about how it would apply since the above is a natural X natural choice space I know cardinality of ...
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Different ways of writing CIES/CARA utility

I frequently encounter the following two versions of writing CIES or CRRA preferences: $$u(c_t) = \frac{c_t^{1-\theta}-1}{1 - \theta}$$ ...and... $$u(c_t) = \frac{c_t^{1-\theta}}{1 - \theta}$$ The ...
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A matter of completeness and preference relations [closed]

I have a question about preference relations and completeness. Prove that: i) There exists a complete relation $\succcurlyeq$, such that $\sim$ is not complete. ii) There exists a complete relation $\...
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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 ...
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Is electoral abstention an example of non-complete preference?

In order for a preference to be rational, it has to be transitive and complete. Complete preference means that any two different bundles can be compared. I.e., a consumer can weakly prefer bundle X ...
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Weak preferences and negative transitivity

Let $ \succ $ be a binary relationship on the set $X$ such that, given any $ x, y, z\in X $, if $x\succ y$: (Asymmetry): $\neg(y\succ x)$, (Negative transitivity): $(x\succ z) \vee (z\succ y)$. ...
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Does x ≽ y imply x > y or x ~ y in preferences?

Mas Collel Micro Theory question: Suppose that X is a set. Let ≽ be a binary preference on X. And ~ represents indifference defined from ≽. If ≽ satisfy completeness, is it okay to assume that: x ≽ y ...
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In a setting with N goods how many combinatorial bits do we need to construct a preference map

I am reading this paper: https://www.researchgate.net/publication/5208445_The_market_for_preferences By P.E Earl and J.Potts On page 3 the following is written: "If we think of individual ...
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Proving properties for preferences?

I have a midterm coming up and I am still not entirely sure on the formal arguments for proving (strict)\convexity, monotonicity, continuity, quasi-concavity e.t.c. I think I have a pretty strong ...
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Violation of Monotonicity of preferences

Hi I am reading Jehle and Reny in my master's course and I have come across a problem in one of the exercises. My instructor herself was a bit confused when a student gave her a counter-example and ...
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Asking for reference of trading strategy without inherent preference?

Han et. al.,2021 mentioned that So in contrast with traditional behavioral finance models, active strategies tend to spread through the population even if investors have no inherent preference for ...
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Part of proof of Gibbard-Satterthwaite Theorem

I'm currently working through Nisan's Algorithmic Game Theory, Chapter 9 (Introduction to Mechanism Design). A part of the proof for the Gibbard-Satterthwaite Theorem is given as "obvious," ...
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Does the second derivative of a concave function need to be negative? [closed]

I have just watched the first part of this video and noticed that the second derivative of the utility function is positive. But I thought the second derivative had to be negative to be a concave ...
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Non well behaved preference

I have to discuss a consumer making a choice between 2 bundles, and the consumer has a non- well-behaved preference. What real example can I use to represent a non-well behaved preference?
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How to explain satiated preferences with algebra?

From looking at the graphical illustrations, I understand that satiated preferences violate the monotonicity assumption. I was wondering if there is algebra which can be used to show this?
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Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?

Is it true that any three of the four vNM axioms (of expected utility theory) can be satisfied without satisfying the fourth? Any examples which support such claim? Basically I'd like to prove that ...
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2 votes
1 answer
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How do you establish uniqueness of a rational preference relation?

Going through a proof in Mas Colell and I am not understanding how (iii) shows uniqueness of the rationalizing preference relation. I understand that well $\beta$ is the power set so it contains all ...
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WARP implies SARP: A 2 Good Case

I am considering an example where there are two goods and three budget sets $(\mathbf{p}^{(n)},w^{(n)}),n=1,2,3$. If we assume $\mathbf{p}^{(n)} \cdot \mathbf{x}(\mathbf{p}^{(n+1)},w^{(n+1)}) \leq w^{(...
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Is this Incomplete or Indifferent? [closed]

Given X = {1,2,..., 100}. For x, y in X, define x # y if and only if x - y is a positive prime number. Is the # relation incomplete? I don't particularly understand the reasoning as of yet, and though ...
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Comparing voting methods when there are only two voters

Consider the Schulze, Kemeny-Young, Ranked Pairs and Borda count voting methods. (The last is obviously the odd one out in this list!) Suppose that there are only two voters. Each voter gives a ...
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