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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21 views

Expenditure function. Prove that this set is bounded

I need to prove that the following set is bounded in order to derive the expenditure function: $e(p,v)=min_x px$ ST $\{x \in R^n_+$ such that $U(x)\geq v\}$. Knowing that $U(x):R^n \longrightarrow R$ ...
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Examples of risk-neutral firms or people in business

I am looking for examples of approximately risk-neutral firms or people in business. Is there an industry where risk-neutrality is common for some agents (firms or people)? Are there perhaps time ...
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2answers
31 views

Remove Linear Good From Quasi-linear Utility Function

Given a quasi-linear utility function: $u(x_1, x_2) = f(x_1) + \beta x_2$, $\beta > 0 $ What would happen if good 2 ($x_2$) is removed from the market? Would the new utility function be: $u(x_1) =...
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Differentiability of the utility function and indifference curves

Comment on the following affirmative: In the traditional consumer model, the hypothesis of differentiability of the utility function and of convexity of preferences, assure the indifference curves ...
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Ordering of bundles for which the axioms of transitivity, continuity, strong monotonicity and convexity are valid

Let there be two bundles which are obtained from the sequences $x^n = (x_1,x_2)= \left( 2 + \frac{1}{n},5 \right)$ and $y^n = (y_1,y_2) = \left(4 + \frac{2}{n}, 5 \right)$. Is it possible to obtain an ...
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37 views

Show that if $\succsim$ is continous on $X$, then the sets $ \precsim (x^0)$ and $\succsim (x^0)$ are closed

For a set to be continuous, it's contour sets must be closed. Since we can define $$ \succsim x^0 = \{x, x^0 \in X: x \succsim x^0 \} $$ and $$ \precsim x^0 = \{x, x^0 \in X: x \precsim x^0\}$$ it ...
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1answer
49 views

Strong monotonicity and weak monotonicity

We say $\succsim$ represents weak monotonic preferences if $$x,y \in X, \,\, y >> x \implies y \succ x $$ where $y >> x$ means that every element of $y$ is greater than every element of $...
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1answer
23 views

Completeness of the strict binary relation

How do I show that considering the preference relation $\succsim$, then $\succ$ is not complete? I tried the following (which I don't know if it's right) but I'd also like to know if it's possible ...
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1answer
44 views

Is the satisficing choice function rationalizable if the ordering isn't observable?

Edited: Say an observer observes only the choices made by the decision maker (and the sets from which these choices are made), but does not know the ordering. Then would the decision maker's choices ...
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1answer
39 views

How to describe a utility function in words?

Suppose I have a utility function of Cobb-Douglas form $$U(x, y) =x^{0.2}*y^{0.8}$$ I want to describe it in words. I would say like: The utility of consumer is captured by number of good x and ...
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37 views

$a\geq 0$, $x\succsim y$ implies $x+a\succsim y+a$ so the preference is linear?

$\succsim$ is a continuous and local non-satiate weak order. $x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (...
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42 views

Continuity of preferences

Let $\succsim$ be a transitive and reflexive relation on a metric space $X$ with closed upper and lower contour sets. If $\succsim$ is not complete, does it hold that: for all converging sequences ...
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1answer
62 views

$a\geq 0$, $x\sim y$ implies $x+a\sim y+a$ so the preference is linear?

$\succsim$ is a countinuous and convex weak order. $x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (or ...
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32 views

Is it possible and logical to have an upwards sloping budget line?

The question I have is, for example, say Garry has two goods, cookies he pays 1 to consume a cookie and a maximum of 10 can be consumed, whilst he gets PAID 2 to consume vegetables. Garry is also ...
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$x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$, then the preference is linear?

$x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (or disprove by counterexample) that: Suppose $x\sim y$ ...
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1answer
73 views

Prove that a preference is linear

Given the following two conditions: $x\succ y$ implies $x+a\succsim y+a$, And, $x\prec y$ implies $x+a\precsim y+a$ We want to prove that $\succsim$ is a linear preference. One of the ...
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Relationship between strictly convex preference and convex preference

Let X be a convex subset of linear topological space and let binary relation >= be a complete preordering. prove: If preference relation is strictly convex and continuous, then it is convex. Since ...
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1answer
266 views

Axiom: More is Better; But when is more better?

I'm taking an introductory microeconomics course and have been introduced to the 3 axioms of economic preferences. These include Completeness Transitivity Non-satiation My understanding of non-...
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1answer
44 views

Cobb-Douglas function homotheticity

I've been given the Cobb-Douglas utility function: $\ u(q_1, q_2)=a\ln q_1+b\ln q_2=q_1^aq_2^b \ $ If I want to prove homothetic preferences, I use the following condition: $\ u(\lambda q_1, \...
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3answers
56 views

locally nonsatiated preferences

what does this symbol mean in the discuss of locally nonsatiated preferences: $\varepsilon > 0$ and $||y-x||<\varepsilon$.
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58 views

For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves

The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x. Possible ...
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Reference request: modeling firms that act to create demand for their products

(I don't have an economics background; apologies if my terminology is confusing.) It seems like there is often a situation where a firm can take actions to create demand that did not already exist. ...
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1answer
110 views

Why are indifference curves (often) of infinite length?

Indifference curves are often of infinite length. Is this implied by monotonicity or non-satiation? If not, what is/are some condition(s) that are sufficient for indifference curves to have infinite ...
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1answer
49 views

Are revealed preferences normative or positive notion?

I always thought from my understanding of the terms normative and positive that revealed preferences are a positive concept. For example, saying Anakin prefers grass to sand (i.e. $U(g)\succ U(s)$) is ...
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1answer
122 views

Can we have a Non-Reflexive Preference Relation?

I've been thinking about preferences alot recently and have been specifically thinking about the reflexivity requirement. That is: $$x \succsim x$$ Though this is apparent and obvious, I have been ...
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1answer
96 views

Linear Utility?

Consider a preference relation $\succeq$ on $X\subseteq\mathbb R^2$. If $\succeq$ satisifies: $$ \begin{align} &1.\mbox{ }(a_1,a_2)\succeq (b_1,b_2)\implies(a_1+t,a_2+s)\succeq (b_1+t,b_2+s),\...
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Question about the relationship between Weak Axiom and Slutsky Matrix

We know that if a differentiable Walrasian demand function $x(p,w)$ satisfies Walras' law ($p^Tx=w$), homogeneity of degree zero ($x(\alpha p,\alpha w)=x(p,w)$), and the weak axiom of revealed ...
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1answer
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Why does Figure 2.F.1(b) (MWG page 30) satisfy the WARP (Definition 2.F.1)?

I can see that Figure 2.F.1(a) satisfies the WARP (Definition 2.F.1) in MWG (page 30). However, as the choice $x(p',w')$ is only feasible under the price-income level $(p',w')$ and $x(p'',w'')$ is ...
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52 views

Cardinal Voting, Incentive Compatibility and Secrecy

Is there any available/feasible/practical way to make a Cardinal Voting both Incentive Compatible and Secret? A method to make a cardinal voting incentive compatible would be to force them to put ...
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1answer
69 views

A question about the property of quasi-linear preference

In case of quasi-linear preference, why would one unit more of the numeraire good (good 1) give the same additional utility as spending an additional amount of wealth equal to the cost of one unit of ...
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2answers
60 views

Why is the nature of graph of utility function different from indifference curve?

I am new to Economics, but I have this doubt. The indifference curve and utility function both have the same equation, so their graph must also be similar, which is true I guess. Then why is it that ...
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29 views

Heckscher-Ohlin with non homothetic preferences [duplicate]

Can someone tell me how I can show with an example that the Heckscher-Ohlin result does not necessarily hold when preferences are not homothetic. I was asked if it similar as a case in which the ...
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53 views

Heckscher-Ohlin with heterogeneous preferences

could someone really help me out I would need to show a situation in which the Heckscher-Ohlin result does not necessarily hold when preferences are heterogeneous. Does someone have an idea how I ...
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1answer
104 views

A question about MWG Exercise 3.D.4

I'm doing exercises of Chapter3 of MWG, there's a problem that I don't understand (I didn't figure out the solution manual either...). It is about exercise 3.D.4, the full statement of the exercise ...
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Does quasilinear preference contain rationality, monotonicity or other assumptions?

I have a question when I'm doing exercise 3.C.5(b) of MWG. The exercise asks to prove that a continuous preference on $(-\infty,\infty)\times R^{L-1}_+$ is quasilinear with respect to the first ...
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Weakly monotone preferences with singleton indifference curves: do any of them admit a utility representation?

Inspired by this question. The original question was answered by Amit with some nice examples. I would like to know the generalized answer: Suppose we have a preference ordering $\succeq$, which is ...
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1answer
38 views

Why might a monotone increasing but nonlinear transformation of a utility function not represent the same preferences?

According to a textbook, a monotone increasing but nonlinear transformation of a utility function might not represent the same preferences. Why is it so? An example of such preference would be ...
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1answer
55 views

Why is a monotone increasing but nonlinear transformation of a utility function not represent the same preferences if the preference is complete?

According to a textbook, in the context of uncertainty (e.g. in lottery), if the preference is complete, a monotone increasing but nonlinear transformation of a utility function would not represent ...
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Kreps Porteus Certainty Equivalent Intuition

In Epstein-Zin recursive preferences, the Kreps-Porteus certainty equivalent is defined by \begin{equation} \mathcal{R}_t(V_{t+1}) = (\mathbb{E}_t V_{t+1}^{1 - \gamma})^{1 /(1 - \gamma)}. \end{...
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Does non-monotonicity imply non-satiation always? Why or why not?

I understand that monotonic preferences imply non-satiation. But I am not sure 100% if non-monotonic functions always have satiation. An intuitive and mathematical explanation would be very helpful.
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Why does quadratic utility function imply $\mu-\sigma$ preference?

Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply $\mu-\sigma$ preference?
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Net Preference Relations

Let say the electorate consists of three segments of voters: 1, 2, and 3 with corresponding weak preference relations defined over the candidates. Let the preferences be given by -- Segment 1: Biden >...
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Given a rational $\succsim$ over a finite set $X$, show that there exists $x \in X$ such that $x \succsim y, \forall y \in X$

I have been able to show this constructively, but would like to prove it by induction. However, I am stuck with the induction step: Consider $\succsim$ defined over $X=\{x_1,...,x_n\}$ and where ...
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80 views

Study whether $\succsim$ represented by $u(x)=\lfloor x \rfloor$ is continuous

Using the following definition of continuity: $\succsim$ is continuous if for any bundles $x,y,z$ such that x$\succ$y$\succ$z, there exists $\alpha \in (0,1)$ such that $\alpha x + (1-\alpha)z \sim y$....
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1answer
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A preference relation $\succ$ is defined as $(x_1,y_1)\succ (x_2,y_2)$ if $x_1>x_2$ and $y_1> y_2$

Does this satisfy completeness property? I need an intuitive explanation of this preference relation as well. I am confused about the way how this relation is defined. The commodity Y in the first ...
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1answer
63 views

Framing Effect Risk-Aversion Risk-Pursuit

I am an economics' graduate seeking to study Law and I want to illustrate the importance of legal certainty. Penalties, Costs are negativelly framed. I am trying to word. 200 dollars with 50% ...
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2answers
105 views

Assumption of sufficient wealth in quasi-linear preferences

Whenever we talk about quasi-linear preferences, we assume that the consumer is sufficiently wealthy. As far as I understand is that we need that assumption in order to obtain an interior solution. ...
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1answer
48 views

Representing preference orderings over a finite set of outcomes by two payoffs

I have read the following statement and I am having difficulty understanding the second part: Any set of preference orderings over a finite set of outcomes can be represented either by ...
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2answers
151 views

Showing that a preference relation admits a utility function representation

Setting: We have two choices of goods $(x_1,y_1)$ and $(x_2,y_2)$ from the set of choices $[-1,1]^2$. Moreover, we have the following preference relation $$(x_1,y_1)\mathcal{R}(x_2,y_2)\iff |x_1|\geq|...
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1answer
282 views

Preference: Convexity and Monotonicity

I need an example of a Convex, non-monotonic preference Non-convex, monotonic preference I figured that an example of non-convex, monotonic utility preference could be $U(x,y)=x^2+y^2$. For convex, ...

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