11
As you stated, transitivity is that overall $x \succeq y$ and $y \succeq z$ implies $x \succeq z$. I will show an example where majority rule isn't transitive and hopefully it will answer your question.
Imagine that we live in a world with three people: Person 1, Person 2, and Person 3. Each of these people have preferences over three outcomes $x$, $y$, and ...
11
I guess you might already know this, but I wanted to add a little detail to the other answers for the sake of any layman who comes here and gets the wrong end of the stick.
What is meant by rationality?
It is important to begin by saying that when economics use the term rational they have in mind a fairly precise definition that does not perfectly coincide ...
8
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 preserved under limits. That is, for any sequence of pairs $\{(x^n, y^n)\}^\infty_{n=1}$ with $x^n \succsim y^n$ for all $n$, $x = \lim_{n \rightarrow \infty} x^n$, ...
7
I would point you firstly to work pioneered by the late Gary Becker on applying the principles of economic optimisation to non-market behaviour.
Becker's insight was that the kinds of trade-offs people face when trading goods and services also affect many other kinds of decisions, including things such as criminal behaviour or marriage and family life. For ...
7
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 probability makes. Given this utility function, not much.
6
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
$$E[u_{RA}(N)] > E[u_{RA}(M)]$$
The fact that a risk-neutral agent ($RN$) could choose $M$ over $N$ implies that
$$E[u_{RN}(N)] < E[u_{RN}(M)] \implies ...
6
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$ so that $x\succeq y$ exactly if either $|x|<|y|$ or $|x|=|y|$ and $x\leq y$. Basically, the closer to the peak of $0$ a number is, the better, and in case of ...
5
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 function $v(p,w)$ is the highest value of the utility function when $u$ is evaluated over all affordable bundles given $(p,w)$. In other words, $v(p,w)=max u(x)$ ...
5
This is the two-period budget constraint:
C1 + C2/(1+r) = Y1 + Y2/(1+r)
Derivation is straightforward. On the LHS, you have the present value of consumption (considered during period 1), and on the RHS you have the present value of income.
Intuitively, think about 1/(1+r) on the LHS as the price ratio between Good 1 and Good 2. Now you can solve the u-...
5
Social choice theory or social choice is a theoretical framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a collective decision or social welfare in some sense.1
A social choice is related to a collective decision. Many different opinions, preferences and interests are taken into account to form a decision ...
5
Amartya Sen, a 1998 Nobel Laureate, has a well cited article on the subject:
"Rational Fools: A Critique of the Behavioral Foundations of Economic Theory", Philosophy & Public Affairs, 1977.
Some other related references:
Persky, Joseph. 1995. “Retrospectives: The Ethology of Homo Economicus”. The Journal of Economic Perspectives 9: 221-231.
...
5
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 problems, examples could be effort, or ability. Since this is a numerical variable, it is an ordered set.
In the paper you mention, the phrase "one dimensional, ...
5
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).
5
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 in $B$ that the decision maker might choose; that is, they are her acceptable alternatives in $B$. In this case, the set $C(B)$ can be thought of as containing ...
5
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. One central theme was product variety and the endogenous determination of the number of product varieties.
There are many models formulated within the ...
4
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 the definition of monotonicity (essentially replacing $y$ with $\mathbf{0}$ above), $x\succeq \mathbf{0}$. I don't think continuity is required (check ...
4
I don't think we have enough data on that matter. They would be irrational, for example, if transitivity of their preferences does not hold. How do we get their preferences? Through the axiom of revealed preferences (WARP).
We can never say whether someone is rational, we can only say whether someone is irrational (read: his actions are not rationalizable)....
4
First off it seems to me you are proceeding in a manner more complicated than necessary. (Perhaps this is intentional because you wish to face a harder exercise.) Since it is given that $v(x)$ represents $\succsim$ you only have to show
$$
\forall x,x' \in X: v(x') \geq v(x) \iff u(x') \geq u(x).
$$
I think this would be a lot simpler than what you are ...
4
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 financing. We show that more durable assets require larger down payments making them harder to finance, because durability affects the price of assets ...
4
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 is $w'$.
It simply does not say anything about case (b) in which $p' \cdot x(p'', w'') > w$ and $p'' \cdot x(p', w') > w''$. Since the first part of the ...
4
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 relation.
A social choice function maps profiles of preference orderings to elements of $X$.
Now let $P$ be a profile of preferences, $f$ a social decision function,...
3
We can not prove terrorists are not rational. We can only have a failure to find a tractable utility function that adequately models their behavior.
On the practical side, one cannot see enough decisions from a person ceteris paribus to confirm much about their rationality. Too much happens over time, without observation, and in a noncontinuous manner. ...
3
Ron Wintrobe has a book on Rational Extremism, which explains how behavior of terrorists, in particular suicide bombers may be "rationalized". He theorizes that the act of blowing up oneself is a form of corner solution to an optimization problem faced by terrorists.
Here's a preview of one of the chapters of the book: http://economics.ca/2005/papers/0708....
3
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 hence non-boundary solutions, are different.
Main Idea
For both a max(x1 x2) and perfect_sub(x1 x2) utility function, the point, say, m/p1 (or m/p2) would maximize ...
3
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 for preference for flexibility." Econometrica: Journal of the Econometric Society (1979): 565-577.
This paper might be directly relevant to the problem you ...
3
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 and is indifferent to changes in variance
A risk loving agent (RL) likes high mean and high variance
From the fact that RN chooses $M$ over $N$, we known that ...
3
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 you only impose convexity on $u$, then any dataset $D$ is rationalisable by the constant (convex) utility function $u(x) = k$. So this has not testable ...
3
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 use of these these linear budgets, we can leverage the results of Matzkin & Richter (JET,1991, Testing strictly concave rationality) to obtain a strictly ...
2
Now that the OP has provided his own answer, let's also give the standard treatment of this problem.
There is no production, the consumer receives windfall endowments in each period, $Y_1, Y_2$, and he can borrow (or lend) during the first period at an exogenous non-negative interest rate $r$.
What is the two-period budget constraint of the consumer? It ...
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