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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 ...

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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 ...

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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 ...

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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 $n$, $x = \lim_{n \rightarrow \infty} x^n$, ...

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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 ... 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 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 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). 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 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 ... 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 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). $$I think this would be a lot simpler than what you are ... 3 What you seem to be interested is the subfield of decision theory that goes by the name of "menu choice." The starting point of this literature is the paper Kreps, David M. "A representation theorem for preference for flexibility." Econometrica: Journal of the Econometric Society (1979): 565-577. This paper might be directly relevant to the problem you ... 3 "1." is wrong and "2." is correct, because (see e.g. Wolfram):$$A \setminus B = \{x: x \in A \text{ and } x \notin B\} Say for example we have $S = \{a, b, c\}$ and $T = \{b,c\}$. Then we'd write $S \setminus \{a\} = T$. We do NOT write $S \setminus a = T$. And so in general, for any set $S$, $S\setminus \emptyset =S$. We write $2^{X}\backslash \{\... 3 Another way of looking at this problem is to consider the means and variances of the lotteries. A risk averse agent (RA) likes high mean and low variance A risk neutral agent (RN) likes high mean and is indifferent to changes in variance A risk loving agent (RL) likes high mean and high variance From the fact that RN chooses$M$over$N$, we known that ... 2 Now that the OP has provided his own answer, let's also give the standard treatment of this problem. There is no production, the consumer receives windfall endowments in each period,$Y_1, Y_2$, and he can borrow (or lend) during the first period at an exogenous non-negative interest rate$r$. What is the two-period budget constraint of the consumer? It ... 2 The basic idea behind scarcity and opportunity cost is fundamental to economics. Basically, if resources are infinite, you have everything. In my opinion, there are no infinite resources because a lifetime is finite. In one way or another, it can be argued all choices are necessitated by scarcity. Eating chips means at that time, you cant be eating a ... 2 There is a long tradition in economics (some 50 years worth of research) called "Hedonic Price Analysis". It consists of "allocating" (through regression estimation) the price of a product to its various qualitative features ("characteristics") quantified, that presumably are valued by the consumers, i.e. those that are utility-enhancing. These ... 2 I think that your example relies on a misunderstanding of the meaning of the choice function.$C(\{x,y,z\})$does not contain the elements that the decision-maker would choose simultaneously if he could. He is only allowed to select one element at a time (otherwise, why not consuming all three items ?). Therefore$C(\{x,y,z\}=\{y,z\}$does not mean that he ... 2 The Weierstrass Extreme Value Theorem guarantees this. However, it's used so frequently that economists take it for granted or omit it in graduate courses and research papers. MWG is a maturity book. Part of mathematical maturity is being able to fill in the details for a proof. 2 If we take the definition of monotonicity to be if$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 ... 2 The answer is there is no single father of rational choice theory. The reason is that rational choice theory is not so much a theory but rather a coherent framework (others would call it a paradigm) resting on the key foundations of methodological individualism and instrumental rationality / self interest. Both of these aspects played important roles in the ... 2 Although it's often left unstated, demand for a good is measured as quantity per unit of time. Broadly, the longer the time period over which demand is measured, the smaller the (proportional) effect on demand of the initial endowment. Whether you have a soda in your fridge one morning may make the difference between your demand for soda that day being$0$... 2 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 ... 2 This assumption is usually considered as reasonable from a normative perspective. For instance, consider the following situations: in situation A, you face an urn with 5 blue balls and 5 red balls. A ball is selected randomly and you can bet on its color. You receive a monetary prize if your bet is correct. in situation B, you face an urn which contains 10 ... 2 Your "counterexample" also violates condition$\alpha$:$y=C(\{x,y,z\})$but$y\ne C(\{y,z\})\$. So it's not really a counterexample.

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