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8 votes
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What is the difference between preferences lacking "completeness" and being indifferent?

As alluded to in the comments the distinction is roughly: Indifference: The decision maker knows she will receive the same utility from the consumption of $x$ or $y$. Incompleteness: The DM does not ...
201p's user avatar
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8 votes
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(Preference Relation/Set) Continuous $\succsim$ imply closedness of upper and lower contour sets

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 ...
Kitsune Cavalry's user avatar
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8 votes
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Has the assumption that individuals' tastes do not change over time been rigorously challenged?

Stigler and Becker's argument is methodological, not philosophical. They do not try to convince us that preferences are indeed identical across individuals and invariant across time as a matter of ...
Alecos Papadopoulos's user avatar
8 votes
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Do discontinuous preferences imply no continuous utility function?

The easiest way to prove it is using the 'old' definition of continuity. $\succ$ is continuous iff whenever $x\succ y$, there exists neighborhoods of $x$ and $y$, $B_x, B_y$, such that all $z\in B(x)$...
Fato's user avatar
  • 550
8 votes
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Why does quadratic utility function imply $\mu-\sigma$ preference?

If you have quadratic preferences then your utility function is: $$ U(W) = W - \lambda W^2$$ this implies your expected utility function looks like: $$ E[U(W)] = E[W - \lambda W^2] = E[W] - \lambda ...
BKay's user avatar
  • 16.3k
8 votes

Rational preferences/individual decision-making theory

(i) Is not complete. For instance, (10,5) is not $\succeq$ (9,6), because $10>9$, but $5<6$. However, (9,6) is also not $\succeq$ (10,5) for the same reason. Hence, there exists a pair of ...
Bayesian's user avatar
  • 5,291
7 votes

How can I tell if 2 different utility functions represent the same preferences?

Recall the definition: The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$ We can show that if $u: X \rightarrow \...
Kitsune Cavalry's user avatar
  • 6,638
7 votes
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How do I represent this indifference curve graphically?

The problem is that there are no indifference "curves" but indifference "areas". Consider the following graph: For a reference bundle $A$ (equivalent to $\{2,3\}$), the gray regions indicate the ...
luchonacho's user avatar
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7 votes
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Homogeneous of Degree Two Utility Functions and Homothetic Preferences.

First of all, in order to provide a counterexample, you need to construct a utility function that is homogeneous of degree two, but is not homothetic. Therefore, the counterexample you gave in your ...
Ziwei Wang's user avatar
7 votes
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Proof of monotonocity of preferences

Let me pick up the discussion in the comments. Consider any diagonal through the origin. Suppose it intersects some IC more than once. Pick two of these points and call them A and B. Because they are ...
Bayesian's user avatar
  • 5,291
7 votes
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Linear Homothetic Utility

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form: $$ u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}. $$ This is a Cobb-...
tdm's user avatar
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7 votes
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Quadratic utility: monotonicity and risk aversion

Quadratic utility is given by $$u(w) = w - b w^2$$ which has derivative $$u'(w) = 1- 2b w$$ such that for high levels of $w, u'(w)<0$. That is, the utility is not everywhere increasing. This may be ...
Bayesian's user avatar
  • 5,291
7 votes

Are Indifference Curve graphs continuous given the preferential definition of continuity?

Given your last comment above it seems that what you are really asking is whether the indifference sets of a continuous preference relation on $\mathbb R^n_+$ are path-connected. The answer is No. Let ...
VARulle's user avatar
  • 6,900
7 votes
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Are Indifference Curve graphs continuous given the preferential definition of continuity?

To follow up on the answer of @VARulle let me give you some conditions for which the indifference curve is path connected. The argument can also be found in the book Mathematical Methods and Models ...
tdm's user avatar
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7 votes
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Why do some game theory textbooks explicitly require preference relations to be reflexive?

Since completeness implies reflexivity, there can be no extremely strong reason. But here are some: Students new to the language of mathematics do not always appreciate that "a pair of outcomes $...
Michael Greinecker's user avatar
7 votes
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Does Debreu's representation theorem of ordinal utility require Hausdorff topology?

No. However, the problem can be reduced to representing preferences on a Hausdorff space. Instead of trying to represent a complete preorder on a set, one can try to represent linear orders on the ...
Michael Greinecker's user avatar
6 votes

Thin indifference curves

To begin with, I think the question is wrongly stated. For if the defininition of a thin indifference curve is such that continuity of a consumer's preferences implies thin indifference curves, then, ...
Elias's user avatar
  • 983
6 votes
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Thin indifference curves

I don't think continuity alone is enough to guarantee thin indifference curves. Consider preferences such that, for any $x$ and $y$ in the choice set, the consumer is indifferent between $x$ and $y$. ...
Ubiquitous's user avatar
  • 16.9k
6 votes
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Existence of utility representation of a rational but discontinuous preference

I think a basic problem is that any utility function defines a preference, and discontinuous utility functions can be used to define discontinuous preferences. Hence there are many discontinuous ...
Giskard's user avatar
  • 29.2k
6 votes

Convex Preference but Convex Utility

It's well known that a convex preference implies quasiconcave utility functions. Since quasiconcavity need not imply concavity, it's easy to find examples of a non-concave utility function ...
Herr K.'s user avatar
  • 15.4k
6 votes
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If a rational preference relation over simple lotteries $\succsim$ are convex then they satisfy independence?

It's well known that if $\succsim$ satisfies independence, then it is also convex. Since $\succsim$ satisfies independence, $L\succsim L^{'} \iff \alpha L+(1-\alpha)L^{''}\succsim \alpha L^{'}+(1-\...
Lorenzo Castagno's user avatar
6 votes

Violation of Monotonicity of preferences

In general, it will not represent the same preferences. There seems to be confusion on what "monotonic transformation" means in this context. It does not have much to do with monotonic ...
Michael Greinecker's user avatar
6 votes
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Is Varian's definition of continuity of preference equivalent to standard definitions?

What Varian (Microeconomic Analysis, p 95) says is that: If $x$ is strictly preferred to $y$ and if $z$ is a bundle that is close enough to $x$ then $z$ must be strictly preferred to $y$. This is a ...
tdm's user avatar
  • 12.2k
6 votes

Is Varian's definition of continuity of preference equivalent to standard definitions?

Here is how one can show that Definition 1 implies Definition 2. We do the contrapositive, we show that if Definition 2 fails then Definition 1 will fail too. Suppose that $x\succ y$, but for every $\...
Michael Greinecker's user avatar
6 votes

Most utility functions under risk and uncertainty generalizes expected utility. What is deadly wrong if a model does not include EU as special case?

Many people accept the axiomatizations of expected utility as normatively appealing, especially in contexts of pure risk. For people with this view, rational decision-makers should behave in ...
Michael Greinecker's user avatar
5 votes
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Continuous rational and monotone preference relation implies $x\succsim0$?

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 ...
Pburg's user avatar
  • 2,218
5 votes
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Are Cobb-Douglas preferences homothetic?

Note that the wikipedia article is very specific: [...] defined a preference to be homothetic, if they CAN be represented by A utility function [...] You chose a specific utility function to ...
Giskard's user avatar
  • 29.2k
5 votes

Existence of utility representation of a rational but discontinuous preference

Consider the (rational) preference on $[0,1]$ defined by $u(0)=0$, $u(x)=1$ for $x>0$. It is not continuous, but it can obviously be represented (by $u$). In fact, many preferences that fail to be ...
john's user avatar
  • 91

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