9
votes
Accepted
Risk Premium in the Expected Utility Theory
Is there any (economic) rational for the first-order expansion of the RHS? And for its different neighborhood evaluation?
As for your first question:
This is a purely mathematical tactic in order to ...
- 33.4k
9
votes
Accepted
Sum of Homothetic Functions
Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$.
Defn: A function is homothetic if it is a ...
- 967
8
votes
Accepted
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 ...
- 956
8
votes
Accepted
(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 ...
- 6,578
8
votes
Accepted
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 ...
- 33.4k
8
votes
Accepted
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:
$$ E[U(W)] = E[W - \lambda W^2] = E[W] - \lambda ...
- 16.3k
8
votes
Rational preferences/individual decision-making theory
(i) Is not complete. For instance, (10,5) is not $\succeq$ (9,6), because $10>9$, but $5<6$. However, (9,6) is also not $\succeq$ (10,5) for the same reason. Hence, there exists a pair of ...
- 5,270
7
votes
Accepted
Strict preference relations and utility representations
Yes it is:
If direction
$$
x \succ y \Rightarrow x \not \precsim y \Rightarrow u(x) > u(y).
$$
Only if direction:
For all $x, y \in X$,
$$
x \succsim y \iff u(x) \geq u(y)
$$
implies
$$
x \sim ...
- 28.7k
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 \...
- 6,578
7
votes
Accepted
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 ...
- 8,581
7
votes
Accepted
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 $z\in B(x)$...
- 540
7
votes
Accepted
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 ...
- 341
7
votes
Accepted
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-...
- 10.3k
7
votes
Accepted
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 ...
- 5,270
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 ...
- 6,550
7
votes
Accepted
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 ...
- 10.3k
7
votes
Accepted
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 $...
- 11.7k
7
votes
Accepted
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 ...
- 11.7k
6
votes
Accepted
Given a Utility function, U(x,y), why is multiplying U(x,y) * x not a monotonic transformation?
Because $x$ is not a constant, and therefore multiplying $U(x,y)$ by $x$ does not necessarily preserve the ordering between bundles.
For instance, consider the function $U(x,y)=x+y$, and the bundles $...
- 3,232
6
votes
Accepted
Are these preferences consistent with rationality?
You implicitly assume that the utility of $n$ units of $Y$ equals $n$ times the utility of 1 unit of $Y$, and there is no reason for that. For instance, if $Y$ is a fridge, the gain in utility from ...
- 3,232
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, ...
- 973
6
votes
Accepted
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 $y$. ...
- 16.9k
6
votes
Accepted
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 ...
- 28.7k
6
votes
Accepted
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 ...
- 5,270
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 ...
- 15.4k
6
votes
Accepted
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-\...
- 278
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 ...
- 11.7k
6
votes
Accepted
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 ...
- 10.3k
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