14
votes
Accepted
Lexicographic preference relation cannot be represented by a utility function
We can say more generally that lexical preferences are not representable using a continuous utility function. Lexical preferences are not continuous. Note the definition of a continuous preference ...
- 1,216
12
votes
What is the importance of Epstein-Zin preferences?
This is only a quick answer, unfortunately. The key intuitive insight for Epstein-Zin is that they separate two distinct properties of preferences: risk aversion ("I'd prefer less uncertainty to more ...
- 1,048
11
votes
Accepted
What is the importance of Epstein-Zin preferences?
I think CompEcon covered most of the points that I was going to mention. Just a few last thoughts:
1) Why are Epstein-Zin preferences important?
The preferences are important because they allow you ...
- 1,553
10
votes
What is the definition of exogenous and endogenous preferences?
Exogenous variables are believed to have some value given by nature. They are not caused by your theory's variables of interest. This is why they are said to be outside the model.
Endogenous ...
- 3,274
9
votes
Are homothetic preferences monotonic?
Are homothetic preferences strictly monotonically increasing?
Homotheticity requires that
$$ \alpha^\gamma U(x,y) = U(\alpha x, \alpha y) $$
This is not defined over the "increasing" part of ...
- 10.4k
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
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 ...
- 31.8k
8
votes
What is an example of a utility function where one good is inferior?
A good cannot be inferior over the entire income range.
The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form:
$$U(x,y) = \alpha_1 \ln(x-\...
- 15.9k
8
votes
Consequences of hyperbolic discounting
As often with models embodying some form of "irrationality" (whatever that means), HD does a great job at matching a whole lot of behaviors, but leaves room for rather annoying Dutch Book situations (...
- 5,157
8
votes
Translog Preferences
The translog function can be used not only in preferences but also in production and cost functions. I am not very familiar with its implications in consumer theory, but from the production point of ...
- 364
8
votes
Accepted
What is the difference between preferences lacking "completeness" and being indifferent?
As alluded to in the comments the distinction is roughly:
Indifference: The decision maker knows she will receive the same utility from the consumption of $x$ or $y$.
Incompleteness: The DM does not ...
- 946
8
votes
Accepted
(Preference Relation/Set) Continuous $\succsim$ imply closedness of upper and lower contour sets
Looking more closely at your question, I think things should not be overly complicated. From Mas-Colell et.al.
Definition 3.C.1:
The preference relation $\succsim$ on X is continuous if it is ...
- 6,409
8
votes
Accepted
Why does quadratic utility function imply $\mu-\sigma$ preference?
If you have quadratic preferences then your utility function is:
$$ 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 ...
- 15.9k
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 ...
- 5,090
7
votes
Accepted
If all indifference curves are parallel lines, then preference has linear representation
The indifference curves are constructed by viewing the utility function as an equation (for a fixed utility index value per curve). So from
$$U = U(x_1,x_2)$$
where the left side is just a symbol, ...
- 31.8k
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
$$
x \sim ...
- 26.1k
7
votes
How can I tell if 2 different utility functions represent the same preferences?
Recall the definition:
The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$
We can show that if $u: X \rightarrow \...
- 6,409
7
votes
Accepted
Has the assumption that individuals' tastes do not change over time been rigorously challenged?
Stigler and Becker's argument is methodological, not philosophical. They do not try to convince us that preferences are indeed identical across individuals and invariant across time as a matter of ...
- 31.8k
7
votes
Accepted
How do I represent this indifference curve graphically?
The problem is that there are no indifference "curves" but indifference "areas". Consider the following graph:
For a reference bundle $A$ (equivalent to $\{2,3\}$), the gray regions indicate the ...
- 8,437
7
votes
Accepted
Do discontinuous preferences imply no continuous utility function?
The easiest way to prove it is using the 'old' definition of continuity.
$\succ$ is continuous iff whenever $x\succ y$, there exists neighborhoods of $x$ and $y$, $B_x, B_y$, such that all $z\in B(x)$...
- 540
7
votes
Accepted
Homogeneous of Degree Two Utility Functions and Homothetic Preferences.
First of all, in order to provide a counterexample, you need to construct a utility function that is homogeneous of degree two, but is not homothetic. Therefore, the counterexample you gave in your ...
- 331
7
votes
Accepted
Linear Homothetic Utility
The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-...
- 8,737
7
votes
Accepted
Quadratic utility: monotonicity and risk aversion
Quadratic utility is given by
$$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 ...
- 5,090
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 ...
- 4,527
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 ...
- 8,737
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 $...
- 9,435
6
votes
What are different ways of specifying utility and decision making?
I'm somewhat surprised that no one has linked to this paper: Backus, Routledge, and Zin (2004) Exotic Preferences for Macroeconomists (this version has some fixed typos, vs the NBER print).
Their ...
- 1,048
6
votes
What are different ways of specifying utility and decision making?
This won't get at individual choice, but how about evolutionary approaches? Perhaps this isn't what you are looking for, but one way to model decisions is to wander from the rational paradigm entirely....
- 2,167
6
votes
What is the difference between "aggregation" and a "representative agent?"
(I cannot say if my answer will respond to your questions, which indeed, are a bit unclear).
If one browses through many-many economic papers, one will get the impression that "representative" just ...
- 31.8k
6
votes
Accepted
Fair voting procedure when there are many issues
That's interesting: the flavor of the frequentist approach to probability used for a socio-political fairness criterion: if my measure as a population group is $0<p<1$, and known, then my ...
- 31.8k
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