14
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 relation.
The preference relation $\succeq$ is continuous if for any sequences of consumption bundles $(x_{i})_{i \in \mathbb{N}}$ and $(y_{i})_{i \in \mathbb{N}}$...
11
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 uncertainty*") and intertemporal substitution ("I may want to shift consumption forward or backwards in time**").
In the very popular Constant Relative Risk ...
10
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 variables have values dependent on your theory's variables of interest. They both cause, and are caused by your topic.
Example: In the study of wages, some ...
10
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 to separate two of the dimensions along which people care about their allocations; namely, risk aversion and intertemporal substitution.
Additionally, one ...
9
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 strictly monotonously increasing. Indeed, you can have decreasing preferences $U(x,y) = -x -y$ for whom homotheticity holds.
Are nomothetic preferences weakly ...
9
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 monotonic transformation of a homogenous function.
Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then
$$
\...
8
A good cannot be inferior over the entire income range.
The paper A Convenient Utility Function with Giffen Behaviour shows that for a person with utility of the form:
$$U(x,y) = \alpha_1 \ln(x-\gamma_x)- \alpha_2 \ln(\gamma_y - y) $$
X is inferior if $\gamma_x$ and $\gamma_y$ are positive, $0<\alpha_1<\alpha_2$, and in the domain $x>\gamma_x$ ...
8
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 (also know as "money pump" situations). These suggest that HD might generate some inaccurate predictions, and induce undesirable behaviors when included in ...
8
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 know her preference between $x$ and $y$. (Note, this could stem from a lack of information, or because no preference exists)
So, from a conceptual vantage, the ...
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$, ...
8
(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 bundles $A,B$ such that neither $A\succeq B$ nor $B \succeq A$. Thus, it is not rational.
(ii) and (iii) are both rational. You can either see this by showing ...
7
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 view, i've seen it widely used.
The Translog Function doesn't impose additivity and homogeneity, and hence Constant Elasticity of Substitution. This is ...
7
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)$$
where the left side is just a symbol, we move to
$$\bar U = U(x_1,x_2)$$
where now the left side is a specific number.
Take the total differential on both sides to obtain
$$0 = U_1dx_1 + U_2dx_2 ...
7
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
$$
x \sim y \iff u(x) = u(y).
$$
Also
$$
u(x) > u(y) \Rightarrow u(x) \geq u(y) \Rightarrow x \succsim y ,
$$
$$
u(x) > u(y) \Rightarrow u(x) \not = u(y) \...
7
Recall the definition:
The function $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$ if for any $x,y \in X$, then $x \succeq y \iff u(x) \geq u(y)$
We can show that if $u: X \rightarrow \mathbb{R}$ represents $\succeq$ on $X$, then for any strictly increasing function, $f: \mathbb{R} \rightarrow \mathbb{R}$, the function $v(x) = f(u(x))$ also ...
7
Stigler and Becker's argument is methodological, not philosophical. They do not try to convince us that preferences are indeed identical across individuals and invariant across time as a matter of reality (the "Rocky Mountains" metaphor is an "as if " approach).
Their point is that any outcome can be rationalized by assuming that "it was preferences that ...
7
The problem is that there are no indifference "curves" but indifference "areas". Consider the following graph:
For a reference bundle $A$ (equivalent to $\{2,3\}$), the gray regions indicate the areas of indifference, based on your definition of preferences (the black lines are part of the indifference areas).
Thus, by selecting any bundle, you can find ...
7
The easiest way to prove it is using the 'old' definition of continuity.
$\succ$ is continuous iff whenever $x\succ y$, there exists neighborhoods of $x$ and $y$, $B_x, B_y$, such that all $z\in B(x)$ and $z'\in B(y)$, $z\succ z'$.
Suppose $x\succ y$. Because $u$ represents $\succ$, $u(x)>u(y)$. Let $2\epsilon=u(x)-u(y)$. Because $u$ is continuous, ...
7
First of all, in order to provide a counterexample, you need to construct a utility function that is homogeneous of degree two, but is not homothetic. Therefore, the counterexample you gave in your solution doesn't work.
To prove the statement directly, let $u(x)$ be a utility representation that is homogeneous of degree two. That is, $u(\alpha x)=\alpha^2 ...
7
The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $x$ and $y$, this takes the form:
$$
u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.
$$
This is a Cobb-Douglas type of utility function where $a$ and $b$ are subsistence levels, i.e. you need to consume at least $a$ from $x$ and $b$ from $y$ to survive. It is the ...
7
Quadratic utility is given by
$$u(w) = w - b w^2$$
which has derivative
$$u'(w) = 1- 2b w$$
such that for high levels of $w, u'(w)<0$. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less.
The second derivative is
$$u'(w) = -2b$$
such that absolute risk aversion is
$$\frac{- u'...
7
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 $n=1$ and let the preference relation be represented by $u(x)=(1-x)^2$. Then the indifference set e.g. for $u=1$ is $\{0\}\cup\{2\}$, which is not path-...
7
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 for Economists by Angel de la Fuente.
Preferences are monotone if $x > y$ implies $x \succ y$ and that preferences are continuous if $x_n \succeq y_n$, $x_n \...
6
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 abstract is concise and extremely descriptive:
We provide a user's guide to 'exotic' preferences: nonlinear time aggregators, departures from expected utility, ...
6
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. All changes in behavior are driven by natural selection, and so an equilibrium is based on stability.
In a symmetric normal form game, an evolutionarily
...
6
(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 means identical. Indeed in large chunks of literature this is the case, for historical reasons.
The drive behind the adoption of the "representative ...
6
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 opinion should be accepted by the whole at the same measure, as number of issues goes to infinity. In other words, current observed acceptance rate should be a ...
6
Consider the locations (1) $(0.000001,1)$ and (2) $(0.0000005,10)$. $U\left(x_1,y_1\right) = 1.1$.$U\left(x_2,y_2\right) = 10.05$. However, $x_2 < x_1$, so this is not a lexicographic ordering. It is only with the additional constraint that the values of $x$ and $y$ be integers $\in[0,1000]$ that the function you proposes has this attribute. Because real ...
6
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 obtain an (approximate) equation for $R$. The expansion of first order on the RHS is motivated by this fact, i.e. to bring $R$ alone "in the surface". The ...
6
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 having 1 fridge compared to 0 is certainly larger than the gain in utility from having 2 fridges compared to 1. The formula $U("nY")=n*U("Y")$ that you use does ...
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