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3 votes
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Question About Proving $\alpha(\cdot)$ is Continuous in the Proof of Proposition 3.C.1 from MWG

The sequence you proposed $0,1,0,2,0,3,\ldots$ i.e. $x_n=\begin{cases}0 &\text{if $n$ is odd} \\ \frac{n}{2} & \text{if $n$ is even} \end{cases}$ is not a bounded sequence and is therefore, ...
Amit's user avatar
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3 votes
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Is my understanding of Arrow's dictatorship correct? The dictator is free to update her preference and the social choice will always follow her taste

Yes, this is correct. If $i=1$ is a dictator for the social choice rule $f$, then this means that for any profile $\langle R_i \rangle_{i=1}^I$, we have $f(\langle R_i \rangle_{i=1}^I)=R_1$. Hence, ...
Marcus Pivato's user avatar
3 votes

Debreu's ordinal representation theorem is unique up to a positive monotonic transformation, what is the source?

The other answer explained why the result is trivial; here is why it is not true without the modifier "on the range" and under the most literal reading. Consider $[0,1]\cup(2,3]$ with the ...
Michael Greinecker's user avatar
2 votes

Debreu's ordinal representation theorem is unique up to a positive monotonic transformation, what is the source?

It is not a 'result' because it is trivial to show that a positive strictly monotonic transformation function applied to the function that represents the preferences leaves the ordering unchanged. Say ...
Surge's user avatar
  • 145
2 votes
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$H$ is a constant? Maximizing: $\int _0^Te^{-t}f(x,u)dt$ st $x_t=g(t,x,u)$ and $g$ is independent of $t$

Consider the problem: $$ \begin{align*} \max_{u} \int_0^T f(x(t), u(t)) dt& \\ \text{ s.t. } &\dot x = g(x(t), u(t)),\\ &\text{ + boundary conditions} \end{align*} $$ Assume that $g(x,u)$ ...
tdm's user avatar
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2 votes

Two step Generalized Method of Moments (Newey 1994). $\hat{W}$ matrix depending on the nuisance parameter

As long as $\hat{W}\rightarrow W$ where $W$ is the inverse of the variance-covariance matrix of moments, then all of the asymptotic properties hold. You seem to be asking if this applies in your case. ...
Michael Gmeiner's user avatar
2 votes

Reference for monotonicity: $x\geq y\implies x\succsim y$ and $x>y\implies x\succ y$

MWG defines two types of montone preferences (definition 3.B.2). For a first they define preferences to be montone if $y \gg x$ implies that $y \succ x$. Here $y \gg x$ means that every component of ...
tdm's user avatar
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2 votes

How to self-study graduate level macroeconomics? Is there even a point?

I am currently a university student who found himself in a similar position this year. I wanted an opportunity to do advanced macroeconomic theory, dynare, etc... so I asked my professor. He ...
David deVyver's user avatar
2 votes

Spiraling decline in labour supply due to small initial wage decreases

How do economists address or account for these feedback loops when advising policymakers based on labor supply elasticity measurements? Estimates refer to an equilibrium context. Policy analysts are ...
Iñaki Viggers's user avatar
2 votes

Mixed gambles and risk aversion

I presume that by "fair gamble" $G$ we mean a gamble whose expected value is zero, $E(G) = 0$. Risk-aversion can be expressed mathematically by showing that the utility function is strictly ...
Alecos Papadopoulos's user avatar
2 votes
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Proof: Let $\epsilon>0$ and $x'\in\mathbb{R}^L_+$ be such that $\|x'-x\|\geq\epsilon$. Then $\alpha(x')$ belongs to some $[\alpha_0,\alpha_1]$

We don't need $X$ (domain of the utility) to be equal to the set of its limit points to use the sequential characterisation of continuity. In other words, the sequential characterisation of continuity ...
Amit's user avatar
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2 votes
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Question About the Step Proving $\alpha(x)$ Represents Preferences in the Proof of Proposition 3.C.1 from MWG

Def 3.B.2: The preference relation $\succsim$ on $X$ is monotone if $x\in X$ and $y\gg x$ implies $y\succ x$. [1] If $\alpha(x)\geq \alpha(y)$, then there are two possibilities: (i) $\alpha(x)>\...
Amit's user avatar
  • 9,196
2 votes
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Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders?

The question in the title seems to differ from the question in the body: all/any. At least the question in the title does have a negative answer: Let $A=\{a,b,c\}$. Consider the sub-correspondence of ...
Michael Greinecker's user avatar
2 votes
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How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?

That is hopeless. The preference order over certain outcomes determines the preferences over every compatible expected utility representation if and only if there are at most two indifference classes. ...
Michael Greinecker's user avatar
2 votes

Can FOSD-transitivity replace the transitivity in utility representation theorem?

Take as example the set of lotteries over the outcomes $(1, 5, 10)$ consider the lotteries $L_1, L_2, L_3$ where, $$ \Pr(L_1 = 1) = 0.5\\ \Pr(L_1 = 5) = 0\\ \Pr(L_1 = 10) = 0.5 $$ $$ \Pr(L_2 = 1) = 0....
tdm's user avatar
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1 vote
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how to reach Continuous Expected Utility (EU)?

I believe we get it from the axioms you have provided with a few technical conditions on the spaces we are working with. Let $X$ be a separable, metrizable space (like your $\mathbb{R}$) and let $\...
Walrasian Auctioneer's user avatar
1 vote
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MWG Exercise 2.E.5

Answer to 1. $$ x_l(\mathbf{p},w) = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l \cdot \frac{w}{p_l} $$ Let $\alpha_l = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l$. This is nice, now use ...
Giskard's user avatar
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1 vote

Axioms Underpinning Game Theory

I'd suggest to have a look at http://brandlf.com/publications/nash/ and the references therein.
VARulle's user avatar
  • 7,044
1 vote

Random Utility Model Multiple Choice Question. Which one is correct?

I believe the correct answer is (C). Both coefficients are identifiable. The cross-sectional variation identifies the alternative-specific coefficients. It is probably easiest to see why if you assume ...
Jesper Hybel's user avatar
  • 3,551
1 vote
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What are the most common axioms to define a strict preference relation?

A relation $P$ is negative transitive if $\neg(xPy)$ and $\neg(yPz)$ imply $\neg(xPz)$. If $\succeq$ is a transitive and complete relation, then the relation $\succ$ defined by $x\succ y\iff(x\succeq ...
Michael Greinecker's user avatar
1 vote
Accepted

Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)

The Marshallian demand correspondence is given by: $$ x(p_x, p_y,m) = \begin{cases} \{m/p_x\} &\text{ if } p_x < p_y\\ \{0, m/p_x\} &\text{ if } p_x = p_y\\ \{0\} &\text{ if } p_x > ...
tdm's user avatar
  • 12.5k
1 vote
Accepted

Goods for resale are considered as producer goods?

Yes, as the definition says, the goods purchased for resale, like carrots purchased by grocer, are intermediate/producer goods.
WilliamT's user avatar
  • 1,847
1 vote

Give bundles $x,y\in \mathbb R^n$, there must exist a budget $B\supset\{x,y\}$ and a demand $D(B)\in[x,y]$?

Let $U$ be a $C^1$, monotone and quasi-concave utility function. Let $\nabla U(z)$ be the gradient of $U$ at the bundle $z$ (If we assume preferences are monotone then $\nabla U(z) \gg 0)$ for all $z \...
tdm's user avatar
  • 12.5k

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