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## Hot answers tagged theory

3 votes
Accepted

### 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, ...
• 9,196
3 votes
Accepted

### 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, ...
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 ...
• 13.4k
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 ...
• 145
2 votes
Accepted

### $H$ 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 $g(x,u)$ ...
• 12.5k
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. ...
• 3,703
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 ...
• 12.5k
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 ...
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 ...
• 1,827
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 ...
• 33.9k
2 votes
Accepted

### 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 ...
• 9,196
2 votes
Accepted

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) $\alpha(x)>\... • 9,196 2 votes Accepted ### Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders? The question in the title seems to differ from the question in the body: all/any. At least the question in the title does have a negative answer: Let$A=\{a,b,c\}$. Consider the sub-correspondence of ... • 13.4k 2 votes Accepted ### How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries? That is hopeless. The preference order over certain outcomes determines the preferences over every compatible expected utility representation if and only if there are at most two indifference classes. ... • 13.4k 2 votes ### Can FOSD-transitivity replace the transitivity in utility representation theorem? Take as example the set of lotteries over the outcomes$(1, 5, 10)$consider the lotteries$L_1, L_2, L_3$where, $$\Pr(L_1 = 1) = 0.5\\ \Pr(L_1 = 5) = 0\\ \Pr(L_1 = 10) = 0.5$$ $$\Pr(L_2 = 1) = 0.... • 12.5k 1 vote Accepted ### how to reach Continuous Expected Utility (EU)? I believe we get it from the axioms you have provided with a few technical conditions on the spaces we are working with. Let X be a separable, metrizable space (like your \mathbb{R}) and let \... • 1,924 1 vote Accepted ### MWG Exercise 2.E.5 Answer to 1.$$ x_l(\mathbf{p},w) = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l \cdot \frac{w}{p_l} $$Let \alpha_l = \frac{\partial x_l(\mathbf{p},w)}{\partial w}p_l. This is nice, now use ... • 29.5k 1 vote ### Axioms Underpinning Game Theory I'd suggest to have a look at http://brandlf.com/publications/nash/ and the references therein. • 7,044 1 vote ### Random Utility Model Multiple Choice Question. Which one is correct? I believe the correct answer is (C). Both coefficients are identifiable. The cross-sectional variation identifies the alternative-specific coefficients. It is probably easiest to see why if you assume ... • 3,551 1 vote Accepted ### What are the most common axioms to define a strict preference relation? A relation P is negative transitive if \neg(xPy) and \neg(yPz) imply \neg(xPz). If \succeq is a transitive and complete relation, then the relation \succ defined by x\succ y\iff(x\succeq ... • 13.4k 1 vote Accepted ### Does duality hold for u(x, y) = x^2 + y^2? (Corner solution) The Marshallian demand correspondence is given by:$$ 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 > ... • 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. • 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 \...
• 12.5k

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