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5

Here is the sketch of a proof. All we need is that every continuous weak order on each $X_i$ admits a continuous utility representation. One sufficient condition is that each $X_i$ is a connected separable topological space by a theorem of Eilenberg. A proof of Eilenberg's theorem is given in Debreu's book Theory of Value. Debreu assumes the domain there ...


4

You can do whatever you want, it's your paper. Will it make it more difficult to publish? Yeah. Referees are fickle and easily annoyed. You would not be the first person to go down this route. For example, the term Maskin monotonicity is common place in implementation theory. However, some authors use another term, I believe Maskin invariance. The claim is ...


4

Decision under uncertainty is sometimes called a "game against chance", and can thus be modeled as a two-player normal form game: the decision-maker vs Nature/Chance. The possible states would form a set of pure strategies for Nature, and Nature commits to a publicly known mixed strategy that randomizes over those pure strategies (assume Nature is ...


3

If I understand correctly you are not interested just in saving but to mathematical approach to normative questions in general. This approach is actually quite common in the whole literature that uses normative economics. For example, the question of redistribution is predominantly normative question, because as opposed to questions of social efficiency, ...


2

In your maximization problem the resident chooses the level of pollution $u$. If he can really do this then the maximization problem has no solution, since $U$ becomes infinite for $u\rightarrow 0$. I guess actually the resident doesn't choose the level of pollution, which rather seems to be an exogenously given parameter here. But then $(x^*,k^*)$ doesn't ...


2

A function $f:D\rightarrow \mathbb{R}$ is said to be quasiconcave if the following set is a convex set for every value of $a\in\mathbb{R}$: $P_a = \{x\in D: f(x) \geq a\}$ To show that $f(x,y) =\min(x, 2y)$ is quasiconcave, we just need to show that $P_a = \{(x,y)\in \mathbb{R}^2: \min(x, 2y) \geq a\}$ is a convex set. For that we consider arbitrary $(x', y')...


2

This has nothing to do with any specific model. For any event $A$, let $I_A$ be the indicator function such that $I_A(\omega)=1$ if $\omega\in A$ and $I_A(\omega)=0$ if $\omega\notin A.$ Then $\mathbb{P}(A)=\mathbb{E}[I_A],$ and here the expectation is given in terms of a density function.


2

This might just be a misunderstanding. The textbook contains various passages like e.g. "An alternative approach holds real income (or utility) constant while examining reactions to changes in $p_x$" (p. 151). When this terminology is introduced, however, "real" stands in quotation marks. This is in the context of explaining the ...


1

I do not have a full answer, but here are my notes when I studied it that hopefully someone can extend to a full answer. Sketch of Proof: Consider the linear space with basis $\cup_{i =1}^N X_i$, and we can identify any $x \in X$ by $\sum_i x_i$. Define the convex cone $D = \{\lambda(x-y): x\succeq y;\lambda > 0\}$ Let $D^{-}$ be the convex hull of $\{...


1

Real income and utility are not the same thing but utility can be expressed as a function of income because income is what allows you to consume goods and services. For example, let us assume there are only two goods $x_1$ an $x_2$ and consumer is given budget $p_1 x_1 + p_2 x_2 = m$, where $p_i$ are prices for good 1 and 2 respectively and $m$ is an income. ...


1

You have the right idea, make a 4 by 3 payoff matrix with fruit at the top and at the side weather/water yes/no, giving 12 different cells, then put the respective dollar payoffs. You could also make another table and find the expected payoff by multiplying each payoff by its probability, although since all the probabilities are the same, the table wouldn’t ...


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