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$L^{\infty}$: the (Banach, usually) space of bounded measurable functions modulo the equality almost everywhere. Like all function spaces not involving holomorphic functions, it's really a space of eq …

What one needs are two facts: Scalarization/Sufficiency for Pareto Optimality Suppose the economy has finitely many agents $u_i$, $i = 1, \cdots, I$, with feasible allocations given by some $X \subs …

It should be pointed out that just because one encounters complex numbers does not mean one is doing "complex analysis", e.g. complex eigenvalues, complex Borel measures, Fourier transforms, etc. wher …

The minimization problem $$\min_{w(\cdot)} \int^{\pi_{max}}_{\pi_{min}}w(\pi)f(\pi|e)d\pi$$ s.t $$\int^{\pi_{max}}_{\pi_{min}}v(w(\pi))f(\pi|e)d\pi-g(e) = \bar{u},$$ is evidently an infinite dimensio …

For a nice, and gentle, functional analytic treatment on this, see the chapter on welfare theorems in Stokey and Lucas. (Although what they call "inner product" is definitely not standard functional a …

This problem is quite specific to economics. The correct statement is: Proposition If $u(\cdot)$ is quasiconcave, strictly increasing, and continuous, then $\forall x$, there exists $p \gg 0$ and $w …

"Definitions-Observation-Lemma-Proof" is not the right perspective for macro, and one will not, and should not, get that in a proper introduction. Start with an undergraduate text, such as Blanchard, …

The equality in question follows from the expression for expectation of a log-normal distribution. For example, if $X \stackrel{d}{\sim} N(\mu, \sigma^2)$, then $$ E[e^{a X}] = e^{a \mu + \frac12 a^2 …

A pretty trivial assumption would do it: If the consumers preferences are strongly increasing, i.e. $$ x \succ_i y, \mbox{ if $x \geq y$ but $x \neq y$ }, $$ and unbounded for all $i$, then no $x$ …

What is the economic interpretation of the identity (apart from giving the law supply for free)? A more basic question is why does one get the expected sign for the law of supply for free? In other …

Mathematically, market completeness in continuous-time models does not follow from discrete-time heuristics. In discrete-time, market completeness replies on only linear algebraic considerations. Thi …

This was too long for comment. "Post 1960" seems an arbitrary and very high bar for an applied field, including micro theory. Most of the topics you name would not be considered contemporary mathemati …

In economics, the Radon-Nikodym density $\frac{d \mathbb Q}{d \mathbb P}$ of the risk-neutral measure $\mathbb Q$ with respect to the physical measure $\mathbb P$ is the price of Arrow-Debreu securiti …

As you say (how did $\sin t$ in the initial problem become $\cos t$?), $\mathbb{Q}$ is a measure under which $W_t$ becomes $\tilde{W}_t - \sin t$, where $\tilde{W}_t$ is a $\mathbb{Q}$-Brownian motion …

(Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let $W$ be standard Brownian motion with respect to filtration $( \ma …