# Questions tagged [economics]

For mathematical problems arising from economics, the social science studying the production, distribution, and consumption of goods and services.

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### An oversimplified model for optimal distribution of wealth

Consider the following, overly simplified, model for determining an optimal wealth distribution for society:
Let $X$ be a random variable, which will model the distribution of wealth in a society.
The ...

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### Can quasi-concave and monotone functions level curves that are not path-connected?

I posted this on MSE [link] and there's been no answer for the past few days. I started a bounty there and decided to post it here as well.
1. For $X = \mathbb{R}^2$, does there exist a quasi-concave ...

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### Minimising risk in dynamical systems

I have been reading the paper of Goerner and Ulancowicz - "Quantifying economic sustainability" in which it is suggested that there is a tradeoff between sustainability and efficiency. ...

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### Is there a version of Arrow's theorem without unrestricted domain?

To recall Arrow's theorem:
Suppose we have a finite set $X$ of voters and a finite set $Y$ of candidates.
An election is a map $\phi: X \rightarrow T$ where $T$ is the space of total orderings of $Y$. ...

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### Conditions for the existence of von Neumann-Morgenstern utility on a Polish space

Let $X$ be a Polish space, i.e. a separable complete metric space. Any Borel probability measure on $X$ must be locally finite, outer regular and tight. Let $\mathcal{P}(X)$ be the set of all Borel ...

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### Mean-preserving spreads and equality of noise in distribution

Let $X$, $Y$ be mean preserving spreads (MPS) of the same random variable $Q$ and assume that $X =_d Y$ in distribution. Then, by the definition of MPS, there exist variables $Z$ and $Z'$ such that $Q ...

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### The reference on Markov chains uncovering the power of the subject in a better way for a working macro-economist

This is by no means a research question. But asking here I hope for the most expert opinion.
A friend of mine, who is a working economist, asked me for advice about a book which uncovers wealth and ...

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### How to prove this "obvious" result from Bernoulli trials / partial Pascal series?

The main problem setup - very brief
I need to prove that the following two equations have a solution with $CAP=v$.
$$0=
( CAP - C ) \sum_{j=1}^J j N \cdot q^{j-1} (1-q)^{N -1-j} \binom{N}{j} ...

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### Calculus of variations

I have the following question and I wasn't sure if I can apply the calculus of variations to it. The control function is $Q$.
$$\max \int_0^1 t Q(t) dt$$
subject to:
$Q$ is weakly increasing
$Q(0) \...

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### Why are financial markets modeled by càdlàg processes?

When opening a book or reading an article on mathematical finance, financial markets (e.g. stock prices) are always modeled by càdlàg semimartingales. I was wondering why it is that these processes ...

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### Can we characterize the set of neoclassical production functions?

INTRODUCTION
The neoclassical production function is the main building block in neoclassical growth theory, and consequently the main building block of modern macroeconomic theory. Mathematically, ...

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### What is the fairest order for stage-striking (and is it the Thue-Morse sequence)?

Here's a fair-sequencing problem that doesn't quite match the usual fair-division problems. I think that, like those, the answer should also be the Thue-Morse sequence ("balanced alternation"), ...

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### Which utility functions are linearly transformed by normal perturbations?

I'll ask this question as pure economics, as pure math, and showing the translation.
Economics (micro):
Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...

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### Can the sum of quasiconcave functions always be made quasiconcave?

Let $f_1,f_2$ be two smooth quasiconcave functions defined on a convex subset of $\mathbb{R}^d$.
It is known that $f_1+f_2$ is not necessarily quasiconcave.
Does there always exist monotonically ...

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### A game-theoretical question in a political economy model

My research question in a dynamic model of political competition boils down to the following conjecture. I am confident that it holds (all simulations work), but I have not been able to prove it yet. ...

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### A new $\ell_p$-metric on the hyperspace of finite sets?

Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=...

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### Mathematical modelling of wealth distribution [closed]

How is the mathematics in modelling of wealth distribution developed? What kind of mathematics is used and how accurately is it able to model this economic phenomena? An example is the Bouchard Mezard ...

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### Stable marriage with contracts: is it known?

Consider the following generalization of the classical Stable Marriage Problem. The rough idea is that instead of merely specifying who marries whom, a matching now chooses a set of "marriage ...

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### An equivariant social choice in Mathematical economics

Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times\cdots\times M}^{\text{$n$ times}}\to ...

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### Cooperation in asymmetric Prisoners Dilemma

There are 2 players, each can choose 2 actions, a or b. The payoffs in each case are given by rules
Actions (a,a) -> payoffs (3,4)
Actions (a,b) -> payoffs (0,5)
Actions (b,a) -> payoffs (4,0)
...

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### Zero lambda, zero constraint in the complementary slackness condition of the Kuhn-Tucker problem

Complementary slackness condition in the KKT theorem states that:
$\lambda_i^*\geq0; \lambda_i^*h_i(x^*)=0 $
The usual reasoning goes like this: either constraint is clack $h_i(x^*)>0$ and then ...

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### Kalman filters and stock price prediction

Could someone be so kind as to direct me to a good source that would explain time series (more specifically) stock price prediction using Kalman filters, Extended kalman filters or particle filters. ...

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### A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version:
https://www....

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### What are Reinert's reproaches to the Ricardo theory?

Economists accuse me in vulgarization of their science, so I'll edit the text from the very beginning to remove the inaccuracies.
Main question
I have just read the book by a norwegian economist, ...

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### Maximizing Expected Utility

I am currently trying to solve a maximization problem given by
$\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$.
Or in other words, I have a utility ...

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### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...

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### Consistent price index

This question came out of a discussion with a colleague from economics about price indices. Here is MattF's formulation of the question which differs somehow from the original problem.
Let $Y=({\...

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### Simple yet interesting applications of Calculus or Linear Algebra to Economics [closed]

This is essentially a vast generalization of my previous question: Examples of separable ordinary differential equations in economics
I'm giving a talk to college-level math teachers on some ...

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### A categorical analogue of Debreu's independent factors theorem

Background
A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...

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### On mathematical aspects of the most recent Nobel Prize in economics winners' work

Can somebody briefly introduce the mathematical aspects, in particular, those related to mathematical finance, of the three economists who were just awarded this year's Nobel Memorial Prize in ...

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### What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility? [closed]

Let $A$ be the set of all possible states of the world, let $G(A)$ be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over $A$. Now consider an individual with ...

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### A kind of economic objective function in assignment

I recently thought about a concept that seems like it should come up in economics, but I don't know if there's a name for it and where people would have encountered it elsewhere: Suppose we have a ...

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### Optimal auction for risk-averse seller

Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim ...

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### To what equal constant in the Gibbs lemma

The Gibbs lemma is broadly used in games theory and in mathematical economics (optimal distributions of resourses, Cournot competition e.t.c.). Here it is:
Lemma (Gibbs). $f_1,f_2,\ldots,f_n$ be ...

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### Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ?

From memories of a quantum mechanics class and Wikipedia:
In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the ...

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### concept of efficiency in auction theory [closed]

I have some confusions about the concept of "efficiency" in auction theory.
One interpretation is that an auction is efficient if it maximizes the social-welfare. But social-welfare is not well ...

2
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### Young transform reference

The Young transform of nonnegative function $f(x)$, $x \in \mathbb R^n_+$ is defined to be
$$
(\mathscr Yf)(y) = \inf \left[ \left. \frac{x_1 y_1 + \ldots + x_n y_n}{f(x)} \; \right|\; x \colon f(...

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### Market-clearing price vector in an "aggregate demand system"

I suppose this is really an economics question, but I'm posting here for want of a more appropriate forum. My question concerns an aggregate demand system in which we have $n$ variants of a product, ...

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### Concise model of modern fiat money and its non-conservation

A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, ...

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### Examples of separable ordinary differential equations in economics

I'm currently teaching an integral calculus course for business students, and we're just about to discuss differential equations. They've worked hard, and I'd like to reward them with some economic ...

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### Equitable division of a contiguous resource

I have come across the following result regarding equitable division of a resource, which is a simple and immediate consequence of linear programming complementarity (in the infinite-dimensional case)....

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### Algebra - Decomposition of a matrix polynomial

Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer.
What is known about a possible ...

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0
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### Why does the OLS estimator simplify as follows for the single regressor case?

I was reading in "A Guide to Econometrics" that given $Y = X \beta + \epsilon$, the variance covariance matrix of $\beta^\text{OLS}$ is given by $\sigma^2 (X' X)^{-1}$ where $\sigma^2$ is the variance ...

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### Optimal tax Rate

Assume you have two countries A and B, with a tax rates $T_A$ and $T_B$. The tax is redistributed to each people equally. Hence if you live in A and you make $I$ as income then you will finally ...

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### Applications of linear fractional relationship

This may be the wrong forum, but are there any natural contexts (physics, economics, etc.) in which one might observe the relationship $y = ax/(bx+c)$ between a pair of variables $x$ and $y$? General ...

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### Reference for Mathematical Economics

I'm looking for a good introduction to basic economics from a mathematically solid(or, even better, rigorous) perspective. I know just about nothing about economics, but I've picked up bits and pieces ...

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### Fairest way to choose gifts

Suppose that a parent brings home from a trip $2n$ gifts of roughly
equal value for his/her two children. The children get to choose one
at a time which gifts they want. What is the fairest way to do ...

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### Continuous optimization

I'm interested in the solution to the following problem:
I have initial capital $C$ which I can invest into $M$ classes of
resources, each unit of a class $m_i$ matures at time $t_i$, has a
return of ...

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### Zero-knowledge proof of positivity

If I have committed to a number x by revealing g^x mod p, can I prove that 0 < x mod (p-1) < (p-1)/2, i.e. that x is positive, without leaking any more information about x?
My bounty is ending ...

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### Is the max of two supermodular functions supermodular?

A function $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is supermodular if for every $x'>x$ and $y'>y$,
$$f(x',y') + f(x,y) > f(x',y) + f(x,y').$$
Suppose $f$ and $g$ are supermodular, ...