Questions tagged [mathematical-economics]

The application of mathematical methods to represent theories and analyze problems in economics.

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66
votes
13answers
18k views

Fundamental equations in economics

For the other sciences it´s easy to point to the most important equations that ground the discipline. If I want to explain Economics to a physicist say, what are considered to be the most important ...
57
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22answers
13k views

Criticism of Math in Economics

I've been reading and speaking to a number of educated economists and economics PhDs who are against the use of intense mathematics and mathematical proof in economic theory. Specifically I've been ...
22
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2answers
27k views

How can I obtain Leontief and Cobb-Douglas production function from CES function?

In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function, $$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho} ]^{-\frac{1}{\rho}}$$ (where the ...
16
votes
5answers
3k views

Topological concepts in economic theory

QUESTION: What are the major or systematic applications of post-1960s mathematics to microeconomics? For example, in the late 19th century, Fisher first used the mathematical ideas of Gibbs to ...
15
votes
2answers
796 views

Complete Markets in Continuous Time

In the standard discrete time economies with a finite number of states, $n$, a complete markets economy is simply an economy with $n$ independent assets (Think Ljunqvist and Sargent Chapter 8). This ...
14
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5answers
5k views

Applications of Trig functions in Economics?

Are there any applications of trig functions (ie $\sin(x)$, $\cos(x)$,$\tan(x)$) in economics?
14
votes
4answers
2k views

Use of mathematics and imprecise definition of terms

As a postgraduate student of economics I've been trying to expand my mathematical "toolset". While doing so I've talked to engineers, physicists and mathematicians, many of which have disdained the ...
13
votes
2answers
378 views

Uses of convex analysis in Economics

I'm taking kind of a crash-course in convex analysis to complement my mathematical skills and was wondering if anyone knew about nice ways in which this kind of tools were used in Economics. To be ...
13
votes
1answer
1k views

Solving the Hamilton-Jacobi-Bellman equation; necessary and sufficient for optimality?

Consider the following differential equation \begin{align} \dot x(t)=f(x(t),u(t)) \end{align} where $x$ is the state and $u$ the control variable. The solution is given by \begin{align} x(t)=x_0 + \...
13
votes
0answers
287 views

How accurate is duality?

In economic theory we know that with the use of some calculus, Hotellings Lemma and Sheppards lemma we can derive a given firms supply function and in term its Profit function. With data of a given ...
12
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4answers
8k views

Is complex analysis used in economics?

It's frequently useful in physics and engineering applications; are there any applications in theoretical economics? (If not, were there any attempts at incorporating CA that just never caught on?) ...
12
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3answers
849 views

Consumer optimum in an economy with a continuum of commodities

Consider an economy with a continuum of commodities, with one commodity for each point in $[0,1]$. Suppose a consumer wants to maximise $$U = \int_0^1 c_i^\theta\,di\qquad 0<\theta<1$$ subject ...
11
votes
4answers
2k views

Why is the Cobb-Douglas production function so popular?

As relatively novice quantitative analyst/ Cost analyst, Ive been asked to estimate the level of a given organizations productivity more than once, and then forecast for the next couple of periods. ...
10
votes
3answers
898 views

What is the calculable effect of counterfeiting on an economy?

I'm curious whether one can numerically calculate the effect that counterfeiting has on an economy. As I understand it, counterfeiting essentially amounts to theft of the wealth of everybody holding ...
10
votes
3answers
1k views

Why is the derivative used to represent marginal cost instead of the difference?

Marginal cost is defined as "the change in the total cost that arises when the quantity produced is incremented by one unit." And given a total cost function $C(q)$ that's differentiable, the marginal ...
10
votes
3answers
1k views

CES Production Function with $\rho>1$

In using CES production functions of the form $f(x_1,x_2)=(x_1^\rho+x_2^\rho)^{1/\rho} $, we always assume that $\rho\leq1$. Why do we make that assumption? I understand that if $\rho>1$, the ...
10
votes
1answer
3k views

Homogenous of degree one in utility function.

Question My solution is as follows. Please check my solution. If I make a mistake, please tell. I am really not sure about my solution. Thank you U(x) is homogenous of degree one i.e. u(...
10
votes
0answers
714 views

Modern theory of integrability of demand?

I am aware of Hurwickz Uzawa work in integability, neatly summarized by Border http://people.hss.caltech.edu/~kcb/Notes/Demand4-Integrability.pdf I am wondering if there is any modern treatment of the ...
9
votes
7answers
2k views

Macroeconomics Textbook on New-Keynesian models

I'm looking for textbooks that explain the New-Keynesian models, without taking shortcuts on the mathematics, that would go in depth on the derivations of the formulas. I appreciate rigour, and ...
9
votes
2answers
961 views

When can one safely talk about decreasing marginal utility?

One thing I hear a lot is talk of decreasing marginal utility—the idea being that additional units of a good become progressively less attractive the more units of that good one has already. However, ...
9
votes
4answers
2k views

Macroeconomics for Mathematicians

I'm coming from a math background (so my mind is set up for Definitions-Observation-Lemma-Proof frameworks) and looking for recommendations of a Macroeconomics textbook mainly devoted to growth models ...
8
votes
2answers
421 views

Why not talk about utility functions on the surreal line when preferences are lexicographic, etc?

I've been reading about the surreal numbers out of curiosity. Though I don't totally understand them, I wonder why I don't see them used more often. Surreal numbers extend the real line to also ...
8
votes
2answers
610 views

Is there a way to link Berge's theorem of maximum to Envelope theorem?

Berge's theorem states Let $X \in \mathbb R^m, \Theta \in \mathbb R^n $, $f : X \times \Theta \to \mathbb R$ be a jointly continuous function, $C : \Theta \rightrightarrows X$ be a continuous(both ...
8
votes
1answer
2k views

Are homothetic preferences monotonic?

I'm trying to understand intuitively what a homothetic preference is, and I am still not quite there. I understand the definition, that a homothetic preference implies that the slope of the ...
8
votes
1answer
250 views

Applications/generalizations of a theorem of Debreu

I would like to know how the last theorem in Debreu's paper "Neighboring economic agents" (La Decision 171 (1969): 85-90; reprinted in G. Debreu, Mathematical Economics: Twenty Papers of Gerard Debreu ...
8
votes
1answer
366 views

For what demand function is a monopoly most harmful?

Consider a firm with zero marginal cost. If it gives the product for free, then all the demand is satisfied and the social welfare increases by the maximum possible amount; call this increase $W$. ...
8
votes
2answers
331 views

Does economic theory support the notion that the wealth of the wealthy is based on the poverty of the poor?

Almost any discussion about poverty and wealth and income inequality at some point includes arguments based on the premise that the wealth of the wealthy is causally related to the poverty of the poor;...
8
votes
1answer
183 views

Rationality and Common Belief in Rationality in Brandenburger & Dekel (1987)

One of the fundamental results in epistemic game theory is that the solution concept of correlated rationalizability gives exactly those action profiles that are compatible with rationality and common ...
8
votes
0answers
499 views

Calculus and Indifference Curves in an Urban Economics Example

I am reading the paper 'The Structure of Urban Equilibria' by Jan Brueckner. It uses a monocentric city model, where all consumers earn income $y$ at the centre of the city. They buy $q$ housing for ...
8
votes
0answers
311 views

Local and Central Wage Bargaining: What Is the Difference?

Consider the following setting: Profit maximizing firms with production functions $\Pi(w,L)$, where $w$ is the wage and $L$ is employment. Unions who want to maximize the expected utility of their ...
8
votes
0answers
263 views

Show that $W_t - \int_0^t \xi_s ds$ is forward-measure-Brownian

Definitions and stuff: Consider a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ where $$T > 0$$ $$\mathbb P = \tilde{\mathbb P}$$ This is risk-...
7
votes
5answers
295 views

Problems with economic modelling: good textbooks/articles/papers?

I am interested in the problems that may arise when one models economical phenomena using mathematics and indeed also the nature of these models. Are there some good and easily accessible textbooks, ...
7
votes
2answers
759 views

Doesnt convexity prevent thick indifference curves aswell?

It is standard in many micro textbooks when analyzing the relationship between preference axioms and the shape of the utility function (and consequently the shape of indifference curves), to ...
7
votes
2answers
827 views

Constant Elasticity of Substitution: Special Cases

Take an $n$-commodity constant elasticity of substitution utility function, $$U = \left[\sum^n_{i=1} \alpha_i x^\rho_i \right]^\frac{1}{\rho}$$ How do we show the following: Show that as $\rho \...
7
votes
2answers
3k views

Why is Roy's Identity so important?

I've been reviewing some of my microeconomics theory and have been reading up on Roy's Identity. Recall that Roy's identity is defined as: $$x^*_{i}(\text{p},m)=-\frac{\frac{\partial v}{\partial p_{i}...
7
votes
2answers
245 views

Envelope Theorem in Keen and Slemrod (2017)

This question pertains to the paper "Optimal Tax Administration" by Slemrod and Keen (2017). The IMF working paper is freely available on SSRN, however, it is not necessary to know the paper in order ...
7
votes
4answers
1k views

Mathematical open problems that (when answered) might unlock MAJOR mathematical (micro)economics/finance/econometrics discoveries

Are there any (specific) math open problems that mathematical (micro)economics / finance / econometrics researchers wish mathematicians could solve for their discoveries to flourish? If positive, ...
7
votes
3answers
131 views

Interchangeability between knowing an event obtains with probability 1 and knowing an event obtains with absolute certainty?

In the literature of interactive epistemology, for a player, knowing an event obtains with probability one and knowing an event obtains with absolute certainty are different. Is there a nontrivial (...
7
votes
4answers
741 views

Simulating a Hamilton-Jacobi-Bellman

Say I have solved an HJB of the form: $\rho V(k) = \max_c g(c) + V'(k)(z - c)$ I have calibrated $\rho$ to monthly parameters. I would like to simulate the development of $k$. I start with $k(0)$. ...
7
votes
1answer
186 views

Karush-Kuhn-Tucker in infinite dimension

Does the Karush-Kuhn-Tucker theorem on sufficient conditions for optimality of a convex program apply in countable dimension? For precisions, see Definition 4.1.1 and Theorem 4.1.4 of this course. ...
7
votes
1answer
269 views

Compute evolution of a distribution over time

We have a population of people with different age $a$, time is indexed with $t$. There is a rate at which people die, $d(a, t)$. For simplicity, ignore births. I want to compute the evolution of the ...
7
votes
1answer
157 views

How to see that upper-semicontinuity and supermodularity are equivalent in a supermodular game context?

One requirement for a supermodular game $(I, \mathbf S, \mathbf u)$ is usually presented in two ways (e.g. in this note): For $i \in I$, $u_i$ is supermodular in $S_i$, when $s_{-i}$ is fixed, i.e. ...
7
votes
1answer
9k views

Understanding Envelope Theorem

Can someone please verify whether or not my knowledge is accurate. From what I understand, the intent of the envelope theorem is to make a shortcut from indirect utility to the expenditure function. ...
7
votes
1answer
86 views

How likely are the Sonnenschein-Mantel-Debreu results?

I have been reading about the SMD results and it's "damning" nature for GET. My understanding of the result is as follows: price changes not only cause a substitution effect but also change the wealth ...
7
votes
1answer
206 views

Optimization: Dynamic Programming vs Kuhn-Tucker

Considering the standard utility maximization of representative household which lives forever, one may use dynamic programming and Kuhn-Tucker in case of discrete time. For instance, one would like to ...
7
votes
1answer
99 views

Uniform bounds on rate of merging for Bayesian learners

Update. Cross posted at Cross Validated. In a well-known paper, Blackwell & Dubins (1962) show that the posterior probabilities of two Bayesian agents, whose priors agree on events of measure $0$,...
6
votes
1answer
392 views

Is there any economic meaning to log returns?

I recently had a discussion with a friend regarding log returns and any economic meaning that could have as a measure in the sense of "assessing the benefit received for compensating work done in a ...
6
votes
2answers
428 views

Which areas of mathematical knowledge are required for understanding different genres of theoretical economics literature (and at which level)?

I would like to know which mathematical skills are most important to understanding different types of theoretical economics literature. An ideal answer would list disciplines used for different '...
6
votes
2answers
102 views

Does every allocation have a maximal Pareto-improvement?

Consider an economy with a finite number of goods and a finite quantity of each good. Each agent $i$ has a preference-relation $\succeq_i$ which is a total, reflexive and transitive relation over the ...
6
votes
2answers
113 views

Comparative dynamics in forward-looking equation

I have set up and solved an optimization problem with time $t$ endogenous state variables, $\alpha_t$ and $\beta_t$ and choice variable $s_t$. After some manipulation, the first-order condition for $...

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