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Questions tagged [mathematical-economics]

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

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Proof of the Gibbard-Satterthwaite theorem in Osborne and Rubinstein

I'm self-studying Osborne and Rubinstein's Models in Microeconomic Theory and I'm struggling to understand a step in their proof of the Gibbard-Satterthwaite theorem. I won't provide the full proof, ...
Viktor Glushkov's user avatar
2 votes
2 answers
76 views

FOCs for the Dixit-Stiglitz aggregator: derivative of an integral w.r.t. integrand at one point

This is perhaps more of a mathematical than an economic question, but since it's from an economic problem, I'm posing it here. Suppose I have, in a macro model, a perfectly competitive final goods ...
chsk's user avatar
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1 vote
0 answers
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Is this a typo in Sydsaeter's book of image definition?

On page 543 of the textbook Further Mathematics for Economic Analysis (second edition), the definition of (direct) image is the following: Let $f: A \rightarrow B$ be a function. The (direct) image ...
Shujun Tan's user avatar
1 vote
0 answers
18 views

A Measure Theory\Probability Question Regarding Model Setting of Ambiguity (Schmeidler (1984, 1989))

In David Schmeidler (1984), "Subjective Probability and Expected Utility without Additivity", "2. Axioms and Background", there is a setting: Let $X$ be a set and let $Y$ be the ...
Chengchuan Liu's user avatar
0 votes
1 answer
75 views

Proof that if the MRS is increasing in one good, then the other is normal

Osborne and Rubinstein (Models in Microeconomic Theory, p. 70) prove the proposition that, if the demand function of a rational consumer has the property that $\text{MRS}(x_1, x_2)$ is increasing in $...
Riccardo Iorio's user avatar
5 votes
1 answer
259 views

Proof that independence implies monotonicity in Osborne and Rubinstein

I'm struggling to understand a proof in Osborne and Rubinstein's Models in Microeconomic Theory (p. 35). The relevant lemma is Let $Z$ be a set of prizes. Assume that $\succeq$ is a preference ...
Riccardo Iorio's user avatar
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0 answers
65 views

Stackelberg Model with inseparable variables: How to find sequential equilibrium?

I have a leader who has to optimise quantity, x, inputting the quantity of follower, y. Problem is quantity equation of leader and follower are not separable. Leader's objective function is - $$ \sqrt{...
Elina Gilbert's user avatar
2 votes
0 answers
53 views

Where does Arrow’s Impossibility Theorem standard proof rely on the existence of at least three alternatives?

I am trying to understand the proof of the world-famous Arrow’s Impossibility Theorem. To do that, I have been relying on Wikipedia’s proof by decisive coalition, as well as on the proof provided by ...
EoDmnFOr3q's user avatar
9 votes
1 answer
324 views

Challenging question on mathematical economics

I'm going through the paper of Acemoglu and Autor (2011, Handbook of Labor Economics) Consider the following system of two equations: \begin{equation} \frac{A_M \alpha_M (I_H) M}{I_H - I_L} = \frac{...
Maximilian's user avatar
2 votes
1 answer
70 views

Properties of Multivariate Frechet Distributions

I was reading an answer from another Economics Stack Exchange post and I had further questions. The highest up-voted answer here utilized two properties of multivariate Frechet distributions, but the ...
nate's user avatar
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1 vote
1 answer
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Can global real GDP be calculated using nominal GDP and inflation rate? If so what is the error?

I am attempting to calculate (in R) the global real GDP using what I assume is the nominal GDP from the World Bank and what I can only identify as the inflation rate, also from the World bank. ...
At_my_wits_end_99's user avatar
1 vote
0 answers
44 views

Kohlberg-Mertens theorem

I am trying to understand the Kohlberg-Mertens theorem. Here is the context and the theorem. We fix $N$ a set of players and $S_i$ the finite set of actions of each players with $\lvert S_i\rvert =m_i$...
G2MWF's user avatar
  • 131
0 votes
1 answer
76 views

Confusion on the Profit Maximization Equation [closed]

So, I found a formula for Profit Maximization on this page. https://en.m.wikipedia.org/wiki/Profit_maximization Listed in the Marginal Revenue- marginal cost perspective section. My question is, what ...
DanceroftheStars's user avatar
1 vote
0 answers
40 views

Flow budget constraint in a paper by Garleanu and Panageas

I am reading this paper and got confused by Equation (6) in said paper. Suppose there is an investor that can trade in riskless bonds that pay interest rate $r_t$, and holds a market portfolio with ...
Wittgenstein's Poker's user avatar
0 votes
1 answer
64 views

Independence Axiom and Expected Utility Theorem Proof

In my micro class we covered the proof of the existence of a Von Neumann–Morgenstern utility representation of preferences $\succeq$ over a set of lotteries $\Delta(Z)$ - where $Z$ is some finite ...
Joseph Basford's user avatar
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0 answers
28 views

Derivation of Euler Equation in presence of the Dixit-Stiglitz Aggregator

I reading the working paper of Sebastian Banz (2012). I have an issue with the derivation of the Euler equation. The author models the demand side of the economy as follows The representative consumer ...
Maximilian's user avatar
1 vote
0 answers
34 views

Reconciling Continuity of Binary/Preference Relations with Continuity of Functions/Correspondences

There are various ways to express the concept of continuity of a binary relation, but one I've come across seems to imply the closed-graph property is sufficient. That is to say: For a nonempty set $X$...
hillard28's user avatar
3 votes
0 answers
88 views

Mean Preserving Spreads as a Markov Kernel

From Blackwell's theorem, it is possible to draw an equivalence between one signal being a garbling of another and the statistical notion of a Mean Preserving Spread. Consequently, at least in the ...
matthewoulton's user avatar
9 votes
3 answers
1k views

Cost Minimization and Karush-Kuhn-Tucker

A firm produces an output $y$ using two inputs $x_1$ and $x_2$, where the production function is given by $y = \sqrt{x_1 x_2}$ for any $(x_1, x_2) \in \mathbb{R}^2_+$. Union agreements obligate the ...
bruno's user avatar
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1 vote
0 answers
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About part of Romer's model. Is my math process correct?

$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\ s.t.\int_{i=0}^{A}L(i)^{\phi}di=1 $$ $$ \begin{aligned} \frac{d\mathcal{L}}{dL}&=\int_{i=0}^{A}p(i)di-\...
Barry Alen's user avatar
6 votes
1 answer
72 views

Question About Proving $\alpha(\cdot)$ is Continuous in the Proof of Proposition 3.C.1 from MWG

See Proposition 3.C.1 from MWG Continue from this post, Microeconomic Theory by Mas-Colell et al. book said the following: What remains is to establish that all convergent subsequences of $\{\alpha(x^...
Champa's user avatar
  • 167
3 votes
1 answer
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Proof: Let $\epsilon>0$ and $x'\in\mathbb{R}^L_+$ be such that $\|x'-x\|\geq\epsilon$. Then $\alpha(x')$ belongs to some $[\alpha_0,\alpha_1]$

See Proposition 3.C.1 from MWG Continue from this post, the book (MWG) then started the proof that $\alpha(x)$ is a continuous function: We now argue that $\alpha(x)$ is a continuous function at all $...
Champa's user avatar
  • 167
5 votes
1 answer
78 views

Question About the Step Proving $\alpha(x)$ Represents Preferences in the Proof of Proposition 3.C.1 from MWG

Continue from this question, the book Microeconomic Theory by Mas-Colell et al. said We now take $\alpha(x)$ as our utility function; that is, we assign a utility value $u(x)=\alpha(x)$ to every $x$. ...
Champa's user avatar
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2 votes
0 answers
54 views

Question About the Step Involving the Connectedness of $\mathbb{R}_+$ in the Proof of Proposition 3.C.1 from MWG

This is a small issue, but it confuses me. In the proof of Proposition 3.C.1 of Microeconomic Theory by Mas-Colell et al, the book said ... $\mathbb{R}_+\subset (A^+\bigcup A^-)$. The nonemptiness ...
Champa's user avatar
  • 167
2 votes
2 answers
124 views

Proving quasi-concavity for a utility function

I have a utility function, and I want to prove that it is a quasi-concave function: $$ u(x_1,x_2)= 2x_1x_2+x_1+2x_2 $$ I do this by showing that the set of points where the utility is larger than or ...
Noah's user avatar
  • 65
2 votes
1 answer
76 views

Using lagrange on a quasi-concave utility function

A consumer has the following utility function $$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$ I have maximized his utility function, and found its demand functions, for $x_1$ and $x_2$, using Lagrange. However, is it ...
Noah's user avatar
  • 65
2 votes
1 answer
213 views

Solving utility maximization, and finding demand function

A consumer has the following utility function $$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$ I want to maximize his utility function. $$max: 2x_1x_2+x_1+2x_2. uc:p_1x_1+p_2x_2=y_A$$ Using Lagrange, I get $$L(x_1,...
Noah's user avatar
  • 65
2 votes
2 answers
27 views

Positive monotonic transformation and identical preferences

Two consumers, A and B have the following utility functions $$ u^{A}(x_1,x_2)=x_1(x_2)^\alpha$$ $$u^{B}(x_1,x_2)=\frac{1}{3}\ln(x_1)+\frac{2}{3}\ln(x_2) $$ I have to find $\alpha$ such that the ...
Jonathan's user avatar
3 votes
1 answer
230 views

When is argmax increasing in some multiplier of the objective function?

Let $f : R_+ \to [0,1]$ be continuously differentiable and strictly increasing with $f(0)=0, \lim_{x\to\infty}f(x)=1$ and let $c : R_+ \to R_+$ be continuously differentiable, strictly increasing, ...
raving-bandit's user avatar
0 votes
1 answer
49 views

What is the formula for price

how to calculate price? Is price just = to demand/supply? I wanted to understand an inflation and wiki says that it is a general increase in the prices. Then we need some assumptions / axioms abut the ...
user avatar
1 vote
1 answer
35 views

Revealed preferences - Commuter preferences

I'm working on a paper about commuter preferences and their level of satisfaction in a South American country. Do they prefer the metro or informal buses? That sort of thing. The data tells me that ...
KEll's user avatar
  • 51
4 votes
2 answers
130 views

Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization

It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign. If the above is ...
hillard28's user avatar
2 votes
0 answers
37 views

Positive monotonic transformation

I want to draw an indifference curve for this preference relation on $ℝ^2$ $$x ≿ y \leftrightarrow (x_1+x_2)^2 ≥ (y_1+y_2)^2$$ Can I just take the square root(and then drop the absolute value sign ...
Jonathan's user avatar
0 votes
1 answer
18 views

Market signaling with a national exam

Question: Consider an economy with a competitive labour market in which firms pay a wage equal to the expected productivity of the employee. There are two types of employees: Good (with a productivity ...
ZZZ's user avatar
  • 25
0 votes
0 answers
28 views

How to find the amount of leisure time and the good consumed in a two-person and one-good economy?

Question: Suppose an economy consists of two price-taking people, “a” and “b.” Person a has available 1 unit of “time” which he divides between rest $R(a)$ and labour $L(a)$. Person b has available 1 ...
ZZZ's user avatar
  • 25
3 votes
0 answers
49 views

Connection between pricing kernels and kernel methods in linear algebra

I'm trying to understand the idea of pricing kernels. I have a maths/computer science background so I understand that kernel methods map information from "data" space into high-dimensional &...
Medulla Oblongata's user avatar
2 votes
2 answers
75 views

What is the rationale behind randomly denying clients for loans

Bank AAA is the only bank in the country and it has a very strange loan policy. It sets a relatively low interest rate for loans (but still high enough to make a profit) so that the demand for loans ...
ZZZ's user avatar
  • 25
5 votes
3 answers
2k views

Economists Working At the Intersection of Pure Math and Economics

As a math student interested in economics I was recently reading some of the papers by Graciela Chichilnisky on social choice theory and some of her papers applying topology to other areas of ...
throwawayaccount's user avatar
2 votes
0 answers
70 views

Berge's theorem of maximum's application to expenditure minimisation problem

Let $f:X\times A \rightarrow \mathbb{R}$ be a continuous function, and $\Gamma:A \rightrightarrows X$ be a non-empty, compact-valued and continuous correspondence, with $X$,$A$ be Euclidean Space. ...
Nonenicht's user avatar
  • 113
2 votes
0 answers
48 views

Proof of Second Welfare Theorem with infinitely many goods

I'm trying to understand the proof of the Second Welfare Theorem and I'm still struggling with the intuition behind the use of a linear functional as a mathematical instrument in order to prove that ...
Pepus's user avatar
  • 93
5 votes
1 answer
104 views

About Two Methods of Computing Bayesian Equilibria

Question I want to compute the Bayesian equilibria for the following Bayesian game: With probability $p$, player 1 would be of type 1.1. With probability $1-p$, player 1 would be of type 1.2. Player ...
Beerus's user avatar
  • 505
0 votes
0 answers
23 views

consumption, income and utility

Question Description: Consider an economy populated by a large number of Farmers (F) and Computer Scientists (CS). Each person divides his 24-hour day into labour and leisure. If a Farmer decides to ...
ZZZ's user avatar
  • 25
1 vote
2 answers
84 views

How to derive Fisher's equation?

Guys help me derive Fisher's equation or suggest references that have the derivation. I am looking for this relation: (1+i)=(1+r)(1+π)
user205152's user avatar
1 vote
0 answers
21 views

Parameter in calibration are not sufficient for pruposed Model. Where is the lacking data coming from?

The paper „social security reforms, capital accumulation and welfare: A notional defined contribution system vs a modified payg system“ by Lin, li Presents a model for analyzing the impact of ...
Felix Neuwirth's user avatar
3 votes
1 answer
92 views

Question About Proof of Proposition 3.C.1 in MWG - Step 1

I have difficulties understanding the first step of the proof of Proposition 3.C.1 in MWG. Proposition 3.C.1$\quad$ Suppose that the rational preference relation $\succsim$ on $X$ is continuous. Then ...
Beerus's user avatar
  • 505
3 votes
1 answer
75 views

Understanding the definition of monotone

In Microeconomic Theory by Mas-Colell, Whinston, and Green, the definition of monotone preference relations is given as follows: Definition 3.B.2$\quad$ The preference relation $\succsim$ on $X$ is ...
Beerus's user avatar
  • 505
3 votes
1 answer
71 views

Demand aggregation with CES

I am reading the following paper by Angeletos and Lian and got stuck on equation (3). The problem is the standard optimal consumption bundle choice. Namely, aggregate consumption of "farmer" ...
Wittgenstein's Poker's user avatar
0 votes
1 answer
50 views

Two candidate voting game on the unit interval ideological spectrum (game theory exercise)

(Politics game) Consider a population of voters uniformly distributed along the ideological spectrum from left $(x = 0)$ to right $(x = 1)$. Each of the candidates for a single office simultaneously ...
Oliver Queen's user avatar
1 vote
1 answer
236 views

How To Publish a Quantitative Economics Paper Without a Degree or Affiliations

I dropped out of a physics degree and do not have a degree. After this, I somehow found myself going from writing marketing software related code to writing quantitative trading code and doing ...
cronius's user avatar
  • 11
0 votes
0 answers
45 views

Chan Park - Fundamentals of Engineering economics - 1st edition - Chapter 5 - Present value - 5.8

hope I could ask some help around here. I'm working with Chan S Park book Fundamentals of Engineering Economics, 1st edition. I'm in the problem 5.8, page 171. Which states: ``Your firm is considering ...
GearKnight95's user avatar

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