# 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, ...
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### 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 ...
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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 ...
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### 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 ...
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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 $... • 167 5 votes 1 answer 78 views ### Question About the Step Proving$\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$. ... • 167 2 votes 0 answers 54 views ### Question About the Step Involving the Connectedness of$\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 ... • 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 ... • 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 ... • 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 $$L(x_1,... • 65 2 votes 2 answers 27 views ### Positive monotonic transformation and identical preferences Two consumers, A and B have the following utility functions$$ u^{A}(x_1,x_2)=x_1(x_2)^\alphau^{B}(x_1,x_2)=\frac{1}{3}\ln(x_1)+\frac{2}{3}\ln(x_2) $$I have to find \alpha such that the ... • 53 3 votes 1 answer 230 views ### When is argmax increasing in some multiplier of the objective function? Let f : R_+ \to [0,1] be continuously differentiable and strictly increasing with f(0)=0, \lim_{x\to\infty}f(x)=1 and let c : R_+ \to R_+ be continuously differentiable, strictly increasing, ... 0 votes 1 answer 49 views ### What is the formula for price how to calculate price? Is price just = to demand/supply? I wanted to understand an inflation and wiki says that it is a general increase in the prices. Then we need some assumptions / axioms abut the ... 1 vote 1 answer 35 views ### Revealed preferences - Commuter preferences I'm working on a paper about commuter preferences and their level of satisfaction in a South American country. Do they prefer the metro or informal buses? That sort of thing. The data tells me that ... • 51 4 votes 2 answers 130 views ### Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign. If the above is ... 2 votes 0 answers 37 views ### Positive monotonic transformation I want to draw an indifference curve for this preference relation on ℝ^2$$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 ... • 53 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 ... • 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 ... • 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 &... 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 ... • 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 ... 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. ... • 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 ... • 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 ... • 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 ... • 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) 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 ... 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 ... • 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 ... • 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" ... 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 ...
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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 ...