Questions tagged [mathematical-economics]
The application of mathematical methods to represent theories and analyze problems in economics.
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Cost Benefit of Publishing Novel Economic Result and Equations
In the course of developing various trading models, I have come across, and further developed equations that deal with predicting states of efficient markets. These may also have potential for risk ...
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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.
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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 ...
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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 ...
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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 ...
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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)
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Demonstrate order and rank conditions for identification with instrumental variables
For an equation in a simultaneous system to be identified two conditions must hold: i) the order condition, and ii) the rank condition.
b_IV=(Z'X)^(-1) Z'Y
How to demonstrate in matrix form that both ...
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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 ...
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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 ...
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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 ...
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Cost function from a weighted CES production function
I want to find the cost function given the CES production function:
$$
Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho}
$$
with $0<\rho<1$.
So far I have set up the ...
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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" ...
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Different elasticities of substitution
I have been reading into generalizations of the concept of elasticity of substitution for more goods/inputs and three main possibilities emerged:
Hicksian EOS
Allen-Uzawa EOS
Morishima EOS
HICKS
As ...
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Ramsey model problem
Explain the no-Ponzi game condition throughout the function.
Does the constrain function: also reflect include the No-Ponzi condition?
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How to test if the effect of one regressor entirely comes from other regressors?
I have a regression model that includes IQ test scores as the dependent variable; my own education, my father's education and my mother's education as independent variables. Suppose I want to know ...
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Proving an additive function is sigma additive (Measure Theory)
I'm here to kindly ask u a check about the proof of the following statement:
Let $\mathcal{A}$ be an algebra and $\mu : A \rightarrow [0,\infty]$ and additive function. Then $\mu$ is $\sigma$ ...
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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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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 ...
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Confusion regarding the Slutsky equation
I'm reading Henderson and Quandt's Microeconomic Theory textbook and in the derivation process of the slutsky equation the final formula confused me a bit. The first term on the right of the equation ...
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Cost Minimization
Guys in the following optimisation problem why are we not considering the partial derivate of K and L in the Lagrangian in the first two equations
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What kind of utility function is needed to derive a "Backwards Bending Labor Supply Curve"
The idea of a backwards bending labor supply curve is often discussed in many introductory labor economics classes where a consumer's labor leisure decision at higher wages (due to a diminishing ...
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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 ...
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Why this optimisation problem cannot be solved with "usual" KT conditions?
I have this optimisation problem:
$$f(x, y, z) = 2xy + yz \qquad \text{subject to} \qquad \begin{cases} x+y+2z \leq 1 \\ x \geq 0, y \geq 0, z\geq 0 \end{cases}$$
I solved it with "a certain ...
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Why is exponential discounting considered rational?
I have repeatedly heard a claim that only an exponential discounting is the rational one, however I've never seen the proof. Why is it so?
Intuitively, we would expect exponential discount since the ...
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Consumer surplus in Logit model should minus actural payment?
In Logit model, as Train(2003) said in his book(page 55) said,
"By
definition, a person’s consumer surplus is the utility, in dollar terms,
that the person receives in the choice situation. The ...
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Judge prosecutor example of Kameinica and Gentzkow
Based on Kamenica and Gentzkow example about the prosecutor and the judge. The utilities of the sender and the receiver are $v(\alpha,\omega)=\alpha$ and $u(\alpha,\omega)=-(\alpha-\omega)^2$.
The ...
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Question About Stable Matrices - MWG Mathematical Appendix M.D Matrices: Negative (Semi)Definiteness and Other Properties
I am self-studying the mathematical appendix of Microeconomic Theory (MWG). They defined a stable matrix as follows:
Definition$\quad$ A matrix is stable if all of its characteristic values have ...
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Is debt eventually negative for nonnegative $y_t\to 0,c_t\not\to 0$?
Suppose $d_t=\sum_{k=0}^\infty \rho^k (y_{t+k}-c_{t+k})$ and $\rho\in (0,1),y_t\geq 0,c_t>0\quad \forall t\geq 0$. Does $y_t\to 0,c_t\not\to 0$ imply $d_t$ is eventually negative?
My initial hunch ...
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How to aggregate goods with different units of measurement to reduce the economy with a Cobb Douglas utility function?
I want to model a economy where consumers have a Cobb Douglas utility function and where X1 = goods that pay a value added TAX (VAT), and X2 = goods that are exempt from this tax.
I am working with ...
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Drawing a Probability simplex
There are 3 possible payoffs - \$4, \$9 and \$36. The utility function for these payoffs is $\sqrt x $.
I have to find all the lotteries preferable over getting $9 with probability 1 in a probability ...
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Proof that utility function is differentiable
after a little problem (I asked my question in the answer section, apologize, all my bad for this), I repost my question here with the same message :
"I'm new on economics stackexchange, and I've ...
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Can anyone help with these calculations?
This is from the this paper in section $3$ about the two period example.
Suppose that we have the following two period, $t=1,2$, sender(S) - receiver(R) model.
For an action path $a=(a_1,a_2)$ and a ...
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Calculating elasticity between terms in a regression equation
Given the following regression: $ln(w_i)=\beta_1+\beta_2age+\beta_3age_i^2+\beta_4Y_i+\beta_5T_i+\beta_6Mar_i+\epsilon_i$ I am asked to calculate the elasticity of wages with respect to age. Is the ...
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Intuition of two Measure theory statements
I'm struggling in getting the intuition of two statements about measure theory:
Given a measure space $(X,F,\mu)$, $f \in M^+ $, where $M^+ = M^+(F) $ is the set of non negative F-measurable functions ...
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Prove: The law of demand holds if WA, Walras' law, homogeneity of degree 0, and homogeneity of degree 1 in wealth hold for Walrasian demand functions
Problem
I am asked to prove the following result (MWG Exercise 2.F.5):
The law of demand always holds if the walrasian demand function $x(\mathbf{p},w)$ satisfies the weak axiom of revealed ...
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CES function: Lagrangian or Kuhn-Tucker
Let's have the following problem:
$$U(\boldsymbol{x}) = \left(x_1^\rho + x_2^\rho \right)^{\frac{1}{\rho}} \qquad s.t. \qquad P_1 x_1 + P_2 x_2 \leq M$$
Is it optimal to solve this problem via pure ...
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Why did Bergemann and Morris chose a different setting with respect to Kamenica and Gentzkow only to reach to the same results?
Why did Bergemann and Morris choose a different setting with respect to Kamenica and Gentzkow only to reach to the same results? However none of both approaches seem to present the problem in a ...
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Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?
Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable.
...
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Mechanism design making the Government reveil its indiferrence price towards goods
In order to Tax non-monetary property and wealth the government sets an arbitrary price (they are not even personalized assesments) and often people transact below that price. I have bought and sold ...
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Question on Isolating The Wealth Effect in Analysis of Changes in Price-Wealth Combinations - MWG Exercise 2.F.3 Parts (e) and (f)
I am doing exercises in Chapter 2 of MWG. I feel I got completely lost in exercise 2.F.3 parts (e) and (f).
$\textbf{Exercise}$
Here is the question:
I have solved parts (a) to (d). In particular,
I ...
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Continuous logit framework
I am reading Dupuy & Galichon (2014), which extend the estimation of matching model in Choo & Siow (2006) to continuous types. The way they build the continuous logit model is based on the ...
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Is this proof correct (measure theory)?
I'm trying to prove the following statement:
I did the following proof but i donno if it makes sense:
Proof of $(\Rightarrow)$ direction:
Assume $\mathcal{F}$ is a $\sigma$-algebra. Assume also $\{...
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Does the Sonnenschein-Mantel-Debreu theorem fundamentally undermine Mises' Economic Calculation Argument?
So recently I have been thinking a lot about this fundamental question: Does the Sonnenschein-Mantel-Debreu theorem disprove the "Law of Demand"?
Basically, the Sonnenschein-Mantel-Debreu ...
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Resources for grad level math courses
I am currently waiting on Econ PhD decisions. I thought I could use this time to start brushing up on some coursework. I wished to know if there are any resources for graduate level mathematical ...
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Alpha interpretation in Solow growth model
Consider the Solow model (without technology):
$Y = F(K, L) = K^\alpha L^{(1-\alpha)}$
What's the economic interpretation of $\alpha$? Prove and argue the result.
I see it as a share that goes into ...
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Can following these marginal conditions have the net utility function converge to a maximum? [Edited]
I have the following net utility function which is made up of one positive utility (with a bliss point) and two negative utility (i.e., disutility) functions;
$$Y(a_1,a_2)=y(a_1)+v(a_1,a_2)+w(a_2).$$
...
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Question on The Weak Axiom of Revealed Preference and The Definition of Revealed Preference Relation
I am solving the following problem (from Exercise 2.F.3 (b) in MWG) and I got confused by the weak axiom of revealed preference and the definition of the revealed preference relation. Here is the ...
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Trouble understanding the equation that lays out the concept of prediction policy problems in Kleinberg et al. 2015
Kleinberg et al. (2015) introduce the concept of prediction policy problems, and claim to unite prediction problems and causal inference problems in a single framework.
In section I of the paper, ...
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Resources for a Deep Dive into the New Keynesian / DSGE models
I would like to deep dive into new Keynesian models and DSGE models. Do you know comprehensive resources that cover both the mathematical and economic requirements?
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Multi dimensional Auction in economics
I am following this paper . They have different suppliers and one buyer and They are using auction to select best suppliers Suppliers will submit. suppliers offer a multidimensional bidding on quality ...
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