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4 votes
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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 ...
Beerus's user avatar
  • 505
-1 votes
1 answer
38 views

Wage setting equation

I'm currently doing multiple choice questions and cant figure this question out. I keep getting 1/9 as an answer when its meant to be 10%. Any help would be great. Thanks Suppose that the mark-up of ...
Jack's user avatar
  • 1
1 vote
0 answers
257 views

Question About Non-Degenerated Constraint Qualification (NDCQ)

I am studying constrained optimization using Mathematics for Economists by Simon and Blume, and I have some difficulties understanding the Non-Degenerated Constraint Qualification (NDCQ). I would like ...
Beerus's user avatar
  • 505
2 votes
2 answers
255 views

Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32

I am studying Implicit Function Theorem and its application on comparative statics using Mathematics for Economists by Simon and Blume. Here is the question: Consider a pure exchange economy with two ...
Beerus's user avatar
  • 505
0 votes
0 answers
30 views

Vector Visualisation of regression model vs scatter plot style

I wrote another post discussing the intuition of visualising OLS regression. I wanted to go a bit deeper here visualising the difference between a vector/matrix approach, and the non-linear algebra ...
CormJack's user avatar
  • 991
0 votes
0 answers
52 views

Intuition of Linear Algebra visualisation for regression

Question: I want to clarify my understanding of the basics of OLS regression in matrix form. Let's assume we have 2 different independent variables $x_1$ and $x_2$. Our 'model' will be the plane that ...
CormJack's user avatar
  • 991
3 votes
0 answers
88 views

Orthogonality of two Subspaces

Note: This is a Linear Algebra Question. I'm posting here because I find this community more helpful than maths stack! And ofc linear algebra is fundamental to econometrics, as well! Martin Anthony ...
CormJack's user avatar
  • 991
0 votes
1 answer
72 views

Quadratic Form Single Summation notation

I'm trying to translate what feels like a double summation notation , compacted into a single summation. It's the summation definition of quadratic forms. $Q(x_1, x_2.....x_n)=\sum_{i \le j}a_{ij}...
CormJack's user avatar
  • 991
2 votes
1 answer
56 views

Stochastic optimal control problem (calculus)

I am reading a paper with a stochastic optimal control problem. At one point the author faces the following Hamilton-Jacobi-Bellman (HJB) equation: $$\rho V(A)=\max_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)...
Alessandro's user avatar
3 votes
1 answer
167 views

$N(A) \oplus R(A) = V \; \forall A$?

If $A$ is a $m•n$ matrix. Question: Is $N(A) \oplus R(A) = V \; \forall A$ ? Update: I now think this question makes sense only for square matrices, as noted below. Terminology $R(A)$ By this I ...
CormJack's user avatar
  • 991
1 vote
0 answers
31 views

Right-to-manage wage bargaining (simple algebraic steps)

I am following (and trying to understand) a paper where the wage of unskilled workers is determined as the outcome of wage bargaining between a single union and a single firm in a right-to-manage ...
Alessandro's user avatar
1 vote
0 answers
77 views

I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?

I consider the following communication game. Suppose that we have $I$ players and each one of them learns a private signal $s_i=(s_{i,1},s_{i,2},...,s_{i,k})$, where $k$ is finite and also, every ...
studen21's user avatar
0 votes
1 answer
157 views

How do you convert or move from a linear cost function to a quadratic cost function?

I am reading a book on electricity cost modelling. I understand equation 2.7 below, which indicates that the total cost for an ith plant is a function of fixed cost(FC), fuel cost(FL), plant ...
od320's user avatar
  • 21
2 votes
0 answers
98 views

The quadratic form of variance and covariance components

I am reading Kline, Saggio, Solvsten 2020 and am confused about some basic econometric stuffs in this paper. They begin their introduction as below: """ Consider the linear model $$y_{i}...
Alalalalaki's user avatar
  • 2,419
4 votes
1 answer
1k views

Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?

When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition). As per the calculus wiki: Link, when the Hessian is negative semi-...
Kinno's user avatar
  • 155
2 votes
1 answer
551 views

Why the definition of productive economy in Leontief Open Model is such?

Note: as an inter-disciplinary question, it has its twin on Mathematics Exchange: https://math.stackexchange.com/questions/4193896/why-the-definition-of-productive-economy-in-leontief-open-model-is-...
Dmitrii Demenev's user avatar
1 vote
2 answers
158 views

Understanding utility function curve and marginal rate of substitution

This example appears in a different question, but there is something I don't understand. Maybe this question is better suitated for algebra stackexchange. John’s utility function for food (f) and ...
Jinpert's user avatar
  • 13
2 votes
1 answer
95 views

Regression Optimization problem under constraints

To estimate a simple linear regression: $$ y = \beta_0 + \beta_1 x + \epsilon $$ I have the assumptions that a researcher $A$ can only sample individuals with a value $y < y^A$. Similarly, a ...
Bazinga's user avatar
  • 155
0 votes
1 answer
83 views

How can difference equations with an infinite summation be represented in matrix form?

I have derived the microeconomic foundations of a dsge model and I've obtained the IS and NKPC. I would like to represent them in matrix form to study the system. However the problem is that both ...
qwerty-qwertz's user avatar
2 votes
0 answers
65 views

Recursive Models of dynamic linear economics (Hansen / Sargent, 2014) - optimal linear regulator problem / solution of bellman equation p. 34 ff

The optimal linear regulator problem according to Hansen/Sargent, 2014, Recursive models of dynamic linear economies, on page 34 ff. is stated as follows: $-E\sum_{t=0}^{\infty}\beta^t[x_t' R x_t+u_t'...
user823's user avatar
  • 121
5 votes
2 answers
1k views

Leontief input output model with column sum greater than 1

In a linear algebra textbook I came across the following question (not included in the answer key): Consider an open economy with a consumption matrix \begin{equation} C = \begin{...
user avatar
1 vote
1 answer
875 views

Calculating natural rate of unemployment

I have sample data on unemployment rate in a market and am looking to calculate the natural unemployment rate. The natural unemployment rate I obtained is constant over a time period, which is not a ...
kms's user avatar
  • 131
0 votes
0 answers
46 views

How can I represent this observation regarding options in a formula?

By observing how an option's expiration P/L changes as its underlying asset price changes, we can discover the following system of equations: $\begin{cases}S_{Long} = C_{Long} + P_{Short} \\ S_{Short} ...
user29918's user avatar
0 votes
2 answers
69 views

What are the concepts in Linear Algebra that model the idea of Identification Strategy in Econometrics?

I just would like to know what concepts one should know before talking about identification strategies in econometrics. I see people studying such concepts but I'm not sure they realize (or even know) ...
Caio Velasco's user avatar
0 votes
1 answer
57 views

$x\sim y$ implies $x+a\sim y+a$ for any $a\geq0$ and $x,y\in\mathbb R^n$, then the preference is linear?

$x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (or disprove by counterexample) that: Suppose $x\sim y$ ...
High GPA's user avatar
  • 1,916
0 votes
1 answer
52 views

Derive the growth rate of an equation

I have the following equation: $$\mu =\left [s_{\pi }-v(s_{\pi }-s_{W})+\zeta \right ]$$ And I have to derive its growth rate, which is: $$\dot \mu =-\frac{v}{\mu } (s_{\pi }-s_{W})\dot v$$ Do ...
Alessandro's user avatar
0 votes
1 answer
32 views

Hep with total differentiation of an AD function [closed]

Is there anyone who can help me with a total differentiation exercize. I am starting with the following formula for AD: $$x=\mu ^{-1}(g+i+e)$$ Where $\mu$ is the Keynesian multiplier. And have to ...
Alessandro's user avatar
6 votes
1 answer
201 views

Econ Intuition for Jacobian inverse in demand system

Consider the following simple linear demand system (in vector notation) with n different products Demand: $\quad\mathbf{q=B\left(a-p\right)}$ Inverse demand: $\quad\mathbf{p=a-B^{-1}q}$ where $\...
bbecon's user avatar
  • 688
0 votes
1 answer
329 views

Calculating the elasticity of substitution between factors of production

Following the work of Lu (1967) (Full text available here!) I got stuck trying to derive the elasticity of substitution between factors. He use the formula developed by Allen, that when the production ...
Alessandro's user avatar
0 votes
1 answer
422 views

How to utilize the projection matrix in econometrics?

When consider the following DGP : $y=X\beta^{*}+\epsilon$ where $\beta^{*}$ is a $\tilde k\times1\ $ vector. Define the projection matrices: $P_{X}=X(X^{T}X)^{-1}X^{T}$ and $M_{X}=I-X(X^{T}X)^{-1}$. ...
Jeffrey's user avatar
  • 63
3 votes
1 answer
229 views

Has this differential calculus inequality approach to optimizing the production possibility curve exist?

I just started micro-economics at my community college and my teacher mentioned the derivative of the PPF for two output resources. I thought about it a while and came up with this approach. Some of ...
FX_NINJA's user avatar
  • 131
2 votes
0 answers
100 views

Visualising eigenvectors/values

This might seem like an odd question but seeing as I haven't had any formal education in solving ratex models yet, it is something I have been thinking about a lot recently. Consider the following ...
user11767's user avatar
  • 661
3 votes
2 answers
269 views

Budget hyperplane in n dimensions

Take the set of all vectors $x = (x_1, \cdots, x_n)$ that are solutions to $p_1x_1 + \cdots + p_nx_n = I > 0$. Show that this set has $n-1$ dimensions. I have somehow managed to get myself ...
Kitsune Cavalry's user avatar
  • 6,638
3 votes
1 answer
571 views

When gradient of utility function is a zero vector

In Advanced Microeconomic theory by Jehle and Reny is said that if $\mathbf{x^*}$ is a solution to the following maximization problem $\max_{\mathbf{x} \in \mathbb{R}_+^n} u(\mathbf{x}) $ subject to $...
Vlad Lev's user avatar
2 votes
0 answers
532 views

Short Run vs Long Run Cost Functions

Let $z_a$ and $z_b$ are two vectors of inputs. $z_a$ is variable in both long run and short run however $z_b$ is only variable in long run. Now let's suppose that the price of one of the inputs in ...
Sher Afghan's user avatar
6 votes
3 answers
365 views

How to deal with a singular Leontiev inverted matrix?

I am currently studying and experimenting the input-output methodology. It is a 1930's method based on national accountings that allows to measure the interindustry flow of good and services. ...
Kataplexy's user avatar
3 votes
1 answer
69 views

How can I write a conditional expectation of finite state markov process in matrix notation

NOTE: This question is related to the econometric method explored in the following two questions: Multiplicative factorization of stochastic growth time series--solving for an eigenfunction/...
jmbejara's user avatar
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