Questions tagged [linear-algebra]
The linear-algebra tag has no usage guidance.
35
questions
1
vote
0
answers
181
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 ...
- 251
2
votes
2
answers
188
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 ...
- 251
0
votes
0
answers
25
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 ...
- 897
0
votes
0
answers
47
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 ...
- 897
3
votes
0
answers
77
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 ...
- 897
0
votes
1
answer
40
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}...
- 897
1
vote
1
answer
52
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:
$$\rho V(A)=\max_{c,w}\left\{ u(c)+V'(A)\left[(r(1-w)...
- 165
3
votes
1
answer
118
views
$N(A) \oplus R(A) = V \; \forall A$?
If $A$ is a $m•n$ matrix.
Question: Is $N(A) \oplus R(A) = V \; \forall A$ ?
Update: I now think this question makes sense only for square matrices, as noted below.
Terminology
$R(A)$ By this I ...
- 897
1
vote
0
answers
30
views
Right-to-manage wage bargaining (simple algebraic steps)
I am following (and trying to understand) a paper where the wage of unskilled workers is determined as the outcome of wage bargaining between a single union and a single firm in a right-to-manage ...
- 165
1
vote
0
answers
76
views
I present a communication game - Could you please make comments on my assumptions, notation and properties that I may have not considered yet?
I consider the following communication game. Suppose that we have $I$ players and each one of them learns a private signal $s_i=(s_{i,1},s_{i,2},...,s_{i,k})$, where $k$ is finite and also, every ...
- 43
0
votes
1
answer
124
views
How do you convert or move from a linear cost function to a quadratic cost function?
I am reading a book on electricity cost modelling. I understand equation 2.7 below, which indicates that the total cost for an ith plant is a function of fixed cost(FC), fuel cost(FL), plant ...
- 21
2
votes
0
answers
97
views
The quadratic form of variance and covariance components
I am reading Kline, Saggio, Solvsten 2020 and am confused about some basic econometric stuffs in this paper.
They begin their introduction as below:
"""
Consider the linear model $$y_{i}...
- 2,339
3
votes
1
answer
934
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-...
- 145
1
vote
1
answer
405
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-...
- 113
1
vote
2
answers
141
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 ...
- 13
2
votes
1
answer
92
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 ...
- 155
0
votes
1
answer
79
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 ...
2
votes
0
answers
64
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'...
- 121
4
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{...
user24655
1
vote
1
answer
632
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 ...
- 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} ...
0
votes
2
answers
66
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) ...
0
votes
1
answer
56
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 $x\sim y$ ...
- 1,816
0
votes
1
answer
45
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 ...
- 165
0
votes
1
answer
31
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 ...
- 165
6
votes
1
answer
187
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 $\...
- 678
0
votes
1
answer
311
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 ...
- 165
0
votes
1
answer
414
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 $M_{X}=I-X(X^{T}X)^{-1}$.
...
- 63
3
votes
1
answer
224
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 ...
- 131
2
votes
0
answers
98
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 ...
- 651
3
votes
2
answers
264
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 ...
- 6,578
3
votes
1
answer
555
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 $...
- 85
2
votes
0
answers
529
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 ...
- 616
6
votes
3
answers
334
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.
...
- 81
3
votes
1
answer
68
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/...
- 9,345