Questions tagged [linear-algebra]
The linear-algebra tag has no usage guidance.
28
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$N(A) \oplus R(A) = V \; \forall A$?
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 ...
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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 ...
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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 ...
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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 ...
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0
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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 $$y_{i}...
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1
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444
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Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?
When a Hessian matrix is negative definite at a critical point then that critical point is a local maximum (Sufficient Condition).
As per the calculus wiki:
Link, when the Hessian is negative semi-...
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Why the definition of productive economy in Leontief Open Model is such?
Note: as an inter-disciplinary question, it has its twin on Mathematics Exchange: https://math.stackexchange.com/questions/4193896/why-the-definition-of-productive-economy-in-leontief-open-model-is-...
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2
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Understanding utility function curve and marginal rate of substitution
This example appears in a different question, but there is something I don't understand.
Maybe this question is better suitated for algebra stackexchange.
John’s utility function for food (f) and ...
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1
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91
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Regression Optimization problem under constraints
To estimate a simple linear regression:
$$ 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 ...
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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 ...
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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'...
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2
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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
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1
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440
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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 ...
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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} ...
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2
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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) ...
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$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$ ...
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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$$
Do ...
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Hep with total differentiation of an AD function [closed]
Is there anyone who can help me with a total differentiation exercize. I am starting with the following formula for AD:
$$x=\mu ^{-1}(g+i+e)$$
Where $\mu$ is the Keynesian multiplier.
And have to ...
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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 $\...
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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 ...
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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 $M_{X}=I-X(X^{T}X)^{-1}$.
...
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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 ...
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92
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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 ...
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247
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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 ...
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1
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534
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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 $...
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526
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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 ...
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308
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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.
...
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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/...
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