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## Hot answers tagged linear-algebra

7 votes
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### Regression Optimization problem under constraints

The situation is given in the following picture The black line is the true conditional mean $E(y|x)$. If we truncate the data, all observations above the truncation $Y^A$ are not observed. For low ...
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7 votes
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### Econ Intuition for Jacobian inverse in demand system

For the 2x2 case being considered, write $$\mathbf{B}=\left[\begin{array}{cc} b_{1,1} & b_{1,2}\\ b_{2,1} & b_{2,2} \end{array}\right].\quad$$ It follows that the element (1,1) in $B^{-1}$ is ...
• 595
5 votes
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### Budget hyperplane in n dimensions

Let matrix $A = \begin{bmatrix} p_1 & p_2 & \ldots & p_n \end{bmatrix}$. Let $\mathbf{x}^*$ be a fixed solution to $A \mathbf{x} = c$. Then for any vector $\mathbf{u}$ that belongs to the ...
• 906
5 votes
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### Why the definition of productive economy in Leontief Open Model is such?

This is a specific terminology employed in the literature on Leontief model. Productive here means that all sectors must be profitable (do not confuse it with notions of productivity used elsewhere in ...
• 57k
4 votes
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### Negative Definite vs Semi-definite Hessian - Sufficient vs Necessary conditions?

The simplest example is $-x^3$ in the single variable case, or $-x_1^3-x_2^3$ in the case of two variables. The Hessian matrix is negative semi-definite at $(0,0)$, but there is no maximum at this ...
• 3,371
3 votes

This answer shows asymptotic convergence to origin/equilibrium if the matrix of coefficients in the homogeneous linear system is diagonalizable. (Continuous time.) Notation I will use the notation $\... • 29.4k 3 votes ### Question About Stable Matrices - MWG Mathematical Appendix M.D Matrices: Negative (Semi)Definiteness and Other Properties You wrote: Definition$\quad$A matrix is stable if all of its characteristic values have negative real parts. I really lack the knowledge about systems of differential equations, stable matrices, and ... • 4,047 3 votes Accepted ###$N(A) \oplus R(A) = V \; \forall A$? Your formula $$N(A) \oplus R(A) = V\qquad (1)$$ is not true in general cases, it is correct in a particular case, if$A$is a symmetric matrix, so that$A^T=A$, where$A^T$is the transpose of$A$. ... • 4,047 3 votes ### How do you convert or move from a linear cost function to a quadratic cost function? Your eq (2.10) is not more general than (2.9), but corresponds to an alternative specification. A more general version would be: $$C_i(Q_i)=FC_i+a_{1,i}Q_i+a_{2,i}Q^2_i.$$ This specification allows ... • 3,371 3 votes Accepted ### Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32 If you assume$e_1=e_2=1$as you have done (but not that$\alpha=1/2), then the solution (as given in equation (48) in Simon and Blume) is: \begin{align*} x_1=y_1&=\alpha\\ x_2=y_2&=1-\... • 691 3 votes ### Calculating natural rate of unemployment The model you fit is simply inadequate to estimate the natural rate of unemployment any results from it will be completely unreliable, so I am not surprised if they make no sense. Furthermore, natural ... • 57k 3 votes Accepted ### When gradient of utility function is a zero vector This concerns the partial derivatives of the utility function with respect to goods, and not the partial derivative of the Lagrangian of the maximization problem. So a zero derivative, and moreover ... • 33.9k 2 votes Accepted ### How to utilize the projection matrix in econometrics? This kind of projections in econometrics are usually employed for partialling out some covariates from a linear regression. Observe that in general P_X\neq I. Consider X=\begin{bmatrix}1&5\\1&... • 354 2 votes ### Derive the growth rate of an equation First, note that the growth rate of \mu is defined as \dot\mu = \frac{ d\mu }{ \mu }. Therefore, you will have to take the total differential of the equation; and divide by \mu. For the first ... • 293 2 votes ### Hep with total differentiation of an AD function For a variable X, let dX denote its total differential. Let k be a constant, and X and Y variables. You'll need the following rules:dk = 0$$(constant rule),$$d(X + Y) = dX + dY$$(... • 293 2 votes ### Understanding utility function curve and marginal rate of substitution This is an expansion on @1muflon1's answer. Motivation and Applications In the utility function, \alpha allows for an alternative to a linear growth function, especially when the parameters are ... 2 votes Accepted ### Quadratic Form Single Summation notation The sum \sum_{i \le j}=a_{ij}x_ix_j goes over all indices (i,j) with i\leq j. We can do that by either summing over all i and then for each i all j\geq i, which would give us$$\sum_{i=1}^... • 13.3k 2 votes ### Wage setting equation Equilibrium should be found equaling price and wage setting equations together. In your case that would be: $$\frac{1}{1+\mu} = z(1-u)$$ where\mu$is markup, and filling in the parameters you will ... • 1,847 1 vote ### Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32 I just want to add an observation, too long for a comment, to the excellent explanations of smcc. You wrote [...]if my answer is correct; especially the step where I write "since the question ... • 4,047 1 vote ### Stochastic optimal control problem (calculus) I think there should be a minus before the final term in the expression you are looking for. In any case, you did not plug in the optimality condition for$w$, which is in your case $$w = -\frac{V'(A)... 1 vote Accepted ### What are the concepts in Linear Algebra that model the idea of Identification Strategy in Econometrics? In econometrics, identification can take several forms depending on the type of model you are working with (see this survey for a more comprehensive description). However, in general, identification ... • 4,188 1 vote Accepted ### x\sim y implies x+a\sim y+a for any a\geq0 and x,y\in\mathbb R^n, then the preference is linear? It is not true. Let us consider \mathbb{R}^2 so bundles are x = (x_1,x_2). Consider the preference: (i) If x_1 \leq 0, preferences are lexicographic, i.e.$$ x \succ y \Leftrightarrow \begin{... • 1,899 1 vote ### How can difference equations with an infinite summation be represented in matrix form? Maybe the differencing approach should work for you. Basically the idea is to reduce this series to a finite expression, using the lag operator. Let me explain with an example from Costa (2016, p.81) (... • 843 1 vote ### Leontief input output model with column sum greater than 1 You are right when saying that mathematically in the cited theorem the condition of column sums being less than 1 is not an "if and only if" condition and thus exceptional circumstances are ... 1 vote Accepted ### Leontief input output model with column sum greater than 1 In terms of if and only if statements according to Peterson & Olinick (1982); A substochastic matrix A is productive if and only if$I-A$is nonsingular. In substochastic matrix the sum of ... • 57k 1 vote ### Calculating the elasticity of substitution between factors of production First of all, I think that 'linear and homogeneous' is a typo of 'linearly homogeneous.' Indeed, it can be shown that if the production function$V$is linearly homogeneous, Allen Elasticity of ... • 66 1 vote ### Has this differential calculus inequality approach to optimizing the production possibility curve exist? I don't really understand your optimization problem as it stands. Just as a few examples, when you write out$$\min \sum_m \sum_v \left( \frac{\partial \vec f_v (\vec Q_v)}{\partial \vec f_m (\vec ... • 6,648 1 vote ### Budget hyperplane in n dimensions I got some outside help for the ending of the proof I was attempting. I'll leave this question if by chance someone else finds it useful. So if we want to show$p_1x_1 + \cdots + p_{n−1}x_{n−1} = 0 \...
• 6,648

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