0
votes
2answers
89 views

Matrix Operation

Let $x$ be a $n \times 1$ vector whose jth element is $x_j$. Show that $A = xx^{T}/x^{T}x$ and $B = I_n - A$ are symmetric idempotent matrices. Note that $x^Tx$ is a scalar (real number)
1
vote
2answers
362 views

Game theory: Nash equilibrium in asymetric payoff matrix

I have a utility function describing the desirability of an outcome state. I weigh the expected utility with the probability of the outcome state occuring. I find the expected utility of an action, a, ...
2
votes
0answers
77 views

Constructing and understanding stock-flow model

Suppose that $\textbf{x} = A\textbf{x} + B\dot{\textbf{x}}$ where $\textbf{x}$ is vector of economic output level, $A$ is input-output matrix, $B$ is stock-flow matrix. The system represents ...
1
vote
1answer
167 views

Input-output economics and stability of general equilibrium

Here, I will start with a simple expression for an input–output system with $x(t)$ representing the vector of outputs and $A$ the input–output matrix. Then, the simplest possible linear ...
1
vote
1answer
767 views

Eigenvalues of matrix and stability

This is about general equilibrium: Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ is determined by the ...
5
votes
1answer
2k views

Understanding the Leontief inverse

What I remember from economics about input/output analysis is that it basically analyses the interdependencies between business sectors and demand. If we use matrices we have $A$ as the input-output ...
1
vote
1answer
272 views

Find a price vector p for various prices of industries.

( Leontief input-output model ) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the $3 \times 3$ consumption matrix A = ...