Chapter 9 Further applications

Question 1

Describe the input output matrix?

Answer 1

Columns of matrix are inputs to a given sector
Rows of matrix are outputs from a given sector
n x n matrix for n sectors in the economy

Question 2

What does aij represent in the input output matrix

Answer 2

The number of units of goods the ith sector need to produce one unit of goods in the jth sector

Question 3

What type of matrix will the input output model be in a closed model - assumption

Answer 3

Column stochastic

Question 4

What does a closed economy model mean

Answer 4

All production is consumed by industries so there is no surplus

Question 5

What does xi represent

Answer 5

Production of goods in sector i is xi

Question 6

In a closed economy how can production vector be found

Answer 6

Solving for solution of homogenous system of equations :

(I -A)x = 0

Question 7

What makes a closed input-output or closed leontief model

Answer 7

If the input output matrix is column stochastic , the system is solvable and solution can be chosen so its nonnegative.

Question 8

What does it mean if economy is open

Answer 8

Some production is consumed by internal industry and the rest satisfies external demand.

Question 9

In words in the open model what is production equal to

Answer 9

Internal consumption plus outside demand

Question 10

What makes a open input-output or closed leontief model

Answer 10

Open economy production vector is given as a solution to the system of linear equations :
(I-A)x = d

Question 11

What is the theorem that explains when there will be a solution to the input output model (As its not homogeneous so there is not a guaranteed solution)

Answer 11

There exists a nonnegative production vector x for every demand vector d : (I-A)x=d if and only if there exists any vector x>=0 st. (I-A)x>0

Question 12

Define column substochastic matrix

Answer 12

Every column sum is less than 1

Question 13

What theorem (state) makes us confident we can find a solution for x based on the matrix A

Answer 13

If A the input output matrix is column sub stochastic and positive then there exists a nonnegative x : (I-A)x=d for every d . 
This means (I-A) is invertible and we can be confident of finding a solution.

Question 14

Define what a markov chain is

Answer 14

Random process. Let (1,2,…,n) be a set of states. The system in every step moves form state i to state j with probability pij. This probability does not change with time and we assume future state depends only on the present state, not on the past.

Question 15

What si the transition matrix P and its properties

Answer 15

Matrix that represents a markov chain. It is positive and stochastic

Question 16

Explain what the distribution vector is and its properties

Answer 16

Vector of system at step k detailing the probability the system is at state i. It is normalised and non negative

Question 17

What si the stationary distribution

Answer 17

If the distribution does not change during the time it is the stationary distribution. It satisfies x^transpose P = x^tranpose and is a left perron eigenvector of the transition matrix at eigenvalue 1

Question 18

State the theorem about the convergence of the distirbution at the kth step

Answer 18

If P is positive the distribution vectors at the kth step x(k) converge to the stationary distribution regardless of x(0) - so the long term state is not depended on initial distribution.