Matrices 2. Slide for MA1203 Business Mathematics II Week 4

Transcription

1 Matrices 2 Slide for MA1203 Business Mathematics II Week 4

2 2.7 Leontief Input Output Model

3 Input Output Analysis One important applications of matrix theory to the field of economics is the study of the relationship between industrial production and consumer demand. The Leontief input output model was pioneered by Wassily Leontief (Nobel Prize in economics 1973). Consider an oversimplified economy consisting of three sectors: agriculture (A), manufacturing (M), and service (S). Part of the output of one sector is absorbed by another sector through interindustry purchases, with the excess available to fulfill consumer demands. The relationship governing both intraindustrial and interindustrial sales and purchases is represented by an input output matrix:

4 Example Refer to the following input output matrix. a. If the units are measured in millions of dollars, determine the amount of agricultural products consumed in the production of $100 million worth of manufactured goods. b. Determine the dollar amount of manufactured goods required to produce $200 million worth of all goods and services in the economy.

5 Total Output & Internal Consumption Matrix Suppose: the total output of goods of the agriculture sector is given by x unit, the total output of goods of the manufacturing sector is given by y unit, and the total output from the service sector of the economy is given by z unit. What is the value of agricultural products consumed in the internal process of producing this total output of various goods and services? This could be obtained by using matrix multiplication. Write the total output of goods and services x, y, and z as a 3 x 1 matrix: Then the product.

6 Consumer Demand Matrix X gives the total production of goods and services in the economy. AX gives the amount of goods and services consumed in the production of these goods and services. Then X - AX gives the net output of goods and services that is enough to satisfy consumer demands. Let matrix D represent the consumer demands. X - AX = D or (I - A) X = D, where I is the identity matrix. If the inverse of (I - A) exists, X = (I A) -1 D

7 Leontief Input Output Model

8 Example: An Input Output Model for a Three- Sector Economy For the three-sector economy with input output matrix given by: a. Find the total output of goods and services needed to satisfy a consumer demand of $100 million worth of agricultural products, $80 million worth of manufactured goods, and $50 million worth of services. b. Find the value of the goods and services consumed in the internal process of production to meet this total output.

9 Example: An Input Output Model for a Three- Product Company TKK Corporation, a large conglomerate, has three subsidiaries engaged in producing raw rubber, manufacturing tires, and manufacturing other rubber-based goods. The production of 1 unit of raw rubber requires the consumption of 0.08 unit of rubber, 0.04 unit of tires, and 0.02 unit of other rubber-based goods. To produce 1 unit of tires requires 0.6 unit of raw rubber, 0.02 unit of tires, and 0 unit of other rubber-based goods. To produce 1 unit of other rubber-based goods requires 0.3 unit of raw rubber, 0.01 unit of tires, and 0.06 unit of other rubber-based goods. Market research indicates that the demand for the following year will be $200 million for raw rubber, $800 million for tires, and $120 million for other rubberbased products. Find the level of production for each subsidiary in order to satisfy this demand.

10 9.1 Determinant of a Matrix

11 Determinant of a 2 x 2 Matrix Recall that A 1 = 1 a 22 a 12 a 22 a 11 a 21 a 12 a 21 a 11 a 22 a 11 a 21 a 12 is called the determinant of the 2 x 2 matrix A. It is denoted by A or det A. If A = 0 then A is singular, if A 0 then A is nonsingular.

12 Properties of Determinant For A = , find A and AT. Theorem. For any matrix A, A T = A. For A = Theorem. and B = , find A and B. If a matrix has two identical rows or columns, its determinant is zero.

13 Properties of Determinant (2) Example. For A = and B = , find A and B. Theorem. If one of the two rows in A is a multiple of the other row, or if one of the two columns in A is a multiple of the other column, then A = 0.

14 Triangular Matrix A matrix that is composed of a nonzero element in the positions above (or below) the main diagonal and zero in the position below (or above) the main diagonal is called a triangular matrix. Example. For A = and B = , find A and B. Theorem. The determinant of a triangular matrix is equal to the product of the main diagonal elements.

15 Determinant of Matrices Product Example. For A = and B = , find A, B and AB. Theorem. If A and B are 2 x 2 matrices, then AB = A. B.

16 Example: The Market for Tea and Coffee Suppose that the market for tea is described by the demand and supply functions D t = 100 5p t + 3p c S t = p t and the market for coffee by D c = 120 8p c + 2p t S c = p c where p t is the price of tea, p c is the price of coffee, D t and S t are the quantities of tea demanded and supplied respectively, and D c and S c are the quantities of coffee demanded and supplied. Solve for the equilibrium prices of tea and coffee.

17 9.2 Determinant of a 3 x 3 Matrix Is it possible to obtain inverse of a 3 x 3 matrix by using its determinant?

18 Steps to Obtain A a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 31 a Define M ij to be the determinant of the 2 x 2 submatrix obtained when the ith row and jth column of A are deleted. M ij is known as a minor. 2. Attach a sign to the minor an define C ij = ( 1) i+j M ij, i = 1,2,3, j = 1,2,3. C ij is known as a cofactor To obtain A, take any row of column of A, multiply its elements by the corresponding cofactors, and add all these products. This is known as a cofactor expansion. Example. Find the determinant of A =

19 Inverse of 3 x 3 Matrix a 11 a 12 a 13 A = a 21 a 22 a 23 a 31 a 31 a Replace each a ij by the corresponding cofactor C ij to get the cofactor matrix of A. C 11 C 12 C 13 C 21 C 22 C 23 C 31 C 31 C Transpose the matrix obtained in 1. The resulting matrix is known as adjoint of A, adj(a). C 11 C 21 C 31 adj A = C 12 C 22 C 32 C 13 C 23 C A 1 = 1 A adj(a) Example. Find the inverse of A =

20 9.3 Determinant of an n x n Matrix

21 Determinant and Inverse of an n x n Matrix Follow the same steps as for a 3 x 3 matrix. 1. Define M ij to be the determinant of the (n-1) x (n-1) submatrix obtained when the ith row and jth column of A are deleted. M ij is known as a minor. 2. Attach a sign to the minor an define C ij is known as a cofactor. C ij = ( 1) i+j M ij, i = 1,2,3, j = 1,2,3. 3. To obtain A, take any row of column of A, multiply its elements by the corresponding cofactors, and add all these products. This is known as a cofactor expansion. 4. Replace each a ij by the corresponding cofactor C ij to get the cofactor matrix of A. 5. Transpose the matrix obtained in 1. The resulting matrix is known as adjoint of A, adj(a). 6. A 1 = 1 A adj(a) Example. Find the inverse of A =

22 10.2 The Eigenvalue Problem

23 The Eigenvalue Problem We have studied system of linear equations Ax = b where A is an n x n matrix, x and b is an n x 1 column matrix (vector). Now we investigate the solution to the following problem: Aq = λq. where A is an n x n matrix, q is an n x 1 column matrix (vector), and λ is an unknown scalar. This problem is known as the eigenvalue problem, arises in many situations in economics and econometrics. In this problem, we have to solve to unknowns: q and λ.

24 Eigenvalue for a 2x2 Matrix a 11 a 12 a 21 a 22 q 1 q 2 = λ q 1 q 2 A λi q = 0 When A λi is nonsingular, then q = 0. If we want q 0, then A λi should be singular. Thus A λi = 0. This is known as the characteristic equation for A. It gives a polynomial of degree 2 in λ, with roots λ 1 and λ 2, called the eigenvalues of A. Each λ 1 and λ 2 can be substituted to the original equation to obtain the eigenvector q of A.

25 Example Find the characteristic equation, eigenvalues, and eigenvectors of: A =

26 Normalized Eigenvectors Sometimes there are infinite eigenvectors corresponding to a particular eigenvalue. In order to find a unique eigenvector, we can choose the eigenvector q whose length is unity. This means q q 2 2 = 1. This is known as the normalization. Example. Find the normalized eigenvectors of: A =

27 Diagonalization Theorem. Let A be a nonsingular matrix and Q is the matrix whose columns are normalized eigenvectors corresponding to different eigenvalues. Then the following diagonalizes the matrix A: Q 1 AQ = Λ where λ Λ = 0 λ λ n Example. Diagonalize the matrix A =

28 Power of a Matrix Let A be a non singular matrix. From the diagonalization: Then Q 1 AQ = Λ or A = QΛ Q 1 A 2 = QΛ Q 1 QΛ Q 1 = QΛ 2 Q 1 A 3 = QΛ Q 1 QΛ 2 Q 1 = QΛ 3 Q 1 A t = QΛ t Q 1

29 Power of a Diagonal Matrix Λ = λ λ λ n Λ 2 = λ λ λ n λ λ 2 0 = 0 0 λ n λ λ n λ 1 2 Λ 3 = λ λ λ n λ λ n λ 1 2 = λ λ n λ 1 3 Λ t = t λ 2 0 t 0 0 λ n λ 1 t

30 Example If A = , find A2017.

31 Markov Model for Employment Unemployment rates change over time as individuals gain or lose their employment. We consider a simple model, called a Markov model, that describes the dynamics of unemployment using transitional probabilities. In this model, we assume: If an individual is unemployed in a given week, the probability is p for this individual to be employed the following week, and 1 - p for him or her to stay unemployed If an individual is employed in a given week, the probability is q for this individual to stay employed the following week, and 1 - q for him or her to be unemployed

32 Markov Model for Employment (2) Let x t be the ratio of individuals employed in week t, and let y t be the ratio of individuals unemployed in week t. Then the week-on-week changes are given by these equations: These equations are linear, and can be written in matrix form as We call A the transition matrix and v t the state vector of the system. What is the long term state of the system? Are there any equilibrium states? If so, will these equilibrium states be reached?

33 Markov Model for Employment (3) The state of the system after t weeks is given by: For white males in the US in 1966, the probabilities where found to be p = and q = If the unemployment rate is 5% at t = 0, expressed by x 0 = 0.95 and y 0 = 0.05, the situation after 100 weeks would be We need eigenvalues and eigenvectors to compute A 100 efficiently.