1560 mathematics for economists
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This document provides an overview of linear models and matrix algebra concepts that are important for economics. It discusses the objectives of using mathematics for economics, including understanding problems by stating the unknown and known variables. The document then covers key topics in linear algebra like the history of matrices, what matrices are, basic matrix operations, and properties of matrix addition and multiplication. It also introduces concepts like the inverse and transpose of a matrix. Finally, it provides an example of how matrices and vectors can represent systems of linear equations used in economic models.
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1560 mathematics for economists
- 1. Mathematics for Economists Chapters 4-5 Linear Models and Matrix Algebra Johann Carl Friedrich Gauss (1777–1855) The Nine Chapters on the Mathematical Art (1000-200 BC)
- 2. Objectives of math for economists To understand mathematical economics problems by stating the unknown, the data and the conditions To plan solutions to these problems by finding a connection between the data and the unknown To carry out your plans for solving mathematical economics problems To examine the solutions to mathematical economics problems for general insights into current and future problems Remember: Math econ is like love – a simple idea but it can get complicated. 2
- 3. 4. Linear Algebra Some history: The beginnings of matrices and determinants goes back to the second century BC although traces can be seen back to the fourth century BC. But, the ideas did not make it to mainstream math until the late 16th century The Babylonians around 300 BC studied problems which lead to simultaneous linear equations. The Chinese, between 200 BC and 100 BC, came much closer to matrices than the Babylonians. Indeed, the text Nine Chapters on the Mathematical Art written during the Han Dynasty gives the first known example of matrix methods. In Europe, two-by-two determinants were considered by Cardano at the end of the 16th century and larger ones by Leibniz and, in Japan, by Seki about 100 years later.
- 4. 4. What is a Matrix? A matrix is a set of elements, organized into rows and columns dc ba rows columns • a and d are the diagonal elements. • b and c are the off-diagonal elements. • Matrices are like plain numbers in many ways: they can be added, subtracted, and, in some cases, multiplied and inverted (divided).
- 5. 4. Matrix: Details Examples: 5 [ ]δβα= − = b d b A ; 1 1 • Dimensions of a matrix: numbers of rows by numbers of columns. The Matrix A is a 2x2 matrix, b is a 1x3 matrix. • A matrix with only one column or only one row is called a vector. • If a matrix has an equal numbers of rows and columns, it is called a square matrix. Matrix A, above, is a square matrix. • Usual Notation: Upper case letters => matrices Lower case => vectors
- 6. 4.1 Basic Operations Addition, Subtraction, Multiplication ++ ++ = + hdgc fbea hg fe dc ba −− −− = − hdgc fbea hg fe dc ba ++ ++ = dhcfdgce bhafbgae hg fe dc ba Just add elements Just subtract elements Multiply each row by each column = kdkc kbka dc ba k Multiply each element by the scalar
- 7. 4.1 Matrix multiplication: Details Multiplication of matrices requires a conformability condition The conformability condition for multiplication is that the column dimensions of the lead matrix A must be equal to the row dimension of the lag matrix B. What are the dimensions of the vector, matrix, and result? [ ] [ ]131211 232221 131211 1211 cccc bb bbb aaaB == = 7 [ ]231213112212121121121111 babababababa +++= • Dimensions: a(1x2), B(2x3) => c(1x3)
- 8. 4.1 Basic Matrix Operations: Examples 222222 117 25 20 13 97 12 xxx CBA =+ = + Matrix addition Matrix subtraction Matrix multiplication Scalar multiplication 8 = − 65 11 32 01 97 12 222222 x 2726 34 32 01 x 97 12 xxx CBA = = = 8143 2141 16 42 8 1
- 9. 4.1 Laws of Matrix Addition & Multiplication ++ ++ = + =+ 22222121 12121111 2221 1211 2221 1211 abaa abba bb bb aa aa BA Commutative law of Matrix Addition: A + B = B + A 9 ++ ++ = + =+ 22222121 12121111 2221 1211 2221 1211 abab abab bb aa bb bb AB Matrix Multiplication is distributive across Additions: A (B+ C) = AB + AC (assuming comformability applies).
- 10. 4.1 Matrix Multiplication Matrix multiplication is generally not commutative. That is, AB ≠ BA even if BA is conformable (because diff. dot product of rows or col. of A&B) − = = 76 10 , 43 21 BA 10 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = +−+ +−+ = 2524 1312 74136403 72116201 AB ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) −− = ++ −+−+ = 4027 43 47263716 41203110 BA
- 11. 4.1 Matrix multiplication Exceptions to non-commutative law: AB=BA iff B = a scalar, B = identity matrix I, or B = the inverse of A -i.e., A-1 11 Theorem: It is not true that AB = AC => B=C Proof: −−−= − = −− −= 132 111 212 ; 011 010 111 ; 321 101 121 CBA Note: If AB = AC for all matrices A, then B=C.
- 12. 4.1 Transpose Matrix − =′=> − = 49 08 13 401 983 AA:Example The transpose of a matrix A is another matrix AT (also written A ′) created by any one of the following equivalent actions: - write the rows (columns) of A as the columns (rows) of AT - reflect A by its main diagonal to obtain AT Formally, the (i,j) element of AT is the (j,i) element of A: [AT ]ij = [A]jj If A is a m × n matrix => AT is a n × m matrix. (A')' = A Conformability changes unless the matrix is square. 12
- 13. 4.1 Inverse of a Matrix Identity matrix: AI = A = 100 010 001 I Notation: Ij is a jxj identity matrix. Given A (mxn), the matrix B (nxm) is a right-inverse for A iff AB = Im Given A (mxn), the matrix C (mxn) is a left-inverse for A iff CA = In
- 14. 4.1 Inverse of a Matrix Inversion is tricky: (ABC)-1 = C-1 B-1 A-1 More on this topic later Theorem: If A (mxm), has both a right-inverse B and a left-inverse C, thenC = B. Proof: AB=Im and CA=In. Thus, C(AB)=C Im = C and C(AB)=(CA)B=InB=B => C(nxm)=B(mxn) Note: - This matrix is unique. - If A has both a right and a left inverse, it is called invertible.
- 15. 4.1 Vector multiplication: Geometric interpretation Think of a vector as a directed line segment in N- dimensions! (has “length” and “direction”) Scalar multiplication (“scales” the vector –i.e., changes length) Source of linear dependence 15 [ ]6 4 2= U [ ]3 2 = U [ ]− ⋅ = − −1 3 2U x2 x1 -4 -3 -2 -1 1 2 3 4 5 6 6 5 4 3 2 1 -2
- 16. 4.1 Vector Addition: Geometric interpretation v' = [2 3] u' = [3 2] w’= v'+u' = [5 5] Note that two vectors plus the concepts of addition and multiplication can create a two-dimensional space. 16 x1 x2 5 4 3 2 1 1 2 3 4 5 w u v u A vector space is a mathematical structure formed by a collection of vectors, which may be added together and multiplied by scalars. (It’s closed under multiplication and addition).
- 17. 4.1 Vector Space Given a field R and a set V of objects, on which “vector addition” (VxV→V), denoted by “+”, and “scalar multiplication” (RxS →V), denoted by “. ”, are defined. If the following axioms are true for all objects u, v, and w in V and all scalars c and k in R, then V is called a vector space and the objects in V are called vectors. 1. u+v is in V (closed under addition). 2. u + v = v + u (vector addition is commutative). 3. θ is in V, such that u+ θ = u (null element). 4. u + (v+w) = (v + u) +w (distributive law of vector addition) 5. For each v, there is a –v, such that v+(-v) = θ 6. c .u is in V (closed under scalar multiplication). 7. c. (k . u) = (c .k) u (scalar multiplication is associative).17
- 18. 4.1 Vector Space 8. c. (v+ u) = (c. v)+ (c. u) 9. (c+k) . u = (c. u)+ (k. u) 10. 1.u=u (unit element). 11. 0.u= θ (zero element). We can write S = {V,R,+,.}to denote an abstract vector space. This is a general definition. If the field R represents the real numbers, then we define a real vector space. Definition: Linear Combination Given vectors u1,...,uk,, the vector w = c1 u1+....+ ckuk is called a linear combination of the vectors u ,...,u . 18
- 19. 4.1 Vector Space Definition: Subspace Given the vector space V and W a sect of vectors, such that W is in V. Then W is a subspace iff: u, v are in W => u+v are in W, and c u is in W for every c in R. u1,...,uk,, the vector w = c1 u1+....+ ckuk is called a linear combination of the vectors u1,...,uk,. 19
- 20. 4.1 System of equations: Matrices and4.1 System of equations: Matrices and VectorsVectors Assume an economic model as system of linear equations in which aij parameters, where i = 1.. n rows, j = 1.. m columns, and n=m xi endogenous variables, di exogenous variables and constants nn n n nm m m nn d d d x x x ax ax ax axa axa axa 2 1 2 22 12 211 22121 12111 = = = + + + + + + 20
- 21. A general form matrix of a system of linear equations Ax = d where A = matrix of parameters x = column vector of endogenous variables d = column vector of exogenous variables and constants Solve for x* dAx dAx d d d x x x aaa aaa aaa nnnmnn m m 1* 2 1 2 1 21 22221 11211 − = = = 21 4.1 System of equations: Matrices and4.1 System of equations: Matrices and VectorsVectors
- 22. 4.1 Solution of a General-equation System Assume the 2x2 model 2x + y = 12 4x + 2y = 24 Find x*, y*: y = 12 – 2x 4x + 2(12 – 2x) = 24 4x +24 – 4x = 24 0 = 0 ? indeterminante! Why? 4x + 2y =24 2(2x + y) = 2(12) one equation with two unknowns 2x + y = 12 x, y Conclusion: not all simultaneous equation models have solutions (not all matrices have inverses). 22
- 23. 4.1 Linear dependence A set of vectors is linearly dependent if any one of them can be expressed as a linear combination of the remaining vectors; otherwise, it is linearly independent. Formal definition: Linear independent The set {u1,...,uk} is called a linearly independent set of vectors iff c1 u1+....+ ckuk = θ => c1= c2=...=ck,=0. Notes: - Dependence prevents solving a system of equations. More unknowns than independent equations. - The number of linearly independent rows or columns in a matrix is the rank of a matrix (rank(A)). 23
- 24. 4.1 Linear dependence Examples: [ ] [ ] // 2 / 1 ' 2 ' 1 ' 2 ' 1 02 2412 105 2410 125 =− = = = vv v v v v 24 [ ] [ ] [ ] 023 54 162216 23 5 4 ; 8 1 ; 7 2 321 3 21 321 =−− == −= − = = = vvv v vv vvv
- 25. 4.2 Application 1: One Commodity Market Model (2x2 matrix) Economic Model 1) Qd = a – bP (a,b >0) 2) Qs = -c + dP (c,d >0) 3) Qd= Qs Find P* and Q* Scalar Algebra form (Endogenous Vars :: Constants) 4) 1Q + bP = a 5) 1Q – dP = -c 25 db bcad Q db ca P + − = + + = * *
- 26. 4.2 One Commodity Market Model (2x2 matrix) 26 dAx c a d b P Q dAx c a P Q d b 1* 1 * * 1 1 1 1 − − = − − = = − = − Matrix algebra
- 27. 4.2 Application II: Three Equation4.2 Application II: Three Equation National Income Model (3x3 matrix)National Income Model (3x3 matrix) Model Y = C + I0 + G0 C = a + b (Y-T) (a > 0, 0<b<1) T = d + t Y (d > 0, 0<t<1) Endogenous variables? Exogenous variables? Constants? Parameters? Why restrictions on the parameters? 27
- 28. 4.2 Three Equation National Income4.2 Three Equation National Income ModelModel Endogenous: Y, C, T: Income (GNP), Consumption, and Taxes. Exogenous: I0 and G0: autonomous Investment & Government spending. Parameters: a & d: autonomous consumption and taxes. t: marginal propensity to tax gross income 0 < t < 1. b: marginal propensity to consume private goods and services from gross income 0 < b < 1. Solution: 28 btb GIbda Y +− ++− = 1 00*
- 29. 4.2 Three Equation National Income Model Parameters & Endogenous vars. Exog. vars. Y C T &cons. 1Y -1C +0T = I0+G0 -bY +1C +bT = a -tY +0C +1T = d Given (Model) Y = C + I0 + G0 C = a + b (Y-T) T = d + t Y Find Y*, C*, T* 29 + = − − − d a GI T C Y t bb 00 10 1 011 dAx dAx 1* − = =
- 30. 4.2 Three Equation National Income Model 30 dAx d a GI t bb T C Y dAx d a GI T C Y t bb 1* 00 1 * * * 00 10 1 011 10 1 011 − − = + − − − = = + = − − −
- 31. 4.3 Notes on Vector Operations = 2 3 12x u An [m x 1] column vector u and a [1 x n] row vector v, yield a product matrix uv of dimension [m x n]. 31 [ ]541 31 =′ x v [ ] = =′ 10 15 8 12 2 3 541 2 3 32x vu
- 32. 4.3 Vector multiplication: Dot (inner), and cross product • The dot product produces a scalar! c’z =1x1=1x4 4x1= z’c 32 44332211 zczczczcy +++= ∑= = 4 1i ii zcy [ ] zc' 4 3 2 1 4321 = = z z z z ccccy
- 33. 4.3 Vectors: Dot Product [ ] cfbead f e d cbaT ++= ==⋅ αββα ccbbaaT ++== 2/1 ][ααα )cos(θβαβα =⋅ Think of the dot product as a matrix multiplication The magnitude (length) is the square root of the dot product of a vector with itself. The dot product is also related to the angle between the two vectors – but it doesn’t tell us the angle. Note: As the cos(90) is zero, the dot product of two orthogonal vectors is zero.
- 34. 4.3 Vectors: Magnitude and Phase (direction) x y ||v|| θ Alternate representations: Polar coords: (||v||, θ) Complex numbers: ||v||ejθ “phase” runit vectoais,1If 1 2 ),, 2 , 1 ( vv n i i xv n xxxv = = = = ∑ T (Magnitude or “2-norm”) (unit vector => pure direction)
- 35. 4.3 Vectors: Cross Product The cross product of vectors A and B is a vector C which is perpendicular to A and B The magnitude of C is proportional to the cosine of the angle between A and B The direction of C follows the right hand rule – this why we call it a “right-handed coordinate system” )sin(θbaba =×
- 36. 4.3 Vectors: Cross Product: Right hand rule
- 37. 4.3 Vectors: Norm • Given a vector space V, the function g: V→ R is called a norm if and only if: 1) g(x)≥ 0, for all xεV 2) g(x)=0 iff x=θ (empty set) 3) g(αx) = |α|g(x) for all αεR, xεV 4) g(x+y)=g(x)+g(y) (“triangle inequality”) for all x,yεV The norm is a generalization of the notion of size or length of a vector. • An infinite number of functions can be shown to qualify as norms. For vectors in Rn , we have the following examples: g(x)=maxi (xi), g(x)=∑i |xi|, g(x)=[∑i (xi)4 ] ¼ • Given a norm on a vector space, we can define a measure of “how far apart” two vectors are using the concept of a metric.
- 38. 4.3 Vectors: Metric • Given a vector space V, the function d: VxV→ R is called a metric if and only if: 1) d(x,y)≥ 0, for all x,yεV 2) d(x,y)=0 iff x=y 3) d(x,y) = d(y,x) for all x,yεV 4) d(x+y)≤d(x,z) + d(z,y) (“triangle inequality”) for all x,y,zεV Given a norm g(.), we can define a metric by the equation: d(x,y) = g(x-y). • The dot product is called the Euclidian distance metric.
- 39. 4.3 Orthonormal Basis Basis: a space is totally defined by a set of vectors – any point is a linear combination of the basis Ortho-Normal: orthogonal + normal Orthogonal: dot product is zero Normal: magnitude is one Example: X, Y, Z (but don’t have to be!) [ ] [ ] [ ]T T T z y x 100 010 001 = = = 0 0 0 =⋅ =⋅ =⋅ zy zx yx • X, Y, Z is an orthonormal basis. We can describe any 3D point as a linear combination of these vectors.
- 40. 4.3 Orthonormal Basis ⋅+⋅+⋅ ⋅+⋅+⋅ ⋅+⋅+⋅ = ncnbna vcvbva ucubua nvu nvu nvu c b a 333 222 111 00 00 00 (not an actual formula – just a way of thinking about it) • To change a point from one coordinate system to another, compute the dot product of each coordinate row with each of the basis vectors. • How do we express any point as a combination of a new basis U, V, N, given X, Y, Z?
- 41. 4.5 Identity and Null Matrices 000 000 000 . 100 010 001 10 01 etc or Identity Matrix is a square matrix and also it is a diagonal matrix with 1 along the diagonals. Similar to scalar “1” Null matrix is one in which all elements are zero. Similar to scalar “0” Both are diagonal matrices Both are idempotent matrices: A = AT and A = A2 = A3 = … 41
- 42. 4.6 Inverse matrix AA-1 = I A-1 A=I Necessary for matrix to be square to have unique inverse If an inverse exists for a square matrix, it is unique (A')-1 =(A-1 )' 42 • A x = d • A-1 A x = A-1 d • Ix = A-1 d • x = A-1 d • Solution depends on A-1 • Linear independence • Determinant test!
- 43. 4.6 Inverse of a Matrix 100 010 001 | ihg fed cba 1. Append the identity matrix to A 2. Subtract multiples of the other rows from the first row to reduce the diagonal element to 1 3. Transform the identity matrix as you go 4. When the original matrix is the identity, the identity has become the inverse!
- 44. 4.6 Determination of the Inverse (Gauss-Jordan Elimination) AX = I I X = K I X = X = A-1 K = A-1 1) Augmented matrix all A, X and I are (nxn) square matrices X = A-1 Gauss elimination Gauss-Jordan eliminationU: upper triangular further row operations [A I ] [ U H] [ I K] 2) Transform augmented matrix
- 45. 4.6 Determinant of a Matrix The determinant is a number associated with any squared matrix. If A is an nxn matrix, the determinant is given by |A| or det(A). Determinants are used to characterize invertible matrices. A matrix is invertible (non-singular) if and only if it has a non- zero determinant That is, if |A|≠0 → A is invertible. Determinants are used to describe the solution to a system of linear equations with Cramer's rule. Can be found using factorials, pivots, and cofactors! More on this later. Lots of interpretations 45
- 46. 4.6 Determinant of a Matrix Used for inversion. Example: Inverse of a 2x2 matrix: = dc ba A bcadAA −== )det(|| − − − =− ac bd bcad A 11 This matrix is called the adjugate of A (or adj(A)). A-1 = adj(A)/|A|
- 47. 4.6 Determinant of a Matrix (3x3) cegbdiafhcdhbfgaei ihg fed cba −−−++= ihg fed cba ihg fed cba ihg fed cba Sarrus’ Rule: Sum from left to right. Then, subtract from right to left Note: N! terms
- 48. 4.6 Determinants: Laplace formula The determinant of a matrix of arbitrary size can be defined by the Leibniz formula or the Laplace formula. The Laplace formula (or expansion) expresses the determinant | A| as a sum of n determinants of (n-1) × (n-1) sub-matrices of A. There are n2 such expressions, one for each row and column of A Define the i,j minor Mij (usually written as |Mij|) of A as the determinant of the (n-1) × (n-1) matrix that results from deleting the i-th row and the j-th column of A. 48 Pierre-Simon Laplace (1749–1827).
- 49. 4.6 Determinants: Laplace formula Define the Ci,jthe cofactor of A as: 49 ||)1( ,, ji ji ji MC + −= • The cofactor matrix of A -denoted by C-, is defined as the nxn matrix whose (i,j) entry is the (i,j) cofactor of A. The transpose of C is called the adjugate or adjoint of A (adj(A)). • Theorem (Determinant as a Laplace expansion) Suppose A = [aij] is an nxn matrix and i,j= {1, 2, ...,n}. Then the determinant njnjjjijij ininiiii CaCaCa CaCaCaA +++= +++= ... ...|| 22 2211
- 50. 4.6 Determinants: Laplace formula Example: 50 −= 642 010 321 A 0)0(x4)3x2-x61)(1()0(x2 0))2x)1((x3)0(x)1(x2)6x1(x1 x3x2x1|| 131211 =−+−+−= =−−+−+−= =++= CCCA |A| is zero => The matrix is non-singular. (Check!)
- 51. 4.6 Determinants: Properties Interchange of rows and columns does not affect |A|. (Corollary, |A| = |A’|.) |kA| = kn |A|, where k is a scalar. |I| = 1, where I is the identity matrix. |A| = |A’|. |AB| = |A||B|. |A-1 |=1/|A|. 51
- 52. 4.6 Matrix inversion: Note It is not possible to divide one matrix by another. That is, we can not write A/B. For two matrices A and B, the quotient can be written as AB-1 or B-1 A. In general, in matrix algebra AB-1 ≠ B-1 A. Thus, writing A/B does not clearly identify whether it represents AB-1 or B-1 A. We’ll say B-1 post-multiplies A (for AB-1 ) and B-1 pre-multiplies A (for B-1 A) Matrix division is matrix inversion. 52
- 53. 4.7 Application IV: Finite Markov Chains Markov processes are used to measure movements over time. 53 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]90110 100*6.100*3.,100*4.100*7. 6.4. 3.7. 100100 PP PP x plant?eachatbewillemployeesmanyhowyear,oneofendAt the 6.4. 3.7. PP PP M yprobabilitknownaw/plantseachbetweenmoveandstayemployeesThe 100100x B&Aplantsover twoddistributeare0at timeEmployees 0000 BBBA ABAA 00 / 011 BBBA ABAA 00 / 0 = ++= = ++= == = = == BBABBAAA PBPBPAPABAMBA BA
- 54. 4.7 Application IV: Finite Markov Chains Associative law of multiplication 54 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ]8711390*6.110*3.90*4.110*7. 6.4. 3.7. 90110 PP PP PP PP x 90110 PP PP x plant?eachatbewillemployeesmanyhowyears,twoofendAt the BBBA ABAA BBBA ABAA 00 2/ 022 BBBA ABAA 00 / 011 =++= = == = == BAMBA BAMBA
- 55. Ch. 4 Linear Models & Matrix Algebra: Summary Matrix algebra can be used: a. to express the system of equations in a compact notation; b. to find out whether solution to a system of equations exist; and c. to obtain the solution if it exists. Need to invert the A matrix to find the solution for x* 55 d A adjA x A adjA A dAx dAx = = = = − − * 1 1* det
- 56. Ch. 4 Notation and Definitions: Summary A (Upper case letters) = matrix b (Lower case letters) = vector nxm = n rows, m columns rank(A) = number of linearly independent vectors of A trace(A) = tr(A) = sum of diagonal elements of A Null matrix = all elements equal to zero. Diagonal matrix = all non-zero elements are in the diagonal. I = identity matrix (diagonal elements: 1, off-diagonal:0) |A| = det(A) = determinant of A A-1 = inverse of A A’=AT = Transpose of A A=AT Symmetric matrix A=A-1 Orthogonal matrix |Mij|= Minor of A 56
- 58. You know too much linear algebra when... You look at the long row of milk cartons at Whole Foods --soy, skim, .5% low-fat, 1% low-fat, 2% low-fat, and whole-- and think: "Why so many? Aren't soy, skim, and whole a basis?"
Editor's Notes
- Numerical examples are demonstrated on black board