econ0702: mathematical methods in economicsyluo/teaching/lecture1.pdf · econ0702: mathematical...

ECON0702: Mathematical Methods in Economics

Yulei Luo

SEF of HKU

January 12, 2009

Luo, Y. (SEF of HKU) MME January 12, 2009 1 / 35

Course Outline

Economics: The study of the choices people (consumers, �rm managers, andgovernments) make to attain their goals, given their scarce resources.

Economic model: Simpli�ed version of reality used to analyze real-worldeconomic situations.

This course will mainly focus on how to use mathematical methods to solveeconomic models.

After solving the models, you may use economic data to test the hypothesisderived from models. If you are interested in testing economic models, youmay take Econ0701: Econometrics.


Course Outline

Static Analysis

Main objective: characterize partial and general equilibrium: demand=supplyand solve for equilibrium prices and quantities.Mathematical tools: matrix algebraEconomic applications: market equilibrium models and national-income models

Comparative-Static Analysis

Main objective: compare di¤erent equilibrium states that are associated withdi¤erent sets of values of parameters or exogenous variables (i.e., they aregiven and not determined by the model structure).Mathematical tools: calculus (di¤erentiation, partial di¤erentiation, etc.)Economic applications: market equilibrium models and national-income models


(Continued.)

Static Optimization Analysis

Main objective: characterize the optimizing behavior of consumers and �rmsbehind equilibrium at one period (static, atemporally).Mathematical tools: unconstrained and constrained static optimizationEconomic applications: utility maximization, pro�t maximization, and costminimization.

Dynamic Analysis

Main objective: characterize the evolution of economic variables over time(dynamic, intertemporally).Mathematical tools: �rst-order and higher-order di¤erence (discrete-time) ordi¤erential (continuous-time) equations.Economic applications: economic growth models, cobweb model, etc.

Dynamic Optimization

Main objective: characterize the optimal behavior of agents over time.Mathematical tools: optimal control theory (the Lagrange multiplier method)or dynamic programming (The Bellman equation method)Economic applications: optimal economic growth models, life-cycle optimalconsumption-saving models, optimal investment models.


The meaning of equilibrium

In essence, an equilibrium for a speci�c model is a situation that ischaracterized by a lack of tendency to change. It is for this reason that theanalysis of equilibrium is referred to as statics.

We will discuss two examples of equilibrium. One is the equilibrium attainedby a market under given demand and supply conditions. The other is theequilibrium of national income under given conditions of total consumptionand investment patterns.


Partial equilibrium model (linear)

In a static equilibrium model, the standard problem is to �nd the set ofvalues of the endogenous variables (i.e., they are not given and aredetermined by the model structure) which will satisfy the equilibriumconditions of the model.Partial equilibrium market model: a model of price determination in a singlemarket. Three variables:

Qd = the quantity demanded of the commodity

Qs = the quantity supplied of the commodity

P = the price of commodity.

And the equilibrium condition is

Qd = Qs . (1)

The model setup is then

Qd = Qs (2)

Qd = a� bP (3)

Qs = �c + dP (4)

where a, b, c , and d are all positive.Luo, Y. (SEF of HKU) MME January 12, 2009 6 / 35

One way to �nd the equilibrium is by successive elimination of variables andequations through substitution. From Qd = Qs , we have

a� bP = �c + dP. (5)

Since b+ d 6= 0, the equilibrium price is then

P� =a+ cb+ d

, (6)

and the equilibrium quantity can be obtained by substituting P� into eitherQd or Qs :

Q� =ad � bcb+ d

, (7)

where we assume that ad � bc > 0 to make the model economicallymeaningful.


Partial equilibrium model (Nonlinear)

The partial equilibrium market model can be nonlinear. The model setup isthen

Qd = Qs (8)

Qd = 4� P2 (9)

Qs = �1+ 4P (10)

As before, the three-equation system can be reduced to a single equation bysubstitution:

P2 + 4P � 5 = 0, (11)

which can be solved by applying the quadratic formula:

P� = 1 or � 5. (12)

Note that only the �rst root is economically meaningful.


General market equilibrium

In the above, we only consider a single market. In the real world, there wouldnormally exist many goods. Thus, a more realistic model for the demand andsupply functions of a commodity should take into account the e¤ects notonly of the price of the commodity itself but also of the prices of othercommodities.As a result, the price and quantity variables of multiple commodities mustenter endogenously into the model. Consequently, the equilibrium conditionof an n�commodity market model will involve n equations, one for eachcommodity, in the form

Ei = Qi ,d �Qi ,s = 0, (13)

where i = 1, � � �, n,Qi ,d = Qi ,d (P1, � � �,Pn) and Qi ,s = Qi ,s (P1, � � �,Pn) (14)

are the demand and supply functions of commodity i . Thus solvingn�equation system for P :

Ei (P1, � � �,Pn) = 0 (15)

if a solution does exist.Luo, Y. (SEF of HKU) MME January 12, 2009 9 / 35

Two commodity market model

The model setup is as follows

Q1,d = Q1,s (16)

Q1,d = a0 + a1P1 + a2P2 (17)

Q1,s = b0 + b1P1 + b2P2 (18)

Q2,d = Q2,s (19)

Q2,d = α0 + α1P1 + α2P2 (20)

Q2,s = β0 + β1P1 + β2P2, (21)

which implies that

(a0 � b0) + (a1 � b1)P1 + (a2 � b2)P2 = 0 (22)

(α0 � β0) + (α1 � β1)P1 + (α2 � β2)P2 = 0 (23)


(Continued.) Denote that

ci = ai � bi ; di = αi � βi , (24)

where i = 0, 1, 2. The above linear equations can be written as

c1P1 + c2P2 = �c0 (25)

d1P1 + d2P2 = �d0, (26)

which can be solved by further elimination of variables:

P�1 =c2d0 � c0d2c1d2 � c2d1

;P�2 =c0d1 � c1d0c1d2 � c2d1

. (27)

Note that for the n�commodity linear market model, we have n linearequations and n unknowns. However, an equal number of equations andunknowns doesn�t necessarily guarantee the existence of a unique solution.Only when the number of unknowns equals to the number of functionalindependent equations, the solution exists and is unique. We thus needsystematic methods to test the existence of a unique solution.


Equilibrium National-Income Model

Consider a simple Keynesian national-income model,

Y = C + I0 + G0 (Equilibrium condition) (28)

C = a+ bY (The consumption function), (29)

where Y and C stand for the endogenous variables national income andconsumption expenditure, respectively, and I0 and G0 are the exogenouslydetermined investment and government spending. Following the sameprocedure, the equilibrium income and consumption can solved as follows

Y � =a+ I0 + G01� b (30)

C � =a+ b (I0 + G0)

1� b . (31)


Why Matrix Algebra?

As more and more commodities are incorporated into the model, it becomesmore and more di¢ cult to solve the model by substitution. A method suitablefor handling a large system of simultaneous equations is matrix algebra.

Matrix algebra provides a compact way of writing an equation system; itleads to a way to test the existence of a solution by evaluation of adeterminant �a concept closely related to that of a matrix; it also gives amethod to �nd that solution if it exists.


Quick Review of Matrix Algebra

Matrix: a matrix is simply a rectangular array of numbers. So any table ofdata is a matrix.

1 The size of a matrix is indicated by the number of its rows and the number ofits columns. A matrix A with k rows and n columns is called a k � n matrix.The number in row i and column j is called the (i , j) entry and is written as aij .

2 Two matrices are equal if they both have the same size and if thecorresponding entries in the two matrices are equal.

3 If a matrix contains only one column (row), it is called a column (row) vector.

Example:�a11 a12a21 a22

�is a 2� 2 matrix.

�a1a2

�is a 2� 1 matrix or a

column vector.


(Continued.)

Addition: One can add two matrices with the same size. E.g.,�a11 a12a21 a22

�+

�1 00 1

�=

�a11 + 1 a12a12 a22 + 1

�.

Subtraction: subtract matrices of the same size simply by subtracting theircorresponding entries. E.g.,�a11 a12a21 a22

��1 00 1

�=

�a11 � 1 a12a12 a22 � 1

�.

Scalar multiplication: matrices can be multiplied by ordinary numbers

(scalar). E.g., α

�a11 a12a12 a22

�=

�αa11 αa12αa12 αa22

�.


(Continued.)

Matrix multiplication:

Not all pairs of matrices can be multiplied together, and the order in whichmatrices are multiplied can matter. We can de�ne the matrix product AB ifand only if

Number of columns of A = Number of rows of B . (32)

For the matrix product to exist, A must be k �m and B must be m � n. Toobtain the (i , j) entry of AB , multiply the ith row of A and the jth column ofB .If A is k �m and B is m � n, then the product AB is k � n. That is, theproduct matrix AB inherits the number of its rows from A and the number ofits columns from B .E.g.,

�a1 a2

� � a1a2

�= a21 + a

22 . (33)�

a11 a12a12 a22

� �α1α2

�=

�α1a11 + α2a12α1a12 + α2a22

�(34)


(Continued.)

Linear dependence of vectors. A set of vectors v1, � � �, vn is said to be linearlydependent if and only if any one of them can be expressed as a linearcombination of the remaining vectors; otherwise, they are linearlyindependent.

For any vectors v1, � � �, vn to be linear independent, for any set of scalars ki ,∑ni=1 ki vi = 0 if and only if ki = 0 for all i .Example: the three vectors

v1 =�27

�, v2 =

�18

�, v3 =

�45

�(35)

are linearly dependent since v3 is a linear combination of v1 and v2:

3v1 � 2v2 = v3 (36)


(Continued.)

Laws of matrix algebra (think of matrices as generalized numbers)

Associative laws:

(A+ B) + C = A+ (B + C ) (37)

(AB)C = A(BC ) (38)

Communication law for addition

A+ B = B + A (39)

Distributive lawsA(B + C ) = AB + AC (40)

Note that Communication law for multiplication does NOT hold:

AB 6= BA in general. (41)


(Continued.)

Transpose: the transpose of a k �m matrix A is the m� k matrix obtainedby interchanging the rows and columns of A. It is also written as AT . E.g.,

�a1 a2

�T=

�a1a2

�(42)

Square matrix: a matrix with the same numbers of rows and columns. E.g.,�a11 a12a21 a22

�.

Determinant of a square matrix: For 1� 1 matrix a, det(a) = a.For 2� 2matrix A =

�a11 a12a21 a22

�,

det(A) = jAj = a11a22 � a12a21 (43)

which is just the product of two diagonal entries minus the product of twoo¤-diagonal entries.


(Continued.)

Inverse: Let A be a square matrix. The matrix B is an inverse for A if

AB = BA = I where I =�1 00 1

�is called identity matrix (a square matrix

with ones in its principal diagonal and zeros everywhere else).If the matrix Bexists, we say that A is invertible and denote B = A�1.

Squareness is a necessary but not su¢ cient condition for the existence of aninverse. If a square matrix A has an inverse, A is said to be nonsingular. If Ahas no inverse, it is said to be a singular matrix.


Some Properties of The Transpose and the Inverse

The operation of matrix transpose and matrix inverse satisfy

(AB)0 = B 0A0 (44)

(A0)0 = A (45)

A0A = 0() A = A0 = 0 (46)

(A+ B)0 = A0 + B 0 (47)

(A�1)0 = (A0)�1 (48)

(A�1)�1 = A (49)

(AB)�1 = B�1A�1 (50)

provided that the objects above are well de�ned.


(Continued.)

System of equations in matrix form: Consider the following two-equation(two unknown variables) system

a11x1 + a12x2 = b1 (51)

a21x1 + a22x2 = b2 (52)

Let A =�a11 a12a21 a22

�be the coe¢ cient matrix of the system, x =

�x1x2

�,

and b =�b1b2

�. The above equations can be written in a compact form:

�a11 a12a21 a22

� �x1x2

�=

�b1b2

�or Ax = b. (53)

The solution to the above equations can be written as

x =A�1b, (54)

if A�1 exists.


Test for the Existence of the Inverse and Find the Inverse

(Continued.)

The necessary and su¢ cient conditions for nonsingularity are that the matrixsatis�es the squareness and linear independence conditions (that is, its rowsor equivalently, its columns are linearly independent.).

An n� n nonsingular matrix A has n linearly independent rows (or columns);consequently, it must have rank n. Consequently, an n� n matrix having rankn must be nonsingular.

The concept of determinant can be used to not only determine whether asquare matrix is nonsingular, but also calculate the inverse of the matrix.


(Continued.)

The value of a determinant of order n can be found by the Laplace expansionof any row or any column as follows

jAj =n

∑j=1

aij (�1)i+j jMij j, (55)

where jMij j is the minor of the element aij and can be obtained by deletingthe ith row and jth column of the determinant jAj and jCij j = (�1)i+j jMij jis called cofactor.

Example: for a 3� 3 matrix A, its determinant has the value:

jAj =

��a11 a12 a13a21 a22 a23a31 a32 a33

�� = a11�� a22 a23a32 a33

�� (56)

�a12�� a21 a23a31 a33

��+ a13 �� a21 a22a31 a32

�� . (57)


Determinantal Criterion for Non-singularity

Given a linear equation system Ax = b, where A is a square coe¢ cientmatrix, we have

jAj 6= 0() A is row or column independent

() A is nonsingular

() A�1 exists

() A unique solution x =A�1b exists.


Finding the Inverse Matrix

Each element aij of n� n matrix A has a cofactor jCij j. A cofactor matrixC =

�jCij j

�is also n� n. The transpose C 0 is referred to as the adjoint of A

and denoted by adjA.The procedure to calculate A�1 :

1 Find jAj2 Find the cofactors of all elements of A and form C =

�jCij j

�3 Form C 0 = adjA4 Determine

A�1 =adjAjAj (58)


Cramer�s Rule

Another way to calculate the inverse of a matrix An�n is to use the Cramerrule. Let Bj be the n� n matrix formed by taking A and substituting itsj�th column, aj , with the constants vector b. For example, for j = 2,

B2 =�a1 b a3 � � � an

�(59)

=

2664a11 b1 a13 � � � a1na21 b2 a23 � � � a2n� � � � � � � � � � � �an1 bn an3 � � � ann

3775 . (60)

Let jAj 6= 0 and jBj j be the determinants of A and Bj , respectively. See anylinear algebra textbook or pages 103-104 in CW for proof.

Cramer rule then says that the j�th element of the solution x =A�1b isgiven by

xj =jBj jjAj 8j = 1, � � �, n. (61)


Reconsider the Two-commodity Model

c1P1 + c2P2 = �c0 (62)

d1P1 + d2P2 = �d0, (63)

which can be rewritten as�c1 c2d1 d2

�| {z }

A

�P1P2

�| {z }

x

=

��c0�d0

�| {z }

b

, (64)

which implies that�P1P2

�=

�c1 c2d1 d2

��1 � �c0�d0

�=

1jAj

�d2 �c2�d1 c1

� ��c0�d0

�=

"c2d0�c0d2c1d2�c2d1c0d1�c1d0c1d2�c2d1

#. (65)


Reconsider Equilibrium National-Income Model

The model

Y = C + I0 + G0 (Equilibrium condition) (66)

C = a+ bY (The consumption function), (67)

can be rewritten in a compact matrix form�1 �1�b 1

�| {z }

A

�YC

�| {z }

x

=

�I0 + G0a

�| {z }

b

, (68)

which implies that�YC

�=

11� b

�1 1b 1

� �I0 + G0a

�=

"a+I0+G01�b

a+b(I0+G0)1�b

#. (69)

Applying Cramer rule gives:

Y =

�� I0 + G0 �1a 1

�� /�� 1 �1�b 1

�� = a+ I0 + G01� b . (70)


IS-LM Model: Closed Economy

Consider another linear model of macroeconomy: the IS-LM model made upof two sectors: the real goods sector and the monetary sector. The goodsmarket involves the following equations:

Y = C + I + G (71)

C = a+ b(1� t)Y (72)

I = d � ei (73)

G = G0, (74)

where Y ,C , I , and i (the interest rate) are endogenous variables, G0 is theexogenous variable, and a, b, d , e, and t are structural parameters.In the newly introduced money market, we have

Md = Ms (Equilibrium condition in the money market) (75)

Md = kY � li (Aggregate money demand) (76)

Ms = M0, (77)

where M0 is the exogenous money supply and k and l are parameters. Notethat these three equations can be reduced to

M0 = kY � li (78)Luo, Y. (SEF of HKU) MME January 12, 2009 30 / 35

(Continued.)Together, the two sectors give the following system of equations

Y � C � I = G0 (79)

b(1� t)Y � C = �a (80)

I + ei = d (81)

kY � li = M0, (82)

which can be written in a compact matrix form:26641 �1 �1 0

b(1� t) �1 0 00 0 1 ek 0 0 �l

3775| {z }

A

2664YCIi

3775| {z }

x

=

2664G0�adM0

3775| {z }

b


(Continued.)

Applying Cramer rule gives:

Y � =

��2664G0 �1 �1 0�a �1 0 0d 0 1 eM0 0 0 �l

3775�� /

��2664

1 �1 �1 0b(1� t) �1 0 0

0 0 1 ek 0 0 �l

3775��

=l (a+ d + G0) + eM0ek + l [1� b(1� t)] . (83)

Similarly, we may derive the expressions for other endogenous variables(C , I , i) in terms of both exogenous variables and structural parameters.

It is clear that Cramer�s rule is elegant, but if you have to invert a numericalmatrix (e.g., a = 1, t = 0.1, b = 0.4, etc.), you�d better to use Matlab,Gauss, Mathematica, etc.


Note on Homogeneous-equation System

The equation system Ax = b considered before can have any constants in thevector b. If b = 0, that is, every elements in the b vector is zero, theequation system becomes

Ax = 0, (84)

where 0 is a zero vector. This special case is referred to as ahomogeneous-equation system. �homogeneous�here means that if all thevariables x1, � � �, xn are multiplied by the same number, the equation systemwill remain valid. This is possible only if b = 0.Note that if the coe¢ cient matrix A is nonsingular (that is, it is invertible), ahomogeneous-equation system can yield only a trivial solution, that is, x = 0.(check it yourself)


(Continued.)

The only way to get a nontrivial solution from the above HE system is to have

jAj = 0, that is, to have a singular coe¢ cient matrix A. (85)

No unique nontrivial solution exists for this HE system. For example, considera 2� 2 case:

a11x1 + a12x2 = 0 (86)

a21x1 + a22x2 = 0,

assume that A =�a11 a12a21 a22

�is singular, so that jAj = 0. This implies that

the row vector�a11 a12

�is a multiple of the row vector

�a21 a22

�;

consequently, one of the two equations is redundant. By deleting the secondequation from the system, we end up with one equation in two variables andthe solution is

x1 = (�a12/a11) x2, (87)

which is well-de�ned if a11 6= 0 and represents an in�nite number of solutions.


Solution Outcomes for a Linear-equation System

Ax = bVector bb 6= 0

jAj 6=0 There exists a unique, nontrivial solution x� 6= 0jAj = 0 In�nite numbers of solutions if equations are dependent

No solution if equations are inconsistentb = 0

jAj 6=0 There exists a unique, trivial solution x�= 0jAj = 0 In�nite numbers of solutions if equations are dependent

Not possible if equations are inconsistent


econ0702: mathematical methods in economicsyluo/teaching/lecture1.pdf · econ0702: mathematical...

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