Basic Quantitative Techniques Shamim S. Mondal Alliance University Class of July 2011 1 BQT - Shamim S. Mondal

Introduction to Numbers System:

Introduction to Numbers System Integers – Positive and Negative Fractions and rational numbers Irrational and Real Numbers The real line R The multi-dimensional R Imaginary numbers 2 BQT - Shamim S. Mondal

Integers:

Integers Integers are whole numbers, such as 1, 2, 3, … etc. Could either be positive, negative, or 0. Positive integers are called natural numbers How many integers are there? Potentially infinite However, it is possible to count them 3 BQT - Shamim S. Mondal

Fractions or Rational Numbers:

Fractions or Rational Numbers Fractions/ rational numbers Could be expressed as a fraction of the form x/y, where both x and y are integers Contains integers Thus, there are more rational numbers than there are integers It is still possible to count them 4 BQT - Shamim S. Mondal

Irrational Numbers:

Irrational Numbers There are numbers that you cannot express as x/y, i.e., ratio of two integers Example: Square root of 2, or for that matter, any higher root of 2 (cubic root etc.) Called irrational numbers Rational and irrational numbers together form the real line , R. 5 BQT - Shamim S. Mondal

Imaginary and Complex Numbers:

Imaginary and Complex Numbers What other types of numbers could exist? For a real number, its square cannot be negative, so a square root of a negative number is not a real number Imaginary numbers are situations where you find the square root of a negative number. Define i = √-1. Numbers of the form ( a+b i ) are called complex numbers. 6 BQT - Shamim S. Mondal

Sets:

Sets 7 BQT - Shamim S. Mondal

Basic Set Theory:

Basic Set Theory A set is a collection of units sharing common characteristics Example: A set of natural numbers N We denote this by N = {x: x>0 & x is an integer} If a set A is completely contained in another set B , then A is called a subset of B , denoted A ⊂ B 8 BQT - Shamim S. Mondal

Special Sets:

Special Sets Null Set : A set without any members, denoted by ø. Unit or Singleton Set : A set containing only one member; A = {x} Complement of a set: A complement of set A , denoted A c is a set that contains everything not in A . 9 BQT - Shamim S. Mondal

Unions and Intersections:

Unions and Intersections The union of two sets, A and B, denoted A U B , is the set that includes all elements of A and all elements of B. The intersection of two sets A and B , denoted A ∩ B , is the set containing elements common to both A and B . 10 BQT - Shamim S. Mondal

Venn Diagrams:

Venn Diagrams Union and Intersection of sets using Venn Diagram: A B A ∩B AUB 11 BQT - Shamim S. Mondal

Mutually Exclusive Sets:

Mutually Exclusive Sets Two sets, A and B , are called mutually exclusive sets if A ∩ B = ø. In other words, these two sets have nothing in common. Example: The sets of rational and irrational numbers. BQT - Shamim S. Mondal 12

Functions:

Functions 13 BQT - Shamim S. Mondal

Functions of Real Numbers:

Functions of Real Numbers A function takes a set of numbers, called input , and for each input, generates one number, called output . The set of all possible values of the inputs is called the domain of a function The set of all possible values of the outputs is called the range or codomain of a function Example: One variable linear function: f(x) = ax + b ; Two variables linear function: f(x, y) = ax + by + c ; Non-linear function: f(x) = ax 2 + bx + c; a≠0 . 14 BQT - Shamim S. Mondal

The Linear Function:

The Linear Function A straight line is represented by a function of the form: y = ax + b ; a is the slope of the line; b is the vertical intercept of the straight line; Slope is interpreted as the rate of change of the y variable for unit change in x variable For a linear function, slope is always equal to a at all points 15 BQT - Shamim S. Mondal

The Linear Function (cont.):

The Linear Function (cont.) Graphically, a line is represented as follows: In this case, a and b are both positive x y 0 y = ax + b b 16 BQT - Shamim S. Mondal

Economics Example:

Economics Example Introduce the linear demand and supply functions Demand: q d = ap + c ( a is typically negative) Supply: q s = bp + d ( b is typically positive) What defines an equilibrium? It is the equalization of demand and supply 17 BQT - Shamim S. Mondal

Supply Function P = 10 + 0.5Q Calculate and plot the supply schedule P = 10 + 0.5Q:

Supply Function P = 10 + 0.5 Q Calculate and plot the supply schedule P = 10 + 0.5 Q Table 2.4 Supply schedule Q : Quantity P : P = 10 +0.5 Q 0 10: P =10 +0.5(0) 20 20: P = 10+0.5(20) 40 30 60 40 80 50 100 60 Intercept (vertical) = 10 Slope = 0.5 Horizontal intercept = - 20 P = 10 + 0.5 Q NOTE: the supply function may be plotted by simply joining the intercepts Figure 2.22 P Q Q P S Slope = 0.5 P = 10 + 0.5 Q

Non-Linear Functions:

Non-Linear Functions These functions cannot be represented as a straight line For a straight line, the slope is always constant For a non-linear function, the slope changes at different points Typically, a non-linear function is represented by a higher order polynomial But, can be a combination of various piecewise linear functions 20 BQT - Shamim S. Mondal

General Polynomial Functions:

General Polynomial Functions In general, a polynomial function of order n in one variable is represented as where n is a natural number, a 1 , a 2 ,…,a n+1 constants When n = 1, you have the straight line When n = 2, you have the quadratic function When n = 3, you have the cubic function, and so on 21 BQT - Shamim S. Mondal

Quadratic Equations:

Quadratic Equations Typically, a quadratic equation in one variable x would be represented by: where a, b and c are constants. For given a, b and c, the goal is to find the values of x such that the equation above holds. 22 BQT - Shamim S. Mondal

Quadratic Equation (cont.):

Quadratic Equation (cont.) To solve it, generally make use of the formula: Notice that there are two possible solutions to this equation (because of the ±) In general, for a polynomial of degree n , there are n number of solutions 23 BQT - Shamim S. Mondal

Quadratic Equations (cont.):

Quadratic Equations (cont.) Notice that depending on the value of the parameters (or constants) a, b and c, we can have real or complex solutions to the equation If b 2 ≥4 ac , then the solutions are both real If, however, b 2 <4 ac , then both solutions are complex In most cases, we focus only on real solutions, and ignore the complex solutions 24 BQT - Shamim S. Mondal

Graphical Representation of Quadratic Functions:

Graphical Representation of Quadratic Functions 25 BQT - Shamim S. Mondal

Slope of a Quadratic Function:

Slope of a Quadratic Function The above curve represents the function: 3x 2 -5x+6 Where does it intersect the x-axis (i.e., what are the solutions?) What happens if it does not intersect the x-axis? What happens to the slope of the curve? Notice, that they are different in two parts of the graph For a quadratic function, the slope depends on x , and is not a constant like in a linear function 26 BQT - Shamim S. Mondal

Economic Example of Quadratic Functions:

Economic Example of Quadratic Functions Usually, in economics, cost functions and revenue functions are quadratic functions The cost function defines the cost of producing quantity Q of a certain good C = f(Q) = aQ + bQ 2 + c Typically, a, b, and c are all positive 27 BQT - Shamim S. Mondal

Properties of Quadratic functions:

Properties of Quadratic functions Over the real line domain, a quadratic function Changes direction Reaches a maximum or a minimum Is symmetric to the vertical line from the point of maximum or minimum Has a variable slope 28 BQT - Shamim S. Mondal

Concave Functions:

Concave Functions Concave Functions are functions that increase or decrease at a decreasing rate Example: y = √x This is an increasing function, but the rate of increase (slope) of the function is decreasing In economics, most utility functions in microeconomics are concave functions The defining relationship for a concave function is: f(λx 1 +(1-λ)x 2 )≥ λf (x 1 )+(1-λ)f(x 2 ) for any λ between 0 and 1. 29 BQT - Shamim S. Mondal

Example of Concave Function:

Example of Concave Function y = √x x y 30 BQT - Shamim S. Mondal

Convex Functions:

Convex Functions Convex Functions are functions that increase or decrease at a increasing rate Example: y = x 2 This is an increasing function, but the rate of increase (slope) of the function is increasing In economics, most cost functions in microeconomics are convex functions The defining relationship for a convex function is: f(λx 1 +(1-λ)x 2 )≤ λf (x 1 )+(1-λ)f(x 2 ) for any λ between 0 and 1. 31 BQT - Shamim S. Mondal

Example of Convex Function:

Example of Convex Function x y y = x 2 32 BQT - Shamim S. Mondal

Series and Sequences:

Series and Sequences 33 BQT - Shamim S. Mondal

Series – Arithmetic Progression:

Series – Arithmetic Progression Arithmetic Progression (AP): A series of numbers where each number is “equidistant” from the other, i.e., the difference between two successive numbers in the series is equal, called the common difference Example: Set of integers form an arithmetic progression, the common difference is 1 In general, the nth term of an arithmetic series is represented as: T n =a+(n-1)d , where a is the initial term and d is the common difference Sum of an arithmetic series: S n =n/2(2a+(n-1)d) 34 BQT - Shamim S. Mondal

Series – Geometric Progression:

Series – Geometric Progression Geometric Progression (GP): A series of numbers where the ratio of two successive numbers in the series is equal, called the common ratio Example: The numbers 1, 2, 4, 8, 16, 32, … form a geometric progression, the common ratio is 2 In general, the nth term of an arithmetic series is represented as: T n =ar n-1 , where a is the initial term and r is the common ratio Sum of an geometric series: S n =a(r n -1)/(r-1) 35 BQT - Shamim S. Mondal

Economic Example: Present Value of a Cash Flow:

Economic Example: Present Value of a Cash Flow Suppose you open a bank account with Rs. A, and the bank promises to pay Rs. C (called the coupon payment) every year for n (>1) years. The market interest rate is r. Payment Scheme: How to evaluate this payment stream today? C C C C C+A 36 BQT - Shamim S. Mondal

Economic Example: Present Value of Cash Flow (Continued):

Economic Example: Present Value of Cash Flow (Continued) Important: If you are paid in the future, you cannot spend it today, or you cannot earn interest rate today. So, if you earn Rs. C one year from now, and the market interest rate is r, the value of that Rs. C to you today is Rs. C/(1+r). This is the amount of money that you could put in the bank today and get paid back Rs. C one year from now. Similarly, present value of Rs. C paid to you 2 years from now is Rs. C/(1+r) 2 37 BQT - Shamim S. Mondal

Economic Example: Present Value of Cash Flow (Continued):

Economic Example: Present Value of Cash Flow (Continued) So, the present discounted value of the cash flow is C/(1+r)+C/(1+r) 2 +C/(1+r) 3 +…+C/(1+r) n +A /(1+r) n Exercise: Using the formula for sum of series, simplify the above expression This is an example of a bond typically issued by corporations (will be dealt with later in macro or finance classes) 38 BQT - Shamim S. Mondal

Special Function: Exponential Function:

Special Function: Exponential Function An exponential function is defined as: y=e x , (or exp(x) ) where e is a constant, approximately equal to 2.718281828, and x is a real number (but could also be complex in some applications) Also represented as An exponential function is a convex function (verify this) 39 BQT - Shamim S. Mondal

Economic Example: Compounding Continuously:

Economic Example: Compounding Continuously Suppose you have 1 rupee, that gives you an annual interest rate of r After 1 year, the sum becomes (1+r) If you are paid r/2 at the end of six month intervals, and reinvest the money without withdrawal, the sum is (1+r/2) 2 using the compounding principle. Now, imagine you will be paid interest of r/n at n intervals, and think of n as a very large number (think that your interest rate is calculated by the second, and every second your money is reinvested) Your money becomes (1+r/n) n When n is a very large number (mathematically, as n tends to infinity, written ), (1+r/n) n = e r 40 BQT - Shamim S. Mondal

Logarithmic Function:

Logarithmic Function Logarithm is the inverse operation of exponentiation with respect to base e . x= ln (exp(x)) Logarithm of a number is defined only for positive real numbers. Logarithm is not defined for a negative number. Log of a variable represents a concave function (verify this) 41 BQT - Shamim S. Mondal

Properties of Logarithmic and Exponential Functions:

Properties of Logarithmic and Exponential Functions ln ( xy ) = ln (x) + ln (y) ln (x/y) = ln (x) – ln (y) ln (x) n = nln (x) exp( x+y ) = exp(x)exp(y) exp(x-y) = exp(x)/exp(y) exp(x) p =exp( px ) 42 BQT - Shamim S. Mondal

Simultaneous Equations and Solutions:

Simultaneous Equations and Solutions 43 BQT - Shamim S. Mondal

Simultaneous Equations:

Simultaneous Equations Sometimes, systems are characterized by equations in many variables To solve them, we try to find values of variables such that all the equations are simultaneously satisfied First Question: Does a solution exist? I.e., could we find numbers that will satisfy all equations? Second question: Is such a solution unique? Could there be multiple solutions to the same problem? 44 BQT - Shamim S. Mondal

Linear Equations:

Linear Equations Generally, we want to find numbers that satisfy linear equations of the form: a 1 x+b 1 y=c 1 a 2 x+b 2 y=c 2 Example: Use Demand and Supply equations to determine equilibrium prices 45 BQT - Shamim S. Mondal

Equilibrium in the Goods Market exists when :

Equilibrium in the Goods Market exists when Going back to our demand and supply equations: Given the demand and supply functions : P = 100 - 0.5 Q and P = 10 + 0.5 Q, respectively, find the values of P and Q for which the market is in equilibrium algebraically

Find the equilibrium price and quantity algebraically:

Find the equilibrium price and quantity algebraically State the equilibrium condition: Q s = Q d and P s = P d Equate prices It is easier to equate prices, since the demand and supply functions have been given in the form P = f( Q )) hence solve for Q :

When the equilibrium quantity is known, find the equilibrium price, algebraically:

When the equilibrium quantity is known, find the equilibrium price, algebraically Now that the equilibrium value of Q has been determined, substitute the equilibrium quantity, Q = 90 into either the supply or demand function hence solve for the equilibrium price, P Hence the equilibrium price and quantity is Q = 90, P = 55

Solutions of Two Quadratic Equations:

Solutions of Two Quadratic Equations Consider a firm’s problem In general, we mentioned that cost functions are quadratic, and typically, we assume them to be a convex function What about revenue functions? Total Revenue = price*quantity sold From demand functions, we know how much would be bought by consumers, and equivalently, sold by firms at each price BQT - Shamim S. Mondal 49

Quadratic Equations Solution - cont:

Quadratic Equations Solution - cont TR = p*q = (100 - 0.5q)q = 100q – 0.5q 2 TC = 5 + 10q + 0.5q 2 Break-Even for a firm is defined as when Total Revenue is equal to Total Cost What is the break-even quantity for the firm? TR = TC implies 100q – 0.5q 2 – 5 - 10q - 0.5q 2 =0 Solve for q BQT - Shamim S. Mondal 50

Matrices:

Matrices 51 BQT - Shamim S. Mondal

Matrix:

Matrix A matrix is a rectangular array of numbers or expressions, consisting of rows and columns Example: This matrix A has 2 rows, and 3 columns. It is called a 2x3 matrix, or we say the dimension of the matrix is 2x3. In general, a matrix with dimensions of k rows and n columns is called a k x n matrix (convention: rows represented first, columns second). Elements of a matrix A are denoted a ij , where i denotes row and j denotes column. 52 BQT - Shamim S. Mondal

Vectors:

Vectors Vectors are matrices with one row or column. Example: From the previous matrix, the first row is the 1 x 3 row vector The first column is the 2 x 1 column vector : Scalars : Individual elements in a vector, such as a 11 . 53 BQT - Shamim S. Mondal

Operations on Matrices:

Operations on Matrices You can add , subtract , multiply and divide a nonzero (applies for division only) scalar to a matrix. What about two matrices? Addition & Subtraction : You can add or subtract two matrices A and B if they have the same dimensions k x n . You simply add the elements of matrix A with the corresponding element of matrix B . Example: If a 11 is the (1, 1) element of A and b 11 is the (1, 1) element of B , then the (1, 1) element of matrix ( A + B ) is ( a 11 + b 11 ). 54 BQT - Shamim S. Mondal

Matrix Multiplication:

Matrix Multiplication Two matrices A and B could be multiplied to form the matrix product AB (in that order) if number of columns of A = number of rows of B . Remember: AB is NOT equal to BA . If A is k x m , and B is m x n , then AB is k x n . Each row of A and each column of B will have m elements, and the product of i th row of A and j th column of B will be the ( i , j ) th element of the resulting matrix AB . 55 BQT - Shamim S. Mondal

Matrix Multiplication (continued):

Matrix Multiplication (continued) Example: Row i of A , multiplied by column j of B : = 56 BQT - Shamim S. Mondal

Matrix Operation: Transpose:

Matrix Operation: Transpose Transpose of a k x n matrix A , denoted A’ or A T , is obtained by interchanging rows and columns. The transposed matrix has dimensions n x k . Example: 57 BQT - Shamim S. Mondal

Rules with Transposition:

Rules with Transposition Add: (A+B) T = A T + B T Multiply: (AB) T = B T A T (Note the change in order) (A T ) T = A 58 BQT - Shamim S. Mondal

Special Types of Matrices:

Special Types of Matrices Square Matrix : When the number of rows and columns are equal ( k = n ) Diagonal Matrix : A square matrix whose nondiagonal entries are 0. Example: Identity Matrix : Diagonal matrix whose all diagonal elements are 1, denoted by I n for a n x n identity matrix Symmetric Matrix : Square matrix with A T = A Idempotent Matrix : A square matrix that multiplied by itself is the same: A.A = A Lower (Upper) Triangular Matrix : All entries above (below) the diagonal are 0. 59 BQT - Shamim S. Mondal

Inverse of a Matrix:

Inverse of a Matrix For a square matrix A , the inverse matrix (denoted A -1 ), if it exists, is defined to be the matrix B such that AB = BA = I . A matrix can be inverted when it is non-singular (more on this later). For two invertible square matrices A and B , (A - 1 ) - 1 =A (A T ) - 1 = (A - 1 ) T (AB) - 1 = B - 1 A - 1 60 BQT - Shamim S. Mondal

Simultaneous Equations in Matrix Forms:

Simultaneous Equations in Matrix Forms Recall the system of equations: a 1 x+b 1 y=c 1 a 2 x+b 2 y=c 2 Represent them in matrix form: Solution to Ax=c is x = A - 1 c 61 BQT - Shamim S. Mondal

Determinant of a Matrix:

Determinant of a Matrix Determinant of a square matrix A is a number, and is denoted by | A |. For a 2 x 2 matrix, the calculation is simple: For a 3 x 3 (or more generally, n x n ) matrix, it is slightly more complicated 62 BQT - Shamim S. Mondal

Determinant of a 3 x 3 Matrix:

Determinant of a 3 x 3 Matrix Take any row or column of 3 x 3 matrix A : Example: Take the first column. To each element of the first column, multiply the determinant of the matrix formed by excluding the row and the column in which the element belongs, and then multiply by (-1) ( i+j ) Example: For a 11 , compute the determinant of the matrix , = ( a 22 a 33 – a 32 a 23 ) Multiply the previous number by this by (-1) (1+1) a 11 Repeat for all elements in the column, add these all, and you have the determinant. 63 BQT - Shamim S. Mondal

Inverse of a Matrix:

Inverse of a Matrix To solve the equation mentioned before, you need to compute the inverse. To compute the inverse, you first have to compute the co-factor matrix, or its transpose, the adjoint matrix A 3 x 3 matrix: Denote Cofactor Matrix by C: 64 BQT - Shamim S. Mondal

Inverse of a 3 x 3 matrix (cont.):

Inverse of a 3 x 3 matrix (cont.) How are the a 11 s and C 11 s related? Computing C ij : Create a matrix from A , without the ith row or the jth column, and compute its determinant. Example: C 11 =(-1) (1+1) ( a 22 a 33 – a 32 a 23 ) Repeat this procedure for all ( i , j ), and you will have the Co-factor Matrix. The inverse of A, or A - 1 is C T /| A |. 65 BQT - Shamim S. Mondal

Applications in Business Economics:

Applications in Business Economics Matrices are used to solve systems of equations in microeconomics, macroeconomics (IS-LM model etc.) Matrices are used in econometrics, particularly regression analysis Matrices are used in operations, particularly linear programming problems Matrices are used in finance, for analysis of arbitrage pricing theory BQT - Shamim S. Mondal 66

Differentiation:

Differentiation 67 BQT - Shamim S. Mondal

Derivatives:

Derivatives Recall the discussion of functions f(x) in one variable x We are interested in finding out the slope of this function For a linear function, this is straightforward However, for a non-linear function, it depends on the specific point at which it is evaluated. The derivative of a function f(x) , at point x 0 , is called the derivative of f at x 0 , and is denoted by f’(x 0 ) , or BQT - Shamim S. Mondal 68

Secant and Tangent:

Secant and Tangent Let’s have a graphical illustration of the derivative Define secant to be a line segment joining two points on a graph of a function The secant between two arbitrarily close points is called the tangent I.e., the tangent is the limiting secant when the distance between two points is extremely close BQT - Shamim S. Mondal 69

Secant and Tangent - Illustration :

Secant and Tangent - Illustration BQT - Shamim S. Mondal 70 f(x 0 ) f(x 0 +h 1 ) f(x 0 +h 2 ) x f(x) 0

Derivatives - Continued:

Derivatives - Continued Mathematically: If f’(x 0 ) is positive (negative), then the slope of the function is positive (negative) at x 0 , and the function is increasing (decreasing) at x 0 . BQT - Shamim S. Mondal 71

Rules of Differentiation:

Rules of Differentiation For two differentiable functions f(x) and g(x) at x 0 : ( f ± g )’( x 0 ) = f’(x 0 ) ± g’(x 0 ) ( kf )’(x 0 ) = kf ’(x 0 ); k is a constant ( f.g )’(x 0 ) = f’(x 0 )g(x 0 ) + f(x 0 )g’(x 0 ) (f/g)’(x 0 ) = {f’(x 0 )g(x 0 ) – f(x 0 )g’(x 0 )}/g(x 0 ) 2 f(g(x))’=f’(g(x))g’(x) – Chain Rule (f(x) n )’ = n(f(x) n-1 )f’(x) ( x n )’ = nx n-1 ( ln (x))’ = 1/x (exp(x))’ = exp(x) BQT - Shamim S. Mondal 72

Higher Order Derivatives:

Higher Order Derivatives It is possible to differentiate some functions more than once The same rules as given before apply, but you have to differentiate the derivative of the function, and not the function itself. Second derivative of a function is computed as BQT - Shamim S. Mondal 73

Higher Order Derivatives (continued):

Higher Order Derivatives (continued) The second derivative indicates whether a function is increasing or decreasing at an increasing or decreasing rate, i.e., whether the slope is increasing or decreasing. Similarly, for the k th order derivative, whenever defined, the formula is Recall the concave and convex functions: for a concave (convex) function, the second derivative is negative (positive) BQT - Shamim S. Mondal 74

Applications of Derivatives - Maximum and Minimum:

Applications of Derivatives - Maximum and Minimum Suppose we are interested in knowing about the maximum of a function f . The local maximum (minimum) would be defined as a point where the function is higher (lower) than any other point in its vicinity Therefore, at this point the function is neither decreasing nor increasing In mathematical terms, its slope, and therefore its derivative, is 0. BQT - Shamim S. Mondal 75

Applications of Derivatives - Maximum and Minimum (cont):

Applications of Derivatives - Maximum and Minimum (cont) This is the necessary condition: for a point x 0 to be a maximum or a minimum, the derivative at that point will have to be 0 But is this sufficient ? I.e., can we be sure that if the derivative at x 0 is 0, it is a maximum? The answer is no. We need to ensure that the function decreases if we move in either direction from x 0 . BQT - Shamim S. Mondal 76

Applications of Derivatives - Maximum and Minimum (cont):

Applications of Derivatives - Maximum and Minimum (cont) If the function reaches maximum value at x 0 , then it must be true that the function value is lower to the left of x 0 , and lower to the right of x 0 . Therefore the slope of the function is positive to the left of x 0 , 0 at x 0 , and negative to the right of x 0 . So, the slope of the function is decreasing at x 0 , or the second derivative is negative. Similarly, for a minimum, the second derivative has to be positive. BQT - Shamim S. Mondal 77

Applications of Derivatives - Elasticity:

Applications of Derivatives - Elasticity Recall the demand function from before. Generally, quantity demanded of a good is expressed as a function of price per unit of that good. Derivative of the demand function gives you the change in quantity demanded of a good in response to a 1 unit increase in the price Sometimes, we are interested in the percentage change in quantity demanded in response to a 1 percent increase in price BQT - Shamim S. Mondal 78

Applications of Derivatives – Elasticity (cont):

Applications of Derivatives – Elasticity (cont) Elasticity of demand: Percentage change in quantity demanded in response to a percent increase in price: Note: A linear demand function has a constant slope (derivative), but at every point, its elasticity varies. BQT - Shamim S. Mondal 79

Multivariate Optimization:

Multivariate Optimization BQT - Shamim S. Mondal 80

Derivatives of Multivariate Functions:

Derivatives of Multivariate Functions Functions containing two arguments are called multivariate functions; z = f(x, y) There are two concepts of derivatives: Partial derivatives Total derivatives Partial derivatives are defined for situations when only one variable varies, keeping all other variables constant Partial derivatives are denoted by f x , or BQT - Shamim S. Mondal 81

Partial Derivative:

Partial Derivative The partial derivative is defined as follows: Notice that the only variable changing is x , the rest are retained at their original values. Example: Suppose the function is f(x, y)= ln (y)x. In this case, the partial derivative of function x with respect to y, f x (x, y) = ln (y) , and the partial derivative with respect to y is x/y. BQT - Shamim S. Mondal 82

Partial Derivatives (cont):

Partial Derivatives (cont) Partial derivatives of all variables are denoted by the vector This vector is also called the gradient vector of f . Gradient is the generalization of slope. As with derivatives of one variables, partial derivatives of higher order are possible; In addition, cross partial derivatives are also possible: . I.e., partial derivative with respect to y of the partial derivative of f with respect to x BQT - Shamim S. Mondal 83

Total Derivatives:

Total Derivatives For total derivatives, all variables are allowed to vary simultaneously. Thus the change in the functional value of function f(x, y) can be approximated by change in x and change in y You can also take second total derivatives and other higher orders, etc. BQT - Shamim S. Mondal 84

Optimization in Multiple Variables:

Optimization in Multiple Variables How to find the maximum and minimum of a function with two or more variables? While it is harder to illustrate graphically, recall that the same intuition from the single variable case carries through Formally, the unconstrained maximization or minimization problems could be stated as: BQT - Shamim S. Mondal 85

Optimization in Multiple Variables (cont):

Optimization in Multiple Variables (cont) As before, we look for points where the rate of change of functional values are 0. These are called critical points . How do we locate the critical points? Recall the total derivative of the function f( x,y ) Thus, for df (x, y) to be equal to zero, both partial derivatives will have to be 0. BQT - Shamim S. Mondal 86

Optimization in Multiple Variables (cont):

Optimization in Multiple Variables (cont) Therefore, the necessary condition for a critical point is that the gradient vector , or vector of derivatives But what do the points that we get from the above represent? They could represent a maximum, a minimum, or neither! The function could attain a maximum viewed from the point of view of one variable, and a minimum viewed from the point of view of the other variable, points called saddle points BQT - Shamim S. Mondal 87

Optimization in Multiple Variables (cont):

Optimization in Multiple Variables (cont) So how do we distinguish between a maximum, a minimum and a saddle point? We need to provide sufficient conditions Recall the second order conditions in the single variable case? For a maximum the second derivative was negative, and for a minimum, the second derivative was positive Similar condition exists in this case, but now we have a matrix of second derivatives! Need to find conditions for positivity and negativity of matrices BQT - Shamim S. Mondal 88

Positive/Negative definiteness of a matrix:

Positive/Negative definiteness of a matrix A square matrix A nxn is called positive (negative) definite , if for any real non-zero vector x 1xn , the product xAx ’ is positive (negative) But what is a functional way to check for positive or negative definiteness? Principal Minors of a Matrix: These are determinants of upper left kxk matrices carved out of A , k going from 1 to n. For a positive definite matrix, all principal minors are positive. For a negative definite matrix, the principal minors alternate in sign, starting negative BQT - Shamim S. Mondal 89

Optimization of multiple variables (cont):

Optimization of multiple variables (cont) Coming back to our optimization problem, let’s define the derivatives of second order for the gradient vector In this case, the resulting derivatives will be a 2x2 matrix, because each first order condition is a function of both x and y Denote the resulting matrix as BQT - Shamim S. Mondal 90

Optimization of multiple variables (cont):

Optimization of multiple variables (cont) The sufficient condition for optimization is: If the matrix of second order derivatives is positive definite, it is a minimum If the matrix of second order derivatives is negative definite, it is a maximum If the matrix of second order derivatives is neither positive definite nor negative definite, it is a saddle point Examples: f(x, y) = x 2 + y 2 BQT - Shamim S. Mondal 91

Constrained Optimization:

Constrained Optimization Most of the times in economics, you are faced with a constrained optimization problem. The constraints arise naturally in economic settings: Your consumption today is limited by how much you earn or how much you can borrow, or how much disposable income you have If you are a manager in a factory, you need to be able to mix inputs in an optimal way to produce a given level of output most efficiently BQT - Shamim S. Mondal 92

Basic Issues:

Basic Issues Imagine you need to maximize or minimize a function f(x, y) You are however, restricted in your choices of x and y We will look at the methods of finding solutions to such class of problems For now, we will confine our attention to one class of such problems BQT - Shamim S. Mondal 93

Problem Statement:

Problem Statement Formally, the problem is stated as: subject to where g(x, y) is denoted as the constraint imposed on the function f(x, y) . I.e., any solution ( x * , y * ) will have to satisfy g(x * , y * )=0. BQT - Shamim S. Mondal 94

Linear Constraints:

Linear Constraints We will confine ourselves to the case of linear constraints, i.e , g(x, y) = ax + by + c We will also confine ourselves to the case where the constraints are satisfied with equality, i.e., g(x, y) = ax + by + c = 0 , without loss of generality An example of this problem is the one of utility maximization with respect to a budget constraint Note: This solution method discussed is completely applicable to non-linear constraints, and can also simultaneously handle multiple constraints BQT - Shamim S. Mondal 95

The Utility Maximization Problem:

The Utility Maximization Problem Suppose you derive satisfaction from consumption of two goods, 1 and 2 Your satisfaction from consumption of x 1 units of good 1 and x 2 units of good 2 is represented by a utility function U(x 1, x 2 ) Utility is increasing in both goods, and the function U is continuous and twice differentiable BQT - Shamim S. Mondal 96

Utility Maximization Problem (cont):

Utility Maximization Problem ( cont ) The consumer, however, has limited budget of income I , and pays p 1 to buy a unit of good 1 and p 2 to buy a unit of good 2 Thus, the budget constraint facing the consumer is p 1 x 1 + p 2 x 2 = I The consumer chooses the quantity of goods 1 and 2 to maximize utility, but respecting his budget constraint BQT - Shamim S. Mondal 97

Utility Maximization Problem (cont):

Utility Maximization Problem ( cont ) Thus, formally, the problem is described as follows: subject to We want to solve for optimal consumption choices in terms of parameter values BQT - Shamim S. Mondal 98

Solution Methods:

Solution Methods We have already discussed solution of an unconstrained problem before. But, now we have a constrained problem How can we turn the constrained problem into an unconstrained problem? We incorporate the constraint explicitly into our objective function The method of Lagrange helps us transform this problem into one of unconstrained optimization, but at the cost of adding a variable BQT - Shamim S. Mondal 99

The Lagrangian Function:

The Lagrangian Function Define the function to be the Lagrangian Function Essentially, we have subtracted λ times the budget constraint from the utility function. The term λ is called the Lagrange multiplier . So, how do we find the optimum points? BQT - Shamim S. Mondal 100

First Order Conditions:

First Order Conditions The first order conditions of maximization (or minimization) equates the gradient vector of Lagrangian function L to 0. Thus, BQT - Shamim S. Mondal 101

Sufficient Conditions for Optimality:

Sufficient Conditions for Optimality The sufficient conditions are a little different from the unconstrained case But, the intuition in the unconstrained case still applies. For a maximum, we want some sort of negative definiteness of the second derivative matrix in the constrained case But rather than checking the signs of all leading principal minors, we check only of a few BQT - Shamim S. Mondal 102

Sufficient Condition (cont):

Sufficient Condition (cont) Let’s write down the second order derivative matrix in this case: If this determinant is positive , we have a maximum If this determinant is negative , we have a minimum BQT - Shamim S. Mondal 103

Integration:

Integration BQT - Shamim S. Mondal 104

Integration:

Integration Integration is the reverse operation of differentiation An antiderivative or integral of a function f(x) is a function F(x) whose derivative is the original function f(x) . Denoted as This is the indefinite integral BQT - Shamim S. Mondal 105

Rules of Integration:

Rules of Integration Power Rule of Integration This is true for all n, except n = -1. C is the constant of an indefinite integral. For n = -1, For exponential function: BQT - Shamim S. Mondal 106

Rules of Integration (cont.):

Rules of Integration (cont.) Addition: Multiplication/Integration by Parts: Example: Integrate ln (x) Application of integration by parts: f(x) = ln (x), g(x) = 1 BQT - Shamim S. Mondal 107

Fundamental Theorem of Calculus:

Fundamental Theorem of Calculus Definite Integral: When calculating the integral between two end points a and b Aggregate the sum of a continuous function (i.e., a function that does not jump suddenly) f(x) in the domain a ≤ x ≤ b. Divide the interval between a and b into N equal intervals Δ = ( b-a )/ N Define x 0 =a , x 1 =a + Δ , x 2 =a + 2 Δ , …, x N =a + N Δ =b . BQT - Shamim S. Mondal 108

Fundamental Theorem of Calculus (cont.):

Fundamental Theorem of Calculus (cont.) Form the products f(x i )(x i - x i-1 ) for all i = 1, …, N , and add them. As Δ gets very close to zero, or equivalently, we make N very large (i.e., the distance between successive points very small), this approaches the integral This is the fundamental theorem of calculus. BQT - Shamim S. Mondal 109

Area under a Graph:

Area under a Graph Alternatively, each one of these f(x i )(x i - x i-1 ) form the area of a rectangle, with base (x i - x i-1 ) and height f(x i ) . We add all these rectangles. With finer and finer subdivision of the interval a and b , we make better and better approximations of the area under the graph. Thus, the area under the graph of function f(x) between the points a and b is BQT - Shamim S. Mondal 110

Example: Consumer Surplus:

Example: Consumer Surplus Recall the demand curve: it is a function that relates how much a consumer is willing to pay for a given quantity of a product The area under the demand curve therefore represents the overall value to the customer Let the (inverse) demand function be represented by p = f ( q ), and the equilibrium price be p 0 and the equilibrium quantity be q 0 . The overall value to the consumer is The total payment by the consumer is p 0 q 0. Therefore, the consumer surplus , or surplus value to consumer is - p 0 q 0 BQT - Shamim S. Mondal 111

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