TX-1037 Mathematical Techniques for Managers Dr Huw Owens Room B44 Sackville Street Building Telephone Number 65891 http://www.manchester.ac.uk/personal/staff/Huw.Owens Dr Huw Owens - University of Manchester : January 06 1 Introduction Graph Theory Linear and quadratic equations Differentiation

Integration Optimisation in management Matrix methods in management Summation techniques Dr Huw Owens - University of Manchester : January 06 2 Reading List Budnick F, 1993, Applied mathematics for business, economics and social sciences, McGraw-Hill Education (ISE Editions). Bostock and Chandler, 2000, Core A-level mathematics, Nelson Thornes. Jacques I, 1999, Mathematics for economics and business, third edition, Addison-Wesley. Jacques I, 2004, Mathematics for economics and business, fourth edition, Addison-Wesley. Soper J, 2004, Mathematics for Economics and Business, An Interactive Introduction, second edition, Blackwell Publishing.

Dr Huw Owens - University of Manchester : January 06 3 Module specific learning outcomes At the end of this module you should : be able to demonstrate familiarity with the basic rules of algebraic manipulations, matrix methods and applications for differentiation and integration; have the ability to deal with unknown quantities; have the ability to estimate order quantities, production planning skills and market forecasting etc; have numerical skills transferable to any discipline. Unseen exam paper worth 100% (10 credits) Dr Huw Owens - University of Manchester : January 06 4 Module delivery 24 hours of lectures 76 hours of private study

Dr Huw Owens - University of Manchester : January 06 5 Lecture Outline Monday, Monday, Monday, Monday, Series

Monday, Monday, Monday, Monday, Monday, Monday, Monday, Monday, 30th January 2006 Functions in Economics 6th February 2006 Equations in Economics 13th February 2006 Macroeconomic Models 20nd February 2006 Changes, Rates, Finance and 27st February 2006 Differentiation in Economics 6th March 2006 Maximum and Minimum Values 13th March 2006 Partial Differentiation 20th March 2006 Constrained Maxima and Minima 27th March 2006 Integration 24th April 2006 Integration 1st May 2006 - Matrices 8th May 2006 - Revision Dr Huw Owens - University of Manchester : January 06

6 Graphs of Linear Equations - Objectives Plot points on graph paper given their coordinates Add, subtract, multiply and divide negative numbers Sketch a line by finding the coordinates of two points on the line Solve simultaneous linear equations graphically Sketch a line by using its slope and intercept Dr Huw Owens - University of Manchester : January 06 7 Graphs of linear equations The horizontal solid line represents the x axis The vertical solid line represents the y axis O is the origin (0,0) P(x,y) y-axis or ordinate y

x O or origin x-axis or abscissa Dr Huw Owens - University of Manchester : January 06 8 Graphs of Linear Equations How do we specify the coordinates? 5 4 3 2 1 A B 0 -5

-4 -3 -2 -1 C 0 1 -1 2 3 4 5

D E -2 -3 -4 -5 Dr Huw Owens - University of Manchester : January 06 9 Graphs of Linear Equations Rules for multiplying negative numbers Negative * negative = positive Negative * positive = negative It does not matter in which order the two numbers are multiplied so Positive * negative = negative These rules produce (-2)*(-3) = 6 (-4)*5 = -20

7*(-5) = -35 Dr Huw Owens - University of Manchester : January 06 10 Fractions Sometimes the functions economists use involve fractions. For example, of peoples income may be taken by the government in income tax. Fraction: a part of a whole. E.g. if a household spends 1/5 of its total weekly expenditure on housing, the share of housing in the households total weekly expenditure is 1/5. If the households total weekly expenditure is 250, the amount it spends on housing is one fifth of that amount. Thus, amount spent on housing = share of housing*total weekly expenditure =1/5*250 = 50 Ratio: one quantity divided by another quantity Dr Huw Owens - University of Manchester : January 06

11 Fractions Numerator: the value on the top of a fraction Denominator: the value on the bottom of a fraction Dr Huw Owens - University of Manchester : January 06 12 Fractions - cancelling When working with fractions we can divide the top and bottom by the same amount to leave the fraction unchanged. 10 is said to be a factor of both the numerator and the denominator, and can be cancelled. 30 (30 / 10) 3 40 (40 / 10) 4 Dr Huw Owens - University of Manchester : January 06

13 Fractions common denominator Which of these fractions is larger? 3/7 or 9/20 In order to compare these fractions we need to find a common denominator. This is the reverse operation to cancelling and leaves the value of the fraction unchanged. 3 20 * 3 60 9 7 *9 63 , while 7 20 * 7 140 20 7 * 20 140 > sign: the greater than sign indicates that the value on its left is greater than the value on its right.

< sign: the less than sign indicates that the value on its left is less than the value on its right Dr Huw Owens - University of Manchester : January 06 14 Fractions Addition and Subtraction If fractions have the same denominators we can immediately add or subtract them. If the denominators are not the same we must find a common denominator for the fractions before adding or subtracting them. 3 1 3 1 4 9 2 9 2 7 , and 7 7 7 7 11 11

11 11 1 2 3 8 11 4 3 12 12 Dr Huw Owens - University of Manchester : January 06 15 Fractions Multiplication and division To multiply two fractions we multiply the numerators and the denominators. 1 3 1* 3 3 * 2 5 2 * 5 10 To divide one fraction by another we turn the divisor upside down and multiply by it. (N.B. You can check that this work by seeing that the reverse operation of multiplication gets you back to the value you started with.)

5 3 5 4 5 * 4 20 * 7 4 7 3 7 * 3 21 Dr Huw Owens - University of Manchester : January 06 16 Graphs of Linear Equations Division is a similar sort of operation to multiplication (it just undoes the result of the multiplication and takes you back to where you started) and the same rules apply when one number is divided by another. ( 15) 5 ( 3) ( 16) ( 8) 2 2

1 ( 4) 2 Dr Huw Owens - University of Manchester : January 06 17 Graphs of Linear Equations Evaluate the following: 5*(-6) (-1)*(-1) (-50)/10 (-5)/-1 2*(-1)*(-3)*6

2 * ( 1) * ( 3) * 6 ( 2) * 3 * 6 Dr Huw Owens - University of Manchester : January 06 18 Graphs of Linear Equations To add or subtract negative numbers it helps to think in terms of a picture of the axis: If b is a positive number then a-b can be thought of as an instruction to start a and to move b units to the left. E.g. 1 - 3 = -2 -4 -3 -2 -1 0

1 2 3 4 0 1 2 3 4 Similarly, -2 - 1 = -3 -4 -3

-2 -1 Dr Huw Owens - University of Manchester : January 06 19 Graphs of Linear Equations On the other hand, a-(-b) is taken to be a+b. This follows from the rule for multiplying two negative numbers since -(-b)=(-1)*(-b) = b Consequently, to evaluate a-(-b) you start at a and move b units to the right (the positive direction). For example (-2)-(-5)= -2+5 = 3 -4 -3 -2

-1 0 1 Dr Huw Owens - University of Manchester : January 06 2 3 4 20 Graphs of Linear Equations

Evaluate the following without using a calculator 1-2 -3-4 1-(-4) -1-(-1) -72-19 -53-(-48) Dr Huw Owens - University of Manchester : January 06 21 Multiplication and division involving 1 and 0 When we multiply and divide by 1 the expression is unchanged, whereas if we multiply or divide by 1 the sign of the expression changes. For example, try y=-(6x3-15x2+x-1) Each term is multiplied by 1, so now we have y = =-6x3+15x2-x+1 When we multiply by 0, the answer is 0

Division divides a value into parts but if there is nothing to begin with the result of the division is 0. For example, 0/4 = 0 division of zero gives zero Division by 0 gives an infinitely large value if it is positive or infinitely small value if it is negative Dr Huw Owens - University of Manchester : January 06 22 Graphs of linear Equations But, in economics we would like to be able to sketch curves represented by equations, to deduce information. Sometimes it is more appropriate to label axes using letters other than x and y. It is convention to use Q (Quantity) and P (Price) in the analysis of supply and demand. We will restrict our attention to graphs of straight lines at this time. Dr Huw Owens - University of Manchester : January 06 23

Graphs of Linear Equations What do you notice about the points (2,5),(1,3),(0,1), (-2,-3) and (-3,-5)? They all lie on a straight line with the equation -2x+y=1 If we substitute the values for x and y into the equation for the point (2,5) -2*2+5=1 We can check the remaining points similarly Point Check (1,3) -2*1+3 = -2+3 = 1 (0,1) -2*0+1 = 0+1 = 1 (-2,-3)

-2*-2-3 = 4-3 = 1 (-3,-5) -2*-3-5 = 6-5 = 1 Dr Huw Owens - University of Manchester : January 06 24 Graphs of Linear Equations The general equation of a straight line takes the form A multiple of x + a multiple of y = a number dx + ey = f for some given numbers d,e and f. Consequently, such an equation is called a linear equation. The numbers d and e are called coefficients. The

coefficients of the linear equation 2x+y = 1, are 2 and 1. Check the points (-2,2),(-4,4),(5,-2),(2,0) all lie on the line 2x+3y = 4 and sketch this line. In general to sketch a line from its mathematical equation , it is sufficient to calculate the coordinates of any two distinct points lying on it. The points can be plotted on paper and a ruler used to draw the line passing through them. Dr Huw Owens - University of Manchester : January 06 25 Graphs of Linear Equations - Example Sketch the line 4x+3y = 11 For the first point, we could choose x=5. Substitution gives: 4*5+3*y=11 20+3y=11 Now we need to determine y but how? We could guess values of y using trial and error. Actually, we only need to apply one simple rule You can apply whatever mathematical operation you like to an equation, provided that you do the same thing to both sides

BUT there is one exception; never divide both sides by zero. Dr Huw Owens - University of Manchester : January 06 26 Graphs of Linear Equations - Example 20+3y = 11 20+3y 20=11-20 3y=-9

3y/3=-9/3 y=-3 Consequently the coordinates of one point on the line are (5,-3). But we need two point to sketch the line. If we choose x=-1 and substitute into the equation 4*-1+3*y = 11 3y=11+4 y=5, therefore the coordinates of the second point are (1,5) Usually we select x=0 and y=0 Dr Huw Owens - University of Manchester : January 06 27 Graphs of Linear Equations

Finding where two lines intersect 4x+3y=11 2x+y=5 y=1 If y=1 4x+3=11 -6 -4 -2 0 4x=11-3 x=2 5 4 3 2 1 0 2

4 6 -1 Series3 Series4 -2 -3 -4 -5 -6 Dr Huw Owens - University of Manchester : January 06 28 Graphs of Linear Equations It can be shown that provided e is non-zero any equation given by dx+ey=f

Can be rearranged into the form y=ax+b Dr Huw Owens - University of Manchester : January 06 29 3 4 Graphs of Linear Equations 1 2 Positive slope 0 Zero slope 1

2 3 4 -2 -1 0 -4 -3 -4 -3 -2 -1 Intercept Dr Huw Owens - University of Manchester : January 06 Negative slope 30

Graphs of linear Equations Use the slope intercept approach to sketch the line 2x + 3y = 12 3y=12-2x y=4-2/3x y=4-2/3x 5 4 3 units 3 2 y 2 units y=4-2/3x

1 0 -2 -1 -1 0 1 2 3 4 5 6

7 -2 x Dr Huw Owens - University of Manchester : January 06 31 Graphs of linear equations Use the slope-intercept approach to sketch the lines y=x+2 4x+2y=1 Dr Huw Owens - University of Manchester : January 06 32 Algebra Algebra is boring!!!!!!! In evaluating algebraic or arithmetic statements certain rules need to be observed about the various operations. E.g. y = 10 + 6x2

If x=3 First substitute the value 3 for x and square it (9) Multiply this by 6 (54) Finally, add the result to the value 10 giving y = 64. Dr Huw Owens - University of Manchester : January 06 33 The order of algebraic operation Brackets - If there are brackets, do what is inside the brackets first Exponentiation exponentiation: raising to a power Multiplication and division Addition and subtraction Acronymn (BEDMAS) Remember An expression in brackets immediately preceded or followed by a value implies that the whole expression in the brackets is to be multiplied by that value. E.g. y = (10+6)x2 If x=3 then y = 144 Dr Huw Owens - University of Manchester : January 06

34 Order within an expression Algebraic expressions are usually evaluated from left to right. Addition or multiplication can occur in any order. In subtraction and division the order is important For example, 8-6 = 2 but 6-8 = -2 8/4 = 2 but 4/8 = 1/2 Dr Huw Owens - University of Manchester : January 06 35 Algebraic solution of simultaneous linear equations - Objectives Solve a system of two simultaneous linear equations with unknowns using elimination Detect when a system of equations does not have a solution Detect when a system of equations have infinitely many solutions

Solve a system of three linear equations in three unknowns using elimination Dr Huw Owens - University of Manchester : January 06 36 The elimination method Why use elimination? The graphical method has several drawbacks How do you decide suitable axes? Accuracy of the graphical solution? Complex problems with > three equations and > three unknowns? Dr Huw Owens - University of Manchester : January 06 37 Example 4x+3y = 11 (1) 2x+y = 5

(2) The coefficient of x in equation 1 is 4 and the coefficient of x in equation 2 is 2 By multiplying equation 2 by 2 we get 4x+2y = 10 (3) Subtract equation 3 from equation 1 to get minus 4x + 3y = 11 4x +

2y = 10 y = 1 Dr Huw Owens - University of Manchester : January 06 38 Example If we substitute y=1 back into one of the original equations we can deduce the value of x. If we substitute into equation 1 then 4x+3(1)=11 4x=11-3

4x=8 x=2 To check this put substitute these values (2,1) back into one of the original equations 2*2+1 = 5 Dr Huw Owens - University of Manchester : January 06 39 Summary of the method of elimination Step 1 Add/subtract a multiple of one equation to/from a multiple of the other to eliminate x. Step 2- Solve the resulting equation for y. Substitute the value of y into one of the original equations to deduce x. Step 4 Check that no mistakes have been made by substituting both x and y into the other original equation. Dr Huw Owens - University of Manchester : January 06 40

Example involving fractions Solve the system of equations 3x+2y =1 (1) -2x + y = 2(2) Solution Step 1 - Set the x coefficients of the two equations to the same value. We can do this by multiplying the first equation by 2 and the second by 3 to give 6x+4y = 2 (3) -6x+3y = 6(4) Add equations 3 and 4 together to cancel the x coefficients 7y = 8 y=8/7 Step three substitute y = 8/7 into one of the original equations 3x+2*8/7=1 Dr Huw Owens - University of Manchester : January 06 41 Example

3x=1-16/7 3x=-9/7 x = -9/7*1/3 x= -3/7 The solution is therefore x= -3/7, y= 8/7 Step 4 check using equation 2 -2*(-3/7)+8/7 = 2 6/7+8/7 = 2 14/7 = 2 2=2 Dr Huw Owens - University of Manchester : January 06

42 Problems 1) Solve the following using the elimination method 3x-2y = 4 x-2y =2 2) Solve the following using the elimination method 3x+5y = 19 -5x+2y = -11 Dr Huw Owens - University of Manchester : January 06 43 Special Cases

Solve the system of equations x-2y = 1 2x-4y=-3 The original system of equations does not have a solution. Why? Solve the system of equations 2x-4y = 1 5x-10y = 5/2 This original system of equations does not have a unique solution Dr Huw Owens - University of Manchester : January 06 44

Special Cases There can be a unique solution, no solution or infinitely many solutions. We can detect this in Step 2. If the equation resulting from elimination of x looks like the following then the equations have a unique solution Any non-zero number * y = Any number If the elimination of x looks like the following then the equations have no solutions zero

* y = Any non-zero number Dr Huw Owens - University of Manchester : January 06 45 Special Cases If the elimination of x looks like the following then the equations have infinitely many solutions zero * y =

zero Dr Huw Owens - University of Manchester : January 06 46 Elimination Strategy for three equations with three unknowns Step 1 Add/Subtract multiples of the first equation to/from multiples of the second and third equations to eliminate x. This produces a new system of the form ?x + ?y + ?z = ? ?y+?z = ? ?y+?z =? Step 2 Add/subtract a multiple of the second equation to/from a multiple of the third to eliminate y. This produces a new system of the form ?x + ?y + ?z = ? ?y+?z = ? ?z = ? Dr Huw Owens - University of Manchester : January 06

47 Step 3 Solve the last equation for z. Substitute the value of z into the second equation to deduce y. Finally, substitute the values of both y and z into the first equation to deduce x. Step 4 Check that no mistakes have been made by substituting the values of x,y and z into the original equations. Example Solve the equations 4x+y+3z = 8 (1) -2x+5y+z = 4 (2) 3x+2y+4z = 9 (3) Step 1 To eliminate x from the second equation multiply it by 2 and then add to equation 1 Dr Huw Owens - University of Manchester : January 06 48 To eliminate x from the third equation we multiply equation

1 by 3, multiply equation 3 by 4 and subtract Step 2 To eliminate y from the new third equation (5) we multiply equation 4 by 5, multiply equation 5 by 11 and add This gives us z = 1. Substitute back into equation 4. This gives us y = 1. Finally substituting y=1 and z=1 into equation 1 gives the solution x=1, y=1, z=1 Step 4 Check the original equations give 4(1)+1+3(1) = 8 -2(1)+5(1)+1=4 3(1)+2(1)+4(1)=9 respectively Dr Huw Owens - University of Manchester : January 06 49 Practice Problems

Sketch the following lines on the same diagram 2x-3y=6 4x-6y=18 x-3/2y=3 Hence comment on the nature of the solutions of the following system of equations A) 2x-3y = 6 x-3/2y=3 B) 4x-6y=18 x-3/2y=3 Dr Huw Owens - University of Manchester : January 06 50 Supply and Demand Analysis At the end of this lecture you should be able to Use the function notation, y=f(x) Identify the endogenous and exogenous variables in the economic model. Identify and sketch a linear demand function.

Identify and sketch a linear supply function. Determine the equilibrium price and quantity for a single-commodity market both graphically and algebraically. Determine the equilibrium price and quantity for a multi-commodity market by solving simultaneous linear equations Dr Huw Owens - University of Manchester : January 06 51 Microeconomics Microeconomics is concerned with the analysis of the economic theory and policy of individual firms and markets. This section focuses on one particular aspect known as market equilibrium in which supply and demand balance. What is a function? A function f, is a rule which assigns to each incoming number, x, a uniquely defined out-going number, y. A function may be thought of as a black-box which performs a dedicated arithmetic calculation.

An example of this may be the rule double and add 3. Dr Huw Owens - University of Manchester : January 06 52 For example, a second function might be g(x) = -3x+10 We can subsequently identify the respective functions by f and g Dr Huw Owens - University of Manchester : January 06 53 We can write this rule as y=2x+3 Or f(x)=2x+3 5 -17 Double and

Add 3 13 f(5)=13 Double and Add 3 -31 f(-17) If in a piece of economic theory, there are two or more functions we can use different labels to refer to each one. Dr Huw Owens - University of Manchester : January 06 54 Independent and dependent variables The incoming and outgoing variables are referred to as the independent and dependent variables

respectively. The value of y depends on the actual value of x that is fed into the function. For example, in microeconomics the quantity demanded, Q, of a good depends on the market price, P. This may be expressed as Q = f(P). This type of function is known as a demand function. For any given formula for f(P) it is a simple matter to produce a picture of the corresponding demand curve on paper. Economists plot P on the vertical axis and Q on the horizontal axis. Dr Huw Owens - University of Manchester : January 06 55 But first a Problem Evaluate f(25) f(1) f(17) g(0) g(48) g(16)

For the functions f(x) = -2x +50 g(x) = -1/2x+25 Do you notice any connection between f and g? Dr Huw Owens - University of Manchester : January 06 56 P=g(Q) Thus the two functions f and g are said to be inverse functions. The above form P=g(Q), the demand function, tells us that P is a function of Q but does not give us any precise details. If we hypothesize that the function is linear P = aQ+b (for some appropriate constants called parameters a and b) The process of identifying real world features and making appropriate simplifications and assumptions is known as modelling. Models are based on economic laws and help to explain the behaviour of real, world situations.

Dr Huw Owens - University of Manchester : January 06 57 A graph of a typical linear demand function may be seen below. Demand usually falls as the price of the good rises and so the slope of the line is negative. In mathematical terms P is said to be a decreasing function of Q. So a<0 a is less than zero and b>0 b is greater than zero P b Q Dr Huw Owens - University of Manchester : January 06 58 Example

Sketch the graph of the demand function P=-2Q+50 Hence or otherwise, determine the value of (a) P when Q=9 (b) Q when P=10 Solution (a) P = 2*9+50, P=32 (b) 10 = -2Q+50, -40 = -2Q, 20 = Q Sketch a graph of the demand function P = -3Q+75 Hence, or otherwise, determine the value of (a) P when Q=23

(b) Q when P=18 Solution (a) P = -69+75, P = 6 (b) 18 = -3Q+75, -57 = -3Q, 19 = Q Dr Huw Owens - University of Manchester : January 06 59 Weve so far looked at a crude model of consumer demand assuming that the quantity sold is based only on the price. In practice other factors are required such as the incomes of the consumers Y, the price of substitute goods PS, the price of complementary goods PC, advertising expenditure A, and consumer tastes T. A substitute good is one which could be consumed instead of the good under consideration. (e.g. buses and taxis) A complementary good is one which is used in conjunction with other goods (e.g. DVDs and DVD players). Mathematically, we say that Q is a function of P, Y,

PS,PC, A and T. Dr Huw Owens - University of Manchester : January 06 60 Endogenous and exogenous variables This is written as Q=f(P,Y,PS,PC,A,T) In terms of our black box diagram P Y PS PT f Q A T Any variables which are allowed to vary and are determined within the model are known as endogenous variables (Q and P).

The remaining variables are called exogenous since they are constant and are determined outside the model. Dr Huw Owens - University of Manchester : January 06 61 Inferior and superior goods An inferior good is one whose demand falls as income rises (e.g. coal vs central heating) A superior good is one whose demand rises as income rises (e.g. cars and electrical goods). Problem Describe the effect on the demand curve due to an increase in (a) the price of substitutable goods, Ps (b) the price of complementary goods, Pc (c) advertising expenditure Dr Huw Owens - University of Manchester : January 06 62

The supply function The supply function is the relation between the quantity, Q, of a good that producers plan to bring to the market and the price, P, of the good. A typical linear supply curve is indicated in the diagram below. Economic theory indicates that as the price rises so does the supply. (Mathematically P is an increasing function of Q) P b Supply curve Demand curve Q Dr Huw Owens - University of Manchester : January 06 63