dr huw owens - university of manchester : january 061 tx-1037 mathematical techniques for managers...

Dr Huw Owens - University of Manchester : January 06 1

TX-1037 Mathematical Techniques for TX-1037 Mathematical Techniques for Managers Managers

Dr Huw OwensRoom B44 Sackville Street Building

Telephone Number 65891http://www.manchester.ac.uk/personal/staff/Huw.Owens


IntroductionIntroduction

• Graph Theory• Linear and quadratic equations• Differentiation• Integration• Optimisation in management• Matrix methods in management• Summation techniques


Reading ListReading 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.


Module specific learning outcomesModule 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)


Module deliveryModule delivery

• 24 hours of lectures• 76 hours of private study


Lecture OutlineLecture Outline

• Monday, 30th January 2006 – Functions in Economics• Monday, 6th February 2006 – Equations in Economics• Monday, 13th February 2006 – Macroeconomic Models• Monday, 20nd February 2006 – Changes, Rates, Finance and

Series• Monday, 27st February 2006 – Differentiation in Economics• Monday, 6th March 2006 – Maximum and Minimum Values• Monday, 13th March 2006 – Partial Differentiation• Monday, 20th March 2006 – Constrained Maxima and Minima• Monday, 27th March 2006 – Integration• Monday, 24th April 2006 – Integration• Monday, 1st May 2006 - Matrices• Monday, 8th May 2006 - Revision


Graphs of Linear Equations - ObjectivesGraphs 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


Graphs of linear equationsGraphs 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)

y-axis or ordinate

x-axis or abscissa O or origin

x

y

P(x,y)


Graphs of Linear EquationsGraphs of Linear Equations

• How do we specify the coordinates?

-5

-4

-3

-2

-1

0

1

2

3

4

5

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

A

B

C

D

E


Graphs of Linear Equations – Rules for multiplying Graphs of Linear Equations – Rules for multiplying negative numbersnegative 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


FractionsFractions

• Sometimes the functions economists use involve fractions. For example, ¼ of people’s 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 household’s total weekly expenditure is 1/5. If the household’s 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


FractionsFractions

• Numerator: the value on the top of a fraction• Denominator: the value on the bottom of a fraction


Fractions - cancellingFractions - 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.

4

3

)10/40(

)10/30(

40

30


Fractions – common denominatorFractions – 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.

140

63

20*7

9*7

20

9,

140

60

7*20

3*20

7

3 while

• > 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


Fractions – Addition and SubtractionFractions – 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.

11

7

11

29

11

2

11

9,7

4

7

13

7

1

7

3

and

12

11

12

83

3

2

4

1


Fractions – Multiplication and divisionFractions – Multiplication and division

• To multiply two fractions we multiply the numerators and the denominators.

10

3

5*2

3*1

5

3*2

1

• 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.)

21

20

3*7

4*5

3

4*7

5

4

3

7

5


Graphs of Linear Equations 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.

5)3(

)15(

)8(2

)16(

2

1

)4(

2



• Evaluate the following:• 5*(-6)• (-1)*(-1)• (-50)/10• (-5)/-1• 2*(-1)*(-3)*6

6*3*)2(

6*)3(*)1(*2



• 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

• Similarly, -2 - 1 = -3

-4 -3 -2 -1 0 1 2 3 4



• 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 2 3 4



• Evaluate the following without using a calculator • 1-2• -3-4• 1-(-4)• -1-(-1)• -72-19• -53-(-48)


Multiplication and division involving 1 and 0Multiplication 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


Graphs of linear EquationsGraphs 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.



• 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

-2*-3-5 = 6-5 = 1(-3,-5)

-2*-2-3 = 4-3 = 1(-2,-3)

-2*0+1 = 0+1 = 1(0,1)

-2*1+3 = -2+3 = 1(1,3)

CheckPoint



• 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.


Graphs of Linear Equations - ExampleGraphs 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.


Graphs of Linear Equations - ExampleGraphs 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



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

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

-6 -4 -2 0 2 4 6Series3

Series4



• 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



-4 -3 -2 -1 0 1 2 3 4

-4-3

-2-1

01

23

4

Positive slope

Zero slope

Negative slope

Intercept


Graphs of linear EquationsGraphs 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

-2

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4 5 6 7

x

y y=4-2/3x

3 units

2 units


Graphs of linear equationsGraphs of linear equations

• Use the slope-intercept approach to sketch the lines• y=x+2• 4x+2y=1


AlgebraAlgebra

• 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.


The order of algebraic operationThe 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


Order within an expressionOrder 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


Algebraic solution of simultaneous linear Algebraic solution of simultaneous linear equations - Objectivesequations - 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


The elimination methodThe 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?


ExampleExample

• 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

4x + 3y = 11

minus 4x + 2y = 10

y = 1


ExampleExample

• 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


Summary of the method of eliminationSummary 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.


Example involving fractionsExample 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


Example 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


ProblemsProblems

• 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


Special CasesSpecial 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



• 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

• If the elimination of x looks like the following then the equations have no solutions

Any non-zero number

Anynumbery =*

zero Any non-zeronumbery =*



• If the elimination of x looks like the following then the equations have infinitely many solutions

zero zeroy =*


Elimination Strategy for three equations with three Elimination Strategy for three equations with three unknownsunknowns

• 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 = ?


• 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


• 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


Practice ProblemsPractice 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


Supply and Demand AnalysisSupply 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


MicroeconomicsMicroeconomics

• 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”.


• For example, a second function might be• g(x) = -3x+10• We can subsequently identify the respective functions

by f and g


• We can write this rule as –• y=2x+3• Or f(x)=2x+3

Double and Add 3

5 13

Double and Add 3

-17 -31

• If in a piece of economic theory, there are two or more functions we can use different labels to refer to each one.

f(5)=13

f(-17)


Independent and dependent variablesIndependent 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.


But first a ProblemBut 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?


• 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.


• 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”

b

P

Q


ExampleExample

• 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


• We’ve 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.


Endogenous and exogenous variablesEndogenous and exogenous variables

• This is written as Q=f(P,Y,PS,PC,A,T)• In terms of our “black box” diagram

• 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.

f Q

P

YPS

PT

AT


Inferior and superior goodsInferior 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


The supply functionThe 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)

b

P

Q

Supply curve

Demand curve

dr huw owens - university of manchester : january 061 tx-1037 mathematical techniques for managers...

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