1. TX-1037 Mathematical Techniques for Managers
Dr Huw Owens
Room B44 Sackville Street Building
Telephone Number 65891
Dr Huw Owens - University of Manchester : January 06

2. Linear and quadratic equations Differentiation Integration
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

3. Bostock and Chandler, 2000, Core A-level mathematics, Nelson Thornes.
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

4. 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

5. Module delivery 24 hours of lectures 76 hours of private study
Dr Huw Owens - University of Manchester : January 06

6. Lecture 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 Matrices
Monday, 8th May Revision
Dr Huw Owens - University of Manchester : January 06

7. 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

8. 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

9. Graphs of Linear Equations
How do we specify the coordinates?
Dr Huw Owens - University of Manchester : January 06

10. 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

11. Fraction: a part of a whole.
Fractions
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
Dr Huw Owens - University of Manchester : January 06

12. Numerator: the value on the top of a fraction
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

13. 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.
Dr Huw Owens - University of Manchester : January 06

14. 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.
> 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

15. 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.
Dr Huw Owens - University of Manchester : January 06

16. Fractions – Multiplication and division
To multiply two fractions we multiply the numerators and the denominators.
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.)
Dr Huw Owens - University of Manchester : January 06

17. 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.
Dr Huw Owens - University of Manchester : January 06

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

19. 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 = -2 -4 -3 -2 -1 1 2 3 4
Similarly, = -3 -4 -3 -2 -1 1 2 3 4
Dr Huw Owens - University of Manchester : January 06

20. 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 1 2 3 4
Dr Huw Owens - University of Manchester : January 06

21. 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

22. 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

23. 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

24. 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
-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)
Check
Point
Dr Huw Owens - University of Manchester : January 06

25. 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

26. 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

27. 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

28. Graphs of Linear Equations
Finding where two lines intersect
4x+3y=11
2x+y=5
y=1
If y=1
4x+3=11
4x=11-3
x=2
Dr Huw Owens - University of Manchester : January 06

29. 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
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30. Graphs of Linear Equations
Positive slope
Zero slope -4 -3 -2 -1 1 2 3 4 -4 -3 -2 -1 1 2 3 4
Intercept
Negative slope
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31. Graphs of linear Equations
Use the slope intercept approach to sketch the line
2x + 3y = 12
3y=12-2x
y=4-2/3x
3 units
2 units
Dr Huw Owens - University of Manchester : January 06

32. Graphs of linear equations
Use the slope-intercept approach to sketch the lines
y=x+2
4x+2y=1
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33. First substitute the value 3 for x and square it (9)
Algebra
Algebra is boring!!!!!!!
In evaluating algebraic or arithmetic statements certain rules need to be observed about the various operations.
E.g. y = x2
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

34. 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

35. 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

36. 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

37. 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

38. By multiplying equation 2 by 2 we get 4x+2y = 10 (3)
Example
4x+3y = (1)
2x+y = (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 = (3)
Subtract equation 3 from equation 1 to get 4x + 3y = 11
minus 2y 10 y 1
Dr Huw Owens - University of Manchester : January 06

39. If we substitute into equation 1 then 4x+3(1)=11 4x=11-3 4x=8 x=2
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

40. 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

41. Example involving fractions
Solve the system of equations
3x+2y = (1)
-2x + y = (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 = (3)
-6x+3y = (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

42. The solution is therefore x= -3/7, y= 8/7
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

43. 1) Solve the following using the elimination method 3x-2y = 4 x-2y =2
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

44. Solve the system of equations x-2y = 1 2x-4y=-3
Special Cases
Solve the system of equations
x-2y = 1
2x-4y=-3
The original system of equations does not have a solution. Why?
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

45. 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
If the elimination of x looks like the following then the equations have no solutions
Any non-zero
number
Any
number * y =
Any non-zero
number
zero * y =
Dr Huw Owens - University of Manchester : January 06

46. 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

47. 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
?z = ?
Dr Huw Owens - University of Manchester : January 06

48. Example – Solve the equations 4x+y+3z = 8 (1) -2x+5y+z = 4 (2)
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 = (1)
-2x+5y+z = (2)
3x+2y+4z = (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

49. Step 4 Check the original equations give 4(1)+1+3(1) = 8
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

50. Sketch the following lines on the same diagram 2x-3y=6 4x-6y=18
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 B)
Dr Huw Owens - University of Manchester : January 06

51. 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

52. An example of this may be the rule “double and add 3”.
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

53. For example, a second function might be g(x) = -3x+10
We can subsequently identify the respective functions by f and g
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54. We can write this rule as – y=2x+3 Or f(x)=2x+35
Double and
Add 3 13
f(5)=13
-17
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

55. 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

56. Do you notice any connection between f and g?
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

57. Thus the two functions f and g are said to be inverse functions.
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

58. 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

59. 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
(a) P = , P = 6
(b) 18 = -3Q+75, -57 = -3Q, 19 = Q
Dr Huw Owens - University of Manchester : January 06

60. Mathematically, we say that Q is a function of P, Y, PS,PC, A and T.
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.
Dr Huw Owens - University of Manchester : January 06

61. Endogenous 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 Y PS PT A T
Dr Huw Owens - University of Manchester : January 06

62. 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

63. A typical linear supply curve is indicated in the diagram below.
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) b P Q
Supply curve
Demand curve
Dr Huw Owens - University of Manchester : January 06