TX-1037 Mathematical Techniques for Managers

1
TX-1037 Mathematical Techniques for Managers

• Dr Huw Owens
• Room B44 Sackville Street Building
• Telephone Number 65891

2
Introduction

• In Lecture 1 we looked at,
• Coordinates and Graphs
• Fractions
• Variables and Functions
• Linear Functions
• Power Functions
• Sketching Functions
• Algebra
• Factors and multiplying out brackets
• Accuracy
• Powers and Indices

3

• Functions of more than One Variable
• Economic Variables and Functions
• Total and Average Revenue
• Total and Average Cost
• Profit
• Production Functions, Isoquants and the average
  Product of Labour

4
Equations in Economics

• Lecture objectives
• Understand how equations are used in economics
• Rewrite and solve equations
• Substitute expressions
• Solve simple linear demand and supply equations
  to find market equilibrium
• Carry out cost-volume-Profit analysis
• Identify the slope and intercept of a line
• Plot the budget constraint to obtain the budget
  line

5
Variables and functions

• Variable a quantity represented by a symbol that
  can take different possible values (variable
  names x and y are often used).
• Constant a quantity whose value is fixed, even
  if we do not know its numerical amount (letters
  commonly used to represent constants are
  a,b,c,k).
• Function a systematic relationship between pairs
  of values of the variables, written yf(x).
• If one variable, y, changes in a systematic way
  as another variable, x, changes we say y is a
  function of x. The mathematical notation for
  this is
• yf(x), where the letter f is used to denote a
  function.
• If there is more than one functional relationship
  we can indicate they are different by using
  different letters, such as g or h.
• For example, yg(x), which is read as y is a
  function of x

6
Substitution of x-values

• A function gives a general rule for obtaining
  values of y from values of x. An example is
• y4x5, where 4x means 4x (by convention we omit
  the multiplication sign).
• To evaluate the function for a particular value
  of x, say x 6
• y465, y29
• Substituting different values into the function
  gives us different points on a graph.
• As the function tells us how to obtain y values
  from any x values, y is said to be dependent on
  x, and x is known as the independent variable.
• The independent variable is plotted on the
  horizontal axis and the independent variable is
  plotted on the vertical axis.

7
Linear functions

• If the relationship between x and y takes the
  form y6x

x 0 5 10
y 0 30 60
8
Linear functions

• Proportional relationship each y value is the
  same amount times the corresponding x value, so
  all points lie on a straight line through the
  origin.
• Linear function a relationship in which all the
  pairs of values form points on a straight line.
• In general, a function of the form ybx
  represents a straight line passing through the
  origin.
• Shift a vertical movement upwards or downwards
  of a line or curve.
• Adding a constant to a function shifts the
  function vertically upwards by the amount of the
  constant.
• For example, y6x20 has y values that are 20
  more than those of the previous function of every
  value of x.

9
Linear functions

• Intercept the value at which a function cuts the
  y-axis.
• Remember a function with a term just in x and
  perhaps a constant is a linear function. It has
  the general form
• yaxb

10
Power functions

• Power an index indicating the number of times
  the item to which is applied is multiplied by
  itself.
• For example, y x2 or z 7x2
• If we evaluate these functions and substitute x5
  we obtain
• yx2, y xx, y 55, y 25
• y 7x2, y 7xx, y 755, y 175
• Functions can have more than one term and one may
  be a constant.
• For example, y 1407x2-2x3 or y25x274
• Quadratic function a function in which the
  highest power of x is two. There may also be a
  term in x and a constant but no other terms.
• Cubic function a function in which the highest
  power of x is 3. There may also be terms in x2,
  x and a constant, but no other terms.

11
Power Functions An example

• Sketch and briefly describe the following
  functions for positive values of x
• y2x3-50 and y 14

12
Some questions

• Sketch the graphs of the functions for values of
  x between 0 and 10
• y0.5x
• y0.5x 6
• yx2
• y3x2
• Which of them is linear? Which is a proportional
  relationship? What is the effect of adding a
  constant term?

13
Some questions

• y0.5x Linear, proportional relationship
• y0.5x6, Linear, non-proportional relationship,
  shifted up by constant
• y x2. Quadratic function.
• y3x2. Quadratic function.

14
Factors and multiplying out brackets

• Factorising writing an expression as a product
  that when multiplied out gives the original
  expression.
• E.g. y6x-3x2, this is a shorthand way of writing
  y(6x)-(3xx)
• If we divide terms on the right-hand side by3x
  we obtain (6x)/3x 2 and (3xx)/(3x) -x.
• The amount we divide by we call a common factor.
• So factorising the expression y6x-3x2 we obtain
• y3x(2-x)
• When two brackets are multiplied together, to
  remove them we multiply each term in the second
  bracket by each term in the first bracket. It is
  then usual to simplify the result by collecting
  terms where possible.

15
Multiplying brackets

• E.g. (a-b)(-cd) -acad- -bc-bd -acadbc-bd
• E.g. (x-7)(4-3x) 4x-3x2-2821x -3x225x-28
• Factorisation is the reverse process to
  multiplying out brackets.
• It might not be obvious and could require some
  intelligent guesswork.
• The following standard results of multiplying out
  brackets are helpful.
• (ab)2 a22abb2
• (a-b)2a2-2abb2
• (ab)(a-b)a2-b2
• NOTE Not every quadratic expression factorises
  to an expression that contains integer values.

16
Powers and indices

• Index or power a superscript showing the number
  of times the value to which it is applied is to
  be multiplied by itself.
• E.g. x3 xxx, x1 x
• When we multiply together two expressions
  comprising the same value raised to a power we
  ADD the indices and raise to that new power.
• E.g. x3x5 (xxx)(xxxxx)x8
• Using our rule, x3x5 x35 x8
• When we divide together two expressions
  comprising the same value raised to a power we
  SUBTRACT the indices and raise to that new power.

17
What is an equation?

• Equation two expressions separated by an equals
  sign such that what is on the left of the equals
  sign has the same value as what is on the right.
• Solving equations lets us discover where lines or
  curves intersect.
• These points are interesting because they often
  indicate information about equilibrium
  situations.
• A graphical solution can be obtained by sketching
  the curves and reading off the x and y values at
  the point where they cross BUT results only have
  limited accuracy.

18
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 gt three equations and gt
  three unknowns?

19
Example

• 4x3y 11 (1)
• 2xy 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
• 4x2y 10 (3)
• Subtract equation 3 from equation 1 to get

4x 3y 11
minus 4x 2y 10
y 1
20
Example

• If we substitute y1 back into one of the
  original equations we can deduce the value of x.
• If we substitute into equation 1 then
• 4x3(1)11
• 4x11-3
• 4x8
• x2
• To check this put substitute these values (2,1)
  back into one of the original equations
• 221 5

21
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.

22
Example involving fractions

• Solve the system of equations
• 3x2y 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
• 6x4y 2 (3)
• -6x3y 6 (4)
• Add equations 3 and 4 together to cancel the x
  coefficients
• 7y 8
• y8/7
• Step three substitute y 8/7 into one of the
  original equations
• 3x28/71

23
Example

• 3x1-16/7
• 3x-9/7
• x -9/71/3
• 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
• 6/78/7 2
• 14/7 2
• 22

24
Problems

• 1) Solve the following using the elimination
  method
• 3x-2y 4
• x-2y 2
• 2) Solve the following using the elimination
  method
• 3x5y 19
• -5x2y -11

25
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

26
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
y


zero
27
Special Cases

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

zero
zero
y


28
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 ?

29

• 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
• 4xy3z 8 (1)
• -2x5yz 4 (2)
• 3x2y4z 9 (3)
• Step 1 To eliminate x from the second equation
  multiply it by 2 and then add to equation 1

30

• 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 y1 and z1 into equation 1
  gives the solution x1, y1, z1
• Step 4 Check the original equations give
• 4(1)13(1) 8
• -2(1)5(1)14
• 3(1)2(1)4(1)9
• respectively

31
Practice Problems

• Sketch the following lines on the same diagram
• 2x-3y6
• 4x-6y18
• x-3/2y3
• Hence comment on the nature of the solutions of
  the following system of equations
• A)
• 2x-3y 6
• x-3/2y3
• B)
• 4x-6y18
• x-3/2y3

32
Supply and Demand Analysis

• At the end of this lecture you should be able to
• Use the function notation, yf(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

33
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.

34

• For example, a second function might be
• g(x) -3x10
• We can subsequently identify the respective
  functions by f and g

35

• We can write this rule as
• y2x3
• Or f(x)2x3

5
13
Double and Add 3
f(5)13
-17
-31
Double and Add 3
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.

36
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.

37
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) -1/2x25
• Do you notice any connection between f and g?

38

• Pg(Q)
• Thus the two functions f and g are said to be
  inverse functions.
• The above form Pg(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 aQb (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.

39

• 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 alt0 a is less than zero and bgt0 b is
  greater than zero

P
b
Q
40
Example

• Sketch the graph of the demand function P-2Q50
• Hence or otherwise, determine the value of
• (a) P when Q9
• (b) Q when P10
• Solution
• (a) P 2950, P32
• (b) 10 -2Q50, -40 -2Q, 20 Q
• Sketch a graph of the demand function P -3Q75
• Hence, or otherwise, determine the value of
• (a) P when Q23
• (b) Q when P18
• Solution
• (a) P -6975, P 6
• (b) 18 -3Q75, -57 -3Q, 19 Q

41

• 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.

42
Endogenous and exogenous variables

• This is written as Qf(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.

P
f
Y
PS
Q
PT
A
T
43
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

44
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
Supply curve
b
Demand curve
Q