Transcript Lecture 2 - Personal Webpages (The University of Manchester)

TX-1037 Mathematical Techniques for Managers
Dr Huw Owens
Room B44 Sackville Street Building
Telephone Number 65891
Huw Owens - University of Manchester : February 06
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Introduction
• In
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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
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•
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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
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Equations in Economics
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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
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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 y=f(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
• y=f(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, y=g(x), which is read as “y is a function of x”
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Substitution of x-values
• A function gives a general rule for obtaining values of y
from values of x. An example is
• y=4x+5, where 4x means 4*x (by convention we omit the
multiplication sign).
• To evaluate the function for a particular value of x, say x=
6
• y=4*6+5, y=29
• 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.
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Linear functions
• If the relationship between x and y takes the form
y=6x
x
0
5
10
y
0
30
60
y=6x
70
60
60
50
40
y
y=6x
30
30
20
10
0
0
0
2
4
6
8
10
12
x
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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 y=bx 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, y=6x+20 has y values that are 20 more
than those of the previous function of every value of
x.
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Linear functions
y=6x+20
90
80
80
70
60
60
50
50
y
y=6x
y=6x+20
40
30
30
20
20
10
0
0
0
2
4
6
8
10
12
x
• 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
• y=ax+b
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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 x=5 we obtain
• y=x2, y = x*x, y = 5*5, y = 25
• y = 7x2, y = 7*x*x, y = 7*5*5, y = 175
• Functions can have more than one term and one may be a
constant.
• For example, y = 140+7x2-2x3 or y=25x2+74
• 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.
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Power Functions – An example
• Sketch and briefly describe the following functions for
positive values of x
y=6x+20
3
• y=2x -50 and y = 14
1600
1500
1400
1200
1008
1000
800
y
y=2x^3-50x
400
336
200
132
14
0
0
-200
y=14
624
600
0
14
14
-48
2
-84
14
14
-96
-72
4
14
0
14
6
14
14
8
14
14
10
12
x
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Some questions
• Sketch the graphs of the functions for values of x
between 0 and 10
• y=0.5x
• y=0.5x + 6
• y=x2
• y=3x2
• Which of them is linear? Which is a proportional
relationship? What is the effect of adding a constant
term?
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Some questions
• y=0.5x – Linear, proportional relationship
• y=0.5x+6, Linear, non-proportional relationship, shifted
up by constant
• y = x2. Quadratic function.
Some questions
• y=3x2. Quadratic function.
350
300
300
250
243
y=0.5x
200
192
y
y=0.5x+6
150
y=x^2
147
y=3x^2
108
100
100
81
75
50
64
48
6
0
0
0
12
7
4
1
6.5
3
1
0.5
2
27
9
7.5
1.5
16
8
2
4
36
25
8.5
2.5
9
3
6
49
9.5
3.5
10
4
8
10.5
4.5
10
11
5
12
x
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Factors and multiplying out brackets
Factorising: writing an expression as a product that
when multiplied out gives the original expression.
E.g. y=6x-3x2, this is a shorthand way of writing
y=(6*x)-(3*x*x)
If we divide terms on the right-hand side by3*x we
obtain (6*x)/3*x = 2 and (–3*x*x)/(3*x) = -x.
The amount we divide by we call a common factor.
So factorising the expression y=6x-3x2 we obtain
y=3x(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.
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Multiplying brackets
• E.g. (a-b)(-c+d) = -ac+ad- -bc-+bd = -ac+ad+bc-bd
• E.g. (x-7)(4-3x) = 4x-3x2-28+21x =-3x2+25x-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.
• (a+b)2 = a2+2ab+b2
• (a-b)2=a2-2ab+b2
• (a+b)(a-b)=a2-b2
• NOTE: Not every quadratic expression factorises to
an expression that contains integer values.
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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 = x*x*x, 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. x3*x5 = (x*x*x)*(x*x*x*x*x)=x8
• Using our rule, x3*x5 = x3+5 = 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.
5
x
x* x* x* x* x x* x
53
2



x

x
x3
x* x* x
1
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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.
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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?
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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
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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
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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.
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Example involving fractions
• Solve the system of equations
• 3x+2y =1
• -2x + y = 2
(1)
(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
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Example
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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
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Problems
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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
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Special Cases
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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
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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
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Special Cases
• If the elimination of x looks like the following then the
equations have infinitely many solutions
zero
*
y
=
zero
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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 = ?
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• 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
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• 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
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Practice Problems
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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
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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
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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”.
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• 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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• 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.
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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.
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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?
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• 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.
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• 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
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Example
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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
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• 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.
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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.
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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
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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
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