Lecture One

1
Lecture One

• Overview of Algebra Numbers, Sets and Functions

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Components of the economic model
Variable
Endogenous
Exogenous
3
Components of the economic model cntnd.
Equations
Definitional
Behavioural
Conditional
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The real-number system used in economic analysis
Integers J
Fractions F
Rational Numbers Q
Irrational Numbers
Real Numbers R
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The concept of sets definition and notation

• A set is a collection of objects
• Examples
• - Sets of numbers, e.g., Q,R,J
• - Sets of words
• - The set of students taking EC1005
• - The set of company profits in Britain

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The concept of sets definition and notation
cntnd.

• Sets can be written up by enumeration or
  description
• Enumeration
• Description

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The concept of sets definition and notation
cntnd.

• Membership/non-membership of a set can be denoted
  by ? (?) , e.g., if A1,2,5 ? 1?A, while 3 ? A.
• Equal Sets contain exactly the same elements
  (order does not matter) A1,2,5, B5,2,1
• Subsets are sets contained in another set if
  A1,2,5 and D1,2,5,6,8, A is a subset of D
• A?D

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Operations with sets

• Union of sets a new set contains all elements
  that belong to either of the original sets, e.g.,
  if A1, 2,5 and D1,2,4,6, 8, then
  A?D1,2,4,5,6,8
• Intersection of sets A new set containing only
  the common elements of the original sets, e.g, if
  A2, 5, 9, D2,4,5,6,8, then A?D2,5
• Complement set the complement set of A is ,
  containing all the elements of the universal set,
  which is not in A, e.g., if A1, 2,3 and U1,
  2, 3, 5, 6, 8, then à 5,6,8

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Operations with sets cntnd.

• Commutative law
• Associative law
• Distributive law
• Example A4,5, B3,6,7, C2,3

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Relations and functions

• Cartesian Product
• For two sets A and B, let A ? B be the set of
  pairs (a,b) where a?A and b ?B.
• That is A ? Ba,b) a?A and b ?B
• A ? B is called the Cartesian product of A and
  B

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Relations and functions cntnd.

• For example, if A1,2,3 and B?,?, then
• A?B is (1, ?), 2, ?),(3, ?),(1, ?),(2,
  ?),(3, ?)

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Relations and functions cntnd.

• The set ???, often denoted by ?2 is
  (x,y)x,y??. This is the Cartesian plane

(x,y)
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Relations and functions cntnd.

• René Descartes
• Born 31 March 1596 in La Haye, Touraine, France
• Died 11 Feb 1650 in Stockholm, Sweden
• Philosopher whose work
• La géometrie includes application of algebra
  to geometry from which we now have Cartesian
  geometry

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Relations and functions cntnd.

• Since any ordered pair associates a y value with
  an x value, any collection of ordered pairs-any
  subset of the Cartesian product-will constitute a
  relation between y and x.
• If the relation is such that for each x value
  there exists only one corresponding y value, y is
  said to be a function of x.

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Relations and functions cntnd.
f

• Each function S T
• has
• A source S (which is a set), also called the
  domain of the function)
• A target T (which is a set), also called the
  co-domain, or target of the function)
  

. . .
. . .
S
T
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Relations and functions cntnd.

• Example 1 A set (x,yy2x is a set of ordered
  pairs including for e.g, (1,2), (0,0) is a
  relationship whose graphical counterpart is the
  set of points lying on the straight line y2x.
• Example2 A set(x,y)y?x which consists of
  ordered pairs (1,0),(1,1) and (1,-4) constitutes
  another relationship.

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Relations and functions cntnd.
y2x
yx
y?x
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Relations and functions cntnd.

• Example The total cost C of a firm per day is a
  function of its daily output Q C1507Q. The
  firm has a capacity limit of 100 units per day.
  What are the domain and the range of the cost
  function?

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Relations and functions cntnd.

• Solution
• DomainQ0?Q ?100
• RangeC150 ?C ?850

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Types of functions

• In this lecture (and for that matter for the rest
  of the course!) we shall explore two types of
  functions
• - Polynomial functions linear and non-linear
• (e.g. quadratic and cubic)
• - Exponential and logarithmic functions

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Linear functions

• Polynomial functions y is the nth degree
  polynomial function of x ya0a1xa2x2a3x3.an
  xn. For the simplest case of a linear function
  with a01 and a12, we have y12x

1
-0.5
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Linear functions cntnd.

• Linear functions are characterised by their slope
  and intercepts

y3x-4
3
1
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Linear functions cntnd.

• They can be explicit or implicit
• 6x-2y80 is an implicit function as both x
  and y
• appear on the same side of the equation.
• The explicit version is y3x4

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Linear functions cntnd.

• Systems of linear equations can be solved
  graphically and algebraically. We can then have
  either a unique solution

y
y22x
M(6,2)
y10-2x
x
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Linear functions cntnd.
.

• Or an inconsistent system

y102x
y
y22x
x
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Linear functions cntnd.

• Or infinite number of solutions

y
y22x 2y44x
x
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Exercises with linear equations

• Draw the graph of 4x3y11
• Find the intercept of
• -2xy2
• 2xy-6
• Solve simultaneously the following system
• x-3y4z5
• 2xyz3
• 4x3y5z1

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Economic application of linear equations

• Market equilibrium example
• qd-8p2000 Since in equilibrium qdqs
• qs12p-200 q1120, p110

qs12p-200
p
qd-8p2000
q
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Economic application of linear equations cntnd.

• Specific/ per unit tax, e.g. t50
  qs12(p-50)-200

q
qs12p-200
qs12(p-50)-200
qd-8p2000
p
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Economic application of linear equations cntnd.

• Ad valorem tax

qs12p-200
q
qd-8p2000
p
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Quadratic functions

• Example 1 of non-linear function quadratic
  function
• ya0a1xa2x2. For a010, a1-7, a21
  y10-7xx2

• If a0a1xa2x20,then x1,2
• If a1-4a0a2gt0, there are 2 real solutions (x1,x2)
• If a1-4a0a20, there is 1 real solution (x1 x2)
• If a1-4a0a2lt0, there is no real solution

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Economic applications of quadratic functions

• Given the supply and demand functions
• Calculate the equilibrium price and quantity.

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Economic applications of quadratic functions
cntnd.

• Example Total Revenue and Total Cost Functions

TR
TC
TC0.02q21.5q100
TR -0.12q210q
Q
Q
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Cubic functions

• For n3, ya0a1xa2x2a3x3, e.g, a05, a17,
  a2-1, a3-2 y57x-x2-2x3

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Example of cubic functions
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Example of cubic functions cntnd.
TC
TR,TC
TR
Q
?TR-TC
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Exponential functions

• Exponential functions yf(t)bt, bgt1

y
1
t
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Exponential functions cntnd.

• The curve of y covers all the positive values of
  y in its range, therefore any positive value of y
  must be expressible as some unique power of
  number b.
• Importantly, even if the base is changed to some
  other real number greater than 1, the same range
  holds. Hence it is possible to express any
  positive number y as a power of any base bgt1.

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Exponential functions cntnd.
y
y
ybt
yb2t
y2bt
2y0
ybt
y0
y0
t
t
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Exponential functions cntnd.

• The preferred base e2.71828
• Motivation 1

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Exponential functions cntnd.

• Motivation 2 economic application

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Exponential functions cntnd.

• Suppose that, starting with a principle of 1 we
  find a hypothetical banker to offer us the
  unusual interest rate of 100 (1 interest per
  year). If the interest is compounded once a year,
  the value of our asset at the end of the year is
  2
• V(1)initial principal(1interest
  rate)1(11)2

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Exponential functions cntnd.

• If the interest is compounded semi-annually
• V(2)(150)(150)(11/2)2
• Analogically V(3)(11/3)3 , V(4)(11/4)4
• In the limiting case, when interest is compounded
  continuously throughout the year, the value of
  our asset at the end of the year will be e.

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Exponential functions cntnd.

• Of course, the assumptions of neither a one
  dollar deposit nor 100 interest rate are
  realistic.
• Our general interest compounding formula is
  therefore
• And we find the asset value in a generalized
  continuous compounding process to be

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Exponential functions cntnd.

• In the reverse case, when we want to find the
  present value of an asset, we have

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Logarithmic functions

• Logarithms
• ybt ?tlogby
• Graphically

yet
tlogey
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Logarithmic functions cntnd.

• Common log and natural log

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Logarithmic functions cntnd.

• One of the main economic applications of
  logarithms is log-linearization
• Qf(K,L)AK?L? ?ln QlnA?lnK?lnL

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Rules of exponents and logarithms

• x01, for x?0 (e.g. 101, 501)
• x1x
• x2x?x, x3x?x?x and so on.
• x-n1/xn, for x ?0 (e.g., x-31/x3, 5-21/52)
• xn? xmxnm (e.g, x2 ? x3x5)
• xn/xmxn-m (e.g., x10/x3x7)
• (xn)mxn ? m
• xm/n
• xn ? yn(xy)n (e.g, 63 ? 23123)
• xn/yn(x/y)n (e.g, 33/23(3/2)3)

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Rules of exponents and logarithms cntnd.

• lne1
• ln10
• ln(xy)lnxlny
• ln(x/y)lnx-lny
• lnxnnlnx

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Problems with exponents and logarithms

• Simplify the following expressions
• Are the following functions homogeneous and if so
  of what degree

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Problems with exponents and logarithms cntnd.

• Simplify the following expressions
• Solve the following equations