Economic Models

1
Chapter 2

• Economic Models

2
Economic Model

• The complexity of the real economy makes it
  impossible for us to understand all the
  interrelationships at once nor, for that matter,
  are all these interrelationships of equal
  importance.
• The sensible procedure is, therefore, to pick
  out what appeals to our reason to be the primary
  factors and relationships relevant our problem
  and focus on these alone.Such a deliberately
  simplified analytical framework is called
  Economic model.

3
Ingredients of Mathematical Model

• A variable is something whose magnitude can
  change, i.e. something that can take on different
  values. Variables used in economics include
  price, profits, demand, supply, cost, GDP,
  inflation rate, investment, and consumption. It
  must be represented by a symbol and not by a
  number, For instance, S 20 represents Quantity
  Supplied or D 40 implies Quantity Demanded. We
  are freezing the variables at specific values in
  appropriate chosen units).
• Endogenous variables (within)
• Variables, whose solution values we seek from
  the model. Exogenous variables (without)
• Variables which are assumed to be determined by
  forces external to the model, and whose
  magnitudes are accepted as given data only.

4
Ingredients of Mathematical Model

• A constant is a magnitude that does not change
  and is therefore, the antithesis of that
  variable. When a constant is joined to a
  variable, we call it coefficient of the variable.
  
• Y 6X. The number 6 is a constant. Coefficients
  can also be symbolic rather than numerical.
• Expression aP. The symbol a is supposed to
  represent a given constant, and yet, since we
  have not assigned to it a specific number, it can
  take virtually any value. Like 5P, a took a value
  of 5.

5
Ingredients of Mathematical Model

• A parameter is a specified term in algebraic
  equation. For example, in y 3x 2, the numbers
  3 and 2 are parameters. As a matter of
  convention, parametric constants are normally
  represented by the symbols a, b, c or their
  counterparts in the Greek alphabet a , ß, and ?
  .
• To distinguish between exogenous and endogenous
  variables, we use subscript 0 to the chosen
  symbol. For example, if P is the Price, then P0
  signifies an exogenously determined price.

6
Equations and Identities

• In economic applications, we distinguish between
  three types of equation
• A definitional equation sets up an identity
  between two alternate expressions that have
  exactly the same meaning. For this purpose, the
  identical equality sign ? is used , read as
  is identically equal to , instead of regular
  equals sign .
• A behavioral equation specifies the manner in
  which a variable behaves in response to changes
  in other variables. For instance, consider the
  two cost functions C 75 10Q
  (2.1) C 110 Q2 (2.2) where
  C is the Cost and Q is the Quantity.In (2.1),
  the fixed cost is 75, whereas in (2.2) the fixed
  cost is 110.In (2.1), for each unit increase in
  Q, there is a constant increase of 10 in C,
  whereas, in (2.2), as Q increases unit after
  unit, C will increase by larger amount.

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Equations and Identities

• A conditional equation states a requirement to
  be satisfied. For example, in a model involving
  the notion of equilibrium, we must setup
  equilibrium condition. The famous equilibrium
  conditions in economics are Qd Qs quantity
  demanded quantity suppliedor S I
  intended saving intended investment
• or MC MR marginal cost marginal revenue

8
The Real-Number System

• Whole numbers such as 1, 2, 3, are called
  positive integers, while -1, -2, -3, are called
  negative integers. Whereas, the number 0 is
  neither positive nor negative. These when put
  collectively put into a single category are then
  referred to as the set of all integers.
• Integers, of course, do not exhaust all the
  possible numbers, for we have fractions, such as,
  ¼ , ½ , ¾ ,Also, we have negative functions such
  as - ½ , and - ¼ . Together, these make up the
  set of all fractions. Any number that can be
  expressed as a ratio of two integers is called a
  rational number.The set of all integers and set
  of all fractions together form a set of all
  rational numbers.

9
The Real-Number System

• An alternative defining characteristic of a
  rational number is that is expressible as either
  a terminating decimal (e.g. ¾ 0.75) or a
  repeating decimal (e.g. 2/3 0.66666.)
• Numbers that cannot be expressed in ratios are
  called irrational numbers. For example, Pi
  3.14215 which is non repeating, and non
  terminating decimal.

10
This figure shows all the number sets, arranged
in interrelationship to one another. Its a
summary of the structure of the real number
system.
11
The Concepts of Sets

• Set Notation A set is simply a collection of
  distinct objects. These objects may be a group of
  (distinct) numbers, persons, food items, or
  something else. The objects in the set are called
  elements.There are two alternative ways of
  writing a set, namely, by description and by
  enumeration. Example of by enumeration would
  be A apple, orange, banana, tomatoes B
  1,2,5,7
• Example of set by description would be S
  y y a positive integer which is read as S
  is the set of all (numbers) y, such that y is a
  positive integer

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The Concepts of Sets

• Relationship between Sets
• When two sets are compared with each other,
  several possible kinds of relationship may be
  observed. Equal sets If two sets A and B
  happen to contain identical elements,
• A a,b,2,g and B a,b,2,gthen A and B
  are said to be equal, i.e., A B.
• Subset S 2,4,6,8,10 and R 2,4,6, then
  we say R is a subset of S because every element
  of R is also an element of S. To denote S and R
  in symbols, we say S R S is contained
  in R or R includes S
• Null set
• The smallest possible subset of S is a set that
  contains no element at all. It is denoted by
  or .

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The Concepts of Sets

• Disjoint setsTwo sets may have no elements in
  common at all. For instance, X 23,65,98,100
  and Y 32, 56, 89, 1 . These two sets are said
  to be disjoint sets because they have nothing in
  common. Finally, when two sets have some
  elements in common but some elements are peculiar
  to each. In that event, the two sets are neither
  equal nor disjoint also, neither set is a subset
  of the other.
• X 23,65,98,100 and Y 23, 65, 89, 1

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Operations on Sets

• Although sets are different from numbers, one
  can perform certain mathematical operations on
  them like. Three principle operations to be
  discussed here involve union, intersection, and
  complement of sets. UnionIf A and B are
  sets, then their union A U B, A union B, is the
  set consisting of all elements that are in A or
  B. (or in both A and B). A 3,5,6,6,8 and B
  2,7,8 ? A U B 2,3,5,6,7,8
• IntersectionThe intersection of two sets A and
  B, on the other hand, is a new set which contains
  those elements and only those elements) belonging
  to both A and B. The intersection set is
  symbolized by A n B, A intersection B . In this
  case, if A -3,-4,5,7,8,9 and B -3,-4, 2 ?
  A n B -3,-4.

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Operations on Sets

• If A 10, 20, 30 and B 11,21,31, then A n
  B F because set A and set B are disjoint
  therefore their intersection is an empty set no
  element in common to A and B. Complement
• A set that contains all the numbers in the
  universal set U that are not in the set A. It is
  symbolized by . For example, U 1,2,3,4,5,6
  and A 4,5,6, then 1,2,3.
• Note that the symbol U has the connotation for
  or and the symbol n means and , the
  complement symbol carries the implication of
  not.

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Operations on Sets

• The three types of sets can be visualized in the
  three diagrams known as Venn Diagrams
• In this figure, the points in the upper circle
  form a set A, and the points in the lower circle
  form set B. The union A and B then consists of
  the shaded area covering both circles.

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Operations on Sets

• In this figure, the same two sets are shown.
  Since their intersection should only comprise the
  points common to both sets, only the (shaded)
  overlapping portion of the two circles satisfies
  the definition.

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Operations on Sets

• In this diagram, let the points in the rectangle
  be the universal set and let A be the set of
  points in the circle then the complement set
  will be the (shaded) area outside the circle.

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Laws of Set Operations

• Commutative law (of unions and intersections)
  A U B B U A and A n B B n AAssociative
  law (of unions and intersections)
• To take the union of three sets, A,B and C, we
  first take the union of any two sets and then
  union the resulting set with the third. A
  similar is procedure is applicable to the
  intersection operation. (Associative law)
• A U (B U C) (A U B) U C
• A n (B n C) (A n B) n C
• Example A 3,4, B 5,7 and C 1,2.
  Left side A U (B U C) 3,4 U 1,2,5,7
  1,2,3,4,5,7
• Right side (A U B) U C 3,4,5,7 U 1,2
  1,2,3,4,5,7
• Hence, Left side Right side Law
  Verified.

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Laws of Set Operations

• The distributive law (of union and intersection)
  
• A U (B n C) (A U B) n (A U C)
• A n (B n C) (A n B) U (A n C)
• These resemble the algebraic law a x (b c)
  (a x b) (a x c).
• Example
• A 9,10 , B 3,4 , and C 5.6
• Left side A U (B n C) 9,10 U
  9,10.Right side (A U B) n (A U C)
  3,4,9,10 n 5,6,9,10 9,10
• Hence, Left side Right side
  Law Verified.

21
Relations and Functions

• Unordered Pair In writing a set a,b, we do
  not care about the order in which the elements a
  and b appear because by definition a,b
  b,a.We call this unordered pair.
• Ordered PairWhen ordering of a and b does
  carry a significance, however, we can write two
  different ordered pairs denoted by (a,b) and
  (b,a), where (a,b) (b,a).
• Example Let t time in minutes and c
  calories burned. We can now form a ordered pair
  (t, c). Then (20, 230) and (230,20) would
  obviously mean different things. Moreover, the
  latter ordered pair would hardly fit any person
  anywhere. Therefore, ordering is important
  

22
Relations and Functions

• Another important example of ordered pair is the
  Cartesian coordinate plane, and dividing the
  plane into four quadrants. In this, the points
  (2,4) and (4,2) are again different. Thus
  ordering is significant here as well.

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Relations and Functions

• Suppose, now from two sets, x 4,5 and y
  7,8, we wish to form all the possible ordered
  pairs with the first element taken from set x and
  second taken from set y. Therefore, we may
  express this Cartesian product alternatively as
  x x y (a,b) a x and b y.
• Therefore, in our example, it would be x x
  y (4,7), (4,8), (5,7), (5,8)
• Similarly, we have ordered triples represented
  by x x y x z (a,b,c) a x, b y, c
  z.

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Relations and Functions

• Since any ordered pair associates a y value with
  an x value, any collection of ordered pairs any
  subsets of the Cartesian product will
  constitute a relation between y and x. Given an x
  value, one or more y values will be specified by
  that relation. For instance, the set (x, y)
  y x, which consists of ordered pairs as (3,2),
  (3,1), and (3,-4) constitutes another relation.
  Observe that, when x value is given, it may not
  always be possible to determine a unique y value
  from a relation. In the example above, when x
  3, then y can take various values, such as 2,
  1,and -4.

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Relations and Functions

• Domain and RangeThe set of all permissible
  values that x can take in a given context is
  known as the domain of the function.
• The y value into which an x value is mapped is
  called the image of that x value. The set of all
  images is called the range of the function.

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Relations and Functions

• Example The total cost C of a firm in per day
  is a function of its daily output
• Q C 75 10Q. The firm has a capacity limit
  of 50 units of output per day. What are the
  domain and range of the cost of
  production? Domain Q 0 Q 50To find
  range, we substitute Q 0 and Q 50 in the Cost
  function to get C 75 and C 575, respectively.
  Therefore, Range C 75 C 575

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Types of Functions

• A function is also called a mapping, or
  transformation both connote the action of
  associating one thing with another.
• Constant functionsA function whose range
  consists of only one element is a called constant
  function. For example, y f(x) 4 or y 4 or
  f(x) 4.

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Types of Functions

• Polynomial functionsThe word polynomial means
  multiterm and a polynomial function of a single
  variable x has the general form y a0 a1x
  a2x2 a3x3 . anxn
• Case of n 0 y a0 constant function
• Case of n 1 y a0 a1x linear
  functionCase of n 2 y a0 a1x a2x2
  quadratic functionCase of n 3 y a0
  a1x a2x2 a3x3 cubic function
• The terms a0 , a1 are called the coefficients
  of x, and the powers of x are called the
  exponents.

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Types of Functions

• Quadratic FunctionsA quadratic function plots
  a parabola, roughly, a curve with a single built
  in-bump or wiggle.
• A parabola opens downwards if a2 lt 0 and the
  shape looks like this

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Types of Functions

• A parabola opens upwards if a2 gt 0 and the shape
  looks like this

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Types of Functions
32

Types of Functions

• Rational Functions
• A function such as y in which
  y is expressed as a ratio of
• two polynomials in the variable x, is known as a
  rational function.
• A special rational function that has interesting
  applications in economics is the function y
  or xy a which plots as
  rectangular hyperbola.

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Types of Functions
34
Types of Functions

• Nonalgebraic FunctionsExponential functions
  such as y bx , in which an independent variable
  appears in the exponent, are Nonalgebraic.Another
  example of nonalgebraic function is the famous
  logarithmic function, y logb x
• However, any function expressed in terms of
  polynomials and/or roots (such as square root) of
  polynomials is an algebraic function. Also, The
  more esoteric name of algebraic functions is
  transcendental functions.

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Types of Functions
36
Types of Functions

• Non Algebraic Functions
• A Digression on Exponents
• In general, we define, for a positive integer n,
  xn x x x x x x x x ? n terms. From the
  general definition, exponents obey the following
  rulesRule I

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Types of Functions
Non Algebraic Functions

• Rule II
• for example,

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Types of Functions
Non Algebraic Functions

• Rule III Rule IV x0 1 ? Anything to
  the power of 0 is always 1
• Rule V

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Types of Functions
Non Algebraic Functions

• Rule VI
• Rule VII
• xm x ym (xy)m

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Functions of Two or More Independent Variables

• So far, we have considered only functions of a
  single independent variable y f(x). Now we
  extend the case to two or more independent
  variables. Given a function z g(x,y)which
  means that given a pair of x and y values will
  uniquely determine a value of the dependent
  variable z. For example,
• z ax byIn this case, the domain is a set
  of ordered pairs (x,y).

41
Functions of Two or More Independent Variables

• Figure (a)
• The function g is a mapping from a point in a
  two dimensional space into a point on a line
  segment, such as from the point (x1,y1) into the
  point z1 or from (x2,y2) into z2.
• Figure (b)
• If a vertical z axis is erected perpendicular to
  the xy plane, there will result a three
  dimensional plane, and the value of the function
  (value of z) is given by the height of a vertical
  line plotted on that point.

42
Levels of Generality

• On a more general level of discussion and
  analysis, there are functions in the form
• y a y a bx y a bx cx2
  (etc)Since parameters are used, each function
  represents not a single curve but a whole family
  of curves. For instance,y a can not only take
  values like y 0, y -1 or y 4. It can also
  take y ¼ or y -10 as well. To attain higher
  level of generality, we may resort to general
  functions like y f(x) or z g(x,y). When
  expressed in this form, the function is not
  restricted to being either linear, quadratic,
  cubic, or exponential.

43
Summary

• The structure of a mathematical economic model
  consist of a system of equations, which may be
  definitional, behavioral, or in the nature of
  equilibrium conditions. The behavioral
  equations are usually in the form of functions,
  which may be linear or nonlinear, numerical or
  parametric, with one independent variable or
  many. It is through these that the analytical
  assumptions adopted in the model are given
  mathematical expressions.