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EES 100: Mathematics for Economists I
Need for Mathematics in Economic Analysis
2. SET THEORY Introduction- Definitions Methods of set representation The Venn Diagram 3. FUNDEMENTAL TECHNIQUES IN ALGEBRA Rules of Algebraic Operations Exponentials and logarithms 4. LINEAR FUNCTIONS The concept of a function Functions of a single independent variable Graphical presentation The slope of linear functions Models in Economic Analysis 5. NON-LINEAR FUNCTIONS AND MULTIVARIATE FUNCTIONS Revenue Functions Cost Functions Profit Functions Multivariate Functions Equations and inequalities 6. LINEAR SIMULTANEOUS EQUATIONS Solutions to linear simultaneous equations Economic Application of linear simultaneous equations 7. DERIVATIVES AND DIFFERENTION Slope of a linear function Slope of a non-linear function Rules of Differentiation Economic Application of derivatives
1.1 Need for Mathematics in Economic Analysis
Mathematics is an invaluable tool at all levels of the study of economics, ranging from the statistical expression of real world trends to the development of fully abstract economic systems
Courtesy of Rodgers Wesonga
universal set will be all the fresh water lakes in that country. The various classifications of the fresh water lakes in the country would be sub sets of the universal set
The notation U is generally used to denote universal sets. Given a universal set we can derive its subsets. for example if U = {1,2,3} the subsets are { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3} and {1,2,3}.
This means that both the Null set, {}, and the Universal set, U, are subsets of U
3. The Null set It is a set that contains no element hence is also referred to as the empty set. It is
Note that the sets { } and {0} are not the same. The former has no element in it while the latter has one element in it.
4. Equal sets Two sets A and B are said to be equal if every member of a set A belongs to B and every element of set B belongs to A. That is the two sets contain the same elements. For example if A = {a, d. c. b} and B = {d, c a, b} are equal sets.
2.2 Methods of Set Representation
Capital letters are used to represent sets.
There are two different methods of representing members of a set. i) Descriptive method ii) Enumerative method
In the descriptive method , members of a set are presented in such a way that one can determine the elements of the set without difficulty. For example,
P x x 0 , 1 , 2 ,..., 7 or P x 0 x 7
The enumerative method requires that one writes all the members of the set within the curly brackets. For example the set P could be written as follows P { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 }
2.3 The Venn diagram
The Venn diagram provides a simple way of representing sets and relations between sets. It consists of a rectangle that represents the universal set. Subsets of the universal set are represented by circles drawn within the rectangle, or the universe. Consider the Venn diagram below;
In the Venn diagram above, U is the universal set, while A, B and C are subsets of U.
2.4 Set Operations
i) The intersection of sets Intersection of sets A and B, is a subset of U containing elements which belong to A and B. it is denoted by the symbol . It can also be represented diagrammatically as below. The shaded area represents the intersection of A and B
Numerical example 1
Given the universal set T and its subsets A and B
Numerical example
Consider the universal set T and its subsets A, B and C below
B bc f
A ad
T abcd e f
Find A B , A C , B C , A B C
Solution
iii) The complement of a set
The complement of as set A is a subset of U containing elements that do not belong to A. it is denoted by A '^ , represented by the shaded area in the figure below
Example
Laws of Set algebra
Given the Venn diagram
Where T is a Universal set and A is its subset, the following laws can be deduced
From the Venn diagram two additional laws can be deduced that
3.1.3 Rules for combined operations
Many problems involve several operations. Sometimes there are brackets used for ordering the operations. For example;
The following rules will be useful in solving such problems.
Numerical example
Solution
Starting with the multiplication and division operations, then to additions and subtraction
Example
Simplify the following
12 2 6 3 6 9 30 15
Solution
X ^ ^ X X X
Example; (^5252533)
3.2.2 Meaning of Logarithms
In the exponential expression y ax , where a is the base and x the exponent, we say that
the logarithm of y to the base of a is x. that is to say that the logarithm of y to the base a is the power to which a must be raised so as to obtain y
Examples;
log 5 log 5 1
log 27 log 3 3
log 100 log 10 2
1 5 5
3 3 3
2 10 10
Laws of logarithmic operations
1. Multiplication law
log (^) a ( x. y )log ax log a y
2. Quotient law
x y y
x log (^) a (^) log a log a
3. Power law
4. Logarithm of a number to its base
log (^) aa 1
4.1 The Concept of a Function
A function is a mathematical relationship in which the values of a single dependent variable are determined from the values of one or more independent variables. That is, for every value assigned to the independent variable, a unique value of the dependent variable is determined. For example in the expression; y a bx
both x and y are variables. This is because they may assume different values throughout the analysis. However, a and b are constants because they are fixed numbers.
The variable y is a dependent variable. It is dependent in that its values are generated from values of x. In this case x is an independent variable.
The collection of all the values of the independent variable for which the function is defined is referred to as the domain of the function, while the collection of all the values of the dependent variable defined by the function.
The functional relationship between variables can also be expressed generally as y f ( x )
The customary symbols used for denoting functions include f, F, g, G. in cases where two variables say y and z are different functions of an independent variable, x, then after denoting the functional relation between y and x as y f ( x ), a different notation should be used to represent the relation between z and x.
Courtesy of Rodgers Wesonga
coordinates.
Since the graph of a function y = f(x) is defined as the totality of all points whose coordinates satisfy the functional relationship. Therefore to draw the graph of a function, we need to obtain a set of ordered pairs (x, y) from the function by; Assigning arbitrary values for the independent variable, x, and then Calculating the corresponding values of the dependent variable. The points are then plotted on a coordinate plane.
Example
Draw the graph for the function y =4-2x
Solution
Assuming arbitrary values for x and calculating corresponding values of y for the function yields the following ordered pairs.
Plotting these ordered pairs of (x, y) on the coordinate plane and joining yields the following graph
x -2 -1 0 1 2 3 4 y 8 9 4 2 0 -2 -
x-axis
y-axis
Note that the graph of a linear function is a straight line. The simplest way to draw a straight line is to determine two ordered pairs of (x,y) that satisfy the function then draw the line. For example for the above example we could have just used two points, e.g. (0,4) and (2,0) to draw the graph thus;
-2 -1 0 1 *2 3 4 x-axis
y-axis
y = 4-2x
-2 -1 0 1 2 3 4 x-axis
y-axis
y = 4-2x
