1. Mathematics for Economics and Business Jean Soper chapter one Functions in Economics1

2. Functions in Economics – Maths Objectives
Appreciate why economists use mathematics
Plot points on graphs and handle negative values
Express relationships using linear and power functions, substitute values and sketch the functions
Use the basic rules of algebra and carry out accurate calculations
Work with fractions
Handle powers and indices
Interpret functions of several variables 2

3. Functions in Economics – Economic Application Objectives
Apply mathematics to economic variables
Understand the relationship between total and average revenue
Obtain and plot various cost functions
Write an expression for profit
Depict production functions using isoquants and find the average product of labour
Use Excel to plot functions and perform calculations 3

4. Axes of Graphs
x axis: the horizontal line along which values of x are measured
x values increase from left to right
y axis: the vertical line up which values of y are measured
y values increase from bottom to top
Origin: the point at which the axes intersect where x and y are both 0 4

5. Terms used in Plotting Points
Coordinates: a pair of numbers (x,y) that represent the position of a point
the first number is the horizontal distance of the point from the origin
the second number is the vertical distance
Positive quadrant: the area above the x axis and to the right of the y axis where both x and y take positive values 5

6. Plotting Negative Values
To the left of the origin x is negative
As we move further left x becomes more negative and smaller
Example: – 6 is a smaller number than – 2 and occurs to the left of it on the x axis
On the y axis negative numbers occur below the origin 6

7. Variables, Constants and Functions
Variable: a quantity represented by a symbol that can take different possible values
Constant: a quantity whose value is fixed, even if we do not know its numerical amount
Function: a systematic relationship between pairs of values of the variables, written y = f(x) 7

8. Working with Functions
If y is a function of x, y = f(x)
A function is a rule telling us how to obtain y values from x values
x is known as the independent variable, y as the dependent variable
The independent variable is plotted on the horizontal axis, the dependent variable on the vertical axis 8

9. Proportional Relationship
Each y value is the same amount times the corresponding x value
All points lie on a straight line through the origin
Example: y = 6x 9

10. Linear Relationships
Linear function: a relationship in which all the pairs of values form points on a straight line
Shift: a vertical movement upwards or downwards of a line or curve
Intercept: the value at which a function cuts the y axis 10

11. General Form of a Linear Function
A function with just a term in x and (perhaps) a constant is a linear function
It has the general form
y = a + bx
b is the slope of the line
a is the intercept 11

12. Power Functions
Power: an index indicating the number of times that the item to which it is applied is multiplied by itself
Quadratic function: a function in which the highest power of x is 2
The only other terms may be a term in x and a constant
Cubic function: a function in which the highest power of x is 3
The only other terms may be in x 2, x and a constant 12

13. To Sketch a Function
Decide what to plot on the x axis and on the y axis
List some possible and meaningful x values, choosing easy ones such as 0, 1, 10
Find the y values corresponding to each, and list them alongside
Find points where an axis is crossed
(x = 0 or y = 0)
Look for maximum and minimum values at which the graph turns downward or upwards
If you are not sure of the correct shape, try one or two more x values
Connect the points with a smooth curve 13

14. Writing Algebraic Statements
The multiplication sign is often omitted, or sometimes replaced by a dot
An expression in brackets immediately preceded or followed by a value implies that the whole expression in the brackets is to be multiplied by that value
Example: y = 3(5 + 7x) = x 14

15. The Order of Algebraic Operations is
1.    If there are brackets, do what is inside the brackets first
2.    Exponentiation, or raising to a power
3.    Multiplication and division
4.    Addition and subtraction
You may like to remember the acronym BEDMAS, meaning brackets, exponentiation, division, multiplication, addition, subtraction 15

16. Positive and Negative Signs
When two signs come together
– + (or + –) gives –
– – gives +
Examples:
11 + (– 7) = 11 – 7 = 4
12 – (– 4) = = 16
(– 9) ´ (– 5) = +45 16

17. Multiplying or Dividing by 1
1  x = x
(– 1)  x = – x
x ¸ 1 = x 17

18. Multiplying or Dividing by 0
Any value multiplied by 0 is 0
0 divided by any value except 0 is 0
Division by 0 gives an infinitely large number which may be positive or negative
0  0 may have a finite value
Example: When quantity produced Q = 0,
variable cost VC = 0
but average variable cost = VC/Q may have a finite value 18

19. Brackets
An expression in brackets written immediately next to another expression implies that the expressions are multiplied together
Example: 5x (7x – 4) = (5x)  (7x – 4) 19

20. Multiplying out Brackets 1
One pair: multiply each of the terms in brackets by the term outside
Example: 5x (7x – 4) = 35x2 – 20x 20

21. Multiplying out Brackets 2
Two pairs: multiply each term in the second bracket by each term in the first bracket
Examples: (3x – 2)(11 + 5x) = 33x + 15x2 – 22 – 10x = 15x x – 22
(a – b)(– c + d) = – ac + ad + bc – bd 21

22. Results of Multiplying out Brackets
(a + b)2 = (a + b) (a + b)
= a2 + 2 ab + b2
(a – b)2 = (a – b) (a – b)
= a2 – 2 ab + b2
(a + b)(a – b) = a2 – ab + ab – b2
= a2 – b2 22

23. Factorizing
Look for a common factor, or for expressions that multiply together to give the original expression
Example:
45x2 – 60x = 15x (3x – 4)
Factorizing a quadratic expression may involve some intelligent guesswork
45x2 – 53x – 14 = (9x + 2) (5x – 7) 23

24. Fractions Fraction: a part of a whole
Amount of an item = fractional share of item  total amount
Ratio: one quantity divided by another quantity
Numerator: the value on the top of a fraction
Denominator: the value on the bottom of a fraction 24

25. Cancelling
Cancelling is dividing both numerator and denominator by the same amount
Examples: 25

26. Inequality Signs
> sign: the greater than sign indicates that the value on its left is greater than the value on its right
< sign: the less than sign indicates that the value on its left is less than the value on its right 26

27. Comparisons using Common Denominator
Example:
To find the bigger of 3/7 and 9/20 multiply both numerator and denominator of each fraction by the denominator of the other:
3/7 = (20  3)/(20  7) = 60/140 9/20 = (7  9)/(7  20) = 63/140
Since 63/140 > 60/ /20 > 3/7 27

28. Adding and Subtracting Fractions
To add or subtract fractions first write them with a common denominator and then add or subtract the numerators
Lowest common denominator: the lowest value that is exactly divisible by all the denominators to which it refers 28

29. Multiplying and Dividing Fractions
Fractions are multiplied by multiplying together the numerators and also the denominators
To divide by a fraction turn it upside down and multiply by it
Reciprocal of a value: is 1 divided by that value 29

30. 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
Exponent: a superscripted number representing a power
Exponentiation: raising to a power 30

31. Working with Indices To multiply, add the indices
To divide, subtract the indices
x – n = 1/ xn
x0 = 1
x1/2 =  x
(ax)n = an xn
(xm)n = xmn 31

32. Functions of More Than 1 Variable
Multivariate function: the dependent variable, y, is a function of more than one independent variable
If y = f(x,z) y is a function of the two variables x and z
We substitute values for x and z to find the value of the function
If we hold one variable constant and investigate the effect on y of changing the other, this is a form of comparative statics analysis 32

33. Total and Average Revenue
TR = P.Q
AR = TR/Q = P
A downward sloping linear demand curve implies a total revenue curve which has an inverted U shape
Symmetric: the shape of one half of the curve is the mirror image of the other half 33

34. Total and Average Cost Total Cost is denoted TC
Fixed Cost: FC is the constant term in TC
Variable Cost: VC = TC – FC
Average Cost per unit output: AC = TC/Q
Average Variable Cost: AVC = VC/Q
Average Fixed Cost: AFC = FC/Q  34

35. Profit Profit =  = TR – TC If TC = 120 + 45Q – Q2 + 0.4Q3
and TR = 240Q – 20Q2
 = TR – TC substitute using brackets
 = 240Q – 20Q2 – ( Q – Q Q3) Taking the minus sign through the brackets and applying it to each term in turn gives
 = 240Q – 20Q2 – 120 – 45Q + Q2 – 0.4Q3 and collecting like terms we find
 = – Q – 19Q2 – 0.4Q3 35

36. Production functions
A production function shows the quantity of output obtained from specific quantities of inputs, assuming they are used efficiently
In the short run the quantity of capital is fixed
In the long run both labour and capital are variable
Plot Q on the vertical axis against L on the horizontal axis for a short-run production function 36

37. Average Product of Labour
Isoquants
An isoquant connects points at which the same quantity of output is produced using different combinations of inputs
Plot K against L and connect points that generate equal output for an isoquant map
Average Product of Labour
Average Product of Labour (APL) = Q  L
Plot APL against L 37

38. Formulae in Excel:
Are entered in the cell where you want the result to be displayed
Start with an equals sign
Must not contain spaces 38

39. Arithmetic Operators are:
( ) brackets
+ add
– subtract
* multiply
/ divide
^ raise to the power of 39

40. Cell References in Formulae
A cell reference, such as A6, tells Excel to use in its calculation the value in that cell
Values calculated using cell references automatically recalculate if the values change
This facilitates ‘what if’ analysis
Relative cell addresses (e.g. A6) change as the formula is copied
Absolute cell addresses (e.g. $B$3) remain fixed as the formula is copied 40