Mathematics for Economics and Business Jean Soper chapter one Functions in Economics

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Mathematics for Economics and Business Jean Soper chapter one Functions in Economics. Functions in Economics – Maths Objectives. Appreciate why economists use mathematics Plot points on graphs and handle negative values

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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
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
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
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
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
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)
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
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
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
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
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
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
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) = 15 + 21x
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

• You may like to remember the acronym BEDMAS, meaning brackets, exponentiation, division, multiplication, addition, subtraction
Positive and Negative Signs
• When two signs come together

– + (or + –) gives –

–– gives +

• Examples:

11 + (– 7) = 11 – 7 = 4

12 – (– 4) = 12 + 4 = 16

(– 9) ´ (– 5) = +45

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

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)
Multiplying out Brackets 1
• One pair: multiply each of the terms in brackets by the term outside
• Example: 5x (7x – 4) = 35x2 – 20x
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= 15x2 + 23x – 22

(a – b)(– c + d) = – ac + ad + bc – bd

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

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
• Example:

45x2 – 53x – 14 = (9x + 2) (5x – 7)

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
Cancelling
• Cancelling is dividing both numerator and denominator by the same amount
• Examples:
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
Comparisons using Common Denominator
• Example:

To find the bigger of 3/7 and 9/20multiply both numerator and denominator of each fraction by the denominator of the other:

3/7 = (20  3)/(20  7) = 60/1409/20 = (7  9)/(7  20) = 63/140

Since 63/140 > 60/140 9/20 > 3/7

• 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
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
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
Working with Indices
• To multiply, add the indices
• To divide, subtract the indices
• x– n = 1/ xn
• x0 = 1
• x1/2 =  x
• (ax)n = anxn
• (xm)n = xmn
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
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
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
Profit
• Profit =  = TR – TC
• If TC = 120 + 45Q – Q2 + 0.4Q3
• and TR = 240Q – 20Q2
•  = TR – TC substitute using brackets
•  = 240Q – 20Q2 – (120 + 45Q – Q2 + 0.4Q3) Taking the minus sign through the brackets and applying it to each term in turn gives
•  = 240Q – 20Q2 – 120 – 45Q + Q2 – 0.4Q3and collecting like terms we find
•  = – 120 + 195Q – 19Q2 – 0.4Q3
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
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
Formulae in Excel:
• Are entered in the cell where you want the result to be displayed
• Must not contain spaces
Arithmetic Operators are:

( ) brackets

– subtract

* multiply

/ divide

^ raise to the power of

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