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

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

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

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

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

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

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

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

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

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

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

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

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

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 14

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

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 16

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

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

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

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

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 21

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

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

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

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

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

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 > 60/140 9/20 > 3/7 27

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

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

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

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

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

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

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

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 – 0.4Q3 and collecting like terms we find  = – 120 + 195Q – 19Q2 – 0.4Q3 35

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

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

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

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

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