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

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1. Mathematics for Economics and BusinessJean Soper chapter oneFunctions in Economics

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

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

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

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

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

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)

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

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

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

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

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

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

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) = 15 + 21x

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

16. 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

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

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

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)

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

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= 15x2 + 23x – 22 (a – b)(– c + d) = – ac + ad + bc – bd

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

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 • Example: 45x2 – 53x – 14 = (9x + 2) (5x – 7)

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

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

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

27. 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

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

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

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

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 = anxn • (xm)n = xmn

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

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

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

35. 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

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

37. 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

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

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

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