TX-1037 Mathematical Techniques for Managers Dr Huw Owens Room B44 Sackville Street Building Telephone Number 65891 Huw Owens - University of Manchester : February 06 1 Introduction In Lecture 1 we looked at, Coordinates and Graphs Fractions Variables and Functions Linear Functions Power Functions Sketching Functions Algebra Factors and multiplying out brackets Accuracy Powers and Indices Huw Owens - University of Manchester : February 06 2

Functions of more than One Variable Economic Variables and Functions Total and Average Revenue Total and Average Cost Profit Production Functions, Isoquants and the average Product of Labour Huw Owens - University of Manchester : February 06 3 Equations in Economics

Lecture objectives Understand how equations are used in economics Rewrite and solve equations Substitute expressions Solve simple linear demand and supply equations to find market equilibrium Carry out cost-volume-Profit analysis Identify the slope and intercept of a line Plot the budget constraint to obtain the budget line Huw Owens - University of Manchester : February 06 4 Variables and functions Variable: a quantity represented by a symbol that can take different possible values (variable names x and y are often used). Constant: a quantity whose value is fixed, even if we do not know its numerical amount (letters commonly used to represent constants are: a,b,c,k). Function: a systematic relationship between pairs of values of the variables, written y=f(x). If one variable, y, changes in a systematic way as another variable, x, changes we say y is a function of x. The

mathematical notation for this is y=f(x), where the letter f is used to denote a function. If there is more than one functional relationship we can indicate they are different by using different letters, such as g or h. For example, y=g(x), which is read as y is a function of x Huw Owens - University of Manchester : February 06 5 Substitution of x-values A function gives a general rule for obtaining values of y from values of x. An example is y=4x+5, where 4x means 4*x (by convention we omit the multiplication sign). To evaluate the function for a particular value of x, say x= 6 y=4*6+5, y=29 Substituting different values into the function gives us different points on a graph. As the function tells us how to obtain y values from any x values, y is said to be dependent on x, and x is known as the independent variable. The independent variable is plotted on the horizontal axis and the independent variable is plotted on the vertical axis. Huw Owens - University of Manchester : February 06

6 Linear functions If the relationship between x and y takes the form y=6x x 0 5 10 y 0 30 60 y=6x 70 60

60 50 40 y y=6x 30 30 20 10 0 0 0 2 4 6

8 10 12 x Huw Owens - University of Manchester : February 06 7 Linear functions Proportional relationship: each y value is the same amount times the corresponding x value, so all points lie on a straight line through the origin. Linear function: a relationship in which all the pairs of values form points on a straight line. In general, a function of the form y=bx represents a straight line passing through the origin. Shift: a vertical movement upwards or downwards of a line or curve. Adding a constant to a function shifts the function vertically upwards by the amount of the constant. For example, y=6x+20 has y values that are 20 more

than those of the previous function of every value of x. Huw Owens - University of Manchester : February 06 8 Linear functions y=6x+20 90 80 80 70 60 60 50 50 y y=6x y=6x+20

40 30 30 20 20 10 0 0 0 2 4 6 8 10

12 x Intercept: the value at which a function cuts the y-axis. Remember a function with a term just in x and perhaps a constant is a linear function. It has the general form y=ax+b Huw Owens - University of Manchester : February 06 9 Power functions Power: an index indicating the number of times the item to which is applied is multiplied by itself. For example, y = x2 or z = 7x2 If we evaluate these functions and substitute x=5 we obtain y=x2, y = x*x, y = 5*5, y = 25 y = 7x2, y = 7*x*x, y = 7*5*5, y = 175 Functions can have more than one term and one may be a constant. For example, y = 140+7x2-2x3 or y=25x2+74 Quadratic function: a function in which the highest power of x is two. There may also be a term in x and a constant but no other terms.

Cubic function: a function in which the highest power of x is 3. There may also be terms in x2, x and a constant, but no other terms. Huw Owens - University of Manchester : February 06 10 Power Functions An example Sketch and briefly describe the following functions for positive values of x y=6x+20 y=2x3-50 and y = 14 1600 1500 1400 1200 1008 1000 800 y y=2x^3-50x

400 336 200 132 14 0 0 -200 y=14 624 600 0 14 -48

2 14 14 -84 -96 14 4 -72 14 0 14 6 14 14

8 14 14 10 12 x Huw Owens - University of Manchester : February 06 11 Some questions Sketch the graphs of the functions for values of x between 0 and 10 y=0.5x y=0.5x + 6 y=x2 y=3x2 Which of them is linear? Which is a proportional relationship? What is the effect of adding a constant term?

Huw Owens - University of Manchester : February 06 12 Some questions y=0.5x Linear, proportional relationship y=0.5x+6, Linear, non-proportional relationship, shifted up by constant y = x2. Quadratic function. Some questions y=3x2. Quadratic function. 350 300 300 250 243 y=0.5x 200 192

y y=0.5x+6 150 y=x^2 147 y=3x^2 108 100 100 75 50 48 6 0

0 0 12 7 4 1 6.5 3 1 0.5 2 27 9 7.5 1.5 16 8 2 4

36 25 8.5 2.5 9 3 6 64 49 9.5 3.5 10 4 8 81 10.5

4.5 10 11 5 12 x Huw Owens - University of Manchester : February 06 13 Factors and multiplying out brackets Factorising: writing an expression as a product that when multiplied out gives the original expression. E.g. y=6x-3x2, this is a shorthand way of writing y=(6*x)(3*x*x) If we divide terms on the right-hand side by3*x we obtain (6*x)/3*x = 2 and (3*x*x)/(3*x) = -x. The amount we divide by we call a common factor. So factorising the expression y=6x-3x2 we obtain y=3x(2-x) When two brackets are multiplied together, to remove them we multiply each term in the second bracket by

each term in the first bracket. It is then usual to simplify the result by collecting terms where possible. Huw Owens - University of Manchester : February 06 14 Multiplying brackets E.g. (a-b)(-c+d) = -ac+ad- -bc-+bd = -ac+ad+bc-bd E.g. (x-7)(4-3x) = 4x-3x2-28+21x =-3x2+25x-28 Factorisation is the reverse process to multiplying out brackets. It might not be obvious and could require some intelligent guesswork. The following standard results of multiplying out brackets are helpful. (a+b)2 = a2+2ab+b2 (a-b)2=a2-2ab+b2 (a+b)(a-b)=a2-b2 NOTE: Not every quadratic expression factorises to an expression that contains integer values. Huw Owens - University of Manchester : February 06 15 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. E.g. x3 = x*x*x, x1 = x When we multiply together two expressions comprising the same value raised to a power we ADD the indices and raise to that new power. E.g. x3*x5 = (x*x*x)*(x*x*x*x*x)=x8 Using our rule, x3*x5 = x3+5 = x8 When we divide together two expressions comprising the same value raised to a power we SUBTRACT the indices and raise to that new power. 5 x x* x* x* x* x x*x 5 3 2 x x x3 x* x* x 1

Huw Owens - University of Manchester : February 06 16 What is an equation? Equation: two expressions separated by an equals sign such that what is on the left of the equals sign has the same value as what is on the right. Solving equations lets us discover where lines or curves intersect. These points are interesting because they often indicate information about equilibrium situations. A graphical solution can be obtained by sketching the curves and reading off the x and y values at the point where they cross BUT results only have limited accuracy. Huw Owens - University of Manchester : February 06 17 The elimination method Why use elimination? The graphical method has several drawbacks How do you decide suitable axes? Accuracy of the graphical solution?

Complex problems with > three equations and > three unknowns? Huw Owens - University of Manchester : February 06 18 Example 4x+3y = 11 (1) 2x+y = 5 (2) The coefficient of x in equation 1 is 4 and the coefficient of x in equation 2 is 2 By multiplying equation 2 by 2 we get 4x+2y = 10 (3) Subtract equation 3 from equation 1 to get minus 4x + 3y

= 11 4x + 2y = 10 y = 1 Huw Owens - University of Manchester : February 06 19 Example

If we substitute y=1 back into one of the original equations we can deduce the value of x. If we substitute into equation 1 then 4x+3(1)=11 4x=11-3 4x=8 x=2 To check this put substitute these values (2,1) back into one of the original equations 2*2+1 = 5 Huw Owens - University of Manchester : February 06 20 Summary of the method of elimination Step 1 Add/subtract a multiple of one equation to/from a multiple of the other to eliminate x. Step 2- Solve the resulting equation for y. Substitute the value of y into one of the original equations to deduce x. Step 4 Check that no mistakes have been made by substituting both x and y into the other original equation. Huw Owens - University of Manchester : February 06

21 Example involving fractions Solve the system of equations 3x+2y =1 (1) -2x + y = 2(2) Solution Step 1 - Set the x coefficients of the two equations to the same value. We can do this by multiplying the first equation by 2 and the second by 3 to give 6x+4y = 2 (3) -6x+3y = 6(4) Add equations 3 and 4 together to cancel the x coefficients 7y = 8 y=8/7 Step three substitute y = 8/7 into one of the original equations 3x+2*8/7=1 Huw Owens - University of Manchester : February 06 22 Example

3x=1-16/7 3x=-9/7 x = -9/7*1/3 x= -3/7 The solution is therefore x= -3/7, y= 8/7 Step 4 check using equation 2 -2*(-3/7)+8/7 = 2 6/7+8/7 = 2 14/7 = 2 2=2 Huw Owens - University of Manchester : February 06 23 Problems

1) Solve the following using the elimination method 3x-2y = 4 x-2y =2 2) Solve the following using the elimination method 3x+5y = 19 -5x+2y = -11 Huw Owens - University of Manchester : February 06 24 Special Cases

Solve the system of equations x-2y = 1 2x-4y=-3 The original system of equations does not have a solution. Why? Solve the system of equations 2x-4y = 1 5x-10y = 5/2 This original system of equations does not have a unique solution Huw Owens - University of Manchester : February 06 25 Special Cases There can be a unique solution, no solution or infinitely many solutions. We can detect this in Step 2. If the equation resulting from elimination of x looks like the following then the equations have a unique solution Any non-zero number

* y = Any number If the elimination of x looks like the following then the equations have no solutions zero * y = Any non-zero number Huw Owens - University of Manchester : February 06 26

Special Cases If the elimination of x looks like the following then the equations have infinitely many solutions zero * y = zero Huw Owens - University of Manchester : February 06 27 Elimination Strategy for three equations with three unknowns Step 1 Add/Subtract multiples of the first equation to/from multiples of the second and third equations to eliminate x. This produces a new system of the form ?x + ?y + ?z = ? ?y+?z = ? ?y+?z =? Step 2 Add/subtract a multiple of the second

equation to/from a multiple of the third to eliminate y. This produces a new system of the form ?x + ?y + ?z = ? ?y+?z = ? ?z = ? Huw Owens - University of Manchester : February 06 28 Step 3 Solve the last equation for z. Substitute the value of z into the second equation to deduce y. Finally, substitute the values of both y and z into the first equation to deduce x. Step 4 Check that no mistakes have been made by substituting the values of x,y and z into the original equations. Example Solve the equations 4x+y+3z = 8 (1) -2x+5y+z = 4 (2) 3x+2y+4z = 9 (3) Step 1 To eliminate x from the second equation multiply it by 2 and then add to equation 1 Huw Owens - University of Manchester : February 06

29 To eliminate x from the third equation we multiply equation 1 by 3, multiply equation 3 by 4 and subtract Step 2 To eliminate y from the new third equation (5) we multiply equation 4 by 5, multiply equation 5 by 11 and add This gives us z = 1. Substitute back into equation 4. This gives us y = 1. Finally substituting y=1 and z=1 into equation 1 gives the solution x=1, y=1, z=1 Step 4 Check the original equations give 4(1)+1+3(1) = 8 -2(1)+5(1)+1=4 3(1)+2(1)+4(1)=9 respectively Huw Owens - University of Manchester : February 06 30 Practice Problems

Sketch the following lines on the same diagram 2x-3y=6 4x-6y=18 x-3/2y=3 Hence comment on the nature of the solutions of the following system of equations A) 2x-3y = 6 x-3/2y=3 B) 4x-6y=18 x-3/2y=3 Huw Owens - University of Manchester : February 06 31 Supply and Demand Analysis At the end of this lecture you should be able to Use the function notation, y=f(x) Identify the endogenous and exogenous variables in the economic model. Identify and sketch a linear demand function. Identify and sketch a linear supply function. Determine the equilibrium price and quantity for a

single-commodity market both graphically and algebraically. Determine the equilibrium price and quantity for a multi-commodity market by solving simultaneous linear equations Huw Owens - University of Manchester : February 06 32 Microeconomics Microeconomics is concerned with the analysis of the economic theory and policy of individual firms and markets. This section focuses on one particular aspect known as market equilibrium in which supply and demand balance. What is a function? A function f, is a rule which assigns to each incoming number, x, a uniquely defined out-going number, y. A function may be thought of as a black-box which performs a dedicated arithmetic calculation. An example of this may be the rule double and add 3. Huw Owens - University of Manchester : February 06 33

For example, a second function might be g(x) = -3x+10 We can subsequently identify the respective functions by f and g Huw Owens - University of Manchester : February 06 34 We can write this rule as y=2x+3 Or f(x)=2x+3 5 -17 Double and Add 3 13 f(5)=13 Double and Add 3

-31 f(-17) If in a piece of economic theory, there are two or more functions we can use different labels to refer to each one. Huw Owens - University of Manchester : February 06 35 Independent and dependent variables The incoming and outgoing variables are referred to as the independent and dependent variables respectively. The value of y depends on the actual value of x that is fed into the function. For example, in microeconomics the quantity demanded, Q, of a good depends on the market price, P. This may be expressed as Q = f(P). This type of function is known as a demand function. For any given formula for f(P) it is a simple matter to produce a picture of the corresponding demand curve on paper. Economists plot P on the vertical axis and Q on the horizontal axis.

Huw Owens - University of Manchester : February 06 36 But first a Problem Evaluate f(25) f(1) f(17) g(0) g(48) g(16) For the functions f(x) = -2x +50 g(x) = -1/2x+25 Do you notice any connection between f and g? Huw Owens - University of Manchester : February 06 37 P=g(Q) Thus the two functions f and g are said to be inverse functions. The above form P=g(Q), the demand function, tells us that P is a function of Q but does not give us any

precise details. If we hypothesize that the function is linear P = aQ+b (for some appropriate constants called parameters a and b) The process of identifying real world features and making appropriate simplifications and assumptions is known as modelling. Models are based on economic laws and help to explain the behaviour of real, world situations. Huw Owens - University of Manchester : February 06 38 A graph of a typical linear demand function may be seen below. Demand usually falls as the price of the good rises and so the slope of the line is negative. In mathematical terms P is said to be a decreasing function of Q. So a<0 a is less than zero and b>0 b is greater than zero P b Q Huw Owens - University of Manchester : February 06

39 Example Sketch the graph of the demand function P=-2Q+50 Hence or otherwise, determine the value of (a) P when Q=9 (b) Q when P=10 Solution (a) P = 2*9+50, P=32 (b) 10 = -2Q+50, -40 = -2Q, 20 = Q

Sketch a graph of the demand function P = -3Q+75 Hence, or otherwise, determine the value of (a) P when Q=23 (b) Q when P=18 Solution (a) P = -69+75, P = 6 (b) 18 = -3Q+75, -57 = -3Q, 19 = Q Huw Owens - University of Manchester : February 06 40 Weve so far looked at a crude model of consumer demand assuming that the quantity sold is based only on the price. In practice other factors are required such as the incomes of the consumers Y, the price of substitute goods PS, the price of complementary goods PC, advertising expenditure A, and consumer tastes T. A substitute good is one which could be consumed instead of the good under consideration. (e.g. buses and taxis) A complementary good is one which is used in conjunction with other goods (e.g. DVDs and DVD players). Mathematically, we say that Q is a function of P, Y,

PS,PC, A and T. Huw Owens - University of Manchester : February 06 41 Endogenous and exogenous variables This is written as Q=f(P,Y,PS,PC,A,T) In terms of our black box diagram P Y PS PT f Q A T Any variables which are allowed to vary and are determined within the model are known as endogenous variables (Q and P). The remaining variables are called exogenous since they are constant and are determined outside the model.

Huw Owens - University of Manchester : February 06 42 Inferior and superior goods An inferior good is one whose demand falls as income rises (e.g. coal vs central heating) A superior good is one whose demand rises as income rises (e.g. cars and electrical goods). Problem Describe the effect on the demand curve due to an increase in (a) the price of substitutable goods, Ps (b) the price of complementary goods, Pc (c) advertising expenditure Huw Owens - University of Manchester : February 06 43 The supply function The supply function is the relation between the quantity, Q, of a good that producers plan to bring to the market and the price, P, of the good. A typical linear supply curve is indicated in the diagram below.

Economic theory indicates that as the price rises so does the supply. (Mathematically P is an increasing function of Q) P b Supply curve Demand curve Q Huw Owens - University of Manchester : February 06 44