TX-1037 Mathematical Techniques for Managers Dr Huw Owens Room B44 Sackville Street Building Telephone Number 65891 Dr Huw Owens - University of Manchester – February 2006 1 • • • • • • 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 • Equations in economics • Rewriting and solving equations • Substitution Dr Huw Owens - University of Manchester – February 2006 2 Functions of more than one 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 statistical analysis. Dr Huw Owens - University of Manchester – February 2006 3 Some definitions (1) • Total revenue, TR, is the amount of money received by the firm from the sale of goods and total cost, TC, is the amount of amount of money that the firm has to spend to produce these goods. • The profit function is denoted by the Greek letter ∏ (pronounced “pie”) and is defined to be the difference between total revenue, TR, and total cost, TC. • Total cost, TC, function relates to the production costs to the level of output, Q. • However, in the short run some of these costs are fixed. • Fixed costs, include the cost of land, equipment, rent and possibly skilled labour. Dr Huw Owens - University of Manchester – February 2006 4 Some definitions (2) • Obviously, in the long run all costs are variable but these particular costs take time to vary, so can be though of as fixed in the short run. • Variable costs vary with output and include the cost of raw materials, components, energy and unskilled labour. Dr Huw Owens - University of Manchester – February 2006 5 Definitions – Factors of production • The output Q, of any production process depends on a variety of inputs, known as factors of production. These include land, capital, labour and enterprise. • For simplicity. We restrict our attention to capital and labour. • Capital, K, denotes all man-made aids to production such as buildings, tools and plant machinery. • Labour, L, denotes all paid work in the production process Dr Huw Owens - University of Manchester – February 2006 6 Total and average revenue • When a firm sells a quantity, Q, of goods each at a price P, its total revenue, TR, is the price that is paid multiplied by the quantity sold. • TR = P*Q • Average return, AR, is the revenue received by the firm per unit of output sold. This is the total revenue divided by the quantity sold. • AR = TR/Q = P • The average return curve shows the average revenue or price at which different quantities are sold. It shows the prices that people will pay to obtain various quantities of output and so is known as the demand curve. Dr Huw Owens - University of Manchester – February 2006 7 Total and average revenue • A market demand curve is assumed to be downward sloping. • Different prices are associated with different quantities being sold at lower prices. • There will also be an associated downward sloping marginal, MR, curve but we will investigate this in later lectures. • For example, If average revenue is given by P=72-3Q sketch this function and also, on a separate graph, the total revenue function. Dr Huw Owens - University of Manchester – February 2006 8 Linear average revenue function • The average revenue function has P on the vertical axis and Q on the horizontal axis. The general form of the linear equation is y=ax+b. We only need to find two points in order to sketch the function. P=72-3Q 80 0, 72 70 60 P 50 10, 42 40 P 30 20 20, 12 10 0 0 5 15 10 20 25 Q Dr Huw Owens - University of Manchester – February 2006 9 • Next we need to find an expression for TR. • TR = P*Q =(72-3Q)Q = 72Q-3Q2 Q=72-3Q 500 450 12, 432 10, 420 14, 420 8, 384 16, 384 400 350 6, 324 TR 300 250 TR 4, 240 200 150 2, 132 100 50 0 -2 0, 0 24, 0 3 8 13 18 23 28 Q Dr Huw Owens - University of Manchester – February 2006 10 • The function is a quadratic, so we must find a number of points. • The graphs shows a curve which at first rises relatively steeply, then flattens out and reaches a maximum Q=12, after which it falls. The curve is symmetric. Its shape to the right of its maximum value is the mirror image of that to the left. • Symmetric: the shape of one half of the curve is the mirror image of the other half. • Some firms sell their output at the same price. This is a feature of firms operating under the market structure known as perfect competition. • These firms face a horizontal demand curve and have a total revenue function which is an upward sloping straight line passing through the origin. Dr Huw Owens - University of Manchester – February 2006 11 Total and Average Cost • A firm’s total cost of production, TC, depends on its output, Q. • The TC function include may a constant term, which represents fixed costs, FC. • The part of the TC that varies with Q is called variable cost, VC. • FC is the constant term in TC • VC = TC-FC • Average Total Cost, AC = TC/Q • Average Variable Cost, AVC = VC/Q • Average Fixed Cost, AFC = FC/Q Dr Huw Owens - University of Manchester – February 2006 12 Example • For a firm with total cost given by, • TC = 120+45Q-Q2+0.4Q3 • Find AC, FC, VC, AVC and AFC functions. List some values of TC, AC and AFC, correct to the nearest integer. Sketch the total cost function and on a separate graph, the AC and AFC functions. • AC = TC/Q = 120/Q+45-Q+0.4Q2 • FC= 120 (the constant term in the TC) • VC = TC-FC = 45Q-Q2+0.4Q3 • AVC = VC/Q = 45-Q+0.4Q2 • AFC = FC/Q = 120/Q Dr Huw Owens - University of Manchester – February 2006 13 Total Cost Total Cost 2500 2000 1500 £ Total Cost 1000 500 0 0 5 10 15 20 Q Dr Huw Owens - University of Manchester – February 2006 14 • The average total cost curve as first falls as output rises, but later the curve rises again. • Average fixed cost is always declining as output increases. 500 Costs (£) 400 300 AC AFC 200 100 0 0 5 10 15 20 Q Dr Huw Owens - University of Manchester – February 2006 15 Profit • Profit is the excess of a firm’s total revenue, TR, over its total cost, TC, and so we calculate it by subtracting TC from TR. Using the symbol ∏ as the variable for profit. • ∏ = TR-TC • A firm has the total cost function • TC = 120+ 45Q-Q2+0.4Q3 • And faces a demand curve given by • P=240-20Q • What is its profit function? • TR = PQ = 240Q-20Q2 • Since TC comprises several terms we enclose it in brackets as we substitute • =240Q-20Q2 – (120+45Q-Q2+0.4Q3) • =-120+195Q-19Q2-0.4Q3 Dr Huw Owens - University of Manchester – February 2006 16 Production functions, isoquants and the average product of labour • A production function shows the quantity of output (Q) obtained from specific quantities of inputs, assuming they are used efficiently. Q=f(L,K) • In the short run the quantity of capital is fixed. • In the long run both labour (L) and capital (K) are variable. • Plot Q on the vertical axis against L on the horizontal for a short-run production function. • Plot K against L and connect points that generate equal output for an isoquant map. • Average Product of Labour (APL) = Q/L • An isoquant connects points at which the same quantity of output is produced using different combinations of inputs. Dr Huw Owens - University of Manchester – February 2006 17 An example • A firm has the production function Q=25(LK)2-0.4(LK)3. • If K=1, find the values of Q for L=2,3,4,6,12,14 and 16. Sketch this short-run production function putting L and Q on the axes of your graph. Production functions for different values of K 15000 14464.8 13737.6 13363.2 10000 Q K=1 K=2 K=3 5767.2 5000 2908.8 1733.4 813.6 0 0 5 10 15 20 L Dr Huw Owens - University of Manchester – February 2006 18 • Now sketch another representation of this production function as an isoquant map. Plot L and K on the axes and look for combinations of L and K amongst the values you have calculated which give the same value of Q. Such points lie on the same isoquant. Isoquant map 6 5 Q=814 Q=2909 Power (Q=814) Power (Q=2909) K 4 3 2 1 0 0 5 10 15 L Dr Huw Owens - University of Manchester – February 2006 19 • For a short-run production function with K = 3, find and plot the average product of labour function. • For K=3 we have Q=25(3L)2-0.4(3L)3=225L2-10.8L3 • APL=Q/L = 225L-10.8L2 The average product of labour function 1200 APL 800 APL 400 0 0 5 10 15 20 L Dr Huw Owens - University of Manchester – February 2006 20 Equations in economics • An equation is a statement that two expressions are equal to one another. • In economic modelling we express relationships as equations and then use them to obtain analytical results. Solving the equations gives us values for which the equations are true. • Solving equations lets us discover where curves intersect. Economists are often interested in these points because they may provide information about equilibrium situations. • Graphical solutions can be obtained by reading off the x and y values at the point or points of intersection BUT the results have limited accuracy. Dr Huw Owens - University of Manchester – February 2006 21 Rewriting and solving equations • Since the x and y values are the same on both curves at intersecting points we can obtain an exact solution using algebra. • At an intersection of the functional relationships y=f(x) and y=h(x), the two y values are equal and therefore f(x)=h(x) • Transposition: rearranging an equation so that it can be solved, always keeping what is on the left of the equals sign equal to what is on the right. • For example, to solve for x the equation • 140+6x = -30x+284 • 6x+140-140=-30x+284-140 • 6x=-30x+144 • 6x+30x=-30x+144+30x • 36x=144 • x=144/36 = 4 Dr Huw Owens - University of Manchester – February 2006 22 • When rewriting equations • Add to or subtract from both sides. • Multiply or divide through the whole of each side (DON’T divide by zero). • Square or take the square root of each side. • Use as many stages as you wish. • Take care to get all the signs correct. Dr Huw Owens - University of Manchester – February 2006 23 Example • Plot the equations y=-5+2x and y=30-3x. At what values of x and y do they cross? Find the algebraic solution by setting the two expressions in x equal to one another. • -5+2x = 30-3x • 5x = 35, x=7 • When x=7, y = 9. Lines intersect at (7,9) 30 25 20 y 15 y=-5+2x y=30-3x 10 5 0 -5 0 2 4 6 8 10 12 -10 x Dr Huw Owens - University of Manchester – February 2006 24 Problem 8x2 • Solve for x, 4 x 16 2x 3 • • • • • (4x-16)(2x+3) = 8x2 8x2+12x-32x-48=8x2 -20x-48=0 -20x = 48 x = -2.4 Dr Huw Owens - University of Manchester – February 2006 25 Solution in terms of other variables • Not all equations will have numerical solutions. Sometimes when you solve an equation for x you obtain an expression containing other variables. • If you are given a relationship of the form y=f(x), rewriting the equation in the form x=g(y) is called finding the inverse function. To be able to find the inverse function there must be just one x value corresponding to each y value. • For non-linear functions there can be difficulties in finding an inverse but it may be done for a restricted set of values (e.g. square roots). • For the linear functions often used in economic models inverse functions can always be found. • One reason for finding the inverse function is if the variable represented by y is conventionally plotted in economics on the horizontal axis. Demand and supply equations are examples of this. Dr Huw Owens - University of Manchester – February 2006 26 • Inverse function: expresses x as a function of y instead of y as a function of x. • Solve for x in terms of z, x=60+0.8x+7z • 0.2x = 7z+60, x = 35z+300 • Solve y = x1/2+5, obtain an expression for x in terms of y. • X1/2=y-5, x = (y-5)(y-5), x = y2-10y+25 Dr Huw Owens - University of Manchester – February 2006 27 Substitution • Substitution: to write one expression in place of another. • When substituting, always be sure to substitute the whole of the new expression and combine it with the other terms in exactly the same way that the expression it replaces was combined with them. • For example, if y=x2+6u and x = 30-u, find an expression for y in terms of u. Substituting 30-u for x we obtain • y=(30-u)2+6u, y=900-60u+u2+6u, y = u2-54u+900 Dr Huw Owens - University of Manchester – February 2006 28 Demand and supply • Demand and supply functions in economics express the quantity demanded or supplied as a function of price, Q=f(P). • According to mathematical convention the dependent variable, Q, should be plotted on the vertical axis. • Economic analysis, however, uses the horizontal axis as the Q axis and for consistency we will follow that approach. • So that we can determine the points on the graph in the usual way, before plotting a demand or supply function we first find its inverse function giving P as a function of Q. Dr Huw Owens - University of Manchester – February 2006 29 Demand and supply • Find the inverse function for the demand equation Q=80-2P and sketch the demand curve. • P=40-Q/2 Demand Curve P=40-Q/2 40 P 30 P=40-Q/2 20 10 0 0 20 40 60 80 100 Q Dr Huw Owens - University of Manchester – February 2006 30 Market equilibrium • Market equilibrium occurs when the quantity supplied equals the quantity demanded of a good. The supply and demand curves cross at the equilibrium price and quantity. • If you plot the demand and supply curves you can read off the approximate equilibrium values from the graph. • Another way is to solve algebraically for the point where demand and supply are equal. • For example, quantity demanded, Qd, is given by • Qd=96-4P, and quantity supplied, Qs, is given by • Qs = 8P • In equilibrium, Qs = Qd so by substitution • 8P = 96-4P • 12P=96, P=8, the equilibrium price • Substitute P into either the supply or demand equation. Using the supply equation gives, Qs=96-4P, Qs=96-32, Qs=64 Dr Huw Owens - University of Manchester – February 2006 31 Changes in demand or supply • The quantity demanded and the quantity supplied of a good are usually modelled as dependent on a number of factors. • All of these except price are assumed to be constant when the demand and supply curves are drawn. • Changes in factors other than price alter the position of the curves. • For example, suppose we have a multivariate demand function where Q, the demand for good X depends on P the price of X, together with M, consumer income and Pz, the price of another good,Z. The demand function is • Q1=80-5P+0.1M+0.3Pz when M and Pz are fixed respectively at 2500 and 60 the demand curve becomes • Q2=80-5P+0.1(2500)+0.3(60) = 298-5P • We get a new demand curve, which is parallel to the old one but shifted down from it. Dr Huw Owens - University of Manchester – February 2006 32 Changes in demand or supply • Demand and supply curves can also change their shapes in other ways. For example, suppose a change in tastes causes twice as much to be demanded at any price. If the original demand is given by • Q=55-5P • Denoting the new quantity demanded Q2, we know that it is twice Q and so, • Q2=2Q = 2(55-5P) = 110-10P • Again the use of subscripts distinguishes two different curves. Dr Huw Owens - University of Manchester – February 2006 33 Changes in demand and supply - Example • For the demand and supply functions given, find the inverse functions giving P as a function of Q, sketch the demand and supply curves and mark the equilibrium position. • Demand: Qd=110-5P • Supply: Qs=6P • If demand increases by 20%, find the new demand function, its inverse and the new equilibrium position on the diagram. Dr Huw Owens - University of Manchester – February 2006 34 Changes in demand or supply - Example • Rewrite the demand equation, adding (5P-Qd) to both sides. This gives • 5P=110-Qd • Dividing both sides by 5 gives the inverse demand function • P=22-Qd/5 • For the inverse of the supply equation we interchange the sides, obtaining • 6P=Qs • And then divide by 6 to get • P=Qs/6 • The demand and supply curves are shown on the following figure. Dr Huw Owens - University of Manchester – February 2006 35 • Equilibrium occurs where they cross, and here the values of P and Q on the demand curve equal those on the supply curve. • This is the point Q=60, P=10. • When demand increases, the quantity demanded is 20% greater than before at every price. Hence the new quantity demanded, Qd2, is given by • Qd2 = 1.2(110-5P)=132-6P • The inverse function is • P=22-Qd2/6 • The new equilibrium position occurs where this curve crosses the supply curve at the point Q=66, P=11 Dr Huw Owens - University of Manchester – February 2006 36 Equilibrium P and Q increase when demand increases by 20% 30 25 20 P 15 P=Qs/6 P=22-Qd/5 P=22-Qd/6 10 5 0 -5 0 50 100 150 200 -10 Q Dr Huw Owens - University of Manchester – February 2006 37 Cost–Volume–Profit (CVP) Analysis • Two simplifying assumptions are made: namely that price and average variable costs are both fixed = P.Q – (FC + VC) = P.Q – FC – VC • Multiplying both sides of the expression for AVC by Q we obtain AVC.Q = VC and substituting this = P.Q – FC – AVC.Q Dr Huw Owens - University of Manchester – February 2006 38 Special Assumptions of CVP Analysis • • • • P is fixed AVC is fixed is a function of Q but P, FC, and AVC are not We can write the inverse function expressing Q as a function of • Adding FC to both sides gives + FC = P.Q – AVC.Q • Interchanging the sides we obtain P.Q – AVC.Q = + FC Dr Huw Owens - University of Manchester – February 2006 39 Solving for Desired Sales Level • • • • • Q is a factor of both terms on the left so we may write Q(P – AVC) = + FC Dividing through by (P – AVC) gives Q = ( + FC)/(P – AVC) If the firm’s accountant can estimate FC, P and AVC, substituting these together with the target level of profit, , gives the desired sales level Dr Huw Owens - University of Manchester – February 2006 40 Dr Huw Owens - University of Manchester – February 2006 41 Linear Equations • Slope of a line: distance up divided by distance moved to the right between any two points on the line • Coefficient: a value that is multiplied by a variable • Intercept: the value at which a function cuts the y axis Dr Huw Owens - University of Manchester – February 2006 42 Representing a Line as y = mx + b • • • • • • The constant term, b, gives the y intercept The slope of the line is m, the coefficient of x Slope = y/x = (distance up)/(distance to right) Lines with positive slope go up from left to right Lines with negative slope go down from left to right Parameter: a value that is constant for a specific function but that changes to give other functions of the same type; m and b are parameters Dr Huw Owens - University of Manchester – February 2006 43 A horizontal line has zero slope as x increases, y does not change 30 y 20 y = 18 slope = 0 10 0 0 5 Dr Huw Owens - University of Manchester – February 2006 x 10 44 Positive slope, zero intercept y = 9x 500 as x increases, y increases y 250 slope = 9 line passes through the origin 0 0 25 Dr Huw Owens - University of Manchester – February 2006 x 50 45 Negative slope, positive intercept y 60 50 larger x values go with smaller y values 40 30 slope = - 4 20 y = 50 - 4x 10 0 0 5 10 Dr Huw Owens - University of Manchester – February 2006 x 15 46 Positive slope, negative intercept y 40 30 20 10 as x increases, y increases 0 -10 -20 -30 y = -25 + 3x slope = 3 x 10 20 line cuts y axis below the origin Dr Huw Owens - University of Manchester – February 2006 47 A vertical line has infinite slope 40 x = 15 y y increases but x does not change 30 20 10 slope = 0 0 5 10 Dr Huw Owens - University of Manchester – February 2006 15 x 20 48 Budget Line • If two goods x and y are bought the budget line equation is x.Px + y.Py = M • To plot the line, rewrite as y = M/Py – (Px/Py )x • Slope = – Px/Py the negative of the ratio of the prices of the goods • Intercept = M/Py the constant term in the equation Dr Huw Owens - University of Manchester – February 2006 49 The Parameters of a Budget Line • Changing Px rotates the line about the point where it cuts the y axis • If Py alters, both the slope and the y intercept change • the line rotates about the point where it cuts the x axis • An increase or decrease in income M alters the intercept but does not change the slope • the line shifts outwards or inwards Dr Huw Owens - University of Manchester – February 2006 50 Constant Substitution Along a Line • The rate at which y is substituted by x is constant along a downward sloping line, but not along a curve Dr Huw Owens - University of Manchester – February 2006 51 Diminishing Marginal Rate of Substitution Along an Indifference Curve • Indifference curve: connects points representing different combinations of two goods that generate equal levels of utility for the consumer • Diminishing marginal rate of substitution: as a consumer acquires more of good x in exchange for good y, the rate at which he substitutes x for y diminishes because he becomes less willing to give up y for a small additional amount of x Dr Huw Owens - University of Manchester – February 2006 52 Quadratic Equations • A quadratic equation takes the form ax2 + bx + c = 0 • You can solve it graphically • or sometimes by factorizing it • or by using the formula where a is the coefficient of x2, b is the coefficient of x and c is the constant term b b2 4ac x 2a Dr Huw Owens - University of Manchester – February 2006 53 Intersection of MC with MR or AVC • Quadratic equations arise in economics where a quadratic function, say marginal cost, cuts another quadratic function, say average variable cost, or cuts a linear function, say marginal revenue • Equate the two functional expressions • Subtract the right-hand side from both sides so that the value on the right becomes zero • Collect terms • Solve the quadratic equation Dr Huw Owens - University of Manchester – February 2006 54 Simultaneous Equations • Simultaneous equations can usually (but not always) be solved if number of equations = number of unknowns Dr Huw Owens - University of Manchester – February 2006 55 Solving Simultaneous Equations • Solution methods for two simultaneous equations include • Finding where functions cross on a graph • Eliminating a variable by substitution • Eliminating a variable by subtracting (or adding) equations • Once you know the value of one variable, substitute it in the other equation Dr Huw Owens - University of Manchester – February 2006 56 Simultaneous Equilibrium in Related Markets • Demand in each market depends both on the price of the good itself and on the price of the related good • To solve the model use the equilibrium condition for each market demand = supply • This gives two equations (one from each market) in two unknowns which we then solve Dr Huw Owens - University of Manchester – February 2006 57 Exponential Functions • Exponential function: has the form ax where the base, a, is a positive constant and is not equal to 1 • The exponential function most used in economics is y = ex • The independent variable is in the power and the base is the mathematical constant e = 2.71828… • Use your calculator or computer to evaluate ex Dr Huw Owens - University of Manchester – February 2006 58 Logarithmic Functions • Logarithm: the power to which you must raise the base to obtain the number whose logarithm it is • Common logarithms denoted log or log10 are to base 10 • Natural logarithms denoted ln or loge are to base e and are more useful in analytical work • Equal differences between logarithms correspond to equal proportional changes in the original variables Dr Huw Owens - University of Manchester – February 2006 59 Working with Logarithms • • • • • log (xy) = log (x) + log (y) log (x/y) = log (x) – log (y) log (xn) = n log (x) ln (ex) = x The reverse process to taking the natural logarithm is to exponentiate Dr Huw Owens - University of Manchester – February 2006 60 Solving a Quadratic Equation in Excel • You can enter formulae in Excel to calculate the two possible solutions to a quadratic equation • First calculate the discriminant b2 – 4ac • To make your formulae easier to understand, name cells a, b, const and discrim and use the names instead of cell references • By default, names are interpreted in Excel as absolute cell references • To name a cell, select it, type its name in the Name Box and press the Enter key Dr Huw Owens - University of Manchester – February 2006 61 Plotting and Solving Equations in Excel • Excel includes many inbuilt functions that you can type in or access by clicking the Paste Function button • Those for exponentials and logarithms are =EXP() and =LN() where the cell reference for the value to which the function is to be applied goes inside the brackets Dr Huw Owens - University of Manchester – February 2006 62 Solving Equations with Excel Solver • Excel includes a Solver tool designed to solve a set of equations or inequalities • Set out the data in a suitable format • Interact with the Solver dialogue box to find the solution • Excel does not solve the equations the same way as you do when working by hand • It uses an iterative method, trying out different possible values for the variables to see if they fit the specified requirements Dr Huw Owens - University of Manchester – February 2006 63