TX-1037 Mathematical Techniques for Managers

Dr Huw Owens - University of Manchester – February 2006 1

TX-1037 Mathematical Techniques for TX-1037 Mathematical Techniques for Managers Managers

Dr Huw OwensRoom B44 Sackville Street Building

Telephone Number 65891


• 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


Functions of more than one variableFunctions 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.


Some definitions (1)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.


Some definitions (2)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.


Definitions – Factors of productionDefinitions – 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


Total and average revenueTotal 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.


Total and average revenueTotal 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.


Linear average revenue functionLinear 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

0, 72

10, 42

20, 12

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25

Q

P P


• Next we need to find an expression for TR.• TR = P*Q =(72-3Q)Q = 72Q-3Q2

Q=72-3Q

0, 0

2, 132

4, 240

6, 324

8, 384

10, 42012, 43214, 420

16, 384

24, 00

50

100

150

200

250

300

350

400

450

500

-2 3 8 13 18 23 28

Q

TR TR


• 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.


Total and Average CostTotal 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


ExampleExample

• 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


Total CostTotal Cost

Total Cost

0

500

1000

1500

2000

2500

0 5 10 15 20

Q

£ Total Cost


0

100

200

300

400

500

0 5 10 15 20

Q

Co

sts

(£

)

AC

AFC

• 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.


ProfitProfit

• 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


Production functions, isoquants and the average Production functions, isoquants and the average product of labourproduct 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.


An exampleAn 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

813.61733.4

2908.8

5767.2

13737.614464.8

13363.2

0

5000

10000

15000

0 5 10 15 20

L

Q

K=1

K=2

K=3


• 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

0

1

2

3

4

5

6

0 5 10 15

L

K

Q=814

Q=2909

Power (Q=814)

Power (Q=2909)


• 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

0

400

800

1200

0 5 10 15 20

L

AP

L

APL


Equations in economicsEquations 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.


Rewriting and solving equationsRewriting 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


• 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.


ExampleExample

• 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)

-10

-5

0

5

10

15

20

25

30

0 2 4 6 8 10 12

x

y

y=-5+2x

y=30-3x


ProblemProblem

• Solve for x,

• (4x-16)(2x+3) = 8x2

• 8x2+12x-32x-48=8x2

• -20x-48=0• -20x = 48• x = -2.4

32

8164

2

x

xx


Solution in terms of other variablesSolution 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.


• 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


SubstitutionSubstitution

• 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


Demand and supplyDemand 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.


Demand and supplyDemand 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

0

10

20

30

40

0 20 40 60 80 100

Q

P P=40-Q/2


Market equilibriumMarket 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


Changes in demand or supplyChanges 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.


Changes in demand or supplyChanges 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.


Changes in demand and supply - ExampleChanges 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.


Changes in demand or supply - ExampleChanges 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.


• 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


Equilibrium P and Q increase when demand increases Equilibrium P and Q increase when demand increases by 20%by 20%

-10

-5

0

5

10

15

20

25

30

0 50 100 150 200

Q

P

P=Qs/6

P=22-Qd/5

P=22-Qd/6


Cost–Volume–Profit (CVP) AnalysisCost–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 obtainAVC.Q = VC and substituting this = P.Q – FC – AVC.Q


Special Assumptions of CVP AnalysisSpecial 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


Solving for DSolving for Desired Sales Levelesired 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


Linear EquationsLinear 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


Representing a Line as Representing a Line as yy = = mxmx + + bb

• 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


A horizontal line has zero slope

0

10

20

30

0 5 10x

yy = 18

slope = 0

as x increases, y does not change


Positive slope, zero intercept

0

250

500

0 25 50x

y

y = 9xas x increases,

y increases

slope = 9

line passes through the origin


Negative slope, positive intercept

0

10

20

30

40

50

60

0 5 10 15x

y

y = 50 - 4x

larger x values go with smaller y values

slope = - 4


Positive slope, negative intercept

-30

-20

-10

0

10

20

30

40

x

y

10 20

y = -25 + 3x

line cuts y axis below the origin

slope = 3

as x increases, y increases


A vertical line has infinite slope

0

10

20

30

40

0 5 10 15 20x

y x = 15

y increases but x does not change

slope =


Budget LineBudget Line

• If two goods x and y are boughtthe budget line equation is x.Px + y.Py = M

• To plot the line, rewrite asy = 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


The Parameters of a Budget LineThe 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


Constant Substitution Along a Line 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


Diminishing Marginal Rate of Substitution Along an Diminishing Marginal Rate of Substitution Along an Indifference CurveIndifference 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


Quadratic EquationsQuadratic 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

aacbb

2

42 x


Intersection of MC with MR or AVCIntersection 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


Simultaneous EquationsSimultaneous Equations

• Simultaneous equations can usually (but not always) be solved if

number of equations = number of unknowns


Solving Simultaneous EquationsSolving 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


Simultaneous Equilibrium in Related MarketsSimultaneous 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


Exponential FunctionsExponential 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


Logarithmic FunctionsLogarithmic 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


Working with Working with LogarithmsLogarithms

• 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


Solving a Quadratic Equation in ExcelSolving 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


Plotting and Solving Equations in ExcelPlotting 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


Solving Equations with Excel SolverSolving 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

TX-1037 Mathematical Techniques for Managers

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