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Mathematics for Economics and Business

Jean Soper

chapter twoEquations in Economics

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Equations in Economics – Objectives 1

• 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• Appreciate there is a constant rate of

substitution as you move along a line

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Equations in Economics – Objectives 2

• Solve quadratic equations• Find the profit maximizing output and also

the supply function for a perfectly competitive firm

• Solve simultaneous equations• Discover equilibrium values for related

markets• Model growth using exponential functions• Use logarithms for transformations and to

solve certain types of equations

• Plot and solve equations in Excel

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Rewriting and Solving Equations

• 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

• 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

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When Rewriting Equations

• Add to or subtract from both sides• Multiply or divide through the whole

of each side (but don’t divide by 0)• Square or take the square root of

each side• Use as many stages as you wish• Take care to get all the signs correct

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Solution in Terms of Other Variables

• Not all equations have numerical solutions• Sometimes when you solve an equation

for x you obtain an expression containing other variables

• Use the same rules to transpose the equation

• In the solution x will not occur on the right-hand side and will be on its own on the left-hand side

• Inverse function: expresses x as a function of y instead of y as a function of x

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Substitution

• Substitution: to write one expression in place of another

• Always 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

• It is often helpful to put the expression you are substituting in brackets to ensure this

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Demand and Supply

• We plot supply and demand with P on the vertical axis

• Before plotting a supply or demand function, write it so that P is on the left, Q is on the right

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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

• You can read off approximate equilibrium values from the graph

• Solving algebraically for the point where the demand and supply equations are equal gives exact values

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Changes in Demand or Supply

• Changes in factors other than price alter the position(s) of the demand and/or supply curves

Effects of a per unit tax

• To obtain the new supply equation when a per unit tax, t, is imposed substitute P – t for P in the supply equation

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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

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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

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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

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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

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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

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A horizontal line has zero slope

0

10

20

30

0 5 10x

yy = 18

slope = 0

as x increases, y does not change

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Positive slope, zero intercept

0

250

500

0 25 50x

y

y = 9xas x increases,

y increases

slope = 9

line passes through the origin

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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

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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

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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 =

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Budget 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

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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  

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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

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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

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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

aacbb

2

42 x

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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

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Simultaneous Equations

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

number of equations = number of unknowns

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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

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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

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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

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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

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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

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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

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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

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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