Mathematics for Economics and
Business
Jean Soper
chapter two
Equations in Economics
1
Equations in Economics –
Objectives 1
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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
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 righthand side and will be on its own on the lefthand 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
as x increases,
y does not change
30
y
20
y = 18
slope = 0
10
0
0
16
5
x 10
Positive slope, zero intercept
y = 9x
500
as x increases,
y increases
y
250
slope = 9
line passes through the origin
0
0
17
25
x
50
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
18
5
10
x 15
Positive slope, negative intercept
y
40
30
20
10
as x increases,
y increases
0
-10
-20
-30
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y = -25 + 3x
slope = 3
x
10
20
line cuts y axis below the origin
A vertical line has infinite slope
40
x = 15
y
y increases but x does not change
30
20
10
slope = 
0
0
20
5
10
15
x
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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
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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
 b  b2  4ac
x
2a
where a is the coefficient of x2, b is the
coefficient of x and c is the constant term
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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
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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
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
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