1. Mathematics for Economics and Business Jean Soper chapter two Equations in Economics1

2. 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 2

3. 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 3

4. 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 4

5. 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 5

6. 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 6

7. 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 7

8. 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 8

9. 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 9

10. 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 10

11. 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 11

12. 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 12

13. 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 13

14. 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 14

15. 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 15

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21. 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 21

22. 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   22

23. 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 23

24. 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 24

25. 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 25

26. 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 26

27. Simultaneous Equations
Simultaneous equations can usually (but not always) be solved if
number of equations = number of unknowns 27

28. 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 28

29. 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 29

30. 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 = …
Use your calculator or computer to evaluate ex 30

31. 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 31

32. 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 32

33. 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 33

34. 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 34

35. 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 35