1
equilibrium
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Linear equations
And it’s application
Break-Even
Analysis
Linear form describes the most simple type of line that you will plot.
When a group of co-ordinates are used to create a graph that is a straight line,
the equation is said to be a linear equation.
A linear equation takes the form:
y = a + bx,
OR y=mx+c where:
y is the dependent variable
x is the independent variable
a or c is the intercept term (when x=0, y = a)
b or m is the slope of the line.
Linear form
When we given the equation of the line and asked to find the values for y by
substituting values for x into the equation.
What if instead you don’t have the equation but are given two sets of coordinates and what to use these to derive the equation of the line?
To do this we need to find the slope, b, and the intercept, a.
The slope is calculated by dividing the change in the value of y by the change
in the value of x:
𝒃 𝒐𝒓 𝒎 =
•
•
Once you have a value for the slope, b, you substitute this into the general equation:
y = a + bx
along with one set of co-ordinates to arrive at an answer for a:
a = y – bx
Determining a Line from Two Points
Example:
Given the co-ordinates (0,2) and (3,4), find the slope of the line, the intercept and
the equation of the line.
Solution:
Slope = b = (4-2)/(3-0) = 2/3
Substituting b=2/3 and x=0, y=2:
a = 2 – (2/3)(0) = 2 y = 2 + (2/3)x
Find the Slope and y-intercept of a Line
Example: Find the slope and y-intercept of the line with equation
Solution:
Rewrite the equation as y=5(3−2x)
y=15−10x
The slope is –10 and the y-intercept is 15.
y = 5(3 – 2x).
y=−10x+15
2
Price-Quantity Relationship
The line in the figure shows the
relationship between the price p of a
widget (in dollars) and the quantity q of
widgets (in thousands) that consumers will
buy at that price. Find and interpret the
slope.
Solution:
The slope is
𝒎=
𝒑𝟏 − 𝒑𝟐 𝟏 − 𝟒
𝟏
=
= −
𝒒𝟏 − 𝒒𝟐 𝟖 − 𝟐
𝟐
Quantity time relationship
the line in the figure shows the relationship
between the time T and the quantity Q
Example:
Suppose that oil exports from Saudi Arabia
grew from 48 billion dollars in 1999 to 75 billion dollars in 2006. Let t = 0
correspond to 1999. a. Find a linear equation to represent this data.
Solution:
Let y be the amount of Saudi Arabia oil exports in billions of dollars, and let t
be the number of years after 1999.
If t = 0 corresponds to 1999 then t = 7 corresponds to 2006. Thus, the given
data are (0, 48) and (7, 75). The slope of the line joining these two points is
m=
=3.86
Using the point (0, 48), we obtain the point-slope form of the equation of the line:
y – 48 = 3.86(t – 0)
y = 3.86t + 48
b. Use this equation to estimate the exports (in billions of dollars) in 2012.
Solution:
Since 2012 corresponds to t = 13, replace t by 13 in the previous equation to get
y = 3.86(13) + 48 = 98.18 billion dollars
Total oil exports from Saudi Arabia in 2012 are estimated at 98 billion dollars.
Linear cost model
Total cost = fixed cost + variable cost
𝑻𝒄 = 𝒎𝒙 + 𝒃
Ex: The variable cost of processing 1 pound of coffee beans is 50¢ and the
fixed costs per day are $300.
a. Given the linear cost equation.
b. Find the cost of processing 1000 pounds of coffee beans in one day.
Solution :
a. Fixed cost (b) = 300 , variable cost per unit (m) = 0.5
Then:
𝑻𝒄 = 𝒎𝒙 + 𝒃
𝑻𝒄 = 𝟎. 𝟓𝒙 + 𝟑𝟎𝟎
b. the cost of processing 1000= 0.5×1000+300 = 800
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Cost(y)
Cost -Quantity Relationship
The line in the figure shows the relationship.
between the cost y and the quantity q
Quantity(q)
Ex: The cost of manufacturing 10 typewriters per day is $350 while it cost
$600 to produce 20 typewriters per day Assuming a linear cost model,
determine the relationship representing the total cost 𝒚𝒄 Of producing X
typewriters per day Solution:
Let 𝒙𝟏 = 𝟏𝟎 ,𝒚𝟏 = 𝟑𝟓𝟎 𝒂𝒏𝒅 𝒙𝟐 = 𝟐𝟎 ,𝒚𝟐 = 𝟔𝟎𝟎
Then
𝒎=
𝒚𝟐 −𝒚𝟏
𝒒𝟐 −𝒒𝟏
=
𝟔𝟎𝟎−𝟑𝟓𝟎
𝒚 = 𝒎𝒙 + 𝒃
𝟐𝟎−𝟏𝟎
=
𝟐𝟓𝟎
= 𝟐𝟓
𝟏𝟎
𝟑𝟓𝟎 = 𝟐𝟓 × 𝟏𝟎 + 𝒃
𝟑𝟓𝟎 = 𝟐𝟓𝟎 + 𝒃
𝒃 = 𝟏𝟎𝟎
𝑻𝒄 = 𝟐𝟓𝒙 + 𝟏𝟎𝟎
Supply and demand
the simplest low is a linear relation of the type:
𝒑 = 𝒎𝑸 + 𝒃
Where: 𝒑 is a price per unit
𝒙 is the quantity.
𝒎 𝒂𝒏𝒅 𝒃 are constant.
Ex:
A dealer can sell 20 electric shavers per day at $25 per Shaver, but he can sell
30 shivers if he changes $20 per shavers. determine the demand equation
assuming it is linear.
Let 𝑸𝟏 = 𝟐𝟎 ,𝒑𝟏 = 𝟐𝟓 𝒂𝒏𝒅 𝑸𝟐 = 𝟑𝟎 ,𝒑𝟐 = 𝟐𝟎
𝒕𝒉𝒆𝒏 ∶ 𝒎 =
𝒑 = 𝒎𝒙 + 𝒃
𝒑𝟐 −𝒑𝟏
𝒒𝟐 −𝒒𝟏
=
𝟐𝟎−𝟐𝟓
𝟑𝟎−𝟐𝟎
= −
𝟓
𝟏𝟎
= −𝟎. 𝟓
𝟐𝟓 = −𝟎. 𝟓 × 𝟐𝟎 + 𝒃
𝟐𝟓 = −𝟏𝟎 + 𝒃
𝒃 = 𝟑𝟓
𝒑 = −𝟎. 𝟓𝑸 + 𝟑𝟓
Production Levels
Suppose that a manufacturer uses 100kg of material to produce products A
and B, which require 4kg and 2kg of material per unit, respectively .
Solution:
If x and y denote the number of units produced of A and B, respectively,
4x+2y =100 where x,y 0
Then Solving for y gives y =−2x+50
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Budget Constraints
• Budget Constraints- an equation that measures the relationship
between prices and demand for certain goods.
• The budget constraint takes the form:
M = PxX + PYY, where:
•
• M = total income or budget
• Px = price of good X
X = quantity of good X demanded
the intercept is:
M/P
X
,
PY = price of good Y
, Y = quantity of good Y demanded
the slope is: – (P /P ).
x
Y
Example
A firm has a budget of £3000 to spend on two goods X and Y. Good X costs
£50 per unit and good Y costs £30 per unit.
Solution
The budget constraint can be written as:
£3000 = £50X + £30Y or dropping the £’s:
3000 = 50X + 30Y the
intercept = M/PX = 3000/50 = 60 the
slope = – (Px/PY) = - 50/30 = - 1.67
Exercises:
1- Find the equation of the line passing through the points (2, 3) and (1, 5).
2- Looking specifically at price and quantity, if price is £5 per unit when
quantity demanded is 20 units, and price is £7 per unit when quantity
demanded is 34 units, find the equation for demand curve.
3- Find the equation of the line passing through each of the given sets of
co-ordinates:
(a)
(2, 4), (6, 8) (b) (-1,4), (3, 8) (c) (-6,-3), (-2, 9)
4- If quantity supplied is 10 units when price is £3 per unit, and quantity
supplied is 15 units when price is £4 per unit, find the equation of the
supply curve.
5- If a consumer has an income of £160 to spend on the two goods X and y
whose prices are £20 and £5 each, respectively :
(a)
Determine the consumer's budget constraint .
(b)
Calculate the slope of the consumer's budget constraint.
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Intersection
•
In mathematics and economics we usually use two or more lines to
illustrate or illustrate the relationship between different factors.
• If we can find the point at which these two lines intersect, then this gives
us more information about the given situation.
• There are four techniques for finding the intersection point between two
linear equations:
▪ Graphically , Equating the equations .The substitution method and Row operation
Example
Find the point of intersection of the following two linear equations:
y = 3x + 6
,
y = -4x + 10.
To arrive at a more accurate answer we could just equate the two equations:
3x + 6 = - 4x + 10
7x = 4, x = 4/7 = 0.571 and then substituting the value of
x in either equation to find y:
y = 3(4/7) + 6
= 12/7 + 6
= 54/7 = 7.714
Exercises
1- Find the point of intersection of the following two linear equations:
y = 3x + 4 , y = -4x + 25
2- Find the point at which the following lines intersect:
y = 3x-14
y = 22x + 5
3- Find the point at which these pairs of lines intersect
Y= 3 x – 14 ,
Y= 22 x + 5
4x + 3 y = 11 ,
2x + y = 5
Y = -4x – 9
,
Y = 4x – 7
Q = -4 + 6 p ,
Q = 11 -4 p
Supply and Demand
•
•
•
The demand curve shows the relationship
between price and quantity demanded,
other things held equal.
The supply curve shows the relationship
between price and quantity supplied,
other things held equal.
It is convention to draw these curves with
the quantity demanded/quantity supplied
on the horizontal axis and price on the
vertical axis.
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Example
Consider the supply and demand for chocolates at different prices.
Price (£’s)
Demand
Supply
0.0
200
0
0.1
160
0
0.2
120
40
0.3
80
80
0.4
40
120
0.5
0
160
Given two sets of co-ordinates on the demand curve we can find the
equation of the line.
e.g. (160, 0.1) and (40,0.4)
The equation of a straight line takes the form:
y = a + bx
𝒑𝟐 − 𝒑𝟏 𝟎. 𝟒 − 𝟎. 𝟏
𝟎. 𝟑
:𝒎 =
=
= −
= −𝟎. 𝟎𝟎𝟐𝟓
𝒒𝟐 − 𝒒𝟏 𝟒𝟎 − 𝟏𝟔𝟎
𝟏𝟐𝟎
𝒚 = 𝒎𝒙 + 𝒃
𝟎. 𝟒 = −𝟎. 𝟎𝟎𝟐𝟓 × 𝟒𝟎 + 𝒃
𝟎. 𝟒 = −𝟎. 𝟏 + 𝒃
𝒃 = 𝟎. 𝟓
As expected the slope term, b, is negative to reflect the downward
sloping nature of the demand curve.
Hence:
y = 0.5 – 0.0025x → P = 0.5 – 0.0025Qd
Now taking two co-ordinates on the supply curve:
(40,0.2) and (160,0.5)
𝒑𝟐 − 𝒑𝟏 𝟎. 𝟓 − 𝟎. 𝟐
𝟎. 𝟑
𝒎=
=
=
= 𝟎. 𝟎𝟎𝟐𝟓
𝒒𝟐 − 𝒒𝟏 𝟏𝟔𝟎 − 𝟒𝟎
𝟏𝟐𝟎
𝒚 = 𝒎𝒙 + 𝒃
𝟎. 𝟓 = 𝟎. 𝟎𝟎𝟐𝟓 × 𝟏𝟔𝟎 + 𝒃
𝟎. 𝟓 = 𝟎. 𝟒 + 𝒃
𝒃 = 𝟎. 𝟏
As expected the slope term, b, is positive to reflect the upward sloping nature of
the supply curve Hence:
y = 0.1 + 0.0025x → P = 0.1 + 0.0025Qs
In equilibrium Qd = Qs:
0.5 – 0.0025Qd = 0.1 + 0.0025Qs
0.4 = 0.005Q* → Q* = 80
Substituting in either the demand or the supply curve: P
= 0.1 + 0.0025Qs
= 0.1 + 0.0025(80) = 0.1 + 0.2 = 0.3
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Exercises:
1- Given the co-ordinates (10,21) and (4,3) find the equation of the line that goes
through these two co-ordinates.
a) y = -9 + 3x
b) y = 3 – 9x
c) y = 1.8 + 3x D) y = 17.67 + 3x
2. Given the co-ordinates (10,-10) and (-10,30) find the equation of the line that goes
through these two co-ordinates.
a) y = -10 + 2x b) y = 10 – 2x c) y = -5 + 0.5x
d) y = 20 – x
3- At what co-ordinates do the following two lines intersect?
7x + 2y = 33 and 4x – 9y = - 42
a) x= 3, y = 6
b) x = -3, y=6 c) x = 6, y = 3
d) x = -3, y = - 6
4. At what co-ordinates do the following two lines intersect?
6x - 8y = 10 and 5x + 3y = 47
a) x= -4, y = -7 b) x = -7, y=-4
c) x = 7, y = 4
d) x = 4, y = 7
5. If a consumer has an income of £320 to spend on two goods X and Y whose
prices are £40 and £10 each respectively what is the slope of the budget
constraint?
a) 0.25
b) 4
c) – 0.25
d) - 4
6. The demand schedule for a good is as follows:
Price Quantity Demanded
2
18
4
14
6
10
8
6
Expressed algebraically the demand schedule is:
a) QD = 22 – P
b) P = 11 + 0.5QD
c) P = 11 – 0.5QD d) P = 22 + QD
7- The supply schedule for a good is as follows:
Price
Quantity Supplied
2
15
4
13
6
11
8
9
Expressed algebraically the demand schedule is:
a) Qs = 17 – P
b) P = 17 + QS c) P = 35 – 2P
d) P = 17 + P
8- Consider the following demand and supply schedules:
Supply: P = 0.4QS + 3 , Demand: P = -0.6QD + 33
What is the equilibrium level of quantity?
a) Q = 10
b) Q = 20
c) Q = 30
d) Q = 40
9- Consider the following demand and supply schedules:
Supply: 4P – QS = 8
, Demand: 4P + 3QD = 88 What is
the equilibrium level of quantity?
a) Q = 10
b) Q = 20
c) Q = 30
d) Q = 40
10Consider the following demand and supply
schedules:
Supply: P = 0.375QS + 1 , Demand: P = 0.5QD + 22 What is the equilibrium price?
a) P = 5
b) P = 10
c) P = 15
d) P = 20
11- If quantity supplied is 10 units when price is $3 per unit, and quantity supplied
is 15 units when price is $4 per unit find the equation of the supply curve.
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Effects of a per unit tax
Information about quantity supplied is based on the prices that suppliers receive.
If a per unit tax is imposed, buyers still pay a price of P for each unit, but suppliers
receive a price of only P-t. Therefore a tax changes the supply equation and
causes the supply curve to shift.
Example
No tax:
Qs = -3 + 4P
Tax:
Qs = -3 + 4 (P – t )
If the value of the tax is t- £2, then
Tax:
Qs = -3+4(P-2)
Qs = -3+4P-8
Qs = -11+4P
Example:
Given the supply and demand curves below:
(a) Find the values of price, P, and quantity, Q, at the
market equilibrium.
(b) Calculate the effect on supply of a per unit tax of t £4.5.
Demand:
Q_d =120-8P
Supply:
Q_s = -6 + 4P
Tax: t=£4.5
Solution:
(a) Find the values of price and quantity at the market equilibrium
Find the market equilibrium by equating the equations
120-8P = -6 + 4P
126=12P
P=10.5
Q = -6 + 4(10.5)
Q = 36
Calculate the effect on supply of a per unit tax of t = £4.5
Demand:
Qd= 120 - 8P , Supply: Qs= -6 + 4(P-t)
Tax: t = £A.5
Supply : Qs= -6+4P-4t
Qs= -6+4P-4(4.5)
Qs = -6 + 4P -18
Qs= -24 + 4P
The new equilibrium can now be calculated as:
-24 + 4P=120-8P
12P=144
P=12
• Note that although the tax is equal to £4.5 per unit, price has risen by only
£1.50(£12 - £10.5) per unit.
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Cost, volume and profit analysis
There are two basic types of costs a company incurs. Variable Costs, Fixed Costs
Variable costs are costs that change with changes in production levels or sales. Examples include: Costs of
materials used in the production of the goods.
Fixed costs remain roughly the same regardless of sales/output levels. Examples: Rent, Insurance and Wages
TOTAL COSTS
Consider the following definitions:
• Profit (π) = Total Revenue (TR) – Total Cost (TC)
• TR = Price (P) x Quantity (Q)
• Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC)
• Average Variable Cost (AVC) = Variable Cost (VC)/Quantity (Q)
• Variable Cost (VC)= Average Variable Cost (AVC) x Quantity (Q)
• Unit Contribution (Unit cont) = Selling Price – Variable Costs
Break-Even Analysis
The Break-even point occurs when Total Costs Equals Revenue (Sales Income)
Revenues (Sales Income) = Total Costs
OR
fixed cost ÷ Unit Contribution
At this point the business is not making a Profit nor incurring a Loss – it is merely covering its Total Costs
Example
A firm has FC= £555, AVC = £12 and is selling a good at a price of £17 per unit.
• Find an expression for profit in terms of level of sales, Q.
• What is the value of Q that will achieve a profit of £195?
• At what sales level does the firm break even?
a) Find an expression for profit in terms of Q.
o π = TR - TC = P.Q – (FC + VC)
o π = P.Q – (FC + AVC x Q)
o π = 17Q – (555 + 12xQ)
o π = 5Q – 555
(b) What is the value of Q that will achieve a profit of £195?
π = 5Q – 555 = 195 → Q = (195+555)/5 = 150
(c) At what sales level does the firm break even?
π = 5Q – 555 = 0 → Q = (0+555)/5 = 111
Example2 : Bannerman Trading Company opens a flower shop.
Fixed Costs (FC) : Rent: £400, Helper (Wages): £200
Variable Costs per unit (VC) : Flowers: £0.50 per bunch
Selling Price per unit(SP): Flowers: £2 per bunch
Answer:
Unit Contribution= SP – VC
•
•
£2.00 - £0.50 = £1.50
For every bunch of flowers sold £1.50 can go towards covering Fixed Costs
Break Even Point = Fixed Costs ÷ Unit Contribution
£600 ÷ £1.50 = 400 Units
Therefore 400 bunches of flowers must be sold to Break Even – at this the point the business is not making
a Profit nor incurring a Loss – it is merely covering its Total Costs
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Break Even can also be used to calculate Profit (or Loss) at a given level of output
For example:
J Bannerman sells Golf Clubs. How much profit/loss is made when 5000 golf clubs are sold?
• Each Golf Club is sold for £20
• Variable Costs per golf club are £10
• Fixed Costs total £24,000
Answer:
Firstly, calculate Unit Contribution
SP – VC = Unit
£20.00 - £10.00 = £10.00
Now calculate Total Contribution when 5,000 golf clubs are sold
Total Contribution = Unit Contribution x no of units
£10.00 x 5,000 = £50,000
Now calculate Net Profit at 5,000 units
Net Profit =Total Contribution – Fixed Costs
=£50,000 - £24,000 = £26,000
Example:
Calculate how many units need to be produced in order to achieve a Net Profit of £25,000 given the following
information
• Fixed Costs £30,000
• Contribution per unit £10
Answer:
Net Profit = Total Contribution – Fixed Cost
£25,000 = Total Contribution - £30,000
therefore, Total Contribution = £55,000
If unit contribution is £10
then 5,500 units will have to be produced in order to achieve a Total Contribution of £55,000.
Therefore, the number of units required to achieve a Net Profit of £25,000 is 5,500 units
Margin of Safety
•
•
•
The difference between budgeted or actual sales and the breakeven point
The margin of safety may be expressed in units or revenue terms
Shows the amount by which sales can drop before a loss will be incurred
Example
Using the following data, calculate the breakeven point and margin of safety in units:
• Selling Price = €50
• Variable Cost = €40
• Fixed Cost = €70,000
• Budgeted Sales = 7,500 units
Solution
• Contribution = €50 - €40 = €10 per unit
• Breakeven point = €70,000/€10 = 7,000 units
• Margin of safety = 7500 – 7000 = 500 units
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Target Profits
• What if a firm doesn’t just want to breakeven – it requires a target profit
• Contribution per unit will need to cover profit as well as fixed costs
• Required profit is treated as an addition to Fixed Costs
Example:
Using the following data, calculate the level of
• sales required to generate a profit of €10,000:
• Selling Price = €35
• Variable Cost = €20
• Fixed Costs = €50,000
Answer:
• Contribution = €35 – €20 = €15
• Level of sales required to generate profit of €10,000:
𝟓𝟎𝟎𝟎𝟎 + 𝟏𝟎𝟎𝟎𝟎
= 𝟒𝟎𝟎𝟎 𝒖𝒏𝒊𝒕𝒔
𝟏𝟓
Exercises
1- At breakeven point there is
A. Profit
B. Loss
C. No profit or loss
D. None of these
2- At breakeven point
A.
Total expenses = Total revenue
C. Total expenses > Total revenue
B. Total expenses < Total revenue
D. Any of the above
3-The breakeven point is obtained at intersection of
A. Total revenue and Total cost line
C. Variable cost and fixed cost line
B. Total cost and variable cost line
D. Fixed cost and total cost line
4- Margin of safety is equal to
A.
Actual sales – Sales at Breakeven point
C. Actual sales + Sales at Breakeven point
B. Actual sales x Sales at Breakeven point
D. Actual sales / Sales at Breakeven point
5- Contribution per unit is equal to
A. Selling price per unit – variable cost per unit
C. Selling cost per unit + variable cost per unit
B. Selling cost per unit x variable cost per unit
D. Selling cost per unit / variable cost per unit
6-An industry is selling a product for $ 10 per unit. The fixed cost for assets is $ 40000 with variable cost of $6 per unit.
How many units should be produced to break even?
A. 8,000
B. 10,000
C. 12,000
D. 14,000
7- Fixed cost is divided by break-even revenues to calculate
A. cost margin B. fixed margin
C. revenue margin
D. contribution margin
8- If break-even number of units are 120 units and fixed cost is $62000, then contribution margin per unit will be
A. $74,400
B. $7,440,000
C. $516.67
D. $51,667
9- If break-even number of units are 200 units and fixed cost is $80000, then contribution margin per unit will be
A. $400
B. $600
C. $800
D. $1,000
10- If contribution margin per unit is $700 per unit and break-even per unit is $40, then fixed cost would be
A. $35,000
B. $28,000
C.$17,500
D. $82,000
11- Janet sells a product for $6.25. The variable costs are $3.75. Janet's break-even units are 35,000. What is the amount
of fixed costs?
a. $ 87,500
B. $ 35,000
C. $131,250
D. $104,750
12- Company A's fixed costs were $42,000, its variable costs were $24,000, and its sales were $80,000 for the sale of
8,000 units. The company's break-even point in units is:
a. 8,000
b. 5,000
c. 6,000
d. 7,000
13- A firm sells widgets for $14 each. The variable costs for each unit is $8. The contribution margin per unit is:
a.
$6
b. $12
c. $14
d. $ 8