The document provides information on linear functions and their applications to production levels, demand and supply curves, demand and supply functions, market equilibrium, break-even analysis, and examples calculating equilibrium points, effects of taxes, total revenue, profit, loss, and break-even points using linear demand and supply functions. It includes graphs of linear demand and supply curves, their intersection at equilibrium, as well as cost and revenue curves to illustrate break-even analysis.

1. ©2007 Pearson Education Asia
Applications and Linear FunctionsApplications and Linear Functions
Example 1 – Production Levels
Suppose that a manufacturer uses 100 lb of material
to produce products A and B, which require 4 lb and
2 lb of material per unit, respectively.
Solution:
If x and y denote the number of units produced of A
and B, respectively,
Solving for y gives
0,where10024 ≥=+ yxyx
502 +−= xy

2. ©2007 Pearson Education Asia
Demand and Supply Curves
• Demand and supply curves have the following
trends:

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3
Demand Function
• Relationship between demand amount of
product and other influenced variables as
product price, promotion, appetite/taste, quality
and other variable.
• Q = f(x1,x2,x3,……xn)

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Demand Function
D : Q = a –b P
Q P
20 100
18 200
16 300
14 400
12 500
10 600
100 200 300 400 500 600
1
0
12
14
16
18
20
22
Q
P

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Linear Demand function
Q = a - b P
Q : amount of product
P : product price
b : slope ( - )
a : value of Q if P = 0 P
Q
0

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Property of Demand function
1. Value of q and p always positive or >= 0
2. Function is twosome/two together, each value
of Q have one the value of P, and each value
of P have one the value of Q.
3. Function moving down from left to the right
side monotonously

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Supply function
• Relationship between Supply amount of product
and other influenced variables as product price,
technology, promotion, quality and other
variable.
• Q = f(x1,x2,x3,……xn)

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Supply Function
S : Q = a +b P
Q P
10 100
12 200
14 300
16 400
18 500
20 600
100200 300400 500 600
1
0
12
14
16
18
20
22

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Linear Function Supply
Q = a + b P
Q : Amount of product
P : product orice
b : slope ( + )
a : value of Q if P = 0 P
Q
0

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Property of Supply Function
1. Value of q and p always positive or >= 0
2. Function is twosome/two together, each value
of Q have one the value of P, and each value
of P have one the value of Q.
3. Function moving up from the left to the right
side monotonously

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The point of market equilibrium
• Agreement between buyer and seller
directly or indrectly to make the
transaction of product with certain price
and amount of quantity.
• In mathematics the same like crossing
between demand and supply function

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Equilibrium
• The point of equilibrium is where demand and
supply curves intersect.

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• D: P = - 2 Q + 10
• S :P = 3/2 Q +3
• A. Determine equilibrium point
• B. Graph D, S function
• C. Determine the interval value of P and
Q in term of D and S function

14. ©2007 Pearson Education Asia
Exercise : Price - Demand
At the beginning of the twenty-first century, the world
demand for crude oil was about 75 million barrels per day
and the price of a barrel fluctuated between $20 and $40.
Suppose that the daily demand for crude oil is 76.1
million barrels when the price is $25.52 per barrel and
this demand drops to 74.9 million barrels when the price
rises to $33.68. Assuming a linear relationship between
the demand x and the price p, find a linear function in the
form p = ax + b that models the price – demand
relationship for crude oil. Use this model to predict the
demand if the price rises to $39.12 per barrel.

15. ©2007 Pearson Education Asia
Exercise : Price - Demand
Suppose that the daily supply for crude oil is 73.4 million
barrels when the price is $23.84 per barrel and this
supply rises to 77.4 million barrels when the price rises to
$34.2. Assuming a linear relationship between the
demand x and the price p, find a linear function in the
form p = ax + b that models the price – demand
relationship for crude oil. Use this model to predict the
supply if the price drops to $20.98 per barrel.
What’s equilibrium point and make a graph in the same
coordinate axes

16. ©2007 Pearson Education Asia
Example 1 – Tax Effect on Equilibrium
Let be the supply equation for a
manufacturer’s product, and suppose the demand
equation is .
a. If a tax of $1.50 per unit is to be imposed on the
manufacturer, how will the original equilibrium price
be affected if the demand remains the same?
b. Determine the total revenue obtained by the
manufacturer at the equilibrium point both before and
after the tax.
50
100
8
+= qp
65
100
7
+−= qp

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Solution:
a. By substitution,
Before tax,
and
After new tax,
and
100
50
100
8
65
100
7
=
+=+−
q
qq ( ) 5850100
100
8
=+=p
70.5850.51)90(
100
8
=+=p
90
65
100
7
50.51
100
8
=
+−=+
q
qq

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Solution:
b.Total revenue given by
Before tax
After tax,
( )( ) 580010058 === pqyTR
( )( ) 52839070.58 === pqyTR

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BREAK EVENT POINT
• BEP is identifying the level of operation
or level output that would result in a
zero profit. The other way thatr the firm
can’t get profit or don’t have loss
• TC= FC + VC
• TC : Total Cost
• FC : Fixed Cost
• VC : Variabel Cost
• VC = Pp x Q = cost production per unit x
• amount of product

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• TR = Pj x Q
• Tr : Total Revenue
• Pj : Selling Price
• Q : Amount of product
Profit = TR –TC
BEP  TR=TC

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BEP
T
R
T
C
F
C
Q
$
0 Q
BEP
p
C
BEP
loss
profit

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Example 2 – Break-Even Point, Profit, and Loss
A manufacturer sells a product at $8 per unit, selling
all that is produced. Fixed cost is $5000 and variable
cost per unit is 22/9 (dollars).
a. Find the total output and revenue at the break-even
point.
b. Find the profit when 1800 units are produced.
c. Find the loss when 450 units are produced.
d. Find the output required to obtain a profit of
$10,000.

23. ©2007 Pearson Education Asia
Break-Even Points
• Profit (or loss) = total revenue(TR) – total cost(TC)
• Total cost = variable cost + fixed cost
• The break-even point is where TR = TC.
FCVCTC yyy +=

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Solution:
a. We have
At break-even point,
and
b.
The profit is $5000.
5000
9
22
8
+=+=
=
qyyy
qy
FCVCTC
TR
900
5000
9
22
8
=
+=
=
q
qq
yy TCTR
( ) 72009008 ==TRy
( ) ( ) 500050001800
9
22
18008 =





+−=− TCTR yy

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BEP Exercise
• A firm produce some products where the cost per unit is
Rp 4.000,- and selling price per unit is
Rp12.000,-.Management developed that fixed cost is Rp
2.000.000,-Determine the amount of product where the
firm should sell amount of product so that the break event
point achieved.
• a. Find the total output and revenue at the break-even
point.
• b. Find the profit when 1600 units are produced.
• c. Find the loss when 350 units are produced.
• d. Find the output required to obtain a profit of
Rp 7,000.