Applications and Linear Functions Example 1 – Production Levels

2007 Pearson Education Asia

Applications and Linear FunctionsApplications and Linear FunctionsExample 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


Demand and Supply Curves

• Demand and supply curves have the following trends:

2007 Pearson Education Asia3

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)


Demand Function

D : Q = a –b P

Q P

20 100

18 200

16 300

14 400

12 500

10 600

100200 300 400 500 600

10

12

14

16182022

Q

P


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


Property of Demand function1. Value of q and p always positif 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


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)


Supply Function

S : Q = a +b P

Q P

10 100

12 200

14 300

16 400

18 500

20 600

100200 300 400 500 600

10

12

14

16182022


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


Property of Supply Function1. Value of q and p always positif 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


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


Equilibrium

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


• D: P = 15 - Q

• S :P = 3 + 0.5Q

• A. Determine equilibrium point

• B. Graph D, S function


Exercise : Price - DemandAt 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.


Exercise : Price - DemandSuppose 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


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.

50100

8 qp

65100

7 qp


Solution:

a. By substitution,

Before tax,

and

After new tax,

and

100

50100

865

100

7

q

qq 5850100100

8p

70.5850.51)90(100

8p

90

65100

750.51

100

8

q

qq


Solution:

b.Total revenue given by

Before tax

After tax,

580010058 pqyTR

52839070.58 pqyTR


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


• TR = Pj x Q

• Tr : Total Revenue

• Pj : Selling Price

• Q : Amount of product

Profit = TR –TC

BEP TR=TC


BEP

TR

TC

FC

Q

$

0 Q bep

C bep

loss

profit


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.


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


Solution:

a. We have

At break-even point,

and

b.

The profit is $5000.

50009

22

8

qyyy

qy

FCVCTC

TR

900

50009

228

q

qq

yy TCTR

72009008 TRy

5000500018009

2218008

TCTR yy


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.

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Applications and Linear Functions Example 1 – Production Levels