Lecture 2. Applying Calculus in Business and Economics II.pdf

Applying Calculus in Business and Economics

1. Applying Calculus in
Business and Economics II
Lecture 2
1

2. •The slope of a straight line is constant while the slopes of a curve
are variable.
•The slope of a curve is defined by a derived function. The slope of a
curve at point ??????=??????
0: ??????′(??????
0)
•A function reaches its turning points where its first–order derivative
is equal to zero.
•The second–order derivative indicates whether a turning point is a
minimum or a maximum.
2
Review Lecture 1

3. 1.Understand and be able to use marginal analysis in economics.
2.Understand and be able to analyze the price elasticity of demand.
3
LearningObjectives

4. Marginal
Analysis

5. •Marginal analysis is the study of the rate of changeof economic
quantities.
•Marginal revenue is the addition to total revenue (TR) from selling
the last unit of output.
•Likewise, marginal cost and marginal profit are the addition to total
cost (TC) and total profit (TP) from producing and selling the last
unit.
Marginal analysis
5

6. The total revenue (TR) function
P = 100 –2Q →TR = P.Q = ?
How can we measure the marginal revenue?)
Marginal revenue
6

7. TR = f(x) = 100Q -2Q
2
, where Q = output
Q (output)
Total Revenue (TR)
Marginal revenue
1 unit approach:
Figure 2.1
7
TR
0
TR
1
Total
revenue if
Q
0 product
is sold
Total
revenue if
Q
0 product
is sold
∆TR
∆Q
Q
0
Q
0 + 1
Total revenue if
Q
0 + 1 product is
sold
Total revenue if
Q
0 + 1 product is
sold

8. TR = f(x) = 100Q -2Q
2
, where Q = output
Q (output)
Total Revenue (TR)
Marginal revenue
Figure 2.2
8
TR
0
TR
1
∆TR
∆Q
Q
0 Q
0 + 1
A
B
Marginal revenue at a
point is equal to the
slope of the tangent
(the derivative of
total revenue with
respect to demand)
at this point.
Marginal revenue at a
point is equal to the
slope of the tangent
(the derivative of
total revenue with
respect to demand)
at this point.()
()
dTR
MRTRQ
dQ
==
The tangent at
point A

9. Given the demand function:
P = 100 –2Q
a.Find the expression for TR in terms of Q
b.Find the value of MR at Q = 20
c.Calculate ∆TR when quantity increases from 20 to 21?
Compare this value with the answer in part b
d.Plot the TR and MR against quantity in a graph
Example
9

10. Example (cont.)
10
Given the demand function
P = 100 –2Q
a.Find the expression for TR in terms of Q
→TR = P.Q = (100 –2Q).Q = 100Q –2Q
2
b. Find the value of MR at Q = 20
c. Calculate ∆TR when quantity increases from 20 to 21?
Compare this value with the answer in part b
d. Plot the TR and MR against quantity in a graph

11. Marginal Revenue and Total Revenue
TR
MR
Total revenue
increases at a
diminishing
rate
Total revenue
increases at a
diminishing
rate
11

12. A firm, or a group of firms, is the only supplier for a particular
product
For example, EVN Vietnam is the only supplier of electricity
✓Monopolies: only one firm controls the market
✓Duopolies: two firms group together and determine the market
price
✓Oligopolies: a collection of firms form a cartel and dominate the
market
Marginal Revenue in Imperfect Competition Markets
12

13. •The only supplier (monopolist) has control over price
•Downward sloping demand curve →price goes up, demand falls
Marginal Revenue in Imperfect Competition Markets
13
The simple model of demand, given the linear demand function:( )0,0PaQb ab=+ 
The total revenue function:( )
2
TRaQbQaQbQ=+=+

14. 14
•The total revenue function is quadratic and
its graph has a parabolic shape in Fig.(a).
Fig. (a)
Fig. (b)
•The marginal revenue curve slopes downhill
exactly twice as fast as the demand curve in
Fig.(b).
•When marginal revenue takes both positive
and negative values, the total revenue curve
is expected to slope both upward and
downward.
•At the maximum point of the TR curve, the
tangent is flat with zero slope (MR=0).
•Demand function
TR is maximized when MR = 0
flat tangent

15. •A large number of suppliers sell
an identical product
•Individual firms are price takers
•The marginal and average
curves are the same
Marginal Revenue in Perfect Competition Markets
AR
MR
Q
b
•Demand function: P = b
•Total revenue: TR = PQ = bQ
•Marginal revenue: MR = b
15

16. Price
elasticity
of Demand

17. Price elasticity of demand
17
Suppose that a demand curve
is downward-sloping
If the firm lowers
the price
Q
P
P1
P2
Q1 Q2
The firm will
receive less for
each item
The number of items
sold increases& ???
TRPQ
PQTR
=


18. Price elasticity of demand
18 The percentage change in price: 100
P
P


•
Q
P
P
1
P
2
Q
1 Q
2the percentage change in demand
the percentage change in price
PED=
The price elasticity of demand (PED) is a measure
of the responsiveness of demand for a product to a
change in its price.100
100
Q
PQQ
P QP
P
PED

= =



 
What quantity
percentage changes
when the price
changes by 1%? The percentage change in quantity/demand: 100
Q
Q


•
∆P
∆Q

19. Price elasticity of demand
•Price elasticity of demand is always negative
•If the percentage change in Q > the percentage change in P, then
|PED|>1. Demand is elastic.
•If the percentage change in Q < the percentage change in P, then
|PED|<1. Demand is inelastic.
•If the percentage change in Q = the percentage change in P, then
|PED|=1. Demand is unit elastic.
19PQ
QP
PED=



20. •The arc elasticity of demand (between two specific points)
•The price elasticity of demand (point elasticity)
20
Price elasticity of demand

21. The arc elasticity of demand (between two specific points)
21PQ
QP
PED=


Q
P
P
1
P
2
Q
1 Q
2( )
( )
12
12
21
21
1

2
1

2


PPP
QQQ
QQQ
PPP
•=+
•=+
•=−
•=−

22. Example: Given the demand function
P = 200 –Q
2
Calculate the arc elasticity as P falls from 136 to 119.
The arc elasticity of demand (between two specific points)
22

23. Example: (con’t)
•At P
A= 136, Q
A= ?
•At P
B= 119, Q
B= ?
•For the arc from A to B;
•∆Q = Q
B–Q
A=?
•∆P = P
B–P
A=?
•Averaging the P values: P = (P
A+ P
B)/2
•Averaging the Q values: Q = (Q
A+ Q
B)/2
•PED = ?
The arc elasticity of demand (between two specific points)
23

24. When there is a small change in price (∆P→0), then
Price elasticity of demand (point elasticity)
24PQ
QP
PED=

 P
P
P
ED
PQPdQ
Q Qd
= =


 : thederivedfunctionofQwithrespecttoP
dQ
dP
• 1dQ
dPdP
dQ
•=

25. Example 1: The quantity demanded (Q) of a product and its
own price (P) are related in the following way:
Calculate this price elasticity at a price of $5.
Price elasticity of demand (point elasticity)
25

26. •P = $5 →Q = 200 –P
2
= 175
•Differentiate Q with respect to P to get
Price elasticity of demand (point elasticity)
26

27. Example 2:Given the demand function
P = –Q
2
–4Q + 96
(a) Find the price elasticity of demand when the price is $51.
(b) If this price rises by 2%, calculate the corresponding
percentage change in demand.
Price elasticity of demand (point elasticity)
27PdQ
QdP
PED= 1dQ
dPdP
dQ
=

28. Elasticity
28
❖Elasticity is independent of the
slope of the demand curve()
11
a
PdQP P P
PED
PbQdP aQ Pb
a
== =
− −
= ( )0,0PaQbab=+ Pb
Q
a
−
=
Consider the standard linear downward-sloping demand function:()
P
PED
Pb
=
−

29. Elasticity
29
|PED∣ = ∞|PED∣ = ∞
|PED∣ = 1|PED∣ = 1
|PED∣ = 0|PED∣ = 0
|PED∣ > 1|PED∣ > 1
|PED∣ < 1|PED∣ < 1
❖Elasticity varies along the linear demand curve
A( )If 0,0PaQbab=+ ()
P
PED
Pb
=
− 0
If 0 then 0
(0)
P PED
b
•= = =
− 1
1
() 2
DemandisunitelasticwhenD
Pb
PED P
P
PE
b
===
−
• =− If then
()
b
PbPED
bb
•= = =
−

30. 30
Marginal revenue and Price elasticity of demand ( )0,0PaQbab=+
()
P
PED
Pb
•=
− ( )
2
TRPQaQbQaQbQ•==+=+ 2
dTR
MR aQb
dQ
•==+ (1)&(2)
P
MRP
PED
=− 1
1MRP
PED

=+

 1
1MRP
PED

=−

 2
Pb
ab
a
−
=+

 ( )2PbPPb=−=+− ( ) (1)MRPPb=−− )( (2)
P
Pb
PED
=−
•The relationship between marginal
revenue, price, and the elasticity of
demand:

31. Marginal revenue and Price elasticity of demand
31
Total revenue is maximized when ∣PED∣ = 1 or MR = 0
•|PED| < 1 (inelastic) →MR is
negative. The total revenue
curve is decreasing.
•|PED| = 1 (unit elastic) →MR is
0. The total revenue curve
reaches its turning point
•|PED| > 1 (elastic) →MR is
positive. The total revenue curve
is increasing.1
1MRP
PED

=−

