Lecture Note #3 Chapter 2-(3)

Business

Mathematics

1

Elasticity

Elasticity is the ratio of the % change in one variable(e.g. Demand or Supply) to a % change in other variable(e.g. Price or Income)

Elasticity: ε= % 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐴

% 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐵=

Δ𝐴

A100

∆𝐵

𝐵100

= ∆𝐴

∆𝐵

𝐵

𝐴

(Elasticity is denoted by the symbol ε.)

Types of Elasticity

Price Elasticity of Demand Point Price Elasticity of Demand Arc(Interval) Price Elasticity of Demand

We can define the similar elasticities for Supply as

is done for Demand.

Price Elasticity of Supply Point Price Elasticity of Supply Arc(Interval) Price Elasticity of Supply

Types of Elasticity

Income Elasticiy of Demand

can also be defined and used.

Price Elasticity of Demand

The concept of Elasticity means responsiveness or sensitivity.

For example, Price Elasticity of Demand measures the responsiveness (or sensitivity) of quantity demanded to changes in price of the product (at various price levels).

Price Elasticity of Demand

• Price Elasticity of Demand(𝜀𝑑)

• 𝜀𝑑 = % 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑

% 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒

=

Δ𝑄

Q100

∆𝑃

𝑃100

= ∆𝑄

∆𝑃

𝑃

𝑄

= 1

∆𝑃

∆𝑄

𝑃

𝑄 =

1

−𝑏

𝑃

𝑄 (when P=a-b×Q)

Price Elasticity of Demand

Suppose the price of a car now is $9,000. Will a 5% increase in price result in a 5% decrease in quantity demanded? Or larger(or smaller) than a 5% decrease in demand?

When the price of a car is $19,200, will a 5% increase in the car price result in a smaller or larger percentage decrease in demand?

Answers to these questions are very valuable information to the automobile manufacturing companies!! Because TR depends on both price(P) and quantity demanded(Q).

Price Elasticity of Demand

• Will the increase in TR generated by the increase in price(P) be offset by the decrease in TR due to decrease in quantity sold(Q)? (Note that TR = P*Q)

• When P increases, it is definite that Q decreases, by law of demand. i.e. Whenever P ↑ Q ↓

• Therefore, the Price Elasticity of Demand is normally negative.

• Now, when P increases, will TR = ( P × Q ) ↑ or ↓ or not change?

Price Elasticity of Demand

Depending on the magnitude(absolute value) of the elasticity(𝜀), we classify into 3 categories.

• when |𝜀| > 1, demand is elastic

• when |𝜀| = 1, demand is unit elastic

• when |𝜀| < 1, demand is inelastic

Price Elasticity of Demand

Point Elasticity and Arc Elasticity

• Point Elasticity is the elasticity at a point on demand curve (or demand function).

• Arc Elasticity is the elasticity over an interval on the demand curve. This is called ether Arc Price Elasticity of Demand or Midpoint Price Elasticity of Demand.

Arc Price Elasticity of Demand

The formula for Arc Price Elasticity of Demand

• 𝜀𝑑 = ∆𝑄

∆𝑃

1

2(𝑃1+𝑃2)

1

2(𝑄1+𝑄2)

= ∆𝑄

∆𝑃

(𝑃1+𝑃2)

(𝑄1+𝑄2)

where (𝑃1, 𝑄1) and (𝑃2, 𝑄2) are two end points of the given interval(or arc) on the demand curve.

Price Elasticity of Demand

Example 2.19

Given demand function as P=2400-0.5Q

(1) Determine the Point Price Elasticity of Demand(𝜀𝑑)at

(i) P=1800, (ii) P=1200, (iii) P=600

(2) If P increases by 12%, calculate % change in Q at

(i) P=1800, (ii) P=1200, (iii) P=600

Price Elasticity of Demand (1) (i) Determine 𝜺𝒅 𝐚𝐭 P=1800

• To calculate Q at P=1800

Substitute P=1800 into the demand function

P=2400-0.5Q 1800=2400-0.5Q ∴ Q=1200

• Now, 𝜀𝑑 at P=1800 can be computed by the formula

𝜀𝑑 = ∆𝑄

∆𝑃

𝑃

𝑄 =

1∆𝑃

∆𝑄

𝑃

𝑄 =

1

−0.5

1800

1200 = -3

(ii) Determine 𝜺𝒅 𝐚𝐭 P=1200

• When P=1200, 1200=2400-0.5Q Q=2400

• 𝜀𝑑 at P=1800 is 𝜀𝑑 = ∆𝑄

∆𝑃

𝑃

𝑄 =

1∆𝑃

∆𝑄

𝑃

𝑄 =

1

−0.5

1200

2400 = -1

Price Elasticity of Demand

(ii) Determine 𝜺𝒅 𝐚𝐭 P=600

• When P=600, 600=2400-0.5Q Q=3600

• 𝜀𝑑 at P=600 is 𝜀𝑑 = ∆𝑄

∆𝑃

𝑃

𝑄 =

1∆𝑃

∆𝑄

𝑃

𝑄 =

1

−0.5

600

3600 = -

1

3 ≈ −0.33

Summary Table P Q 𝜀𝑑 |𝜀𝑑| Demand is

1800 1200 -3 3>1 Elastic

1200 2400 -1 1=1 Unit elastic

600 3600 -0.33 0.33<1 Inelastic

Price Elasticity of Demand

• Example 2.19 (continued)

(2) 𝜀𝑑 = %∆𝑸

%∆𝑷 %∆𝑸 = 𝜀𝑑 %∆𝑷

(i) At P=1800 𝜺𝒅=-3, which indicates that 1% increase (decrease) in P will cause a 3% decrease (increase) in Q. Therefore, 12% increase in P will cause 36% decrease in Q i.e. %∆𝑸 = 𝜀𝑑 %∆𝑷 = -3× 𝟏𝟐% = −𝟑𝟔%

(ii) At P=1200 𝜺𝒅=-1. 12% increase in P will cause a 12% decrease in Q.

(iii) At P=600 𝜺𝒅=- 𝟏

𝟑. 12% increase in P will cause

a 4% decrease in Q.

Price Elasticity of Demand

𝑷𝟎 𝑸𝟎 𝑷𝒏 𝜺𝒅 𝑸𝒏 𝑻𝑹𝟎 𝑻𝑹𝒏 ∆𝑻𝑹

𝑄0=4800-2𝑃0 𝑃𝑛=𝑃0 × 1.12 𝜺𝒅 =

- 1

0.5∙

𝑃0

𝑄0

𝑄𝑛=4800-2𝑃𝑛 or

𝑄𝑛=𝑄0(1+𝜺𝒅 ×0.12)

𝑇𝑅0 =𝑃0 × 𝑄0

𝑇𝑅𝑛 =𝑃𝑛 × 𝑄𝑛

𝑇𝑅𝑛

− 𝑇𝑅0

1800 1200 2016 -3 768 2,160,000 1,548,288 -611,712

1200 2400 1344 -1 2112 2,880,000 2,838,528 -41,472

600 3600 672 - 1

3 3456 2,160,000 2,322,432 162,432

Demand function: P=2400-0.5Q Q= 4800-2P 𝑃0 = initial price 𝑃𝑛 = new price after increasing by 12% 𝑄0 = initial demand quantity 𝑄𝑛 = new demand quantity at new price 𝑇𝑅0 = initial TR 𝑇𝑅𝑛 = new TR

Price Elasticity of Supply Price Elasticity of Supply(𝜀𝑠)

When supply function is given as P=c+d×Q ,

where P= price, Q=supply quantity

𝜀𝑠 = % 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑

% 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑝𝑟𝑖𝑐𝑒

= %∆𝑄

%∆𝑃=

Δ𝑄

Q100

∆𝑃

𝑃100

= ∆𝑄

∆𝑃

𝑃

𝑄

= 1

∆𝑃

∆𝑄

𝑃

𝑄 =

1

𝑑

𝑃

𝑄

Price Elasticity of Supply

• By the law of supply, supply function has positive slope. i.e. d>0 in the supply function of P=c + dQ.

• Hence, 𝜀𝑠= 1

𝑑

𝑃

𝑄 is always positive.

• Supply is said to be elastic, when 𝜀𝑠>1

unit elastic, when 𝜀𝑠=1

inelastic, when 𝜀𝑠<1