Managerial Economics, Measures Of Elasticity By Tarun Das

Managerial Economics Module 2: Analysis of Market Demand and Supply

Module 2: Analysis of Market Demand and Supply Lecture 3:  Elasticity of Demand and its Determinants  Measurement of Elasticity – Point and Arc Elasticity  Other Elasticities of Demand and Its Implications  Price Elasticity and Revenue - Implication for Business  Business Application of Income and Cross Elasticities

Demand Elasticities 

Look at the demand function: QD=f(P, PY, I, Tastes, Expect., Buyers, Govt, ε ) 



Can examine responsiveness along the demand curve. 

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Aug-06

Causality goes from right to left in function.

Price elasticity: ε =%∆ Q/%∆ P

In words, this is the percent change in quantity demanded brought about by the percent change in price. S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Price elasticity: ε =%∆ Q/%∆ P  



Causality: denominator

An elastic response is one where numerator is greater than denominator. 

i.e., %∆ Q>%∆ P so ε < −1



Imagine extreme example.

An inelastic response is one where numerator is smaller than denominator. i.e., %∆ Q<%∆ P so ε > −1 

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numerator!

Again, imagine extreme example.

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Look at the Extremes 

Perfectly Elastic D ε



Perfectly Inelastic D

= −infinite P

P

D ε

=0

D

Q

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Relatively Elastic vs. Relatively Inelastic Demand Curves

P

D’ is relatively more elastic than D P1 P2 D Q1 Q2

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Q2’

D’ Q

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Determinants of ε 

Number of close substitutes 

Market vs. brand level



Nature of goods – Necessities/ Luxuries



Importance of goods in budget



Time taken to readjust habit, usage behaviour, etc.



No. of uses of a product



Price Level

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Measurement of Elasticity: Point Elasticity Formula 

Point elasticity 

Point elasticity is responsiveness at a point along the demand function; ε = ∆ Q/Q

Price responsiveness at this point

∆ P/P 

Simplifying gives:

ε 

= (∆

Q/∆ P) ∗ (P /Q)

In limiting case:

ε

= δ ε

Q/ δ P ∗ (P /Q)

P1 D

= 1/ (Slope) ∗ (P /Q)

Q1 Aug-06

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Q

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Look at our mathematical demand function 



Assume equilibrium P and Q: Q=13,750 and P=190 Demand function QDX =15000 - 25PX + 10PY+2.5*I 

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Derive demand curve by holding PY and I constant (e.g., at PY=100, and I=1000) giving: QDX =18500-25PX

Derive ε

= (δ Q/δ P)* P /Q



What is P and Q?

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What is δ Q/δ P? S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Elasticity calculation

ε ε

= (∆ Q/∆ P)* P /Q

= -25*190/13750 = -0.34 What is the interpretation?

Price Elasticity varies along Linear Demand Curve  

∆Q /∆P = 1/ (slope of dd line) = QN/ RQ and At R; P/Q = OP/OQ

Price

QN OP QN QN ε= × = = QR OQ OQ RP (Since Since RQ = OP

M

Demand Line P

R

P-∆P

∆P

R’ ∆Q

O

Q

Q+∆Q

Quantity Aug-06

Angle of slope of demand line

N

and OQ = RP) ∆MPR and ∆RQN are similar triangles;∴ Ratio of their respective sides are equal. QN RN Or, = RP RM RN Lower Segment ε= = RM Upper Segment

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For Constant Elasticity Demand Curve, ε = constant 

Price M

In case of demand curve (refer Fig 2 b), when ∆P approaches zero, point R’ will approach R and 

Demand Curve Tangent to DD curve 

R

P P-∆P

Angle of slope of R’ tangent to DD curve

In this limiting case, slope of the demand curve at R is the slope of tangent MN at point R of the demand curve ∆Q/∆P= dQ/dP =1/ (slope of dd curve) = QN/ RQ



Rest of derivation remains same, here also ε = RN/RM =

=Lower Segment/ Upper Segment 

O

Q

Q+∆Q

Quantity Aug-06

N

For Const. Elasticity dd-curve: eg. Q=aP-b, the ratio is constant & equal to –b; ε = -b

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Price Elasticity varies along Linear Demand Curve 

P



|ε | > 1 elastic |ε | = 1 PM unit |ε | < 1 elastic inelastic QM Aug-06

At low prices, %∆ Q< %∆ P so | ε | < 1



i.e., Quantity unresponsive to Price

At high prices, %∆ Q>%∆ P so |ε | > 1 

i.e., Quantity responsive to Price

Q S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Price Elasticity and Revenues 

Suppose we look at P increase along D curve.



Revenues = P*Q 



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Impact on expenditure depends on which effect is greater.

For elastic responses, | ε | %∆ P

> 1 so %∆ Q>



Thus, when P increases, Q decreases by more!



Revenues = P*Q falls



Vice versa when P decreases revenue goes up

For inelastic response, | ε | < 1 so %∆ Q< %∆ P 

Thus, when P increases, Q decreases by less!



Revenues = P*Q rises

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Vice versa when P decreases, revenue falls S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Elasticity and Revenue: When demand is elastic (| ε | >1) - Graphical Illustration 

Decrease in price leads to increase in revenue – when elastic demand 



Rs

(Illustration) the total revenue rectangle becomes larger when price is lowered as depicted in the figure

Vice versa rise in P leads to fall in R Aug-06

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Elasticity and Revenue: When demand is inelastic (| ε | <1) - Illustration 

Decrease in price will lead to decrease in revenue - when inelastic demand 



Rs.

(Illustration) decrease in price decreases the total revenue rectangle, as seen in the figure.

Vice versa rise in P leads to rise in R Aug-06

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Price Elasticity and Revenues: Mathematical Derivation 

R= P * Q



Therefore, dR/dQ = P+Q*dP/dQ ⇒ MR= P[1+ (Q/P)*dP/dQ]= P (1 + 1/ep) ep is negative therefore ep= -|ep| ⇒ MR = P (1 - 1/|ep|) (Note: P is non-negative entity)



Marginal revenue (MR) is the rate of change in revenue with change in quantity. 

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Decrease in P means increase in Q (i.e. moving left to right along the demand curve); +ve MR will result in rise in R and negative MR will result in fall in R.

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Price Elasticity and Revenues Elasticity of Marg. Effect of Fall Effect of Rise Demand Reven. in Price on R in Price on R Elastic demand | ep|> 1

Revenue rises.

Revenue falls.

Unit Elastic, MR is |ep|=1 zero

Revenue is unchanged

Revenue is unchanged

Inelastic Demand | ep|<1

Revenue falls.

Revenue rises.

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MR is +ve

MR is -ve

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Price Elasticity & Revenues  

P | ε | >1 

|ε |

PM

= 1

|ε | < 1 

QM Aug-06

Q

Look at price increase. In inelastic range (lower segment of demand line i.e. when P< PM or alternatively Q> QM), revenue rises as P increases. At unit elastic point (at midpoint, when P= PM or Q= QM , revenue is unchanged as P increases. In elastic range (upper segment of demand line i.e. when P> PM or alternatively Q< QM), revenues fall as P increases.

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Implication 1: Firms prefer to operate in elastic region of demand 

Rs.

Demand Line 

MR Aug-06

In inelastic range, firms would find their revenues increasing if they reduce output i.e. increase the price - they will do so until they are out of the inelastic range. The result is that each firm seeks to produce in the elastic range of demand

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More Implication for Businesses 



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Where in the elastic region depends upon production and cost side of story A company may sell more of output at lower prices i.e. in inelastic range of prices but only to get lower revenues. More of luxury goods (since they have ep > 1) can be sold at lower prices resulting in increase in total revenue. 

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Firms dealing in luxury goods do manipulate prices through promotional schemes and often indulge in price wars with competition S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Arc Elasticity and Its Measurement Arc elasticity is simply an average elasticity along a range of the demand curve. Price  Arc elasticity: Responsiveness along a range of demand average function; ε = P2 response ∆ Q/((Q1+ Q2)/2) 

∆ P/ ((P1+ P2)/2) 

Simplifying:

∆Q P1 + P2 ε= × ∆P Q1 + Q2 Aug-06

}

P1

D

Q2 Q1

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Q

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Comments 

Don’t forget the economics behind your calculations 

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i.e., Don’t lose the forest for the trees!

Know how to calculate these, and how to manipulate them.

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Look at an Example 

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Suppose the price elasticity of demand is ε = -3.6, and you expect a 5% price increase next year. What should happen to the quantity demanded? Answer: ε = %∆ Q/∆ %P −3.6 = %∆ Q/(+5) Solving for %∆ Q=5*(-3.6)=-18% S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Constant Elasticity Demand Function: QX=α PXβ 1 ∗ Ι β 2 ∗ PYβ 3

P



One of the special type of functions.



QX=α PXβ 1 ∗ Ι



δ QX/δ PX = α β =

⇒

D 

Q

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β 2

∗ PY β 1

PX β

β 1 (α PXβ 1 ∗ Ι

δ

3 1 −1

∗ Ι

β 2

β 2

∗ PY β

3

∗ PYβ 3)/PX

QX/δ PX = β 1 (QX/ PX)

⇒ β

1

⇒ β

1

= (δ QX/δ PX) (PX/QX) =ε

Therefore β

P 1

is price

elasticity of demand.

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Constant Elasticity Demand Function: QX=α PXβ 1 ∗ Ι β 2 ∗ PYβ 3 (Contd.) 

Similarly β 2 and β 3 are interpreted respectively as income and cross elasticities.

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These don’t vary as you move along the demand curve

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How do revenues vary along the curve? (We investigated this in slide nos. 17-19.)

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Other Elasticity Concepts 

Value of ε



Concept can be easily generalized. 



determines type of good.

Indeed formula looks very similar

Our focus is on Demand and Supply elasticities, although these also exist for costs. 

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I

Also, you will need to interpret and calculate the elasticity.

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Income Elasticity of Demand 

Recall demand function is:

Q=f (P, I, Prelated , Tastes, #Buyers, Expectations, Govt. Influence, χ ) 

Change in I causes shift in demand.



Size of shift depends on income elasticity.

ε 

I

= %∆ Q/%∆ I

Can derive using point or arc formula.

By Point Formula: ε By Arc Formula:

Aug-06

I

∂Q I = × ∂I Q

(=

1 slope of Engel curve

I × ) Q

∆Q I εI = × ∆I Q

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Values for Income Elasticity (ε I) 

Sign indicates normal or inferior ε I >0 implies normal good.

ε 



I

Normal goods may be necessity or luxury. Can look at the Engel's curve (Income Vs Quantity Demanded Plot). 

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<0 implies inferior good.

For normal goods, slope of Engel’s curve is positive For inferior goods, slope of Engel’s curve is negative S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Engel's Curves: Normal Goods (Positively Sloped Engel’s Curves) 

Necessity: Income inelastic I

0< ε I< 1



Luxury: Income elastic I

ε

Q

I

> 1

Q

NOTE: These are not demand Aug-06

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Engel's Curve: Inferior Good (Negatively Sloped Engel’s Curves) 

Inferior Goods have negative income elasticities: ε I < 0

I

Q 

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Goods may be both normal (low income levels) and become inferior (high income levels).

I

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ε I<0

ε I>0 Q 31

Uses of Income Elasticity in Business Decisions 

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

While price and cross elasticities are of greater significance in pricing and revenue maximizing decisions in the short run, income elasticity assumes significance in long term demand forecasting, production planning and management, as businesses try to adjust to different phases of a business/ economic cycle. In forecasting demand only the relevant concept of income and data should be used. The concept of income elasticity is also used  

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to define normal and inferior goods and classify normal goods into the category of necessities, comforts and luxuries. S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Cross Price Elasticity (ε

XY

)

QX = f (PX , I, PY, Tastes, #Buyers, Expectations, Govt. Influence, χ ) 

Change in PY causes shift in demand for X.



Size of shift depends on cross-price elasticity.

ε 



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= %∆ QX / %∆ PY

XY

Can be measured using point elasticity formula or arc elasticity formula Sign indicates relationship between goods ε

XY

> 0 implies goods are substitutes.

ε

XY

< 0 implies goods are complements. S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Magnitude shows size of shift in Demand (assume Psubst increases) 

ε

XY

>1



ε

<1

XY

PX PX

D’

D’ D

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D QX

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Application of Cross Elasticity in Business Decisions 



Used to identify/ distinguish between substitute and complimentary goods Magnitude of cross elasticity is also important in pricing decisions. 

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If a commodity is elastic to the price changes of substitute (cross elasticity > 1), it is advisable to decrease the price rather than increasing the price. In case of complimentary goods, it is better to reduce the price to maintain the demand if price of complimentary good rises. S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Application of Cross Elasticity in Business Decisions (contd.) 

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The firms can forecast the demand for its products and can be prepared to take necessary measures against fluctuating prices if accurate measures of cross elasticities of substitutes and compliments are available.

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Can also look at Supply Elasticity (η ) 

Recall the supply function: QS =f (P, PINPUT, Technology, POTHER, #Sellers, Govt. Influence,µ )



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Aug-06

Can examine responsiveness along the supply curve.  Price elasticity: η =%∆ Q/%∆ P Elastic responses represent flat supply curves, inelastic responses give steep supply curves. S. Karna/PG06/IILM/Mgrl Eco/Mod2/Lec3&4

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Elastic vs. Inelastic Supply P

S(η < 1 ) S (η > 1 )

Q Aug-06

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