ELASTICITY

1

ELASTICITY

Principles of Microeconomic Theory, ECO 284

John Eastwood CBA 247 523-7353 e-mail address:

[email protected]

2

Learning Objectives

Define and calculate the price elasticity of demand

Explain what determines the price elasticity of demand

Use the price elasticity to determine whether a price change will increase or decrease total revenue

3

Learning Objectives (cont.)

Define, calculate and interpret the income elasticity of demand

Define, calculate and interpret the cross-price elasticity of demand

Define and calculate the elasticity of supply

Use elasticities to analyze tax incidence.

4

Learning Objectives

Define and calculate the price elasticity of demand

Explain what determines the price elasticity of demand

Use the price elasticity to determine whether a price change will increase or decrease total revenue

5

Elasticity

Elasticity measures the response of one variable to changes in some other variable.

Civil Engineers need to know the elasticity of construction materials.

Economists need to know the elasticity of quantities demanded (and supplied).

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Elasticity of Demand

How does a firm go about determining the price at which they should sell their product in order to maximize profit?

– Profit = total revenue – total cost = TR - TC– Total Revenue = Price Quantity =PQ

How does the government determine the tax rate that will maximize tax revenue?

7

Price Elasticity of Demand, ed

ed measures the responsiveness of quantity

demanded of a product to a change in its own price, ceteris paribus.

ed = (percentage change in Qd) divided by

(the percentage change in the Px)

8

Example

Assume that the price of crude oil has increased by 100%, and that the quantity demanded has fallen by 10%

ed = -10% / 100% = -0.1

For every 1% increase in price, the quantity demanded fell by 0.1%

9

Computing Elasticity Using the “Arc Formula”

where P1 represents the first price, P2 the second price, and Q1 and Q2 are the respective quantities demanded.

Elasticity is dimensionless (units divide out).

ed

Q Q

Q QP P

P P

2 1

1 2

2 1

1 22 2( ) / ( ) /

10

Arc Formula Notation

Some people prefer to write delta for change, and “overbar” for average.

e

e

d

d

Q Q

Q QP P

P P

Q

Q

P

P

2 1

1 2

2 1

1 22 2( ) / ( ) /

11

Calculating Elasticity

The changes in price and quantity are expressed as percentages of the average price and average quantity.– Avoids having two values for the price

elasticity of demand

ed is negative; its sign is ignored

12

Calculating the Elasticity of Demand

Quantity (millions of chips per year)

Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

Originalpoint

13Quantity (millions of chips per year)

Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

Originalpoint

Newpoint



Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

= $20P

= 8Q

Originalpoint

Newpoint



Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

Originalpoint

Newpoint

Pave = $400

= $20P

= 8Q


Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

Originalpoint

Newpoint

Pave = $400

Qave = 40

= $20P

= 8Q


ed = ?


Pri

ce (d

olla

rs p

er c

hip)

36 40 44

390

400

410

Da

Originalpoint

Newpoint

Pave = $400

Qave = 40

= $20P

= 8Q


ed = 20/5 = 4

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Example -- Crude Oil

Assume P1 = $15/bbl, Q1 = 105 bbl/day, and that P2 = $25/bbl, Q2 = 95 bbl/day

Calculate ed using this formula:

ed

Q Q

Q QP P

P P

2 1

1 2

2 1

1 22 2( ) / ( ) /

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Answer:

For every 1% increase in price, Qd fell 0.2%.

e

e

d

d

( )

( ) /

( )

( ) /

.

95 105

105 95 2

25 15

15 25 2

10

100

10

2010% 50%

1

50 2

20

Elasticity and Slope

ed and slope are inversely related.

e

e

d

d

Q

Q

P

P

Q

Q

P

P

Q

P

P

Q

PQ

P

Q Slope

P

Q

1 1

21

Discussing ed

Note that ed is always negative (or zero)

because of the law of demand. However, when discussing the value of

ed , economists almost always use the

absolute value. Using | ed |, a larger value

means greater elasticity.

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Elastic Demand, | ed |>1

If the percentage change in quantity demanded is greater than the percentage change in price, demand is said to be price elastic.

The demand for luxury goods tends to be price elastic.

Examples – see page 99 of McEachern.

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Inelastic Demand, | ed |< 1

If the percentage change in quantity demanded is smaller than the percentage change in price, demand is said to be price inelastic.

The demand for necessities tends to be price inelastic.

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Perfectly Elastic D, ed = infinity

If quantity demanded drops to zero in response to any price increase, demand is said to be perfectly elastic.

This corresponds to a horizontal demand curve.

Sounds unlikely, doesn’t it? Example: Demand for a small country’s

exports

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Inelastic and Elastic Demand

6

12

Pri

ce

Quantity

D3

Elasticity =

Perfectly Elastic

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Perfectly Inelastic D, ed =0

If quantity demanded is completely unresponsive to a change in price, demand is said to be perfectly inelastic.

This corresponds to a vertical demand curve.

Can you think of a vertical demand curve?

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6

12

Pri

ce

Quantity

D1

Elasticity = 0

Perfectly Inelastic

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Unit Elastic D, | ed |= 1

If the percentage change in quantity just equals the percentage change in price, demand is said to be unit elastic.

While there are many goods that could be unit elastic, there aren’t any we can identify without statistical evidence.

Example:

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6

12

Pri

ce

Quantity

D2

1 2 3

Elasticity = 1

Unit Elasticity

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ed and Total Revenue (TR)

Note that TR = P times Q = PQ. Will a change in price raise or lower total

revenue? It all depends on the price elasticity of

demand!

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When Demand is Elastic, P and TR vary inversely. Since | ed | > 1, the percentage change in Qd

is greater than the percentage change in P. If P rises by, say, 1%, Qd will fall by more

than 1%. Therefore, if price is increased, total revenue

will decrease. If price is reduced, then TR will rise.

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When Demand is Inelastic, P and TR vary directly. Since | ed | < 1, the percentage change in Qd

is smaller than the percentage change in P. If P rises by, say, 1%, Qd will fall by less

than 1%. Therefore, if price is increased, total revenue

will increase. If price is reduced, then TR will fall.

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When Demand is Unit Elastic, TR does not change. Since | ed | = 1, the percentage change in

Qd equals the percentage change in P. If P rises by, say, 1%, Qd will fall by

exactly 1%. Therefore, if price is increased, total

revenue will stay the same. If price is reduced, TR will not change.

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Some Real-World Price Elasticities of Demand

Good or Service ElasticityElastic Demand

Metals 1.52Electrical engineering products 1.30Mechanical engineering products 1.30Furniture 1.26Motor vehicles 1.14Instrument engineering products 1.10Professional services 1.09Transportation services 1.03

Inelastic DemandGas, electricity, and water 0.92Oil 0.91Chemicals 0.89Beverages (all types) 0.78Clothing 0.64Tobacco 0.61Banking and insurance services 0.56Housing services 0.55Agricultural and fish products 0.42Books, magazines, and newspapers 0.34Food 0.12

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Example: Demand for Oil and Total Revenue

Assume demand is p = 60 - q TR = price x quantity =PQ Substituting 60-q for p gives, TR=(60-q)q Multiply through by q to get an equation for

TR, TR = 60q - q2

TR will graph as a parabola. Let’s calculate TR and graph it with D.

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Computing Total Revenue

Q 0 10 20 30 40 50 60

P 60 50 40 30 20 10 0

TR 0 500 800 900 800 500 0

Unit Analysis:Q (bbl/day)

P ($/bbl)

TR = P ($/bbl.) times Q (bbl. /day) = TR ($/day)

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Demand (P), Total Revenue (TR), and Marginal Revenue (MR)

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

0100200300400500600700800900

P=ARMRTRQuantity (bbl./day)

Pri

ce

($/b

bl.

)To

tal R

even

ue ($/d

ay)

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Total Revenue as an Area

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

P=AR

Quantity (bbl./day)

Pri

ce (

$/b

bl.

)

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Linear Demand and Point Elasticity

ed can be illustrated with geometry.

With a linear D, the slope is constant. We don’t need an arc to get the slope. Elasticity is inversely related to slope.

ed

Q

P

P

Q PQ

P

Q Slope

P

Q

1 1

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ed and Linear Demand

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

D

Quan

P

M

P

M TO

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ed Using Line Segments

The formula for ed may be rewritten in terms

of the length of line segments. O is the origin, T is the x-intercept, and M is a

point between O and T.

ed

Q

P

P

Q

MT

MP

MP

OM

MT

OM

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Elasticity at the Midpoint

| ed | =MT/OM

for any linear demand curve.

If M is the middle, then MT=OM.

ed = | -1| = 1

Unit Elastic at the midpoint.

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

D

Quan

P

M

P

M TO

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Elasticity at Higher Prices

If M is left of the middle, then MT>OM.

| ed | =MT/OM.

| ed | > 1

Demand is elastic at higher prices.

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

D

Quan

P

M

P

M TO

45

Elasticity at Lower Prices

If M is right of the middle, then MT<OM.

| ed | =MT/OM.

| ed | < 1

Demand is inelastic at lower prices.

0

10

20

30

40

50

60

0 5 10 15 20 25 30 35 40 45 50 55 60

D

Quan

P

M

P

M TO

46

Two Extremes

At the point where the demand curve

intercepts the vertical axis, ed is infinite or

perfectly elastic. At the point where the demand curve

intercepts the horizontal axis, ed = 0, that

is, demand is perfectly inelastic.

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Determinants of ed

Number of substitutes– quality– availability

Budget proportion Time

– to respond– to consume

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Other Elasticity Concepts

Income Elasticity of Demand, ey

Cross Price Elasticity of Demand, ex,z

Price Elasticity of Supply, es

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Income Elasticity of Demand, ey

ey measures the change in demand for a

good (X) in response to a change in income (Y), ceteris paribus.

If ey > 0, X is a normal good.

If ey < 0, X is an inferior good.

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Computing Income Elasticity

With Q1 and Q2, find the change in quantity and the average quantity .

Given Y1 and Y2, find the change in income and the average income.

ey

QQ

YY

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Example Computations

Median annual family income rose from $39,000 to $41,000 per year.

The demand for electricity rose from 79,000 GWh to 81,000 GWh.

Normal or inferior?

ey

QQ

YY

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Cross Price Elasticity of D, ex,z

ex,z measures the responsiveness of the demand for one good to a change in the price of another good, ceteris paribus.

ex,z = (% change in demand for X ) divided by (% change in PZ)

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Using Cross Price Elasticity

ex,z > 0 tells us the goods X and Z are substitutes.

ex,z < 0 tells us the goods X and Z are complements.

ex,z = 0 tells us the goods X and Z are unrelated.

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Computing Cross-Price Elasticity

With QX1 and QX2, find the change in quantity and the average quantity .

Given PZ1 and PZ2, find the change in price and the average price.

x z

X

X

Z

Z

e

QQ

PP

,

57


The price of gasoline rose from $.75 to $1.25/gal.

The demand for Subarus rose from 9/day to 11/day.

The demand for Cadillacs fell by 10%.

x z

X

X

Z

Z

e

QQ

PP

,

58

Subaru Example

Let X = Subarus, and Z = gasoline.

Find ex,z .

Are Subarus and gasoline related goods? If so, are they complements or substitutes?

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Cadillac Example

Let X = Cadillacs, and Z = gasoline.

Find ex,z .

Are Cadillacs and gasoline related goods? If so, are they complements or substitutes?

62

Price Elasticity of Supply, es

es measures the

responsiveness of quantity supplied to a change in the good’s price.s

s

se

Q

QP

P

63


The price of corn fell from $3/bu. to $1/bu.

The quantity supplied of corn fell from 101,000 bu to 99,000 bu.

Compute the price elasticity of supply.

s

s

se

Q

QP

P

66

es Along a Supply Curve

-10

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50 55 60

SuSiSe

Quan

Q

i

O

u

e

x

y

P

67

es Using Line Segments

Rewrite the formula for es in terms of

point elasticity. Note the relationship with the slope. Use length of line segments to get

es

seQ

P

P

Q Slope

P

Q

1

68

A Supply Curve with es = 1

Find es at point u:

Note that Su is unit elastic at any point.

seQ

P

P

Q

Q

uQ

uQ

Q

0

01

69

A Supply Curve with es <1

Find es at point i.

Si is inelastic at any point.

seQ

P

P

Q

xQ

iQ

iQ

Q

xQ

Q

0 0

1

70

Supply Curves May Not Touch the x-axis or y-axis. Si is unrealistic.

It implies that the firm would supply positive quantities of its product at a price of zero (or at a negative price)!

As we will learn later, a firm will shut down if the price of its product falls too low. Thus, we should draw supply curves that begin at a positive (Q, P).

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A Supply Curve with es >1

Find es at point e. Se is elastic at any point.

As y gets larger, esgets larger.

As P gets larger, esapproaches 1.

seQ

P

P

Q

Q

yP

P

Q

P

yP

0 0

0

01

72

Perfectly Elastic Supply

es is infinite when

the slope is zero. Cost per unit is

constant. Example: One

consumer may buy as many apples as s/he wishes at the going price.

0

10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50 55 60

Quantity (units/time)

Pri

ce (

$/u

nit

)

S

D

73

Perfectly Inelastic Supply

es is zero when the

slope is infinite. Price has no effect on the

quantity supplied. e.g.: Once the crop is

ready to harvest, the farmer will do so as long as s/he can earn at least the cost of harvesting it.

PPminmin

PP

QQ00

SSmarket periodmarket period

DD

QsQs

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Determinants of es:

The degree of substitutability of resources among different productive activities.

Time -- Given more time, producers are able to make more adjustments to their production processes in response to a given change in price.

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Elasticity and the Burden of a Tax

The economic incidence of taxation falls on the persons who suffer reduced purchasing power because of the tax.

The legal incidence falls on the persons who are required by law to pay the tax to the government.

76

Tax Burden

Demand for Tonic: P = $42 - 3Q Let Supply be: P = -3 + 2Q. (es <1.)

Solve for equilibrium quantity:– -3 + 2Qe = 42 - 3Qe

– 5Qe = 45

– Qe = 9 pints per day (|ed|<1 if Q>7.)

Solve for equilibrium price: – Pe = 42 - 3Qe = 42 - 27 = $15 per pint.

77

Legal incidence on seller:

Add the tax to Supply: P= -3+2Q+10=7+2Q Solve for new quantity:

– 7 + 2Qn = 42 - 3Qn es>1

– 5Qn = 35

– Qn = 7 pints per day (|ed|=1 if Q=7.)

Solve for gross & net price: – Pgross = 42 - 3Qn = 42 - 21 = $21 per pint.

– Pnet = - 3 + 2Qn = -3 + 14 = $11 per pint.

78

Specific Tax on the Seller

0

510

1520

2530

3540

45

0 2 4 6 8 10 12 14

Demand

Supply

S + Tax

Quantity (pints/day)

Pri

ce (

$/p

int)

79

Legal incidence on buyer:

Subtract tax from Demand: P= 42-3Q-10 Solve for new quantity:

– -3 + 2Qn = 32 - 3Qn

– 5Qn = 35

– Qn = 7 pints per week (|ed|=1 if Q=7.)

Solve for gross & net price: – Pgross = 42 - 3Qn = 42 - 21 = $21 per pint.

– Pnet = 32 - 3Qn = 32 - 21 = $11 per pint.

80

Specific Tax on the Buyer

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14

DemandSupplyD - Tax

Quantity (pints/day)

Pri

ce (

$/p

int)

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Compute|ed| and es

Before the tax Pe = $15/pint and Qe = 9 pints/week The slope of D = -3, while the slope of S = 2.

56.09

15

3

11

Q

P

Slopeed

83.09

15

2

11

Q

P

Slopees

82

Now Who Pays the Tax?

Consumers now pay $21 per pint– $6 / pint more than before the tax

Vendors now receive $21 per pint,– but must pay the $10 per pint tax.– Sellers keep only $11 per pint.– $4 / pint less than before

Buyers respond less to a change in price, so they pay more of the tax.

ELASTICITY

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