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Lecture 4 A G S M 2004 Page 1

LECTURE 4: ELASTICITYTodays Topics1. The Price Elasticity of Demand: total

revenue , determinants, formul, a bestiary,total revenue , estimation of price elasticity ofdemand.

2. The Income Elasticity of Demand, and theCross-Price Elasticity of Demand.

3. The Elasticity of Supply: determinants,formula.

4. Tw o Applications: the OPEC cartel tries tokeep the price of oil up, farmers adoptionslower their profits.

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REVENUE AND PRICEManagers make decisions at the margin a littlemore , a little less and only have reasonableinformation about a small region on the demandcur ves they face .Q: How does Revenue (Price Quantity) chang ewhen we raise the price we sell at?

QD

P ......................................................................................................................................................................................................... D

P

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ALGEBRAIC DERIVATIONA: It depends on how much the quantity demandedfalls as we move up the demand curve toP+P ,Q+Q .Old Revenue: R = P QNew Revenue: R = (P + P )(Q + Q)

= P Q + P Q + QP + P QIgnoring P Q , the chang e in revenue is:

R R = P Q + QP= QP

PQ

QP + 1

= QP ( + 1),Is = PQ

QP greater than, equal to, or less than 1?

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INTUITION OF THE REVENUE CHANGE

Q /QP /P is the price elasticity of demand.

That is: if the percentage fall in quantity demandedQ /Q is greater than the percentage rise in pricecharged P /P , then the Revenue will fall.So:

< 1 elastic || > 1 R < 0 > 1 inelastic || < 1 R > 0 = 1 unitar y || = 1 R = 0

Taxes on what?

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To summariz e:Total

|| Price Expenditure(Revenue)

Up DownElastic> 1demand

Down UpUp ConstantUnitar y

= 1elasticityDown Constant

Up UpInelastic< 1demand

Down Down

Price Elasticity of Demand and Revenue Changes

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PRICE ELASTICITY OF DEMANDElasticity is a dimensionless measure of thesensitivity of one variable to chang es in another,cet. par.

The price elasticity of demand is the percentagechang e in quantity demanded Q divided by thepercentage chang e in the price, P .

Because of The Law of Demand, if P is positive(price rises), then Q cannot be positive, and ingeneral is negative.

Since the price elasticity of demand is neverpositive , we usually ignore its sign (or use itsabsolute value ||).

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FOUR DETERMINANTS OF Four determinants of a goods own-price elasticityof demand :

1. Necessities v. discretionar y goods (orluxuries): necessities tend to have inelasticdemands; luxuries have elastic demand.Depends on the buyers preferences.Examples?

2. Av ailability of close substitutes: the greaterthe number of available substitutes, themore elastic the demand.Examples?

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DETERMINANTS3. Definition of the market: broad definitions

(e .g. food) have less elastic demands thando narrowly defined markets (e.g. Nestlschocolate) which have more substitutes.Examples?

4. Time horizon: the greater the time horizon,the easier for consumers to find substitutes,or make do without, so the more elastic thedemand.Examples?

(These proper ties do not follow from the axiomsand definitions; they have been observed in themarket.)

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ARC OR POINT MEASUREMENTS

The arc elasticity: = Q /QP /P =

PQ

QP 0

using mid-points: P 12 (P1 + P2), Q 12 (Q1 + Q2)

Q

P............................................................................................................................................................................................................................................................................................................. D

PP2

P1

QQ2 Q1

QP

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A LINEAR DEMAND SCHEDULEElasticity is not equal to the slope of the demandcur ve. Indeed, we can calculate the priceelasticities along a linear demand curve . (Arcelasticities, midpoint convention.)

Price Purchase Value of Sales ||($/t) (tonnes) ($) Elasticity

2 2500 50003 2000 6000 5/9 = 0.5564 1500 6000 15 1000 5000 9/5 = 1.8

eg.59=

(2, 500 2, 000) / 2, 250(3 2) / 2. 5 =

Q/QP/P

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A LINEAR DEMAND CURVE

2

4

6

7

1000 2000 3500Q

P

elastic demand (>1)

inelastic demand (


POINT ELASTICITY

The point elasticity: = P1Q1QP , where

PQ is the

slope of the curve at the point P1, Q1. (The partialderivatives imply ceteris paribus: only price P ischanging.)

Q

P............................................................................................................................................................................................................................................................................................................. D

Q1

P1

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POINT ELASTICITY FORMULAA linear demand function: Q = 3500 500P , orP = 7 Q /500. The slope is P /Q = 1/500.

= 500 P /QNB: elasticity varies along a straight line.

7 D

3500

P

Q

Elasticityat point =

the slope of the ray through the originthe slope of the demand curve

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A BESTIARY OF DEMAND CURVESWhen || > 1 we have elastic demand

= 1 we have unitar y elastic demand< 1 we have inelastic demand= 0 we have perfectly inelastic demand perfectly elastic demand

|| =

P

Q

D

Horizontal demand: perfectly elastic.

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A BESTIARY 2 =0P

Q

D

Vertical demand: perfectly price-inelastic.

P

Q

D..................................................................................................................................

|| = : inelastic demand

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A BESTIARY 3

P

Q

D.................................................................................................................................. || = 1 unitar y

(A rectangular hyperbola.)P

Q

D.................................................................................................................................. || = 2: elastic demand

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INCOME ELASTICITY OF DEMAND The propor tional chang e in the amount demandedin response to a 1 percent chang e in income I .Or algebraically:

(arc) QD /QDI /I

(point) QD

II

QD

Normal goods have positive income elasticities.Inferior goods have negative .Luxuries have > 1.Necessities have < 1, but positive.Examples?

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CROSS-PRICE ELASTICITY OF DEMANDThe percentage chang e in the demand X D for goodX in response to a 1 percent chang e in the price PYof good Y .Arc measure:

X ,Y X D/X D

PY /PYPoint measure:

X ,Y X DPY

PYX D

,

where X D = X D1 X D2 , PY = PY 1 PY 2,using midpoints: X D = 12 (X D1 + X D2 ), andPY = 12 (PY 1 + PY 2).

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SUBSTITUTES AND COMPLEMENTSIf X ,Y > 0 then X and Y are substitutes

< 0 then X and Y are complements= 0 then X and Y are unrelated

Examples?of substitutes?of complements?

Note: in general X ,Y Y ,X (see Coke and Pepsibelow) because of income effects (GKSM p.472).

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ESTIMATING ELASTICITYA constant-elasticity demand function can bewritten as

Q = APwhere is the price elasticity of demand, and A isa constant.

Taking logarithms:logQ = log A + logP ,

ory = a + x

which means that we can use linear regression toestimate the elasticity (assuming our data comefrom an unshifting demand curve).

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MARKET DATAPrice , Cross-Price , and Income Elasticities

of Demand for Coca-Cola and Pepsi

Elasticity Coca-Cola PepsiOwn Price elasticity 1.47 1.55Cross-price elasticity X ,Y 0.52 0.64Income elasticity 0.58 1.38

Source: Besanko & Braeutigam, Microeconomics.

So a 1% increase in Cokes price led to a 1.47% fallin Cokes quantity sold and a 0.64% increase inPepsis sales.(Perhaps estimated using X D = APX I PX ,YY ).

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PRICE ELASTICITY OF SUPPLY(Only for price-taking suppliers monopolists donot have supply cur ves.)The price elasticity of supply is the percentagechang e in quantity supplied QS per percentagechang e in price P :

=QS /QSP /P

Can be perfectly inelastic ( = 0, ver tical), perfectlyelastic ( = , horizontal), inelastic ( < 1), andelastic ( > 1).Depends mainly on the time horizon: the longer,the more elastic, in general, because firms havemore time to adjust their production processes inorder to increase their profits.

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OPEC AND THE OIL PRICEIn the early 1980s the OPEC cartel squeezedsupply and pushed up the world price of oil. Whathappened?

In the short run, both supply and demand for oilare relatively inelastic:

changing capacity and proving up morereser ves is relatively slow;

old guzzlers and old habits of use are slow tochang e: demand adjusts only slowly.

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LONG-RUN MARKET ADJUSTMENTIn the long run, the higher price affected bothsupply and demand:

there was increased exploration increasedproduction, especially in the non-OPEC oilproducers, such as?

New R&D more fuel efficient vehicles andindustrial processes and householdmachiner y, and these were eventually boughtand installed to cut fuel bills lower demand(than otherwise).

The initial high price fell, although only slowly, andnot (at first) back to the pre-squeeze price .

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GRAPHICALLY

P

QS1

D1

P1

Q1

P2

Q2

P3

Q3

D3

S3

Over time, both supply and demand become moreelastic: the later price P3 is lower than the earlierprice P2, and the later quantity Q3 is lower than theearlier quantity Q2. OPEC cannot long sustain thehigh price P2.

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ARE FARMERS IRRATIONAL?Farmers adopt new technology which reduces theircosts. But such technology, when all adopt it andmarket supply expands, lowers their output prices.Why do they adopt it?

P

Q

D

S1P1

Q1

S2

P2

Q2

The industry view: downwards-sloping demand.With inelastic demand, revenues fall with price.

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THE PRICE-TAKING FARMERFrom the small (price-taking) farmers view, themarket price is a given: she faces an infinitelyelastic (horizontal) demand curve , the going price.She adopts the new technology to improve her netreturns or profits, by reducing her costs. Hersupply cur ve expands.

P

q

D1

s1

P1

q1

s2

q2

D2P2

q3

As all farmers adopt the technology, price will fall.No single farmer, however, can prevent this. < >

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