Chapter 7

MARKET DEMAND AND ELASTICITY

Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONS

EIGHTH EDITION

WALTER NICHOLSON

Elasticity• Suppose that a particular variable (B)

depends on another variable (A)

B = f(A…)

• We define the elasticity of B with respect to A as

B

A

A

B

AA

BB

A

Be AB

/

/

in change %

in change %,

– The elasticity shows how B responds (ceteris paribus) to a 1 percent change in A

Price Elasticity of Demand• The most important elasticity is the price

elasticity of demand– measures the change in quantity demanded

caused by a change in the price of the good

Q

P

P

Q

PP

QQ

P

Qe PQ

/

/

in change %

in change %,

• eQ,P will generally be negative

– except in cases of Giffen’s paradox

Distinguishing Values of eQ,P

Value of eQ,P at a Point Classification of Elasticity at This Point

eQ,P < -1 Elastic

eQ,P = -1 Unit Elastic

eQ,P > -1 Inelastic

Price Elasticity and Total Expenditure

• Total expenditure on any good is equal to

total expenditure = PQ

• Using elasticity, we can determine how total expenditure changes when the price of a good changes

Price Elasticity and Total Expenditure

• Differentiating total expenditure with respect to P yields

P

QPQ

P

PQ

• Dividing both sides by Q, we get

PQeQ

P

P

Q

Q

PPQ,

/

11

Price Elasticity and Total Expenditure

• Note that the sign of PQ/P depends on whether eQ,P is greater or less than -1

– If eQ,P > -1, demand is inelastic and price and total expenditures move in the same direction

– If eQ,P < -1, demand is elastic and price and total expenditures move in opposite directions

PQeQ

P

P

Q

Q

PPQ,

/

11

Price Elasticity and Total Expenditure

Responses of PQ

Demand Price Increase Price Decrease

Elastic Falls Rises

Unit Elastic No Change No Change

Inelastic Rises Falls

Income Elasticity of Demand• The income elasticity of demand (eQ,I)

measures the relationship between income changes and quantity changes

Q

QQeQ

IIII

in change %

in change %,

• Normal goods eQ,I > 0

– Luxury goods eQ,I > 1

• Inferior goods eQ,I < 0

Cross-Price Elasticity of Demand

• The cross-price elasticity of demand (eQ,P’) measures the relationship between changes in the price of one good and and quantity changes in another

Q

P'

P'

Q

P'

Qe PQ

in change %

in change %',

• Gross substitutes eQ,P’ > 0

• Gross complements eQ,P’ < 0

Relationships Among Elasticities

• Suppose that there are only two goods (X and Y) so that the budget constraint is given by

PXX + PYY = I

• The individual’s demand functions are

X = dX(PX,PY,I)

Y = dY(PX,PY,I)

Relationships Among Elasticities

• Differentiation of the budget constraint with respect to I yields

1

IIY

PX

P YX

• Multiplying each item by 1

1

Y

YYP

X

XXP YX III

III

Relationships Among Elasticities

• Since (PX · X)/I is the proportion of income spent on X and (PY · Y)/I is the proportion of income spent on Y,

sXeX,I + sYeY,I = 1

• For every good that has an income elasticity of demand less than 1, there must be goods that have income elasticities greater than 1

Slutsky Equation in Elasticities• The Slutsky equation shows how an

individual’s demand for a good responds to a change in price

I

XX

P

X

P

X

UXX constant

• Multiplying by PX /X yields

X

XXP

X

P

P

X

X

P

P

XX

U

X

X

X

X

1

constant

I

Slutsky Equation in Elasticities

• Multiplying the final term by I/I yields

X

XXP

X

P

P

X

X

P

P

X X

U

X

X

X

X

III

constant

Slutsky Equation in Elasticities• A substitution elasticity shows how the

compensated demand for X responds to proportional compensated price changes– it is the price elasticity of demand for

movement along the compensated demand curve

constant,

U

X

X

SPX X

P

P

Xe

X

Slutsky Equation in Elasticities• Thus, the Slutsky relationship can be

shown in elasticity form

I,,, XXS

PXPX eseeXX

• It shows how the price elasticity of demand can be disaggregated into substitution and income components– Note that the relative size of the income

component depends on the proportion of total expenditures devoted to the good (sX)

Homogeneity

• Remember that demand functions are homogeneous of degree zero in all prices and income

• Euler’s theorem for homogenous functions shows that

0

II

X

XP

P

XP

P

XY

YX

Homogeneity

• Dividing by X, we get

0

X

X

X

P

P

X

X

P

P

X Y

YX

II

X

• Using our definitions, this means that

0 I,,, XPXPX eeeYX

• An equal percentage change in all prices and income will leave the quantity of X demanded unchanged

Cobb-Douglas Elasticities• The Cobb-Douglas utility function is

U(X,Y) = XY

• The demand functions for X and Y are

XPX

I

YPY

I

• The elasticities can be calculated

11

2

,

X

X

X

X

X

XPX

P

PX

P

PX

P

P

Xe

X I

II

Cobb-Douglas Elasticities• Similar calculations show

1I,Xe 0YPXe ,

1 , YPYe 1I,Ye 1YPYe ,

• Note that

IXP

s XX

IYP

s YY

Cobb-Douglas Elasticities• Homogeneity can be shown for these elasticities

• The elasticity version of the Slutsky equation can also be used

0101 I,,, XPXPX eeeYX

I,,, XXS

PXPX eseeXX

(1) , SPX X

e1

)-(1 ,S

PX Xe

Cobb-Douglas Elasticities• The price elasticity of demand for this

compensated demand function is equal to (minus) the expenditure share of the other good

• More generally

)-(1 , XS

PX seX

where is the elasticity of substitution

Linear Demand

Q = a + bP + cI + dP’

where:Q = quantity demanded

P = price of the good

I = income

P’ = price of other goods

a, b, c, d = various demand parameters

Linear Demand

Q = a + bP + cI + dP’

• Assume that: Q/P = b 0 (no Giffen’s paradox) Q/I = c 0 (the good is a normal good) Q/P’ = d ⋛ 0 (depending on whether

the other good is a gross substitute or gross complement)

Linear Demand

• If I and P’ are held constant at I* and P’*, the demand function can be written

Q = a’ + bP

where a’ = a + cI* + dP’*– Note that this implies a linear demand

curve– Changes in I or P’ will alter a’ and shift the

demand curve

Linear Demand• Along a linear demand curve, the slope

(Q/P) is constant– the price elasticity of demand will not be

constant along the demand curve

Q

Pb

Q

P

P

Qe PQ ,

• As price rises and quantity falls, the elasticity will become a larger negative number (b < 0)

Linear Demand

Q

P

-a’/b

a’

eQ,P < -1

eQ,P = -1

eQ,P > -1

Demand becomes moreelastic at higher prices

Constant Elasticity Functions• If one wanted elasticities that were

constant over a range of prices, this demand function can be used

Q = aPbIcP’d

where a > 0, b 0, c 0, and d 0.⋛• For particular values of I and P’,

Q = a’Pb

where a’ = aIcP’d

Constant Elasticity Functions• This equation can also be written as

ln Q = ln a’ + b ln P

• Applying the definition of elasticity,

bPa

PPba

Q

P

P

Qe

b

b

PQ

'

' ,

1

• The price elasticity of demand is equal to the exponent on P

Important Points to Note:• The market demand curve is negatively

sloped on the assumption that most individuals will buy more of a good when the price falls– it is assumed that Giffen’s paradox does not

occur

• Effects of movements along the demand curve are measured by the price elasticity of demand (eQ,P)

– % change in quantity from a 1% change in price

Important Points to Note:• Changes in total expenditures on a good

caused by changes in price can be predicted from the price elasticity of demand– if demand is inelastic (0 > eQ,P > -1) , price and

total expenditures move in the same direction

– if demand is elastic (eQ,P < -1) , price and total expenditures move in opposite directions

Important Points to Note:• If other factors that enter the demand

function (prices of other goods, income, preferences) change, the market demand curve will shift– the income elasticity (eQ,I) measures the effect

of changes in income on quantity demanded

– the cross-price elasticity (eQ,P’) measures the effect of changes in another good’s price on quantity demanded

Important Points to Note:• There are a number of relationships among

the various demand elasticities– the Slutsky equation shows the relationship

between uncompensated and compensated price elasticities

– homogeneity is reflected in the fact that the sum of the elasticities of demand for all of the arguments in the demand function is zero