chapter 7 market demand and elasticity copyright ©2002 by south-western, a division of thomson...
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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
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