Transcript Elasticity ecn5402.ch07

Chapter 7
MARKET DEMAND AND
ELASTICITY
MICROECONOMIC THEORY
BASIC PRINCIPLES AND EXTENSIONS
EIGHTH EDITION
WALTER NICHOLSON
Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.
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
% change in B B / B B A
eB,A 



% change in A A / A A B
– 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
eQ,P
% change in Q Q / Q Q P




% change in P P / P P Q
• eQ,P will generally be negative
– except in cases of Giffen’s paradox
Distinguishing Values of eQ,P
Classification of
Value of eQ,P at a Point
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
PQ
Q
QP
P
P
• Dividing both sides by Q, we get
PQ / P
Q P
 1
  1  eQ,P
Q
P Q
Price Elasticity and Total
Expenditure
PQ / P
Q P
 1
  1  eQ,P
Q
P Q
• 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
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
eQ,I
% change in Q Q I



% change in I
I Q
• 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
% change in Q Q P'
eQ,P ' 


% change in P' P' Q
• 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
X
Y
PX
 PY
1
I
I
• Multiplying each item by 1
PX  X X I PY  Y Y I

 

 1
I
I X
I
I Y
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
X
X

PX PX
U constant
X
X
I
• Multiplying by PX /X yields
X PX
X PX



PX X PX X
U constant
X 1
 PX  X 

I X
Slutsky Equation in Elasticities
• Multiplying the final term by I/I yields
X PX
X PX



PX X PX X
U constant
PX  X X I



I
I X
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
e
S
X ,PX
X PX


PX X
U constant
Slutsky Equation in Elasticities
• Thus, the Slutsky relationship can be
shown in elasticity form
S
eX ,PX  eX ,PX  s X eX ,I
• 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
X
X
X
 PX 
 PY 
I 0
PX
PY
I
Homogeneity
• Dividing by X, we get
X PX X PY X I




 0
PX X PY X I X
• Using our definitions, this means that
e X ,PX  e X ,PY  e X ,I  0
• 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) = XY
• The demand functions for X and Y are
I
X
PX
I
Y
PY
• The elasticities can be calculated
e X ,PX
X PX
I PX
I
1


 2 


 1
PX X
PX X
PX  I 


 PX 
Cobb-Douglas Elasticities
• Similar calculations show
e X ,PY  0
e X ,I  1
eY ,PY   1
eY ,I  1
eY ,PY  1
• Note that
PX X
sX 

I
PYY
sY 

I
Cobb-Douglas Elasticities
• Homogeneity can be shown for these
elasticities
e X ,PX  e X ,PY  e X ,I  1  0  1  0
• The elasticity version of the Slutsky
equation can also be used
S
eX ,PX  eX ,PX  s X eX ,I
1  e
e
S
X ,PX
S
X ,PX
 (1)
  (1- )   
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
eSX ,PX   (1- s X )
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
eQ,P
Q P
P

  b
P Q
Q
• As price rises and quantity falls, the
elasticity will become a larger negative
number (b < 0)
Linear Demand
Demand becomes more
elastic at higher prices
P
-a’/b
eQ,P < -1
eQ,P = -1
eQ,P > -1
a’
Q
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,
eQ,P
Q P ba' P b 1  P

 
b
b
P Q
a' P
• 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