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Elasticity and Consumer Surplus
Elasticity Introduction
• Elasticity
• Price Elasticity
y,x
% y
% x
xx,p
% x
% p
Elasticity• Principles
– Arc elasticity– Mid-point method
• Intermediate– Point elasticity
Q,PP 0
Q,P
QdQ PQ
P dP QP
1 P
slope Q
lim
Sometimes as
, ...but remember, this is the slope
of the inverse demand curve!
2 1
1 2
Q,P2 1
1 2
Q Q
Q Q
2
P P
P P
2
Elasticity• What we add in ECON 5340
x
xx,p
x x
d ln xpdx
dp x d ln p
Elasticity: Proportionate Change in Q for a Proportionate change in P
D(p)
x p
d ln x
d ln p
Let demand be defined: xSay we want to know the % for a % .
So we need the slope of this function... a.k.a.
ln p
ln D(p)
ln p
d ln xln q ln f (ln p)
d ln p
We cannot simply take the log of both sides and substitute for p to get
and take derivatives to get
Elasticity: Proportionate Change in Q for a proportionate change in P
v
v v
v v
x(p)
d ln xln x x(ln p).
d ln p
u ln x u ln x(p) v ln p p e
d ln x du d ln x(e ) dx(e )
d ln p dv dx(e ) de
Let demand be defined: x
We want to find , but cannot simply substitute in:
Instead:
Define: so and define , so v
vv
v v
de
dv
d ln x 1 dx(e )e
d ln p x(e ) de
d ln x 1 dx(p)p
d ln p dx(p) dp
d ln x 1 dxp
d ln p x dp
=
vP e
d ln x dx p
d ln p dp x =
x(p) q
dx(p) dq
dp dp
Lots of chain rule
Empirical Estimation of Elasticity• Estimating elasticity with regression is common:
• With this specification:
x p
x,p
x
p
p
x
Empirical Estimation of Elasticity• How about a linear-log model:
• In this case:
x ln(p)
x,p
x 1
p p
xp
p
x
Empirical Estimation of Elasticity• Log-linear:
ln(x) p
p
p
x,p
ln(x) p
e e
x e
xe
p
xx
p
x 1
p x
p
x 1
p x
xxp
x
Which is the % change in given a absolute change in p
Empirical Estimation of Elasticity• Much more common is the log-log specification:
ln(x) ln(p) ln(p)d
ln(p)
ln(x) ln(p)
e e Q e
xe
p p
xx
p p
x p
p x
x
, or
, So =.25 means a .25% change
in x for a 1% change in p
Elasticity• Marshallian Demand Elasticities
– Own price– Cross price– Income– Price elasticity and expenditure– Expenditure Elasticity– Budget Share elasticity
• Compensated Demand Elasticity• Slutsky Equation in Elasticity Form• Constant Elasticity Demand
Elasticity, Elasticity, Elasticity!!! With respect to a % change in:
Percentage change in:
px M py
x
xc
x y
M
x y
x p ,p ,M M
M x p ,p ,M
y
x y yx,p
y x y
x p , p ,M p
p x p , p ,M
x x xp x,p p 1 xp x,M M
x y yp x,p x,p
x x xS ,p p 1 xS ,M M 1
x y yS ,p x,p
c
y
cx y yc
x ,p cy x y
x p ,p , U p
p x p ,p , U
cx
cx yc x
x ,p cx x y
x p ,p , U p
p x p ,p , U
x
x y xp
x x y
x p ,p ,M p
p x p ,p ,M
Demand Elasticities• Most of the commonly used demand elasticities
are derived from the Marshallian demand function x(px,py,M)
• Price elasticity of demand (ex,px)
x
x
x y xp
x x x x y
x x y
p
x x y
x p ,p ,M px / x
p / p p x p ,p ,M
p p ,p ,M1
xp p ,p ,M
x
x yOr start with the inverse demand function: p =p x, p , M
Price Elasticity of Demand
• The own price elasticity of demand is always negative (excepting for Giffen goods)
• The size of the elasticity is important– if epx
= -∞, demand is perfectly elastic
– if epx < -1, demand is elastic
– if epx = -1, demand is unit elastic
– if epx > -1, demand is inelastic
– if epx = 0, demand is perfectly inelastic
Income and Cross-price Elasticities
• Income elasticity of demand (ex,M)
• Cross-price elasticity of demand (ex,py)
x y
M
x y
x p ,p ,Mx / x M
M / M M x p ,p ,M
y
x y yx,p
y y y x y
x p ,p ,M px / x
p / p p x p ,p ,M
Own Price Expenditure Elasticity• Expenditure on x = px*x• What is the % change in expenditure on x
when the price of x changes?
x x
x x
x x x
x x y xp x,p
x x
x x y x y
x x yx x
xp x,p x
x x
xp x,p x,p
x
p • x p ,p ,M p
p p x
p • x p ,p ,M x p ,p ,Mp x p ,p ,M
p p
px: p x
p p x
px1 1
p x
Where:
So
Own Price Expenditure Elasticity
• One of the most important implications of elasticity is the relationship between elasticity and total revenue or expenditure.
Price Elasticity and Expenditure
p
x
p1
p2
x2 x1
1 1 1
2 1 2 1
2 1 1
2 1 1 1 1
2 1 1 1 1 1 1 1
1 1
1 1
1 1 1 1 1 1
1 1 1
E p x
p p p ; x x x
E p p x x
E p x p x x p p x
E E p x p x x p p x p x
E p x x p; p x 0
p x x pE
p x p x p x
E x p
E x p
p
x
% p % x % E
E1
E2
Price Elasticity and Expenditure
p
x
p
x
% E % x % p
Elasticity Price change % Changes Expenditure
Inelastic Increase %ΔP > %ΔX Rises
Decrease %ΔP < %ΔX Falls
Unit Elastic Either %ΔP = %ΔX No change
Elastic Increase %ΔP < %ΔX Falls
Decrease %ΔP > %ΔX Rises
Price Elasticity and Expenditure• Total expenditure on x is = pxx(px, py, M)
• Using elasticity, we can determine how total spending on x changes when the price of x changes
x x y x y
x x yx x
p x p ,p ,M x p ,p ,Mp x p ,p ,M
p p
Price Elasticity and Expenditure
x
x
x x
x x
x x
x x
xp
x
xp
x
(p x) p xxx
p x p
(p x) p xx x
p x p
(p x)x x
p
(p x)x[ 1]
p
• Multiply by x/x:
Price Elasticity and Expenditure
• If px > -1, demand is inelastic
– price rises, so does total expenditure on x
• If px < -1, demand is elastic
– price rises, and total expenditure on x falls
x
xp
x
(p x)x 1
p
Income Expenditure Elasticity• Expenditure on x = px*x• What is the % change in expenditure on x
when income changes?
x
x
x
x x y
p x,Mx
x x y x y
x
p x,M xx
p x,M M
p • x p ,p ,M M
M p x
p • x p ,p ,M x p ,p ,M: p
M Mx M
: pM p x
x M
M x
Where
So
Cross Price Expenditure Elasticity• Expenditure on x = px*x• What is the % change in expenditure on x then
the price of y changes?
x y
x y
x y y
x x y yp x,p
y x
x x y x y
xy y
yp x,p x
y x
yp x,p x,p
y
p • x p ,p ,M p
p p x
p • x p ,p ,M x p ,p ,M: p
p p
px: p
p p x
px
p x
Where
So
Own Price Budget Share Elasticity
x x
x x
x x x
x x y
xS ,p
xx
xx y
x y x yx
x x
x xS ,p
xx
xS ,p p
x
p x p ,p ,M
M pp xp
Mp
x p ,p ,M x p ,p ,M x p ,p ,MpMp M p M
p px xp xM p M
M
p x x1
x p x
Where:
So:
Income Budget Share Elasticity
x
x
x x y
S ,Mx
x x y x yx x x y
x2
x
x xx
S ,M 2x
p x(p ,p , M)
M Mp xM
Mp x(p , p , M) x(p ,p , M)
p M p x(p , p , M)M p
p M
xp M p x
p Mp xM
M
Where:
So:
Income Budget Share Elasticity (cont.)
• Start reducing:
x
x
x
x xx
S ,M 2x
x x 2x
S ,M 2x
x xx
S ,Mx
xp M p x
p Mp xM
M
xp M p x
p M
M p x
xp M p x
p
p x
Income Budget Share Elasticity (cont.)
• Almost there
x
x
x
xx x
S ,Mx x
S ,Mx
S ,M x,M
xp M
p p x
p x p x
M x1
x p
1
Cross Price Budget Share Elasticity
x y
x y
x y y
x x y
yS ,p
xy
xx y
x yx
y y
yxS ,p
xy
yS ,p x,p
y
p x(p ,p ,M)
pMp xp
Mp
x p ,p ,M x p ,p ,MpMp M p
pp xp xM p
M
p x
x p
Where:
So:
Compensated Price Elasticities• It is also useful to define elasticities based on
the compensated demand function• If the compensated demand function is
x = xc(px, py, U)
we can calculate– compensated own price elasticity of demand– compensated cross-price elasticity of demand
Compensated Price Elasticities• The compensated own price elasticity of
demand is
• The compensated cross-price elasticity of demand is
c
x
cx yc x
x ,p cx x y
x p ,p , U p
p x p ,p , U
c
y
cx y yc
x ,p cy x y
x p ,p , U p
p x p ,p , U
Slutsky Equation in Elasticity Form• The relationship between Marshallian and
compensated price elasticities can be shown using the Slutsky equation
cx y x y x y
x yx x
x p ,p ,M x p ,p , U x p ,p ,Mx p ,p ,M
p p M
Slutsky Equation in Elasticity Form• Multiply both sides by px/x• Gets us a few elasticities
x yx
x
cx y x yx x
x yx
x p ,p ,Mp
x p
x p ,p , U x p ,p ,Mp px p ,p ,M
x p x M
• Hmmm? Let’s focus on the last part
xp
cx
c
x ,p
Slutsky Equation in Elasticity Form• Multiply that part by
x yxx y
x x y x y
x p ,p ,Mp Mx p ,p ,M
MAnd then let and switch places
p x p
MM
M
,p ,M x p ,p ,M M
x
x
M x
MeSx, the shareOf income spent on X
M
M
x yx
x
cx y x yx x
x yx
x p ,p ,Mp
x p
x p ,p , U x p ,p ,Mp px p ,p ,M
x p x M
Slutsky Equation in Elasticity Form• So now we have:
x MS
xp
cx
c
x ,p
Slutsky Equation in Elasticity Form
• So the own price elasticity = the compensated own price elasticity – the share of income spent on the good times the income elasticity
x x
cp x,p X Me S
Slutsky Equation in Elasticity Form• The Slutsky equation shows that the
compensated and uncompensated price elasticities will be similar if– the share of income devoted to x is small– the income elasticity of x is small
Elasticity Case Studies
• Linear demand• Constant elasticity demand• Cobb-Douglas
Linear Demand
x
x
x
x
x
cx
x xx,p
x x
c 1 xx,p x c
x
cx
x,p cx
x,p
x,p
x a bp
x(p ) p
p x(p )
pbcp
a bp
bcp
a bp
a xbc
bx
c x a
x
cx
cx
x a bp
a xp
b
Note that
Linear Demand
p
x
p
x
elastic
inelastic
=-1
Constant Elasticity Demand
q Ap
Let demand be specified as
where 0Then is the price elasticity of demand...and it does not vary with q.
Constant Elasticity
p
x
p
x
=-0.8
Constant Elasticity Demand
ln x ln A ln p
ln A ln p
ln A ln p
x Ap
ln x ln A ln p
e e
x e
xe x
p p p
x p
p x
Taking natural log of each side
So the log-log empirical specification assumesconstant demand elasticity.
Cobb-Douglass Demand• The Cobb-Douglas utility function is
U(x,y) = xy, where (+=1)• The demand functions for x and y are
x
Mx
p
y
My
p
xxp x
SM
y
yp yS
M
Cobb-Douglass Own Price Elasticity
x
2x x
p 2 2x x
x
p pM MMp p M
p
x
x xp 2
x x
p px M
p x p x
x
Mx
p
xp 1
Cobb-Douglass Cross Price and Income Elasticities
y
y yx,p
y
p px0 0
p x x
Mx x
x M M1
M x p M p
Cross price elasticity
Income elasticity
x
Mx
p
Cobb-Douglass and Slutsky Equation in Elasticity Form
• We can also use the Slutsky equation to derive the compensated price elasticity
• The compensated price elasticity depends on how important other goods (y in this case) are in the utility function
x x
x x
x
x
x
cx,p x,p x x,M
cx,p x,p x x,M
cx,p
cx,p
cx,p
s
s
1 (1)
1
Simply rearrange
Engle Aggregation Condition x x x y x x
x y
x y
yx
x x,M y y,
M p x p ,p ,M p y p ,p ,M
x y1 p p
M M
x xM y yM1 p p
M xM M yM
p yMp xM x M y1
x M M y M yM
1 S S
Partially differentiate each side w.r.t. M
Then multiply each term on the right by a version of 1
M
Share weighted income elasticities sum to 1.
Cournot Aggregation Condition x x x y x x
x yx x
x x x
x x xx y
x x
x
x
M p x p ,p ,M p y p ,p ,M
x y0 p x p
p p
p p px y
M M x M y
p p px x y y0 p x p
p M x M p M y
p x0
x p
xPartially differentiate each side w.r.t. p
Then multiply each term on the right by , , or
x x
x x
yx x x
x
x x,p x y y,p
x x,p y y,p x
x
p yp x p x p y
M M y p M
0 S S S
S S S
The size of the cross price effect of a change in p on yconsumed is limited by the budget constraint.
Consumer Surplus
• Measuring the welfare impacts of policy is one of the things economists like to do.
• Consumer surplus is how we look at the impact of policies on demanders.
– The special case of quasi-linear utility– Compensating variation– the area under the demand curve
Quasi-linear Preferences
• The MRS at any x is the same, no matter the y• General form:
• Two most common derive from the following utility functions.
• Assume that py = 1
U x y
U ln x y
U v(x) y
No income effect
y
x
Reservation Price - General• If we consider buying discrete units of a good,
the reservation price, r1, is the price at which you decide to buy one more.– You are indifferent between buying one more or not.– So r1 satisfies the equation U(0, M) = U(1, M-r1)
– And r2 satisfies the equation U(1, M-r2) = U(2, M-r2)
Reservation Price – Quasi-linear• Now U(x,y) = v(x)+y and y = M-pxx (as py=1)
– So r1 satisfies the equation v(0) + M = v(1) + M - r1
– Then r1 = v(1) - v(0)
– And r2 satisfies the equation
v(1) + m - r2 = v(2) + m-2r2
– Which gets us: r2 = v(2) –v(1)
– And so on: r3 = v(3) –v(2)
– That is, rn = v(n) – v(n-1)
Reservation Price – Quasi-linear• So, rn = v(n) –v(n-1)
• So the reservation price, rn, is telling us the increase in v(n) consuming the nth unit.
• If U = v(x)+y, then v(n) is the contribution to utility from consuming n units of x.• For total utility, we must add in y: y* = M - pxx
• U(x*,y*) = v(x*) + M – pxx*
And the willingness to pay• r1 = v(1) - v(0)
• r2 = v(2) –v(1)
• r3 = v(3) –v(2)• Therefore,
v(3) = r3 + v(2)
v(3) = r3 + r2 + v(1)
v(3) = r3 + r2 + r1 , assuming v(0) = 0.
• So U(x*=n, y*) = rn + rn-1 + rn-2 + … + r1 + M - pxx
Utility
• If v(n) = rn + rn-1 + rn-2 + … + r1 • Then
U(x*=n, y*) = rn + rn-1 + rn-2 + … + r1 + M – pxx*
Utility and Consumer Surplus
px
x
px
x*
pxx*
v(x*)=U(x*,y*) = v(x*) + y*
U(x*,y*) = v(x*) + M - pxx*C.S. from x* = v(x*) - pxx*
C.S.
Every extra $1 spent on x means 1 fewer unit of y consumed.
Quasi-Linear Preferences, Consumer Surplus, and Utility
• With this preference assumption, consumer surplus and utility are directly linked.
• Since px does not affect M, a change in price directly affects utility by reducing the amount of y consumed. And each unit of y provides 1 unit of utility.
• Outside of quasi-utility, consumer surplus and utility are not directly linked.
Now let’s look at a change in CS when there is a price change with less specific
preference assumptions.• When the price of a good rises, the individual
would have to increase expenditure to remain at the initial level of utility.
• So if px rises from px to px’
ΔE = E(px’, py, U1)-E(px, py, U1)
• That change in the expenditure function measures the monetary value of the reduction in utility… i.e. the loss in consumer surplus.
Compensating Variation
• That change in expenditure needed to maintain a certain level of utility is called the compensating variation.
CV= E(px’, py, U1)-E(px, py, U1)
Consumer Welfare
Quantity of x
Quantity of y
U1
A
Suppose the consumer is maximizing utility at point A.
U2
B
If the price of good x rises, the consumer will maximize utility at point B.
The consumer’s utility falls from U1 to U2
Consumer Welfare
Quantity of x
Quantity of y
U1
A
U2
B
The consumer could be compensated so that he can afford to remain on U1
C
CV is the amount that the individual would need to be compensated -- that is, the change in Expenditure for a change in px. Note, py=$1 on this graph
CV
Consumer Welfare• At the margin, the CV is the change in Expenditure
needed to maintain utility.• So for a price increase of $1, you need xc more
income to maintain your utility.
x y 1 cx y 1
x
E p ,p ,Ux p ,p ,U
p
px
x
xc= xc(px,py,U1)
xc*
Consumer Welfare• For a bigger price change, we need to calculate an
area.• So how much consumer surplus is lost for this price
increase?
px
x
px
px’
xc= xc(px,py,U1) x y 1 cx y 1
x
E p ,p ,Ux p ,p ,U
p
Loss due to reduced
consumption of x
Loss due to higher px
Consumer Welfare
• The amount of CV required can be found by integrating from px to px
’
x x
x x
p px y 1 c
x x y 1 xp px
E p ,p ,UCV dp x p ,p ,U dp
p
– this integral is the area to the left of the compensated demand curve between px and px
’
Consumer Welfare• The area under the curve is the amount of compensation
we WOULD needed to maintain utility. Since we are never really compensated, it is simply the welfare loss of a price change.
• Note, it is larger than the area under the ordinary demand curve.
px
x
px
px’ x
x
pc
x y 1 xp
x p ,p ,U dpLost Surplus
x= x(px,py,M)
xc= xc(px,py,U1)
Consumer Welfare• However, a price change involves both
income AND substitution effects– should we use the compensated demand
curve for the original target utility (U1) or the new level of utility after the price change (U2)?
• That is, a change in price from px’ to px would yield a different magnitude of welfare loss because utility would be lower to start (different Hicksian).
Consumer Welfare• The area under the curve is the amount of compensation
needed to maintain utility. Since price fell in this case, maintaining utility means reducing income and the area under the curve measures the gain in welfare.
• In this case, the area under the ordinary demand curve is larger.
px
x
px
px’
x= x(px,py,M)
With a price decrease, because of the lower starting utility, you need less compensating variation to maintain U2 .
xc= xc(px,py,U1)
xc= xc(px,py,U2)
Consumer Welfare• The area under the Ordinary demand curve is a good
compromise that is close to an average of the other two.
px
x
px
px’
x= x(px,py,M)
xc= xc(px,py,U1)
xc= xc(px,py,U2)
One more way to think about Consumer Surplus
x
y
x*
CS from x*
The entire area of consumer surplus can be measured as the CV from consuming x = 0 rather than x = x*.Requires indifference curves to intersect the y axis.
Elasticity: My first draft
v
v v v
v v
Q D(P)
d ln Qln Q ln D(P)
d ln P
u ln Q u ln D(P) v ln P P e
d ln Q du d ln D(e ) dD(e ) de
d ln P dv dD(e ) de d
Let demand be defined:
Take the natural log of each side: , but want to find
Define: so and define , so
v
d ln Q d ln D(P) dD(P) dP
d ln P dD(P) dP d ln P
d ln Q d ln Q dQ dP
d ln P dQ dP d ln P
d ln Q 1 dQ 1d ln Pd ln P Q dP
dPd ln Q 1 dQ 1
1d ln P Q dPP
=
=
by the inverse function rule
vP e
ln P v
d ln Q dQ P
d ln P dP Q =
D(P) Q
dD(P) dQ
dP dP
