### Elasticity

```Elasticity and Consumer Surplus
Elasticity Introduction
• Elasticity
 y,x
% y

% x
• Price Elasticity
 x,px
%x

%p
Elasticity
 Q 2  Q1 
  Q1  Q 2  
• Principles
– Arc elasticity
– Mid-point method
 Q,P


• Intermediate

2

 P2  P1 
  P1  P2  


2


– Point elasticity
Q
Q,P  lim
P 0
P
Q
P

dQ P

dP Q
Sometimes as
1
P
Q,P 
 , ...but remember, this is the slope
slope Q
of the inverse demand curve!
Elasticity
• What we add in ECON 5340
x,px
dx px d  ln x 


dpx x d  ln px 
Elasticity: Proportionate Change in Q
for a Proportionate change in P
Let demand be defined: x  D(p)
Say we want to know the %x for a %p.
d ln x
So we need the slope of this function... a.k.a.
d ln p
ln D(p)
ln p
We cannot simply take the log of both sides and substitute ln p for p to get
d ln x
ln q  ln f (ln p) and take derivatives to get
d ln p
Elasticity: Proportionate Change in Q
for a proportionate change in P
Let demand be defined: x  x(p)
d ln x
We want to find
, but cannot simply substitute in: ln x  x(ln p).
d ln p
Define: u  ln x so u  ln x(p) and define v  ln p, so p  e v
d ln x du d ln x(e v ) dx(e v ) de v




v
v
d ln p dv
dx(e )
de
dv
d ln x
1 dx(e v ) v


e
v
v
d ln p x(e ) de
d ln x
1
dx(p)


p
d ln p dx(p) dp
d ln x 1 dx
=  p
d ln p x dp
Lots of chain rule
P  ev
x(p)  q
dx(p)
dq

dp
dp
d ln x dx p
= 
d ln p dp x
Empirical Estimation of Elasticity
• Estimating elasticity with regression is common:
x    p  
• With this specification:
x

p
 x,p
p

x
Empirical Estimation of Elasticity
• How about a linear-log model:
x     ln(p)  
• In this case:
x
1

p
p
x
p
p

 x ,p 
x
Empirical Estimation of Elasticity
• Log-linear:
ln(x)      p  
e
ln(x )
e
p 
x  ep 
x
 ep  
p
x
 x 
p
x 1

p x
 x,p    p
x 1

p x
x
x 
p
Which is the % change in x given
a absolute change in p
Empirical Estimation of Elasticity
• Much more common is the log-log specification:
ln(x)      ln(p)  
e
ln(x )
e
ln(p) 
, or Q d  e
ln(p) 
x
ln(p)  

e
p
p

x
 x
p
p
x p
, So =.25 means a .25% change

p x
in x for a 1% change in p x
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:
px
M
py
Percentage
change in:
x
xc
 px 

c
x c ,p x
x  p x , p y , M 
p x


px
x  px , py , M 
M 
x  p x , p y , M 
M

M
x  px , py , M 
px x,px  px  1
px x,M  M
Sx ,px  px  1
Sx ,M  M 1
x c  p x , p y , U 
p x
px
 c
x  px , py , U 
 x,py 
x  p x , p y , M 
p y

py
x  px , py , M 
px x,py  x,py
Sx ,py  x,py

c
x c ,p y

x c  p x , p y , U 
p y

py
xc  px , py , U 
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  p x , p y , M 
px
x / x
 px 


p x / p x
p x
x  px , py , M 
Or start with the inverse demand function: px =p  x, py , M
px  px , py , M 
1
 px 

x
p x  p x , p y , M 
x
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)
x / x x  p x , p y , M 
M
M 


M / M
M
x  px , py , M 
• Cross-price elasticity of demand (ex,py)
 x,py
x  p x , p y , M 
py
x / x



p y / p y
p y
x  px , py , M 
Own Price Expenditure Elasticity
• Expenditure on x = px*x
• What is the % change in expenditure on x
when the price of x changes?
 px x,px 
Where:

 px • x  px , py , M 

p x
 px • x  px , py , M 
p x

px
px  x
 p

 px
x
So :  px x,px   p x 
 x
p x

 px  x
 x p x 
 px x,px  
 1   x,px  1
 p x x

x

x  p x , p y , M 
p x
 x  px , py , M 
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  %E
E1  p1  x1
p
p  p 2  p1; x  x 2  x1
E2
E 2   p1  p  x1  x 
E 2  p1x1  p1x  x1p  px
p2
E1
p
E 2  E1  p1x1  p1x  x1p  px  p1x1
E  p1x  x1p; px  0
E p1x x1p


p1x1 p1x1 p1x1
p1
E x p


E1 x1 p1
x2
x
x1
x
Price Elasticity and Expenditure
%E  %x  %p
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
p
x
x
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

 px  x  px , py , M 
p x
 p
x

x  p x , p y , M 
p x
 x  px , py , M 
Price Elasticity and Expenditure
• Multiply by x/x:
 (p x  x) x p x  x
 
x
p x
x p x
 p x x 
 (p x  x)
 x 
x
p x
 x p x 
 (p x  x)
 x   px  x
p x
 (p x  x)
 x[ px  1]
p x
Price Elasticity and Expenditure
(p x  x)
 x px  1
p x
• If
px
> -1, demand is inelastic
px
< -1, demand is elastic
– price rises, so does total expenditure on x
• If
– price rises, and total expenditure on x falls
Income Expenditure Elasticity
• Expenditure on x = px*x
• What is the % change in expenditure on x
when income changes?
 px x,M 
Where :

 px • x  px , py , M 
 px • x  px , py , M 
So :  px x,M
 px x,M

M

M
x  M

  px 

M  p x  x

 x M 

  M
 M x 
M
px  x
 p
x

x  p x , p y , M 
M
Cross Price Expenditure Elasticity
• Expenditure on x = px*x
• What is the % change in expenditure on x then
the price of y changes?
  p • x  p , p , M  p
x
 px x,py 
Where :

x
y
p y
 px • x  px , py , M 
p y

x  p y
So :  px x,p y   p x 


p y  p x  x

 x p y 
 px x,p y  

 p y x  x,p y



y
px  x
 p
x

x  p x , p y , M 
p y
Own Price Budget Share Elasticity
 px  x  px , py , M  



 p
M

 x
Sx ,px 
px  x
p x
M
p

  x  x  px , py , M  
p x x  p x , p y , M  x  p x , p y , M 
M


Where:
 

p x
M
p x
M
 p x x  p x
So: Sx ,px   x 
 
 M p x M  p x  x
M
 p x x x 
Sx ,px   
    px  1
 x p x x 
Income Budget Share Elasticity
 p x  x(p x , p y , M) 


M
M


Sx ,M 

px  x
M
M
x(p x , p y , M)
 p x x(p x , p y , M) 

 M  p x x(p x , p y , M)
 px 
M
p x


Where:
p x
M2
So: Sx ,M
x


p
M


p
x
x 
 x p
M
x


2
M

 px  x

 M


Income Budget Share Elasticity (cont.)
• Start reducing:
Sx ,M
Sx ,M
Sx ,M
x


p
M


p
x
x 
 x p
M
x


M2

 px  x

 M


x


p
M


p
x
x 
2
 x p
M
x


2
M

 px x




x
px M 
 px x
p x

px x
Income Budget Share Elasticity (cont.)
• Almost there
Sx ,M
Sx ,M
x
px M 
p x p x x


px x
px x
M x
 
1
x p x
Sx ,M   x,M  1
Cross Price Budget Share Elasticity
 p x  x(p x , p y , M) 

 p
M
 y
Sx ,p y  
px  x
p y
M
p

  x  x  px , py , M  
p x x  p x , p y , M 
M


Where:
 
p y
M
p y
 p x x  p y
So: Sx ,p y   
 M p y  p x  x


M
 p y x 
Sx ,py   

 x p y  x,p y


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

c
x c ,p x

x c  p x , p y , U 
p x
px
 c
x  px , py , U 
• The compensated cross-price elasticity of
demand is

c
x c ,p y

x c  p x , p y , U 
p y

py
x  px , py , U 
c
Slutsky Equation in Elasticity Form
• The relationship between Marshallian and
compensated price elasticities can be
shown using the Slutsky equation
x  p x , p y , M 
p x

x c  p x , p y , U 
p x
 x  px , py , M  
x  p x , p y , M 
M
Slutsky Equation in Elasticity Form
• Multiply both sides by px/x
• Gets us a few elasticities
p x x  p x , p y , M 


x
p x
c
p x x  p x , p y , U 

x
p x
c
 x c ,p
x
•
px
x  p x , p y , M 
px
  x  px , py , M  
x
M
Hmmm? Let’s focus on the last part
Slutsky Equation in Elasticity Form
• Multiply that part by
M
M
x  p x , p y , M  M
px
  x  px , py , M  

x
M
M
And then let x and M switch places

px  x  px , py , M 
M
Sx, the share
Of income spent on X
x  p x , p y , M  M


M
x
eM
Slutsky Equation in Elasticity Form
• So now we have:
p x x  p x , p y , M 


x
p x
px

x
c
 x c ,p
x
x c  p x , p y , U 
p x
px
x  p x , p y , M 
px
  x  px , py , M  
x
M
Sx M
Slutsky Equation in Elasticity Form
epx  
c
x,px
 SX M
• So the own price elasticity = the
compensated own price elasticity – the
share of income spent on the good times
the income elasticity
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  a  bp x c
 x,px 
x(p x ) p x

p x x(p x )
 x,px   bcp x c 1 
 x,px
 x,px
 x,px
px
a  bp x c
 bcp x c

a  bp x c
Note that
ax
 bc 

b



x
cx  a

x
ax
px 
b
x  a  bp x c
c
Linear Demand
p
elastic
=-1
p
inelastic
x
x
Constant Elasticity Demand
Let demand be specified as
q  Ap where   0
Then  is the price elasticity of demand...
and it does not vary with q.
Constant Elasticity
p
=-0.8
p
x
x
Constant Elasticity Demand
x  Ap
Taking natural log of each side
ln x  ln A   ln p
eln x  eln A  ln p
x  eln A  ln p
x

ln A  ln p 
e
  x
p
p
p
x p

p x
So the log-log empirical specification assumes
constant demand elasticity.
Cobb-Douglass Demand
• The Cobb-Douglas utility function is
U(x,y) = xy,
where (+=1)
• The demand functions for x and y are
M
x
px
px x
Sx 

M
M
y
py
Sy 
py y
M

Cobb-Douglass Own Price Elasticity
x
M
px
x p x
M p x
px 
   2 
p x x
px x
M p x
M p 2x
 px   2 
 2 
p x M
p x M
px
px  1
Cobb-Douglass
Cross Price and Income Elasticities
M
x
px
Income elasticity
x M

M
M 
 

1
M x
p x  M p x 
Cross price elasticity
x,py
py
x py

  0  0
py x
x
Cobb-Douglass and Slutsky Equation in
Elasticity Form
• We can also use the Slutsky equation to derive the
compensated price elasticity
 x,p x   cx,p x  s x  x,M
Simply rearrange
 cx,p x   x,p x  s x  x,M
 cx,p x  1   (1)
 cx,p x    1  
 cx,p x  
• The compensated price elasticity depends on how
important other goods (y in this case) are in the utility
function
Engle Aggregation Condition
M  px x  px , px , M   py y  px , px , M 
Partially differentiate each side w.r.t. M
x
y
 py
M
M
Then multiply each term on the right by a version of 1
x xM
y yM
1  px

 py

M xM
M yM
M x p x x M y p y yM
1



x M M
y M yM
1  Sx  x,M  Sy  y, M
1  px
Share weighted income elasticities sum to 1.
Cournot Aggregation Condition
M  px x  px , px , M   py y  px , px , M 
Partially differentiate each side w.r.t. px
 x

y
0  px
 x   py
p x
 p x

Then multiply each term on the right by
0  px
0
px px x
p y
,
 , or x 
M M x
M y
p
x p x x
y p x y
   x  x  py
 
p x M x
M
p x M y
p x x p x x p x x p x y p y y




x p x M
M
y p x M
0  Sx   x,px  Sx  Sy  y,p x
Sx   x,px  Sy  y,px  Sx
The size of the cross price effect of a change in p x on y
consumed 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:
U  v(x)  y
• Two most common derive from the following
utility functions.
U xy
U  ln  x   y
• Assume that py = 1
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
– 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
U(x*,y*) = v(x*) + y*
U(x*,y*) = v(x*) + M - pxx*
C.S. from x* = v(x*) - pxx*
px
v(x*)=
Every extra \$1 spent on x means 1
fewer unit of y consumed.
C.S.
px
pxx*
x*
x
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
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 y
Suppose the consumer is maximizing
utility at point A.
If the price of good x rises,
the consumer will maximize
utility at point B.
The consumer’s
utility falls from U1
to U2
A
B
U1
U2
Quantity of x
Consumer Welfare
Quantity of y
The consumer could be compensated so that
he can afford to remain on U1
CV
C
A
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
B
U1
U2
Quantity of x
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
px

E p x , p y , U1
p x
  xc
 px , p y , U1 
xc= xc(px,py,U1)
xc*
x
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
px’
px
xc= xc(px,py,U1)

E p x , p y , U1
p x
Loss due to
reduced
consumption of x
Loss due to
higher px
x
  xc
 px , p y , U1 
Consumer Welfare
• The amount of CV required can be found by
integrating from px to px’

p x
E p x , p y , U1
px
p x
CV  
 dp
p x
x


  x c p x , p y , U1 dp x
px
– 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
px’
xc= xc(px,py,U1)
p x


Lost Surplus   x c p x , p y , U1 dp x
px
px
x= x(px,py,M)
x
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
xc= xc(px,py,U1)
With a price decrease, because of the
lower starting utility, you need less
compensating variation to maintain U2 .
px’
px
x= x(px,py,M)
xc= xc(px,py,U2)
x
Consumer Welfare
• The area under the Ordinary demand curve is a good
compromise that is close to an average of the other
two.
px
xc= xc(px,py,U1)
px’
px
x= x(px,py,M)
xc= xc(px,py,U2)
x
One more way to think about
Consumer Surplus
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.
y
CS from x*
x*
x
Elasticity: My first draft
Let demand be defined: Q  D(P)
Take the natural log of each side: ln Q  ln D(P), but want to find
Define: u  ln Q so u  ln D(P) and define v  ln P , so P  e v
v
du d ln D(e v ) dD(e v ) de v
P

e




dv
dD(e v )
de v
dv
ln P  v
d ln D(P) dD(P) dP



dD(P)
dP d ln P
D(P)  Q
d ln Q dQ dP
dD(P)
dQ




dQ dP d ln P
dP
dP
1 dQ
1
by the inverse
= 

function rule
Q dP d ln P
dP
d ln Q 1 dQ 1
d ln Q dQ P
= 

=

d ln P Q dP 1
d ln P dP Q
P
d ln Q
d ln P
d ln Q
d ln P
d ln Q
d ln P
d ln Q
d ln P
d ln Q
d ln P
```