Download Finance 510: Microeconomic Analysis

Finance 510: Microeconomic
Analysis
Consumer Demand Analysis
Suppose that you observed the following consumer behavior
P(Bananas) = $4/lb.
Q(Bananas) = 10lbs
P(Apples) = $2/Lb.
Q(Apples) = 20lbs
P(Bananas) = $3/lb.
Q(Bananas) = 15lbs
P(Apples) = $3/Lb.
Q(Apples) = 15lbs
Choice A
Choice B
What can you say about this consumer?
Choice B
Is strictly
preferred to
Choice A
How do we know this?
Consumers reveal their preferences through their observed choices!
Q(Bananas) = 10lbs
Q(Bananas) = 15lbs
Q(Apples) = 20lbs
Q(Apples) = 15lbs
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
P(Bananas) = $3/lb.
Cost = $80
Cost = $90
Cost = $90
Cost = $90
P(Apples) = $3/Lb.
B Was chosen even though A was the same price!
What about this choice?
Choice C
P(Bananas) = $2/lb.
Q(Bananas) = 25lbs
P(Apples) = $4/Lb.
Q(Apples) = 10lbs
Q(Bananas) = 15lbs
Choice B
Cost = $90
Cost = $90
Q(Apples) = 15lbs
Q(Bananas) = 10lbs
Choice A
Choice C
Cost = $100
Q(Apples) = 20lbs
Is strictly
preferred to
Choice B
Is choice C
preferred to choice
A?
Choice B
Choice C
Is strictly
preferred to
Choice A
Is strictly
preferred to
Choice B
C>B>A
Choice C
Is strictly
preferred to
Choice A
Rational preferences exhibit transitivity
Consumer theory begins with the assumption that every
consumer has preferences over various consumer
goods. Its usually convenient to represent these
preferences with a utility function
U : AB
U
A
Set of possible
choices
B
“Utility Value”
Using the previous example (Recall, C > B > A)
Choice A
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Choice B
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Choice C
Q(Bananas) = 25lbs
Q(Apples) = 10lbs
U (25,10)  U (15,15)  U (10,20)
We only require a couple restrictions on Utility functions
 For any two choices (X and Y), either U(X) > (Y), U(Y) > U(X), or
U(X) = U(Y) (i.e. any two choices can be compared)
 For choices X, Y, and Z, if U(X) > U(Y), and U(Y) > U(Z), then U(X)
> U(Z) (i.e., the is a definitive ranking of choices)
However, we usually add a couple additional restrictions
to insure “nice” results
If X > Y, then U(X) > U(Y) (More is always better)
If U(X) = U(Y) then any combination of X and Y is preferred to
either X or Y (People prefer moderation to extremes)
Suppose we have the following utility function
U  U ( x, y )
U = 20
Imagine taking a “cross section” at some utility level.
The “cross section” is called an indifference curve
(various combinations of X and Y that provide the same
level of utility)
Any two choices can be compared
U ( A)  U ( B)  20
y
There is a definite ranking of all
choices
U (C )  U ( A)  U (C )  U ( B)
A
C
U ( x, y )  25
B
U ( x, y )  20
x
The “cross section” is called an indifference curve
(various combinations of X and Y that provide the same
level of utility)
More is always better!
y
U (C )  U ( A)
C
A
B
U ( x, y )  20
x
The “cross section” is called an indifference curve
(various combinations of X and Y that provide the same
level of utility)
People Prefer Moderation!
y
U (C )  U ( A)
A
C
B
U ( x, y )  20
x
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
U ( x*  x, y* )  U ( x* , y* )
+ U ( x, y*  y)  U ( x* , y* )
y
x
y*
y
Suppose you are given a little
extra of good X. How much Y is
needed to return to the original
indifference curve?
U ( x, y )  k
x
x*
=0
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
U ( x*  x, y* )  U ( x* , y* ) 
U ( x, y *  y)  U ( x* , y* ) 
y = 0

 x + 

x
y




y
x
y*
Now, let the change in X become
arbitrarily small
y
U ( x, y )  k
x
x*
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
U x ( x* , y* )dx  U y ( x* , y* )dy  0
y
y
Marginal Utility of X
Marginal Utility of Y
U x ( x* , y * )
dy
MRS 

dx
U y ( x* , y * )
*
U ( x, y )  k
x
x*
The marginal rate of substitution (MRS) measures the amount of Y
you are willing to give up in order to acquire a little more of X
MRS ( x* , y* )  MRS ( x' , y' )
y
If you have a lot of X relative to Y, then X is
much less valuable than Y MRS is low)!
y*
U ( x, y )  k
y'
x
*
x'
x
An Example

U ( x, y)  x y
U x ( x, y )  x
 1

U y ( x, y)  x y
y

 1

U x ( x* , y * ) x 1 y    y 
   1   
*
*
U y ( x , y ) x y
 x
The elasticity of substitution measures the curvature of the
indifference curve
y
 y
 
x
 y
%  
x

%MRS
'
 y
d 
MRS
x

 y  d MRS 
 
x
 y
 
x
x

U ( x, y)  x y
An Example
U x ( x, y )  x
 1

U y ( x, y)  x y
 y
d 
x  
d MRS  
y


 1
U x ( x* , y * ) x 1 y    y 
   1   
*
*
U y ( x , y ) x y
 x
   y 
  
    x 

1
  y
 
x
Consumers solve a constrained maximization – maximize
utility subject to an income constraint.
max U ( x, y )
x 0, y 0
subject to
px x  p y y  I
As before, set up the lagrangian…
 ( x, y ,  )  U ( x , y )   ( I  p x x  p y y )
 ( x, y ,  )  U ( x , y )   ( I  p x x  p y y )
First Order Necessary Conditions
 x ( x, y,  )  U x ( x, y)  px  0
 y ( x, y,  )  Uy( x, y )  p y  0
U x ( x, y) Px

U y ( x, y) Py
  ( x, y ,  )  I  p x x  p y y  0
px x  p y y  I

U y ( x, y )
py
U x ( x, y )

px
max U ( x, y )
x 0, y 0
subject to
px x  p y y  I
y
x *  x( p x , p y , I )
I
py
y
y*  x( px , p y , I )
*
x
*
I
px
x
.5
max x y
.5
x 0, y 0
subject to
px x  p y y  I
( x, y,  )  x y   ( I  px x  p y y)
.5
.5
U x ( x, y ) .5 x .5 y .5 Px


.5 .5
U y ( x , y ) .5 x y
Py
 Px
y
P
 y

x


px x  p y y  I
.5
max x y
.5
x 0, y 0
subject to
px x  p y y  I
px x  p y y  I
 Px
p x x  Py 
P
 y
y

x  I


I
x
2 px
I
y
2 py
Suppose that we raise the price of X
Can we be sure that demand for x will
fall?
y
I
py
y*
x
*
I
px
x
Suppose that we raise the price of X, but at the same time,
increase your income just enough so that your utility is
unchanged
y
U x ( x, y) Px

U y ( x, y) Py
I
py
Substitution effect
y*
x
*
I
px
x
Now, take that extra income away…
y
I
py
px x  p y y  I
Income effect
y*
x
*
I
px
x
Demand Curves present the same information in a
different format
px
y
p' x
px
x'
x*
x
D
x'
x*
x
Demand Curves present the same information in a
different format
y
px
 y
%  
x

%MRS
% x
x 
% p x
px
x
x
*
x*
x
Elasticity of Substitution vs. Price Elasticity
y
px
 is small
 x is small
x
x
px
y
 is large
x
 x is large
x
Perfect Complements vs. Perfect Substitutes
y
px
 0
x  0
(Almost)
x
x
px
y
 
x
x  
x
.5
max x y
.5
x 0, y 0
subject to
px
px x  p y y  I
%x
dx p x
x 

%p x dp x x
I
x
2 px
dx
I
 2
dp x
2 px
px
x*
x
px
I
x   2
 1
2 px  I 


 2 px 
Suppose that we raise the price of Y…
U x ( x, y) Px

U y ( x, y) Py
y
px x  p y y  I
I
py
y
Substitution effect (+)
Income effect (-)
Net Effect = ????
*
x
*
I
px
x
Cross Price Elasticity
%x
dx p y
y 

%p y dp y x
px
%x
px
x*
x
.5
max x y
.5
x 0, y 0
subject to
I
x
2 px
I
y
2 py
px x  p y y  I
%x
dx p y
y 

0
%p y dp y x
Income and Substitution effects
cancel each other out!!
Suppose that we raise Income
U x ( x, y) Px

U y ( x, y) Py
y
Substitution effect = 0
px x  p y y  I
I
py
y*
*
x x'
I
px
x
Income effect (-)
Income Elasticity
%x dx I
I 

%I dI x
px
%x
px
x*
x
.5
max x y
.5
x 0, y 0
subject to
I
x
2 px
I
y
2 py
px x  p y y  I
%x dx I
I 

%I
dI x
1
I

1
2 px  I 


 2 px 
Willingness to pay
P
Q  200  2 P
Suppose that we have the
following demand curve
$100
A demand curve tells you the maximum
a consumer was willing to pay for every
quantity purchased.
$50
D
100
Q
For the 100th sale of this
product, the maximum anyone
was willing to pay was $50
Willingness to pay
P
Q  200  2 P
Suppose that we have the
following demand curve
$100
$75
$50
D
50
100
Q
For the 50th sale of this product,
the maximum anyone was
willing to pay was $75
Consumer Surplus
Consumer surplus measures the
difference between willingness to pay
and actual price paid
Q  200  2 P
P
$100
$75
Whoever purchased the 50th unit of
this product earned a consumer
surplus of $25
$50
D
50
100
Q
For the 50th sale of this product,
the maximum anyone was
willing to pay was $75
Consumer Surplus
Consumer surplus measures the
difference between willingness to pay
and actual price paid
Q  200  2 P
P
$100
If we add up that surplus over all consumers,
we get:
CS = (1/2)($100-$50)(100-0)=$2500
$2500
$50
Total Willingness to Pay ($7500)
- Actual Amount Paid ($5000)
$5000
D
100
Q
Consumer Surplus ($2500)
A useful tool…
In economics, we are often interested in elasticity as a measure
of responsiveness (price, income, etc.)
% x
x 
% p x
dx
%x 
 d ln x 
x
dp x
%p x 
 d ln p x 
px
d (ln x)
x 
d (ln p x )
Estimating demand curves
Given our model of demand as a function of income, and prices,
we could specify a demand curve as follows:
xd  a0  a1 p x  a2 I  a3 p y  
%x
dx px
 px 
x 

 a1  
%px dpx x
 x 
xd  a0  a1 p x  a2 I  a3 p y  
%x
dx px
 px 
x 

 a1  
%px dpx x
 x 
px
High Elasticity
Low Elasticity
x
Linear demand has a
constant slope, but a
changing elasticity!!
Estimating demand curves
We could, instead, use a semi-log equation:
xd  a0  a1 ln p x  a2 ln I  a3 ln p y  
%x
dx 1 a1
x 


%p x d ln p x x x
Estimating demand curves
We could, instead, use a semi-log equation:
ln xd  a0  a1 p x  a2 I  a3 p y  
%x d ln x
x 

p x  a1 p x
%p x
dp x
Estimating demand curves
The most common is a log-linear demand curve:
ln xd  a0  a1 ln p x  a2 ln I  a3 ln p y  
%x
d ln x
x 

 a1
%p x d ln p x
Log linear demand curves are not straight lines, but have
constant elasticities!
.5
max x y
.5
x 0, y 0
subject to
px x  p y y  I
If we assumed that this was the maximization problem underlying
a demand curve, what form would we use to estimate it?
ln xd  a0  a1 ln p x  a2 ln I  a3 ln p y  
H 0 : a1  1
a2  1
a3  0
Estimating demand curves
px
Suppose you observed the
following data points.
Could you estimate the
demand curve?
D
x
Estimating demand curves
A bigger problem with estimating demand curves is the
simultaneity problem.
xd  a0  a1 px  a2 I   d
px
S
Market prices are the result
of the interaction between
demand and supply!!
px
D
xd  xs
x
Estimating demand curves
Case #1: Both supply and
demand shifts!!
px
S
Case #2: All the points are due
to supply shifts
px
S’
S’’
S
S’
S’’
D
D’
D’’
D
x
x
An example…
Demand
Supply
Equilibrium
Suppose you get a random shock
to demand
xd  a0  a1 px  a2 I   d
xs  b0  b1 px   s
xs  xd
The shock effects quantity
demanded which (due to the
equilibrium condition
influences price!
Therefore, price and the error
term are correlated! A big
problem !!
Suppose we solved for price and quantity by using the
equilibrium condition
xs  xd
a0  a1 px  a2 I   d  b0  b1 px   s
 a2    d   s 
 I  

p x  
 b1  a1   b1  a1 
 a2   b1  d  a1 s 
 I  

x  b2 
 b1  a1   b1  a1 
We could estimate the following equations
p x   1 I  1
x   2 I  2
The original parameters are related as follows:
 a2 

 1  
 b1  a1 
 a2 

 2  b2 
 b1  a1 
2
b2 
1
We can solve for the
supply parameter, but
not demand. Why?
xd  a0  a1 px  a2 I   d
xs  b0  b1 px   s
px
By including a demand shifter (Income),
we are able to identify demand shifts
and, hence, trace out the supply curve!!
S
D
D
D
x