Transcript Indifference curves

Indifference Analysis
Shape of ‘Clive’s’ an indifference curve
30
a
Notice that this curve is
downward sloping
28
26
b
24
22
c
Pears
20
18
16
d
14
Why is this and does it
have any economic
meaning?
12
e
10
f
8
g
6
4
2
0
0
2
4
6
8
10
12
Oranges
14
16
18
20
22
INDIFFERENCE ANALYSIS
• To see the special significance of its shape
we need to look at a concept known as the
– diminishing marginal rate of substitution
INDIFFERENCE ANALYSIS
• The rate of Substitution is the rate we are
prepared to exchange Pears (Y) for
Oranges (X).
• Or: If I give ‘Clive’ one more orange how
many pears must I take away to leave him
just as happy…
MRSYX
Y1  Y0
Y


X 1  X 0 X
The MRS of Y for X is:
•
Pears Oranges
MRSYX
Y1  Y0
Y


X 1  X 0 X
MRSYX
24  30  6


 6
76
1
30
24
20
14
10
8
6
6
7
8
10
13
15
20
Point
a
b
c
d
e
f
g
Deriving the marginal rate of substitution (MRS)
30
a
b
Units of good Y
24
20
10
0
0
67
10
Units of good X
20
Deriving the marginal rate of substitution (MRS)
a
30
MRS = Y/ X= - 6
b
Y = - 6
24
Units of good Y
X = 1
20
10
0
0
67
10
Units of good X
20
a
30
MRS = Y/ X= - 6
b
Y = - 6
24
Units of good Y
X = 1
Consider instead
the MRS at 13
oranges
Pears Oranges
20
30
24
20
14
10
9
6
10
0
0
67
10
Units of good X
6
7
8
10
13
14
20
20
Point
a
b
c
d
e
f
g
Deriving the marginal rate of substitution (MRS)
30
a
Pears Oranges
MRS = - 6
b
Y = - 6
24
30
24
20
14
10
9
6
Units of good Y
X = 1
20
10
9
Y = - 1
c
6
7
8
10
13
14
20
MRS = - 1
d
X = 1
0
0
67
10
13 14
Units of good X
20
Point
a
b
c
d
e
f
g
INDIFFERENCE ANALYSIS
• From an Indifference curves to an
indifference Map
• An indifference curve tells us the bundles which
give Clive the same happiness as 13 pears and 10
oranges.
• An indifference map shows us a whole series of
different curves showing which bundles give
different levels of happiness
An indifference map
Units of good Y
30
20
10
I1
I1
0
0
10
Units of good X
20
An indifference map
Units of good Y
30
20
10
I2
0
0
10
Units of good X
I1
20
An indifference map
Units of good Y
30
20
10
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
20
10
I4
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
20
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
This is basically a
map of the happiness
mountain
20
The further up
we are, the
happier we are
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
‘We’re Climbing up the Sunshine Mountain’
• We can think of picking bundles of goods
as an attempt to climb the happiness
mountain.
‘We’re Climbing up the Sunshine Mountain’
• A mountain of course is three dimensional
and it is difficult to view on a screen.This
might be a side view.
• Putting in contours the mountain looks like
this:
• Putting in contours the mountain looks like
this:
• Gradually rotating and looking down from
above it looks like this:
• Gradually rotating and looking down from
above it looks like this:
• Gradually rotating and looking down from
above it looks like this:
• Gradually rotating and looking down from
above it looks like this:
• Gradually rotating and looking down from
above it looks like this:
• Looking down from above the mountasin
looks like this:
• Gradually rotating and looking down from
above it looks like this:
So from above the happiness mountain looks
like this
Putting two axes in:
Y
X
For a given Y, the more X we have the
happier we are:
Y
Y0
X
Similarly, for a given X, the more Y we have
the happier we are:
Y
Y0
X
X0
So the more X and Y we have the happier we
are :
Y
Y0
X
Of course the issue about the happiness
mountain is that we never get there
Y
Y0
X
Of course the issue about the happiness
mountain is that we never get there.
So we are interested in just one
corner of the mountain
Y0
This is basically
the indifference
map we started
with a while
ago
X0
X
An indifference map
Units of good Y
30
20
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
Note for a given
quantity of x, a rise
in y increases our
utility, from I1 to I4
20
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
Similarly, for a given
quantity of y, a rise
in x increases our
utility, from I2 to I5
20
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
An indifference map
Units of good Y
30
So to labour the
point, anything to
the right an
upwards increases
our utility
20
Down and left
decreases it
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
Could two indifference curves ever cross?
Units of good Y
30
20
10
I1
0
0
10
Units of good X
20
Could two indifference curves ever cross?
30
I2
Units of good Y
Could we get this?
20
10
I1
0
0
10
Units of good X
20
Could two indifference curves ever cross?
30
Could we get this?
I2
Units of good Y
Consider points a,b and c
Point a is indifferent to b
20
And point a is indifferent to c
=> b is indifferent to c
a
10
b
c
I1
0
0
10
Units of good X
20
Could two indifference curves ever cross?
Could we get this?
I2
30
Units of good Y
Consider points a,b and c
But point b has more X and Y
than c so b is preferred to C
20
a
Contradiction, so indifference
curves can never cross
10
b
c
I1
0
0
10
Units of good X
20
An indifference map
Units of good Y
30
So just like the map of a
mountain, our contours
of happiness (the
indifference curves)
never cross.
20
10
I5
I4
I1
0
0
10
Units of good X
I2
20
I3
INDIFFERENCE ANALYSIS
• We now have a map which represents peoples’
choices. What else do we need to make an
economic decision about what to consume?
• Answer: Information on prices and Income
• The budget line:
• Suppose the Price of X is Px
• And the Price of Y is Py
• And Income is represented by I
Budget Line
• Our expenditure on X (PxX) and Y (PyY)
must sum to our total income:
PX X  PY Y  I
e.g .
PX  2
PY  1
I  30
2 X  1Y  30
Budget Line
• Our expenditure on X and Y must sum to
our total income:
e.g .
PX  2
If Y =0, then
PY  1
2X = 30
I  30
2 X  1Y  30
and
Max Consumption of
X = 15
Budget Line
• Our expenditure on X and Y must sum to
our total income:
e.g .
PX  2
If X= 0, then 1Y =30
PY  1
and
I  30
2 X  1Y  30
Max Consumption of Y
= 30
A budget line
30 a
Units of good Y
Units of Units of Point on
Units of Units of Point on
good X good Y budget line
good X good Y budget line
0
0
15
15
20
10
30
30
0
0
a
b
Assumptions
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
We join these two points to get the budget line
30 a
Units of good Y
Units of
good X
Units of Point on
good Y budget line
0
15
20
10
30
0
Assumptions
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
A budget line
a
30
Units of good Y
Units of
good X
Units of Point on
good Y budget line
0
15
20
30
0
a
Assumptions
10
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
A budget line
a
30
Units of good Y
Units of
good X
Units of Point on
good Y budget line
0
5
10
15
20
30
20
10
0
a
Assumptions
10
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
A budget line
a
30
Units of good Y
Units of
good X
0
5
10
15
b
20
Units of Point on
good Y budget line
30
20
10
0
a
b
Assumptions
10
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
A budget line
a
30
Units of good Y
Units of
good X
0
5
10
15
b
20
Units of Point on
good Y budget line
30
20
10
0
c
10
a
b
c
Assumptions
PX = £2
PY = £1
Budget = £30
0
0
5
10
Units of good X
15
20
A budget line
a
30
Units of good Y
Units of
good X
0
5
10
15
b
20
Units of Point on
good Y budget line
30
20
10
0
c
10
a
b
c
d
Assumptions
PX = £2
PY = £1
Budget = £30
d
0
0
5
10
Units of good X
15
20
Budget Line
• So all these points satisfy the equation:
PX X  PY Y  I
becomes
2 X  1Y  30
when
PX  2, PY  1, I  30
A budget line
a
30
Units of good Y
Units of
good X
0
15
b
20
Units of Point on
good Y budget line
30
0
Note the
Maximum x
we can
consume is
30/Px
c
10
a
d
d
0
0
5
10
Units of good X
15
20
=30/Px=30/2
A budget line
30/PY =
a
30
Units of good Y
Units of
good X
0
15
b
20
Units of Point on
good Y budget line
30
0
And the
maximum Y
we can
consume is
30/PY = 30/1
c
10
a
d
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
a
30
Units of good Y
So Height is
30/Py
b
20
And length is
30/Px
c
10
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
So (minus) the
slope of the
curve is Height /
Length
c
10
30
Py

30
Px
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
So (minus) the
slope of the
curve is Height /
Length
30 Px

.
Py 30
c
10
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
So (minus) the
slope of the
curve is Height /
Length
Px
30
X

.
Py 30
X
c
10
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
So (minus) the
slope of the
curve is Height /
Length
Px

Py
c
10
d
0
0
5
10
Which is the
relative price of
x in terms of
good y
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
Actually the slope is a
negative number since
we must GIVE UP X to
get Y.
c
10
Px
 
Py
d
0
0
5
10
15
20
=30/Px=30/2
A budget line
30/PY =
Units of good Y
30 a
b
20
So the slope of the
budget constraint is the
relative price of X in
terms of Y
c
10
Px
 
Py
d
0
0
5
10
15
20
=30/Px=30/2
INDIFFERENCE ANALYSIS
• We now combine the indifference curve we
found before and the budget constraint to
find:
• The optimum consumption point
Units of good Y
Finding the optimum consumption
I5
I4
I1
O
Units of good X
I2
I3
Units of good Y
Finding the optimum consumption
Budget line
I5
I4
I1
O
Units of good X
I2
I3
Finding the optimum consumption
r
Consider the Points
r and v
Units of good Y
What indifference
curve are they on?
Could we do better?
I5
I4
v
I1
O
Units of good X
I2
I3
Finding the optimum consumption
Clearly s and u are
better.
r
s
Units of good Y
They are on I2
u
I5
I4
v
I1
O
Units of good X
I2
I3
Finding the optimum consumption
At point t we have
reached the highest
point possible and
this is our optimum
consumption point
r
Units of good Y
s
t
u
I5
I4
v
I1
O
Units of good X
I2
I3
Finding the optimum consumption
Utility is maximised
at the point of
tangency between
the indifference
map and the budget
constraint.
r
Units of good Y
s
Y1
t
u
I5
I4
v
I1
O
X1
Units of good X
I2
I3
INDIFFERENCE ANALYSIS
• The optimum consumption point
• Recall that the slope of the indifference
curve is
• the marginal rate of substitution
• While the slope of the budget constraint is:
• (Minus) the relative price of X and Y
Same slope at t of indifference curve and budget line
So when the
consumer chooses
optimally:
r
Units of good Y
s
Y1
Px
MRS  
Py
t
u
I5
I4
v
I1
O
X1
Units of good X
I2
I3
Same slope at t of indifference curve and budget line
If
r
Units of good Y
s
Y1
Px
MRS  
Py
At s MRS is above the
market price. ‘Clive’ is
prepared to give up more Y
than market requires to get
X – so he does.
t
u
I5
I4
v
I1
O
X1
Units of good X
I2
I3
Same slope at t of indifference curve and budget line
If
r
Units of good Y
s
Y1
Px
MRS  
Py
At u MRS is below the
market price. ‘Clive’ is
prepared to give up more X
than market requires to get
Y – so he does.
t
u
I5
I4
v
I1
O
X1
Units of good X
I2
I3
Same slope at t of indifference curve and budget line
The optimum must
therefore satisfy:
r
Units of good Y
s
Y1
Px
MRS  
Py
t
u
I5
I4
v
I1
O
X1
Units of good X
I2
I3
INDIFFERENCE ANALYSIS
• So now we have established what Clive will
choose to consume IF the price of X is 2,
the price of Y is 1 and his income is 30.
• So XD = f(px,py,I) = f(2,1,30) = x1
• What can we say about X1?
Good Y
We have found the demand for X at particular set of prices
and Income
Y1
I3
B1
X1
B2
I1
I2
B3
Good X
Good Y
We have found the demand for X at particular set of prices
and Income
Y1
I3
B1
X1
B2
I1
I2
B3
Good X
Price of X
Suppose we draw a
diagram below with
Price on one axis and
quantity on the other
X1
Good X
Good Y
We have found the demand for X at particular set of prices
and Income
Y1
I3
B1
Price of X
X1
B2
I1
I2
B3
Good X
And now we project
the demand for x
downwards at the
Price, Px=2
PX=2
X1
Good X
Good Y
We have found the demand for X at particular set of prices
and Income
Y1
I3
B1
Price of X
X1
B2
I1
I2
B3
Good X
What does this point
represent?
PX=2
Ans: A point on the
demand curve!
X1
Good X
INDIFFERENCE ANALYSIS
• So now we have established what Clive will
choose to consume IF the price of X is 2,
the price of Y is 1 and his income is 30.
• So XD = f(px,py,I) = f(2,1,30) = x1
• What can we say about X1?
• What would happen if Px were to change?
• or Py
• Or I
•
It is to these issues we now turn.