Slides on Concavity versus
Quasi-Convexity
This supplements the material in
lecture 4 and the best reference is
Appendix 4 in Microeconomics by
Layard & Walters
Convexity
• A function is convex if
U(x 3 , y 3 )  U(0.5x 10.5y1,0.5x 2 0.5y 2 )
1
1
 U(x 1, y1 )  U(x 2 , y 2 )
2
2
U
This is a representation of the
three-dimensional utility
mountain
y
There are two aspects to the
mountain. Going up it (from
one indifference curve to the
next) and going around it
(staying on the same
indifference curve)
x
y
In two dimensions
our indifference
curves look like this.
x
y
Going Up the
Mountain means
moving from u1 to u2
What is the
shape of the
mountain as
we go up it?
u2
u1
x
U(x,y)
U
y
Is it like this ?
Steep at the
bottom and
flattening out
x
U(x,y)
U
y
Or this ?
Flat at the
bottom and
getting steeper
x
U(x,y)
U
y
Or even this ?
Flat at the
bottom, gets
steeper and
then flattens
out
x
U(x,y)
U
y
The first of these is
Concave?
Looking into cave like
shape
x
U(x,y)
U
y
The second is
Convex (if you
were under the
curve it would
be sloping in
on you!
x
U(x,y)
U
y
And this is
convex at the
bottom before
becoming
concave!
x
y
But Going Up the
Mountain is only
one part of the
problem.
u2
u1
x
y
B
But Going Up the
Mountain is only
one part of the
problem.
What about moving
around it from A to
B say. What shape is
that?
A
u2
u1
x
The mountain might be nice
and rounded but have crosssections like this.
y1
u1
Concavity and Quasi-Convexity
• We can rule out all these problems if
the Utility function is Concave
– (looking into cave from below)
• and if the indifference curves are
quasi-convex
– (that is the cross-sections look convex
looking from the origin of the x,y graph).
• What does this mean in terms of our
diagrams?
U(x,y)
U
U1
The utility we get
from consuming
x1 and y1
y
y1
U1
x
x1
U(x,y)
U
U1
y
Consider
U2(x2,y2)
y1
x
x1
U(x,y)
U
U1
y
Consider
U2(x2,y2)
U2
y1
y2
x
x2
x1
U(x,y)
U
U1
y
If the Utility Function
is Concave then:
U2
y1
U(x 3 , y 3 ) 
U(0.5x 10.5x 2 ,0.5y10.5y 2 )
y2
x2
1
1
 U(x 1, y1 )  U(x 2 , y 2 )
2
2
x
x1
U(x,y)
U
U1
y
y3
U2
Pick x3,y3 halfway between
x1,y1 and x2,y2
y1
y2
x2
x3
x
x1
U(x,y)
U
U1
y
U2
y1
1
1
U(x 1, y1 )  U(x 2 , y 2 )
2
2
y2
x2
x3
x
x1
U(x,y)
U
U1
U3
y
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
U2
y1
So the utility function
is concave
y2
x2
x3
x1
x
Concave Utility Function
• So if this property holds then the Utility
function looks like the top quarter of a
football
• What will the cross-sections look like?
y
If the utility function is
concave everywhere
then the indifference
curve looks like this
We say it is Quasiconvex because the
cross-sections look
convex from the x,y
origin
x
And this special Quasi-convexity
property holds along the
indifference curve:
U(x 3 , y3 )  U(0.5x 10.5x 2 ,0.5y10.5y 2 )
1
1
 U(x 1, y1 )  U(x 2 , y 2 )  U(x 1, y1 )
2
2
Where U(x1,y1) = U(x2,y2)
What does Quasi-convex mean?
• Suppose we take a weighted average of two
bundles on the same indifference curve and
compare the utility we get from this new
bundle compared with the utility we got
from the originals.
• If it is higher we say that the function is
quasi-convex.
y
U(x1,y1) = U(x2,y2)
y1
U(x2,y2)
y2
x1
x2
x
Consider a new
bundle: (x3, y3) where
y
U(x1,y1)
y1
x3= half of x1 and x2
and
y3= half of y1 and y2
y3
U(x2,y2)
y2
x1
x3
x2
x
What is the utility
associated with this
U(x1,y1) new bundle?
y
y1
y3
U(x2,y2)
y2
x1
x3
x2
x
y
y1
U(x3,y3)
y3
y2
x1
x3
x2
x
y
If
y1
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
Then we say the indifference
curve is quasi-convex
y3
U(x3,y3)
y2
x1
x3
x2
x
U(x 3 , y 3 )
y
1
1
 U(x 1, y1 )  U(x 2 , y 2 )
2
2
 U(x 1, y1 )  U(x 2 , y 2 )
y1
y3
y2
x1
x3
x2
x
y
U(x 3 , y3 )
 U(x 1, y1 )
y1
y3
y2
x1
x3
x2
x
Note The bundle need not be x3, y3,
but any point on the red line. That is,
we could use any fraction l instead of
1/2. If the indifference curve is quasiconvex the condition
y
y1
would still hold
y3
U(x 4 , y 4 )
 U(x 1, y1 )
y2
x1
x3
x2
x
y
But this
indifference curve is
convex, since
y1
y3
y2
x1
x3
x2
x
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
 U(x 1, y1 )
y
y1
U(x3,y3)
y3
But not Strictly
convex
y2
x1
x3
x2
x
Strict Convexity
• So we really need Strict convexity
• And it is STRICTLY convex if
U( lx1(1 - l )x 2 , ly1(1 - l )y 2 ) 
lU(x 1, y1 )  (1  l ) U(x 2 , y 2 )
 U(x 1, y1 )
Where l lies between 0 and 1
1
1
U(x 3 , y3 )  U(x 1, y1 )  U(x 2 , y 2 )
2
2
y
y1
Strictly Convex
y3
y2
x1
x3
x2
x
Strict Convexity rules out every case here except case (b)
y
y
x
x
(a)
(b)
y
y
(c)
x
(d)
x