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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
x
y
U
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)
This is a representation of the three-dimensional utility
mountain
y Going Up the Mountain means
moving from u1 to u2
x
u1
u2
What is the shape of the mountain as we go up it?
yBut 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?
x
u1
u2A
B
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?
x
y
U
x1
y1
U(x,y)
x2
y2
U2
U1
)y,U(x2
1)y,U(x
2
1
)0.5y,0.5y0.5xU(0.5x
)y,U(x
2211
2121
33
If the Utility Function is Concave then:
x
y
U
x1
y1
U(x,y)
x2
y2
U2
U1
)y,U(x2
1)y,U(x
2
1)y,U(x 221133
x3
U3
So the utility function is concave
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?
yIf the utility function is
concave everywhere then the indifference curve looks like this
We say it is Quasi-convex because the cross-sections look convex from the x,y
origin
x
And this special Quasi-convexity property holds along the
indifference curve:
)y,U(x)y,U(x2
1)y,U(x
2
1
)0.5y,0.5y0.5xU(0.5x)y,U(x
112211
212133
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
x
U(x1,y1)
x1
y1
x2
y2
U(x2,y2)
Consider a new bundle: (x3, y3) where
x3= half of x1 and x2 and
y3= half of y1 and y2
x3
y3
y
xx1
y1
y3
x2x3
y2
)y,U(x2
1)y,U(x
2
1)y,U(xIf 221133
U(x3,y3)
Then we say the indifference curve is quasi-convex
Note The bundle need not be x3, y3,
but any point on the red line. That is, we could use any fraction instead of 1/2. If the indifference curve is quasi-
convex the condition
would still hold
y
xx1
y1
y3
x2x3
y2
)y,U(x
)y,U(x
11
44
Strict Convexity
• So we really need Strict convexity
• And it is STRICTLY convex if
)y,U(x
)y,U(x)1()y,U(x
))y-(1y,)x-(1xU(
11
2211
2121
Where lies between 0 and 1