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

)y,U(x2

1)y,U(x

2

1

)0.5y,0.5x0.5yU(0.5x)y,U(x

2211

221133

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

In two dimensions our indifference

curves look like this.

x

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?

x

y

U U(x,y)

Is it like this ?

Steep at the bottom and flattening out

x

y

U U(x,y)

Or this ?

Flat at the bottom and getting steeper

x

y

U U(x,y)

Or even this ?

Flat at the bottom, gets steeper and then flattens out

x

y

U U(x,y)

The first of these is Concave?

Looking into cave like shape

x

y

U U(x,y)

The second is Convex (if you were under the curve it would be sloping in on you!

x

y

U U(x,y)

And this is convex at the bottom before becoming concave!

yBut Going Up the Mountain is only

one part of the problem.

x

u1

u2

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

y1

u1

The mountain might be nice and rounded but have cross-sections like this.

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

The utility we get from consuming

x1 and y1

U(x,y)

U1

U1

x

y

U

x1

y1

U(x,y)

Consider U2(x2,y2)

U1

x

y

U

x1

y1

U(x,y)

Consider U2(x2,y2)

x2

y2

U2

U1

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

x3

Pick x3,y3 half- way between x1,y1 and x2,y2

y3

x

y

U

x1

y1

U(x,y)

x2

y2

U2

U1

)y,U(x2

1)y,U(x

2

12211

x3

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) = U(x2,y2)

x1

y1

x2

y2

U(x2,y2)

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

x

U(x1,y1)

x1

y1

x2

y2

U(x2,y2)

What is the utility associated with this new bundle?

x3

y3

y

xx1

y1

y3

x2x3

y2

U(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

y

xx1

y1

y3

x2x3

y2

)y,U(x)y,U(x

)y,U(x2

1)y,U(x

2

1

)y,U(x

2211

2211

33

y

xx1

y1

y3

x2x3

y2

)y,U(x

)y,U(x

11

33

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

y

xx1

y1

y3

x2x3

y2

But this indifference curve is

convex, since

y

xx1

y1

y3

x2x3

y2

)y,U(x

)y,U(x2

1)y,U(x

2

1)y,U(x

11

221133

U(x3,y3)

But not Strictly convex

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

y

xx1

y1

y3

x2x3

y2

)y,U(x2

1)y,U(x

2

1)y,U(x 221133

Strictly Convex

y

x

y

x

y

x

y

x

Strict Convexity rules out every case here except case (b)

(a) (b)

(c) (d)