Slides on Concavity versus Quasi Convexity This supplements

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 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 u 1 to u 2 What is the shape of the mountain as we go up it? u 2 u 1 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. u 2 u 1 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 u 2 u 1 x

The mountain might be nice and rounded but have crosssections like this. y 1 u 1

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 quasiconvex – (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 U 1 The utility we get from consuming x 1 and y 1 y y 1 U 1 x x 1

U(x, y) U U 1 y Consider U 2(x 2, y 2) y 1 x x 1

U(x, y) U U 1 y Consider U 2(x 2, y 2) U 2 y 1 y 2 x x 2 x 1

U(x, y) U U 1 y If the Utility Function is Concave then: U 2 y 1 y 2 x x 2 x 1

U(x, y) U U 1 y y 3 U 2 Pick x 3, y 3 halfway between x 1, y 1 and x 2, y 2 y 1 y 2 x 3 x x 1

U(x, y) U U 1 y U 2 y 1 y 2 x 3 x x 1

U(x, y) U U 1 U 3 y U 2 y 1 So the utility function is concave y 2 x 3 x 1 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: Where U(x 1, y 1) = U(x 2, y 2)

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(x 1, y 1) = U(x 2, y 2) y 1 U(x 2, y 2) y 2 x 1 x 2 x

y U(x 1, y 1) y 1 Consider a new bundle: (x 3, y 3) where x 3= half of x 1 and x 2 and y 3= half of y 1 and y 2 y 3 U(x 2, y 2) y 2 x 1 x 3 x 2 x

What is the utility associated with this U(x 1, y 1) new bundle? y y 1 y 3 U(x 2, y 2) y 2 x 1 x 3 x 2 x

y y 1 U(x 3, y 3) y 3 y 2 x 1 x 3 x 2 x

y y 1 Then we say the indifference curve is quasi-convex y 3 U(x 3, y 3) y 2 x 1 x 3 x 2 x

y y 1 y 3 y 2 x 1 x 3 x 2 x

y y 1 y 3 y 2 x 1 x 3 x 2 x

Note The bundle need not be x 3, y 3, 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 y 1 would still hold y 3 y 2 x 1 x 3 x 2 x

y But this indifference curve is convex, since y 1 y 3 y 2 x 1 x 3 x 2 x

y y 1 U(x 3, y 3) y 3 But not Strictly convex y 2 x 1 x 3 x 2 x

Strict Convexity • So we really need Strict convexity • And it is STRICTLY convex if Where l lies between 0 and 1

y y 1 Strictly Convex y 3 y 2 x 1 x 3 x 2 x

Strict Convexity rules out every case here except case (b) y y x y (a) (c) x y x (b) (d) x