3.4 Quasiconcavity and quasiconvexity

Definitions and basic properties

Think of a mountain in the Swiss Alps: cows grazing on the verdant lower slopes, snow capping the majestic peak.

Now forget about the cows and the snow. Ask yourself whether the function defining the surface of the mountain is concave. It is if every straight line connecting two points on the surface lies everywhere on or under the surface.

If, for example, the mountain is a perfect dome (half of a sphere), then this condition is satisfied, so that the function defined by its surface is concave. The condition is satisfied also if the mountain is a perfect cone. In this case, every straight line connecting two points on the surface lies exactly on the surface.

Now suppose that the mountain is a deformation of a cone that gets progressively steeper at higher altitudes---call it a "pinched cone". (Many mountains seem to have this characteristic when you try to climb them!) That is, suppose that when viewed from far away, the mountain looks like this:

In this case, a straight line from the top of the mountain to any other point on the surface does not lie on or under the surface, but rather passes through clear air. Thus the function defined by the surface of the mountain is not concave.

The function does, however, share a property with a perfect dome and a cone: on a topographic map of the mountain, the set of points inside each contour---the set of points at which the height of the mountain exceeds any given number---is convex. In fact, each contour line of this mountain, like each contour line of a perfect dome and of a cone, is a circle. If we draw contour lines for regularly-spaced heights (e.g. 50m, 100m, 150m, ...), then topographic maps of the three mountains look like this:

The spacing of the contour lines differs, but the set of points inside every contour has the same shape for each mountain---it is a disk. In particular, every such set is convex.

If we model the surface of the mountain as a function  f  of its longitude and latitude (x, y), then a contour is a level curve of  f . A function with the property that for every value of a the set of points (x, y) such that  f (x, y) ≥ a---the set of points inside every contour on a topographic map---is convex is said to be quasiconcave.

Not every mountain has this property. In fact, if you take a look at a few maps, you'll see that almost no mountain does. A topographic map of an actual mountain is likely to look something like this:

The three outer contours of this mountain definitely do not enclose convex sets. Take, for example, the one in red. The blue line, connecting two points in the set enclosed by the contour, goes outside the set.

Thus the function defined by the surface of this mountain is not quasiconcave.

Let  f  be a multivariate function defined on the set S. We say that  f  (like the function defining the surface of a mountain) is quasiconcave if, for any number a, the set of points for which  f (x) ≥ a is convex. For any real number a, the set

Pa = {xS:  f (x) ≥ a}
is called the upper level set of  f  for a. (In the case of a mountain, Pa is the set of all points at which the altitude is at least a.)

Example
Let  f (x, y) = −x2y2. The upper level set of  f  for a is the set of pairs (x, y) such that −x2y2a, or x2 + y2 ≤ −a. Thus for a > 0 the upper level set Pa is empty, and for a < 0 it is a disk of radius a1/2.

Definition
The multivariate function  f  defined on a convex set S is quasiconcave if every upper level set of  f  is convex. (That is, Pa = {xS:  f (x) ≥ a} is convex for every value of a.)

The notion of quasiconvexity is defined analogously. First, for any real number a, the set

Pa = {xS:  f (x) ≤ a}
is the set of all the points that yield a value for the function of at most a; it is called the lower level set of  f  for a. (In the case of a mountain, Pa is the set of all points at which the altitude is at most a.)

Definition
The multivariate function  f  defined on a convex set S is quasiconvex if every lower level set of  f  is convex. (That is, Pa = {xS:  f (x) ≤ a} is convex for every value of a.)

Note that  f  is quasiconvex if and only if − f  is quasiconcave.

The notion of quasiconcavity is weaker than the notion of concavity, in the sense that every concave function is quasiconcave. Similarly, every convex function is quasiconvex.

Proposition
A concave function is quasiconcave. A convex function is quasiconvex.
Proof
Denote the function by  f , and the (convex) set on which it is defined by S. Let a be a real number and let x and y be points in the upper level set Pa: xPa and yPa. We need to show that Pa is convex. That is, we need to show that for every λ ∈ [0,1] we have (1 − λ)x + λyPa.

First note that the set S on which  f  is defined is convex, so we have (1 − λ)x + λyS, and thus  f  is defined at the point (1 − λ)x + λy.

Now, the concavity of  f  implies that

 f ((1−λ)x + λy) ≥ (1−λ) f (x) + λ f (y).
Further, the fact that xPa means that  f (x) ≥ a, and the fact that yPa means that  f (y) ≥ a, so that

(1−λ) f (x) + λ f (y) ≥ (1−λ)a + λa = a.
Combining the last two inequalities, we have

 f ((1−λ)x + λy) ≥ a,
so that (1−λ)x + λyPa. Thus every upper level set is convex and hence  f  is quasiconcave.

The converse of this result is not true: a quasiconcave function may not be concave. Consider, for example, the function  f (x, y) = xy defined on the set of pairs of nonnegative real numbers. This function is quasiconcave (its upper level sets are the sets of points above rectangular hyperbolae), but is not concave (for example,  f (0, 0) = 0,  f (1, 1) = 1, and  f (2, 2) = 4, so that  f ((1/2)(0, 0) + (1/2)(2, 2)) =  f (1, 1) = 1 < 2 = (1/2) f (0, 0) + (1/2) f (2, 2)).

Some properties of quasiconcave functions are given in the following result. (You are asked to prove the first result in an exercise.)

Proposition
  • If the function U is quasiconcave and the function g is increasing then the function  f  defined by  f (x) = g(U(x)) is quasiconcave.
  • If the function U is quasiconcave and the function g is decreasing then the function  f  defined by  f (x) = g(U(x)) is quasiconvex.
In an
exercise you are asked to show that the sum of quasiconcave functions may not be quasiconcave.

Why are economists interested in quasiconcavity?

The standard model of a decision-maker in economic theory consists of a set of alternatives and an ordering over these alternatives. The decision-maker is assumed to choose her favorite alternative---that is, an alternative with the property that no other alternative is higher in her ordering.

To facilitate the analysis of such a problem, we often work with a function that "represents" the ordering. Suppose, for example, that there are four alternatives, a, b, c, and d, and the decision-maker prefers a to b to c and regards c and d as equally desirable. This ordering is represented by the function U defined by U(a) = 3, U(b) = 2, and U(c) = U(d) = 1. It is represented also by many other functions---for example V defined by V(a) = 100, V(b) = 0, and V(c) = V(d) = −1. The numbers we assign to the alternatives are unimportant except insofar as they are ordered in the same way that the decision-maker orders the alternatives. Thus any function W with W(a) > W(b) > W(c) = W(d) represents the ordering.

When the decision-maker is a consumer choosing between bundles of goods, we often assume that the level curves of the consumer's ordering---which we call "indifference curves"---look like this

and not like this or like this

That is, we assume that every upper level set of the consumer's ordering is convex, which is equivalent to the condition that any function that represents the consumer's ordering is quasiconcave.

It makes no sense to impose a stronger condition, like concavity, on this function, because the only significant property of the function is the character of its level curves, not the specific numbers assigned to these curves.

Functions of a single variable

The definitions above apply to any function, including those of a single variable. For a function of a single variable, an upper or lower level set is typically an interval of points, or a union of intervals. In the following figure, for example, the upper level set for the indicated value a---that is, the set of values of x for which  f (x) ≥ a---is the union of the two blue intervals of values of x: the set of all values that are either between x' and x" or greater than x'''.

By drawing some examples, you should be able to convince yourself of the following result.

Proposition
A function  f  of a single variable is quasiconcave if and only if either
  • it is nondecreasing,
  • it is nonincreasing, or
  • there exists x* such that  f  is nondecreasing for x < x* and nonincreasing for x > x*.

Note that this result does NOT apply to functions of many variables!

Another characterization of quasiconcavity

The following alternative characterization of a quasiconcave function (of any number of variables) is sometimes useful.

Proposition
The function  f  is quasiconcave if and only if for all xS, all x' ∈ S, and all λ ∈ [0,1] we have

if  f (x) ≥  f (x') then  f ((1−λ)x + λx') ≥  f (x').

That is, a function is quasiconcave if and only if the line segment joining the points on two level curves lies nowhere below the level curve corresponding to the lower value of the function. This condition is illustrated in the following figure, in which a' > a: all points on the green line, joining x and x', lie on or above the indifference curve corresponding to the smaller value of the function (a).

Strict quasiconcavity

This characterization of quasiconcavity motivates the following definition of a strictly quasiconcave function.
Definition
The multivariate function  f  defined on a convex set S is strictly quasiconcave if for all xS, all x' ∈ S with x' ≠ x, and all λ ∈ (0,1) we have

if  f (x) ≥  f (x') then  f ((1−λ)x + λx') >  f (x').

That is, a function is strictly quasiconcave if every point, except the endpoints, on any line segment joining points on two level curves yields a higher value for the function than does any point on the level curve corresponding to the lower value of the function.

The definition says that a quasiconcave function of a single variable is strictly quasiconcave if its graph has no horizontal sections.

For a function of two variables, it says that no level curve of a strictly quasiconcave function contains a line segment. (Take x = x' in the definition.) Two examples of functions that are not strictly quasiconcave, though the level curves indicated are consistent with the functions' being quasiconcave, are shown in the following figure. In both cases, the red level curve contains a line segment. (In the diagram on the right, it does so because it is "thick"---see the earlier example.)

How can we tell if a twice-differentiable function is quasiconcave or quasiconvex?

To determine whether a twice-differentiable function is quasiconcave or quasiconvex, we can examine the determinants of the bordered Hessians of the function, defined as follows:
Dr(x) =
 0     f 1'(x    f 2'(x  ...     f r'(x
  f 1'(x    f 11"(x    f 12"(x  ...     f 1r"(x
  f 2'(x    f 12"(x    f 22"(x  ...     f 2r"(x
 ...    ...    ...   ...    ... 
  f r'(x    f 1r"(x    f 2r"(x  ...     f rr"(x

Notice that a function of n variables has n bordered Hessians, D1, ..., Dn.

Proposition
Let  f  be a function of n variables with continuous partial derivatives of first and second order in an open convex set S.
  • If  f  is quasiconcave then D1(x) ≤ 0, D2(x) ≥ 0, ..., Dn(x) ≤ 0 if n is odd and Dn(x) ≥ 0 if n is even, for all x in S. (Note that the first condition is automatically satisfied.)
  • If  f  is quasiconvex then Dk(x) ≤ 0 for all k, for all x in S. (Note that the first condition is automatically satisfied.)
  • If D1(x) < 0, D2(x) > 0, ..., Dn(x) < 0 if n is odd and Dn(x) > 0 if n is even for all x in S then  f  is quasiconcave.
  • If Dk(x) < 0 for all k, for all x in S then  f  is quasiconvex.
Another way to state this result is to say that "D1(x) ≤ 0, D2(x) ≥ 0, ..., Dn(x) ≤ 0 if n is odd and Dn(x) ≥ 0 if n is even, for all x in S" is a necessary condition for quasiconcavity, whereas "D1(x) < 0, D2(x) > 0, ..., Dn(x) < 0 if n is odd and Dn(x) > 0 if n is even for all x in S" is a sufficient condition, and similarly for quasiconvexity.

Note that the conditions don't cover all the possible cases, unlike the analogous result for concave functions. If, for example, Dk(x) ≤ 0 for all k, for all x, but Dr(x) = 0 for some r and some x, then the result does not rule out the possibility that the function is quasiconvex, but it does not tell us that it is.

Example
Consider the function  f (x) = x2 for x > 0. We have D1(x) = −4x2 < 0 for all x > 0, so we deduce that this function is both quasiconcave and quasiconvex on the set {x: x > 0}.

Example
Consider the function  f (x) = x2 for x ≥ 0. We have D1(0) = 0, so this function does not satisfy the sufficient conditions for either quasiconcavity or quasiconvexity, although it is in fact both quasiconcave and quasiconvex.

Example
Consider the function  f (x1,x2) = x1x2. For x > 0 the sufficient conditions for quasiconcavity are satisfied, while the necessary conditions for quasiconvexity are not. Thus the function is quasiconcave and not quasiconvex on the set {x: x > 0}. For x ≥ 0 the sufficient conditions for quasiconcavity are not satisfied, but the necessary conditions are not violated. (The function is in fact quasiconcave on this domain.)

Exercises


Copyright © 1997–2014 by Martin J. Osborne. This version 2014.3.12.