Quasiconcavity And Quasiconvexity

Last Updated on Sat, 13 Aug 2022 | Time Path

In Sec. 11.5 it was shown that, for a problem of free extremum, a knowledge of the concavity or convexity of the objective function obviates the need to check the second-order condition. In the context of constrained optimization, it is again possible to dispense with the second-order condition if the surface or hyper-surface has the appropriate type of configuration. But this time the desired configuration is quasiconcavity (rather than concavity) for a maximum, and quasiconvexity (rather than convexity) for a minimum. As we shall demonstrate, quasiconcavity (quasiconvexity) is a weaker condition than concavity (convexity). This is only to be expected, since the second-order sufficient condition to be dispensed with is also weaker for the constrained optimization problem (d2z definite in sign only for those dx, satisfying dg = 0) than for the free one (d2z definite in sign for all dxt).

Geometric Characterization

Quasiconcavity and quasiconvexity, like concavity and convexity, can be either strict or nonstrict. We shall first present the geometric characterization of these concepts:

Let u and v be any two distinct points in the domain (a convex set) of a function /, and let line segment uv in the domain give rise to arc MN on the graph of the function, such that point N is higher than or equal in height to point M. Then function /is said to be quasiconcave (quasiconvex) if all points on arc MN other than M and N are higher than or equal in height to point M (lower than or equal in height to point N). The function / is said to be strictly quasiconcave (strictly quasiconvex) if all the points on arc MN other than M and N are strictly higher than point M (strictly lower than point N).

It should be clear from this that any strictly quasiconcave (strictly quasiconvex) function is quasiconcave (quasiconvex), but the converse is not true.

For a better grasp, let us examine the illustrations in Fig. 12.3, all drawn for the one-variable case. In diagram a, line segment uv in the domain gives rise to arc MN on the curve such that N is higher than M. Since all the points between M and N on the said arc are strictly higher than M, this particular arc satisfies the condition for strict quasiconcavity. For the curve to qualify as strictly quasiconcave, however, all possible (w, t>) pairs must have arcs that satisfy the same condition. This is indeed the case for the function in diagram a. Note that this function also satisfies the condition for (nonstrict) quasiconcavity. But it fails the condition for quasiconvexity, because some points on arc MN are higher than N, which is forbidden for a quasiconvex function. The function in diagram b has the opposite configuration. All the points on arc M'N' are lower than N', the higher of the two ends, and the same is true of all arcs that can be drawn. Thus the function in diagram b is strictly quasiconvex. As you can verify, it also satisfies the condition for (nonstrict) quasiconvexity, but fails the condition for quasiconcavity. What distinguishes diagram c is the presence of a horizontal line segment M"N", where all the points have the same height. As a result, that line segment—and hence the entire curve—can only meet the condition for quasiconcavity, but not strict quasiconcavity.

Generally speaking, a quasiconcave function that is not also concave has a graph roughly shaped \ike a bell, or a portion thereof, and a quasiconvex function has a graph shaped like an inverted bell, or a portion thereof. On the bell, it is admissible (though not required) to have both concave and convex segments. This more permissive nature of the characterization makes quasiconcavity (quasicon-

Not Quasi Convex Function Quasiconcavity

Figure 12.4

Figure 12.4

vexity) a weaker condition than concavity (convexity). In Fig. 12.4, we contrast strict concavity against strict quasiconcavity for the two-variable case. As drawn, both surfaces depict increasing functions, as they contain only the ascending portions of a dome and a bell, respectively. The surface in diagram a is strictly concave, but the one in diagram b is certainly not, since it contains convex portions near the base of the bell. Yet it is strictly quasiconcave; all the arcs on the surface, exemplified by MN and M'N\ satisfy the condition that all the points on each arc between the two end points are higher than the lower end point. Returning to diagram^, we should note that the surface therein is also strictly quasiconcave. Although we have not drawn any illustrative arcs MN and M'N' in diagram a, it is not difficult to check that all possible arcs do indeed satisfy the condition for strict quasiconcavity. In general, a strictly concave function must be strictly quasiconcave, although the converse is not true. We shall demonstrate this more formally in the paragraphs that follow.

Algebraic Definition

The geometric characterization above can be translated into an algebraic definition for easier generalization to higher-dimensional cases:

f quasiconcave)

A function / is . iff, for any pair of distinct points u and v in

(quasiconvex j the (convex) domain of /, and for 0 < 6 < 1,

To adapt this definition to strict quasiconcavity and quasiconvexity, the two weak

You may find it instructive to compare (12.20) with (11.20).

From this definition, the following three theorems readily follow. These will be stated in terms of a function /(x), where x can be interpreted as a vector of variables, x = (x,,..., xn).

Theorem I (negative of a function) If/(x) is quasiconcave (strictly quasicon-cave), then — /(x) is quasiconvex (strictly quasiconvex).

Theorem II (concavity versus quasiconcavity) Any concave (convex) function is quasiconcave (quasiconvex), but the converse is not true. Similarly, any strictly concave (strictly convex) function is strictly quasiconcave (strictly quasiconvex), but the converse is not true.

Theorem III (linear function) If f(x) is a linear function, then it is quasiconcave as well as quasiconvex.

Theorem I follows from the fact that multiplying an inequality by — 1 reverses the sense of inequality. Let /(x) be quasiconcave, with f(v) > /(h). Then, by (12.20), f[6u + (1 - 6)v] >/(«). As far as the function -/(x) is concerned, however, we have (after multiplying the two inequalities through by - 1) -/(h) > -/(v) and -f[6u + (1 - 6)v] < -/(h). Interpreting -/(h) as the height of point N, and —f(v) the height of M, we see that the function —/(x) satisfies the condition for quasiconvexity in (12.20). This proves one of the four cases cited in Theorem I; the proofs for the other three are similar.

For Theorem II, we shall only prove that concavity implies quasiconcavity. Let f(x) be concave. Then, by (11.20),

Now assume that f(v) > f(u); then any weighted average of f(v) and f(u) cannot possibly be less than /(h), i.e., which satisfies the definition of quasiconcavity in (12.20). Note, however, that the condition for quasiconcavity cannot guarantee concavity.

Once Theorem II is established, Theorem III follows immediately. We already know that a linear function is both concave and convex, though not strictly so. In view of Theorem II, a linear function must also be both quasiconcave and quasiconvex, though not strictly so.

inequalities on the right should be changed into strict inequalities f[6u + (1 - 0)t>] > 6f(u) + (1 - 6)f(v)

In the case of concave and convex functions, there is a useful theorem to the effect that the sum of concave (convex) functions is also concave (convex). Unfortunately, this theorem cannot be generalized to quasiconcave and quasicon-vex functions. For instance, a sum of two quasiconcave functions is not necessarily quasiconcave (see Exercise 12.4-3).

Sometimes it may prove easier to check quasiconcavity and quasiconvexity by the following alternative definition:

(quasiconcave)

A function fix), where x is a vector of variables, is { } iff, for

, quasiconvex any constant k, the set ^ '

The sets S~ and S~ were introduced earlier (Fig. 11.10) to show that a convex function (or even a concave function) can give rise to a convex set. Here we are employing these two sets as tests for quasiconcavity and quasiconvexity. The three functions in Fig. 12.5 all contain concave as well as convex segments and hence are neither convex nor concave. But the function in diagram a is quasiconcave, because for any value of k (only one of which has been illustrated), the set S~ is convex. The function in diagram b is, on the other hand, quasiconvex since the set S1- is convex. The function in diagram c—a monotonic function—differs from the other two in that both and S- are convex sets. Hence that function is quasiconcave as well as quasiconvex.

Note that while (12.21) can be used to check quasiconcavity and quasiconvexity, it is incapable of distinguishing between strict and nonstrict varieties of these properties. Note, also, that the defining properties in (12.21) are in themselves not sufficient for concavity and convexity, respectively. In particular, given a concave function which must perforce be quasiconcave, we can conclude that S~ is a

Concave Function Economics

convex set; but given that S~ is a convex set, we can conclude only that the function / is ¿/«ohconcave (but not necessarily concave).

Example 1 Check z = x2 (x > 0) for quasiconcavity and quasiconvexity. This function is easily verified geometrically to be convex, in fact strictly so. Hence it is quasiconvex. Interestingly, it is also quasiconcave. For its graph—the right half of a U-shaped curve, initiating from the point of origin and increasing at an increasing rate—is, similarly to Fig. 12.5c, capable of generating a convex S~ as well as a convex S~ .

If we wish to apply (12.20) instead, we first let u and v be any two distinct nonnegative values of x. Then f{u) = u2 f(v) = v2 and f[du + (1 - 6)v] = [6u + (1 - 6)v}2

Suppose that f(v) > f(u), that is, v2 > w2; then v > u, or more specifically, v > u (since u and v are distinct). Inasmuch as the weighted average [Ou + (1 — S)v] must lie between u and v, we may write the continuous inequality v2 > [Ou + (1 - 6)v]2 > u2 forO < 6 < 1

or f(v) > f[0u + (1 - 6)v] > f(u) for 0 < # < 1

By (12.20), this result makes the function / both quasiconcave and quasiconvex—indeed strictly so.

Example 2 Show that z = /(x, y) = xy (x, y > 0) is quasiconcave. We shall use th>criterion in (12.21) and establish that the set S~ = {(x, y) | xy > k) is convex for any k. For this purpose, we set xy = k to obtain an isovalue curve for each value of k. Like x and y, k should be nonnegative. In case k > 0, the isovalue curve is a rectangular hyperbola in the first quadrant of the xy plane. The set S~ , consisting of all the points on or above a rectangular hyperbola, is a convex set. In the other case, with k = 0, the isovalue curve as defined by xy = 0 is L-shaped, with the L coinciding with the nonnegative segments of the x and y axes. The set S> , consisting this time of the entire nonnegative quadrant, is again a convex set. Thus, by (12.21), the function z = xy (x, y > 0) is quasiconcave.

You should be careful not to confuse the shape of the isovalue curves xy = k (which is defined in the xy plane) with the shape of the surface z = xy (which is defined in the xyz space). The characteristic of the z surface (quasiconcave in 3-space) is what we wish to ascertain; the shape of the isovalue curves (convex in 2-space for positive k ) is of interest here only as a means to delineate the sets S in order to apply the criterion in (12.21).

Example 3 Show that z = f(x,y)= (x — a)2 + (y — ft)2 is quasiconvex. Let us again apply (12.21). Setting (x - a)2 + (y - ft)2 = k, we see that k must be nonnegative. For each k, the isovalue curve is a circle in the xy plane with its center at (a, ft) and with radius ]/k . Since = {(x, y) \ (x - a)2 + (y - ft)2 < k) is the set of all points on or inside a circle, it constitutes a convex set. This is true even when k = 0—when the circle degenerates into a single point, {a, b)—since by convention a single point is considered as a convex set. Thus the given function is quasiconvex.

Differentiable Functions

The definitions (12.20) and (12.21) do not require differentiability of the function /. If / is differentiable, however, quasiconcavity and quasiconvexity can alternatively be defined in terms of its first derivatives:

• ,, r • , , ■ /■ • I quasiconcave, .

A differentiable function of one variable, f(x), is { ) iff, for any

pair of distinct points u and v in the domain, v

Quasiconcavity and quasiconvexity will be strict, if the weak inequality on the right is changed to the strict inequality > 0. When there are two or more independent variables, the definition is to be modified as follows:

I quasiconcave)

A differentiable function fix,_____ x„) is { > iff, for any two

\ quasiconvex j distinct points u = («,,____un) and v = (u,,..., vn) in the domain,

where / = df/dx to be evaluated at u or v as the case may be.

Again, for strict quasiconcavity and quasiconvexity, the weak inequality on the right should be changed to the strict inequality > 0.

Finally, if a function z = /(x,_____ xn) is twice continuously differentiable, quasiconcavity and quasiconvexity can be checked by means of the first and second partial derivatives of the function, arranged into the bordered determinant

Continue reading here: Utility Maximization And Consumer Demand

Was this article helpful?