In this case the condition for concavity can be expressed as. A graphic illustration. Figure 2. Notice that in all three cases thelevel curves of the function have hyperbolic, convex shapes. That is, for any fixed value of ythe functions are quite similar. This shows the quasi-concavity of the function. The primarydifferences among the functions are illustrated by the way in which the value of y increases as. In Figure 2. This gives the function a rounded, teacuplike shape that indicates its concavity.
This is the borderline between concavity and convexity. The spine of the function looks convex to reflect such increasing returns. A careful look at Figure 2. You are asked to prove that this is indeed the case in Problem 2. This example shows that the converse of this statement is not true—quasi-concave functions need not necessarily be concave. Most functions we will encounter in this book will also illustrate this fact; most will be quasi-concave but not necessarily concave.
Ask your question! Definition The function f of many variables defined on a convex set S is quasiconvex if every lower level set of f is convex.
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. We need to show that P a is convex.
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. 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. 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. 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.
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 a function of a single variable. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Linked 0. Related 1.
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Line segments connecting any two points must lie inside the shape. Here I draw two lines, both of them are in the interior. Let's plot a quasi-concave function now. To graphically illustrated this, let's draw a couple of levels into our quasi-concave function and then look at the corresponding contour plot i. See below. Next, here's the contour plot.
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