# Quasiconcave Utility Functions

"Quasiconcave" is a mathematical concept that has several applications in economics. To understand the significance of the term's applications in economics, it is useful to begin with a brief consideration of the origins and meaning of the term in mathematics.

## Origins of the Term

The term "quasiconcave" was introduced in the early part of the 20th century in the work of John von Neumann, Werner Fenchel and Bruno de Finetti, all prominent mathematicians with interests in both theoretical and applied mathematics, Their research in fields such as probability theory, game theory and topology eventually laid the groundwork for an independent research field known as "generalized convexity." While the term "quasiconcave: has applications in many areas, including economics, it originates in the field of generalized convexity as a topological concept.

## Definition of Topology

Wayne State Mathematics Professor Robert Bruner's brief and readable explanation of topology begins with the understanding that topology is a special form of geometry. What distinguishes topology from other geometrical studies is that topology treats geometric figures as being essentially ("topologically") equivalent if by bending, twisting and otherwise distorting them you can turn one into the other.

This sounds a little strange, but consider that if you take a circle and begin squashing from four directions, with careful squashing you can produce a square. Thus, a square and a circle are topologically equivalent. Similarly, if you bend one side of a triangle until you've created another corner somewhere along that side, with more bending, pushing and pulling, you can turn a triangle into a square. Again, a triangle and a square are topologically equivalent.

## Quasiconcave as a Topological Property

Quasiconcave is a topological property that includes concavity. If you graph a mathematical function and the graph looks more or less like a badly made bowl with a few bumps in it but still has a depression in the center and two ends that tilt upward, that is a quasiconcave function.

It turns out that a concave function is just a specific instance of a quasiconcave function—one without the bumps. From a layperson's perspective (a mathematician has a more rigorous way of expressing it), a quasiconcave function includes all concave functions and also all functions that overall are concave but that may have sections that are actually convex. Again, picture a badly made bowl with a few bumps and protrusions in it.

## Applications in Economics

One way of mathematically representing consumer preferences (as well as many other behaviors) is with a utility function. If, for example, consumers prefer good A to good B, the utility function U expresses that preference as:

U(A)>U(B)

If you graph out this function for a real-world set of consumers and goods, you may find that the graph looks a bit like a bowl—rather than a straight line, there's a sag in the middle. This sag generally represents consumers' aversion to risk. Again, in the real world, this aversion isn't consistent: the graph of consumer preferences looks a bit like an imperfect bowl, one with a number of bumps in it. Instead of being concave, then, it's generally concave but not perfectly so at every point in the graph, which may have minor sections of convexity.

In other words, our example graph of consumer preferences (much like many real-world examples) is quasiconcave. They tell anyone wanting to know more about consumer behavior—economists and corporations selling consumer goods, for instance—where and how customers respond to changes in good amounts or cost.

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