Concave function

What Is a Concave Function?

A concave function is a mathematical function characterized by a downward curve, resembling an inverted "U" shape or a bowl opening downwards. In the realm of finance, economics, and optimization problems, concave functions are crucial for modeling scenarios where the rate of change of an output decreases as its input increases. This property is often associated with concepts like diminishing returns or risk aversion.⁴⁶, ⁴⁷

Specifically, for any two points on the graph of a concave function, the line segment connecting these points will lie entirely below or on the graph itself.⁴⁴, ⁴⁵ This characteristic distinguishes it from its counterpart, the convex function. Understanding the behavior of a concave function is fundamental in various analytical frameworks, including utility theory and portfolio optimization.

History and Origin

The mathematical concepts of concavity and convexity have roots in calculus and geometry, developing as tools to describe the curvature of functions. While the precise attribution of the term "concave function" to a single individual is challenging, the underlying mathematical principles were formalized and applied over centuries.

A significant application of concave functions emerged with the development of expected utility theory in the 18th century. Daniel Bernoulli, in 1738, used the concept of diminishing marginal utility to explain the St. Petersburg Paradox. He proposed that the additional utility gained from an extra unit of wealth decreases as one's wealth increases, implying a concave relationship between wealth and utility.⁴³ This idea, later formalized by economists like John von Neumann and Oskar Morgenstern in the mid-20th century, laid the groundwork for modern financial economics and the understanding of individual decision-making under uncertainty.⁴²

Key Takeaways

A concave function exhibits a downward curve, where the slope of the function decreases as the input increases.⁴¹
In economic contexts, concave functions often represent scenarios such as diminishing marginal utility or diminishing returns to scale.⁴⁰
For a twice-differentiable function, concavity is identified when its second derivative is non-positive across its domain.³⁹
Any local maximum of a concave function within a convex set is also its global maximum, simplifying optimization problems.
Concave utility functions are central to modeling risk-averse behavior in financial economics.³⁸

Formula and Calculation

A function (f(x)) is defined as concave over an interval (or a convex set) if, for any two points (x_1) and (x_2) in the interval and any scalar (\lambda) such that (0 \le \lambda \le 1), the following inequality holds:

f(\lambda x_1 + (1 - \lambda)x_2) \ge \lambda f(x_1) + (1 - \lambda)f(x_2)

This formula mathematically captures the idea that the function's value at a weighted average of two points is greater than or equal to the weighted average of the function's values at those points.

For functions that are twice-differentiable, an equivalent and often more practical test for concavity involves examining its second derivative. A function (f(x)) is concave if and only if its second derivative, (f''(x)), is less than or equal to zero for all (x) in its domain:³⁶, ³⁷

f''(x) \le 0

Where:

(f(x)) represents the function.
(x), (x_1), (x_2) are points in the function's domain.
(\lambda) is a scalar weight between 0 and 1, inclusive.
(f'(x)) is the first derivative, representing the slope of the function.
(f''(x)) is the second derivative, indicating the rate of change of the slope.

Interpreting the Concave Function

Interpreting a concave function hinges on understanding its diminishing rate of change. When a function is concave, its slope is non-increasing, meaning that as the input variable increases, the output increases at a progressively slower rate.³⁵ This "bending downwards" characteristic has profound implications in various economic and financial models.

In microeconomics, for example, a concave production function illustrates diminishing marginal product. This means that adding more units of a single input (e.g., labor) to a fixed amount of other inputs will increase total output, but each additional unit of input will contribute less to the total output than the previous one.³⁴ Similarly, in utility theory, a concave utility function signifies diminishing marginal utility. This implies that as an individual consumes more of a good or service, the additional satisfaction or utility derived from each extra unit decreases.³², ³³ Such an interpretation is central to understanding consumer behavior and preferences, particularly regarding risk.

Hypothetical Example

Consider an individual, Sarah, who is evaluating the utility she derives from wealth. Her utility function can be represented as a concave function, demonstrating her risk-averse nature. Let's assume her utility (U) from wealth (W) is given by (U(W) = \sqrt{W}).

Suppose Sarah currently has $100,000. Her utility is (U(100,000) = \sqrt{100,000} \approx 316.23).

Now, imagine she has a choice:

Certainty: Keep her $100,000.
Gamble: A 50% chance of gaining $50,000 (ending with $150,000) and a 50% chance of losing $20,000 (ending with $80,000).

Let's calculate the expected utility of the gamble:

Utility if she wins: (U(150,000) = \sqrt{150,000} \approx 387.30)
Utility if she loses: (U(80,000) = \sqrt{80,000} \approx 282.84)

Expected Utility (Gamble) = (0.50 \times U(150,000) + 0.50 \times U(80,000))
Expected Utility (Gamble) = (0.50 \times 387.30 + 0.50 \times 282.84)
Expected Utility (Gamble) = (193.65 + 141.42 = 335.07)

Comparing the outcomes:

Utility of Certainty: (316.23)
Expected Utility of Gamble: (335.07)

In this specific example, the square root function, being strictly concave, implies that Sarah would slightly prefer the gamble, as it offers a higher expected utility. However, the concavity itself highlights that the increase in utility from gaining $50,000 is less than the decrease in utility from losing $20,000, even though the monetary gain ($50,000) is larger than the monetary loss ($20,000). This illustrates the concept of diminishing marginal utility of wealth, where each additional dollar provides less additional satisfaction. If the gamble had a slightly lower expected monetary value, her risk aversion due to the concave utility function would lead her to prefer the certain outcome.

Practical Applications

Concave functions are foundational across various domains in finance, economics, and decision-making due to their ability to model situations involving diminishing returns or decreasing marginal effects.

Utility Theory: In expected utility theory, the concavity of a utility function is synonymous with risk aversion.³⁰, ³¹ Individuals with concave utility functions prefer a certain outcome to a risky one with the same expected value, reflecting a preference for stability over volatility. This concept is critical in understanding investor behavior and portfolio choices.
Portfolio Optimization: Concave functions play a role in models that seek to maximize expected return for a given level of risk or minimize risk for a target return. While risk itself is often modeled with convex functions (e.g., variance), the overall objective function in a maximization problem might incorporate elements of concavity, particularly when considering expected utility from investment outcomes. The presence of concave functions helps ensure that local maximum solutions found are also global maximum solutions in optimization problems.²⁸, ²⁹
Production and Cost Functions: In microeconomics, a concave production function illustrates diminishing marginal product. This means that as more units of a single input are added (while others are held constant), the additional output gained from each successive unit of input will eventually decrease. This property often leads to convex cost functions for firms.²⁶, ²⁷
Welfare Economics: Concavity of social welfare functions is central to discussions on income redistribution and inequality. A concave social welfare function implies that society values an additional dollar more when it goes to a poorer individual than to a richer one, supporting policies aimed at reducing income disparities.²⁵
Optimal Taxation: The design of progressive tax systems often implicitly relies on the concept of diminishing marginal utility of income, which is described by concave utility functions, to justify taxing higher incomes at higher rates. Research by economists like Louis Kaplow has explored the implications of utility function concavity on optimal income taxation.²⁴

Limitations and Criticisms

While concave functions are widely used and powerful analytical tools in financial and economic modeling, they come with certain limitations and criticisms.

One primary critique stems from the foundational assumptions, particularly in utility theory. The assumption of a consistently concave utility function for wealth implies universal risk aversion. However, empirical evidence suggests that individuals often exhibit mixed risk attitudes—being risk-averse for large gains but risk-seeking for small losses, a phenomenon explored by behavioral economics in theories like Prospect Theory. T²³his deviation from strict concavity highlights that human decision-making is more complex than a single mathematical form can always capture.

Furthermore, applying concave functions to real-world scenarios sometimes requires simplifying assumptions that may not fully reflect market complexities. For instance, while a concave production function is useful for illustrating diminishing returns, actual production processes can exhibit increasing returns to scale over certain ranges before diminishing returns set in. The strict mathematical definition of a concave function might not perfectly capture these nuanced shifts.

²²Another limitation arises in optimization problems. While concavity ensures that a local maximum is also a global maximum, many real-world financial or economic problems involve non-concave functions. Solving such "non-convex" problems becomes significantly more challenging, often requiring more complex computational methods or relying on heuristics that do not guarantee a globally optimal solution.

²¹## Concave Function vs. Convex Function

Concave and convex functions are opposite concepts in mathematics, crucial for describing the curvature of a function and its implications in optimization and economic modeling.

Feature	Concave Function	Convex Function
Shape (Graphical)	Curves downwards, like an inverted "U" or frown.	Curves upwards, like a "U" shape or smile.
Chord Property	Line segment connecting any two points on the graph lies below or on the graph.	²⁰ Line segment connecting any two points on the graph lies above or on the graph.
Slope	Non-increasing (decreasing) as input increases.	¹⁸ Non-decreasing (increasing) as input increases.
Second Derivative	Non-positive ((f''(x) \le 0)). ¹⁴, ¹⁵	Non-negative ((f''(x) \ge 0)). ¹², ¹³
Optimization Use	Often associated with maximization problems (e.g., maximizing utility).	¹⁰, ¹¹ Often associated with minimization problems (e.g., minimizing cost or risk).
Economic Analogy	Diminishing marginal utility, diminishing returns.	⁸ Increasing marginal costs, economies of scale (in reverse).

The confusion between the two often arises because a function (f(x)) is concave if and only if ( -f(x) ) is convex. T⁷his duality means that a maximization problem with a concave objective function can be reformulated as a minimization problem with a convex objective function. Both types of functions are fundamental to mathematical economics and the study of decision theory.

FAQs

What is a simple definition of a concave function?

A concave function is a mathematical curve that bends downwards, similar to an inverted bowl. As you move along the curve from left to right, its slope becomes less steep.

⁶### Where are concave functions used in finance?
In finance, concave functions are primarily used in utility theory to model risk aversion. A concave utility function means that each additional unit of wealth provides less additional satisfaction, leading individuals to prefer less risky investments.

⁵### Can a function be both concave and convex?
Yes, a function can be both concave and convex if and only if it is a linear (or affine) function. For example, a straight line has both properties simultaneously.

³, ⁴### How does concavity relate to diminishing returns?
In economics, a concave production function illustrates diminishing returns. This means that beyond a certain point, adding more of a single input (like labor) to production will still increase output, but at a continually slower rate.

²### Is the natural logarithm function concave?
Yes, the natural logarithm function, (f(x) = \ln(x)), is a classic example of a concave function over its domain. Its second derivative is always negative.¹