Riemann and Riemann-Stieltjes Integrals in Economics: Meaning, Geometry, and Conditions

Introduction
Integration plays a vital role in economics, especially when modeling accumulated values like total cost, revenue, or surplus. Two foundational concepts in integration theory are the Riemann integral and the more general Riemann-Stieltjes integral. While both are used to evaluate areas under curves, the Stieltjes approach allows for integration with respect to a function other than $x$ — an important distinction when modeling weights, distributions, or discontinuous data.

This blog post explores both integrals, their geometric interpretations, conditions for integrability, and relevance to economics.

1. The Riemann Integral: Basics and Intuition
The Riemann integral approximates the area under a curve $f(x)$ over an interval $[a, b]$ by dividing it into rectangles.

Definition:

A function $f$ is Riemann integrable over $[a, b]$ if the limit of its upper and lower sums (based on partitioning the interval) exists and is equal:
$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x_i$

Where:

• $x_i^*$ is a sample point in the $i$-th subinterval
• $\Delta x_i = x_i – x_{i-1}$

Geometric Interpretation:

The Riemann integral gives the net area under the curve, accounting for sign (area below the x-axis is negative).

2. Conditions for Riemann Integrability
A function $f(x)$ is Riemann integrable on $[a, b]$ if:

• It is bounded
• The set of its discontinuities has measure zero (e.g., jump discontinuities at finite points are allowed)

Examples of integrable functions:

• Continuous functions
• Piecewise continuous functions

Non-integrable:

• Dirichlet function: too many discontinuities

3. The Riemann-Stieltjes Integral: Generalizing the Concept
The Riemann-Stieltjes integral extends the Riemann integral by replacing $dx$ with $dg(x)$:
$\int_a^b f(x) \, dg(x)$

Here, $g(x)$ is a monotonic (non-decreasing) function, often interpreted as a weight, distribution, or cumulative change.

Geometric Meaning:

Rather than computing area under $f(x)$, we compute a weighted sum of $f(x)$ based on how $g(x)$ changes. When $g(x)$ has jumps, the integral captures those discontinuous effects — useful for modeling taxes, benefits, or sudden pricing shifts.

4. Conditions for Riemann-Stieltjes Integrability
A function $f$ is Riemann-Stieltjes integrable with respect to $g$ if:

• $f$ is bounded on $[a, b]$
• $g$ is monotonic
• $f$ is continuous at all points where $g$ is not constant

This makes the Riemann-Stieltjes integral more flexible than the Riemann integral.

5. Applications in Economics

• Probability-weighted expectations:
  • $\int x \, dF(x)$ where $F$ is a distribution function (expected value)
• Piecewise taxation/benefits:
  • $g(x)$ represents stepped tax brackets or subsidy jumps
• Utility with preferences:
  • $g(x)$ encodes behavioral biases or consumption weights

These scenarios require a framework that supports discontinuous or non-uniform accumulations — precisely what Riemann-Stieltjes provides.

6. Comparison Table

Conclusion
Understanding the Riemann and Riemann-Stieltjes integrals deepens our ability to model real-world economic systems. The Riemann integral captures continuous accumulation; the Riemann-Stieltjes integral accommodates discontinuities and non-uniform weights — making it ideal for modern, data-driven economic modeling.

Whether you’re studying marginal-to-total transitions or cumulative effects under complex rules, these integrals provide the mathematical foundation for nuanced, accurate results.

Further Reading

• Bartle & Sherbert, Introduction to Real Analysis
• Chiang & Wainwright, Fundamental Methods of Mathematical Economics