Area Between Curves: A Graphical and Analytical Tool in Welfare Economics

Introduction
In microeconomics, visual models often communicate efficiency, welfare, and market dynamics more powerfully than equations alone. One such concept is the area between curves, a central analytical and graphical tool used to measure economic surplus, deadweight loss, and the effects of policy interventions.

This blog post explains how to calculate the area between curves using definite integrals and illustrates its relevance to welfare economics.

1. Conceptual Definition
The area between curves represents the difference in values between two functions over a given interval. If $f(x) \geq g(x)$ on $[a, b]$, then:

$\text{Area} = \int_a^b [f(x) – g(x)] dx$

In economics, this typically reflects:

• $f(x)$: Demand function (willingness to pay)
• $g(x)$: Supply function (cost)

The integral measures the total net benefit to society or a specific group.

2. Graphical Interpretation

• The area under the demand curve above the market price is consumer surplus.
• The area above the supply curve below the market price is producer surplus.
• The area between supply and demand curves reflects total gains from trade.

This visual understanding reinforces core welfare analysis.

3. Practical Example: Total Economic Surplus
Let:

• Demand: $D(q) = 100 – 2q$
• Supply: $S(q) = 20 + q$
• Market equilibrium: Set $D(q) = S(q)$:
  $100 – 2q = 20 + q \Rightarrow q = 26.67$

Then:
$\text{Total Surplus} = \int_0^{26.67} [(100 – 2q) – (20 + q)] dq = \int_0^{26.67} (80 – 3q) dq$
$= [80q – 1.5q^2]_0^{26.67} = 2133.6 – 1066.8 = 1066.8$

So, the area between the demand and supply curves represents €1066.80 in total welfare gains.

4. Applications in Policy and Welfare Analysis

• Deadweight Loss (DWL): The area between curves lost due to taxes, quotas, or monopolies.
• Subsidies and Price Floors: Create triangles/trapezoids between supply and demand that indicate inefficiency or redistribution.
• Comparative Statics: Changing one curve (e.g., shift in demand) alters the area and reflects welfare change.

5. Integrating With Respect to Price
In some cases, especially in demand-side analysis:

• Express quantity as a function of price: $q = D(p)$
• Calculate area between $D(p)$ and $p_m$ to compute consumer surplus:
  $CS = \int_{p_m}^{p_{max}} D(p) dp$

This is particularly useful in policy models or monopolistic pricing.

6. Summary Table

Conclusion
The area between curves is not just a geometric concept—it represents real economic value. It quantifies the benefits of market exchange, the costs of inefficiencies, and the distribution of gains among participants. Through definite integration, economists can evaluate policy, compare outcomes, and make informed decisions grounded in visual and analytical rigor.

Understanding this concept is essential for students and professionals engaged in welfare economics and public policy design.

Further Reading

• Pindyck & Rubinfeld, Microeconomics
• Chiang & Wainwright, Mathematical Economics