Public Goods: Aggregate Demand

Introduction

In our last lesson, we focused on correcting market failures caused by externalities using precisely calculated Pigouvian taxes and subsidies. We now turn our attention to another significant cause of market failure: public goods. While markets can be guided to produce the right amount of a private good with an externality, they often fail to provide public goods at all.

This lesson will explore the unique characteristics that define public goods and, crucially, how we derive the total societal demand for them. You'll find that the method is fundamentally different from the one used for private goods. Your quantitative background will be invaluable as we move from the conceptual definition to the mathematical technique of vertical summation to construct the aggregate demand curve for a public good, also known as the marginal social benefit curve.

What Makes a Good "Public"?

For private goods like coffee or clothing, markets generally work well. If you buy a shirt, someone else cannot buy that same shirt. And if you don't pay for it, the store can prevent you from taking it. These two properties are called rivalry and excludability. Public goods are defined by the absence of these properties.

Let's clarify these definitions with a reference sheet.

[PDF] Public good: characteristics & aggregating demand curves

This reference sheet from Claremont McKenna College provides clear definitions and examples of public goods.

Please read the sections titled "Definition" and "Intuition". Pay close attention to the two defining characteristics: Non-rivalry: One person's consumption does not diminish another's ability to consume it. Non-excludability: It is impossible or prohibitively expensive to prevent someone from consuming the good, even if they don't pay. Also, review the "Real-world aspects" section on national defense to see these concepts applied.

As the resource explains, national defense is a classic example. Your protection by the military doesn't reduce your neighbor's protection (non-rival), and it's not feasible to exclude specific citizens from that protection (non-excludable). This leads to the free-rider problem: since people can benefit from the good without paying, they have little incentive to contribute to its cost, leading the market to under-provide or not provide it at all.

Aggregating Demand: A Tale of Two Summations

To determine how much of a good society wants, we must aggregate individual demands. The method of aggregation depends critically on whether the good is private or public.

Private Goods (Horizontal Summation): For a private good like pizza, consumption is rival. To find the total market demand at a price of $15, we ask how many pizzas Person A wants, how many Person B wants, and so on, and then sum these quantities horizontally. We add up the individual quantities demanded at each price level.
Public Goods (Vertical Summation): For a public good like a fireworks display, consumption is non-rival. Everyone in the vicinity sees the same number of fireworks. The question isn't how many fireworks each person consumes, but rather what is the value each person places on an additional firework. To find the total social value, we must sum each individual's willingness to pay (the price they would be willing to pay) for each given quantity.

This is the principle of vertical summation. We are summing the individual demand curves vertically to derive the marginal social benefit (MSB) curve.

A formal treatment of this concept from a Berkeley economics course explains the rationale. In their "Vertical Addition of Demand Curves" section, the authors note that for a public good, an individual's inverse demand curve represents their marginal rate of substitution (MRS) between the public good and a private good. Therefore, finding the social value involves summing these individual MRS values.

Let's visualize this process.

This graph illustrates the vertical summation of demand for a public good (rockets). For any given quantity, such as 20 rockets, Leah's willingness to pay is \$6 and Tara's is \$4. The total social willingness to pay for the 20th rocket is the vertical sum of these amounts, \$10. This process is repeated for every quantity to trace out the aggregate demand curve.

From Graphics to Equations: Deriving the MSB Curve

While the concept of adding up willingness to pay is straightforward, the mathematical execution requires careful attention, especially since individual demand can drop to zero at different quantities. This creates "kinks" in the aggregate demand curve, meaning it is often a piecewise function.

The following video provides an excellent walkthrough of both the graphical and mathematical derivation for three consumers.

Efficient Provision of a Public Good with Three Different Consumers

This video from the "Economics in Many Lessons" channel demonstrates how to derive the marginal social benefit (MSB) curve by vertically summing individual inverse demand curves.

First, watch the initial setup from the beginning, where the presenter graphs the three individual demand curves. Next, focus on the graphical summation from this segment. Notice how the aggregate curve has kinks where an individual consumer's willingness to pay becomes zero. Finally, and most importantly for your quantitative focus, watch the mathematical derivation from this part. The key is to see how the presenter constructs a piecewise function for the MSB based on the different ranges of quantity (Q).

As the video demonstrates, deriving the MSB curve involves these steps:

Start with the individual demand curves, expressed in their inverse form (i.e., Price as a function of Quantity, $P = f (Q)$ ). These represent each individual's marginal benefit (MB).
Identify the quantities at which each individual's willingness to pay drops to zero (the Q-intercepts). These points will define the segments of your piecewise MSB curve.
For each segment of quantity, sum the inverse demand functions of all individuals who still have a positive willingness to pay in that range.

Practice Problem

Let's apply this method to a problem. This is adapted from Question 4 in the reference sheet we looked at earlier.

Problem: Jack and Jill's demand for public park maintenance (Good A), measured in hours per week, are given by the following functions:

Jack's demand: $Q_{J} = 20 - 2 P$
Jill's demand: $Q_{j i l l} = 10 - P$

Derive the equation for the marginal social benefit (MSB) of park maintenance.

Solution:

Find the Inverse Demand Functions (MB curves):
We need to solve each equation for P to find each person's marginal benefit.
- Jack: $2 P = 20 - Q_{J} ⟹ P = 10 - 0.5 Q$ . So, $M B_{J a c k} = 10 - 0.5 Q$ .
- Jill: $P = 10 - Q$ . So, $M B_{J i l l} = 10 - Q$ .
Identify the Kinks:
We find where each person's willingness to pay becomes zero.
- For Jack, $M B_{J a c k} = 0$ when $10 - 0.5 Q = 0 ⟹ Q = 20$ .
- For Jill, $M B_{J i l l} = 0$ when $10 - Q = 0 ⟹ Q = 10$ .
  This tells us our MSB function will have a kink at $Q = 10$ .
Construct the Piecewise MSB Function:
We sum the MB curves over the relevant ranges.
- For $0 \leq Q \leq 10$ : Both Jack and Jill have a positive willingness to pay. $M S B = M B_{J a c k} + M B_{J i l l} = (10 - 0.5 Q) + (10 - Q) = 20 - 1.5 Q$
- For $10 < Q \leq 20$ : Jill's willingness to pay is now zero, so only Jack's demand contributes to the MSB. $M S B = M B_{J a c k} = 10 - 0.5 Q$
- For $Q > 20$ : Neither person is willing to pay for more maintenance. $M S B = 0$

Therefore, the complete marginal social benefit curve is the piecewise function:

M S B (Q) = ⎩ ⎨ ⎧ 20 - 1.5 Q 10 - 0.5 Q 0 if 0 \leq Q \leq 10 if 10 < Q \leq 20 if Q > 20

This rigorous approach of constructing a piecewise function is the correct way to aggregate demand for public goods.

Conclusion

This lesson has introduced the defining features of public goods—non-rivalry and non-excludability—and demonstrated why these features demand a unique analytical approach. Unlike with private goods, we cannot simply add up quantities. Instead, we must sum individual willingness to pay for each unit of the good.

Key Takeaways:

Public goods are non-rival and non-excludable, leading to the free-rider problem.
The aggregate demand for a public good is its marginal social benefit (MSB) curve.
The MSB curve is derived by vertically summing the individual inverse demand (or marginal benefit) curves.
This process often results in a piecewise function with "kinks" that occur where an individual's willingness to pay drops to zero.

Now that you can derive the demand for a public good, our next step will be to determine the efficient amount to provide. In the upcoming lesson, we will combine the MSB curve with the marginal cost of provision to calculate the socially optimal quantity of a public good, applying a key principle in public finance known as the Samuelson condition.

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