Positive externalities — underproduction of good things — Economics

Before this

This leads to

A benefit nobody pays for

Ana decides whether to get a flu vaccination. She thinks about the cost: the price, the time off work, a sore arm. She thinks about the benefit: a smaller chance of a week in bed. She compares the two and decides.

But her decision also affects other people. If Ana does not catch flu, she cannot pass it to her colleagues, her grandmother or the people on her bus. That protection is real, and it is valuable. Yet nobody pays Ana for it, so it plays no part in her decision. Some people like Ana will decide that the vaccination is not quite worth it for them, even though it is clearly worth it for everyone together.

The last lecture studied goods that harm people outside the market, and found that the market makes too much of them. It ended with a promise to turn the picture round. This lecture looks at goods whose benefits spill over to others. The market makes too little of these good things, and this lecture shows exactly why, how much, and what can be done about it.

Where the market stops too early

Use the market from the first lecture in this topic. Marginal private benefit (MPB, the demand curve) is $P = 30 - 0.2 Q$ . Marginal private cost (MPC, the supply curve) is $P = 6 + 0.1 Q$ . The market settles where they meet: $Q^{*} = 80$ units a week at a price of $14.

Now suppose each unit also gives $6 of benefit to people outside the market. Then the marginal social benefit (MSB) is the private benefit plus the external benefit, so MSB lies $6 above MPB: $P = 36 - 0.2 Q$ . The cost of making the good falls only on the producer, so the marginal social cost (MSC) equals MPC. The best quantity for society is where MSB meets MSC:

36 - 0.2 Q 30 Q = 6 + 0.1 Q = 0.3 Q = 100

So $Q_{opt} = 100$ , but the market stops at 80. Think about unit 90. Society values it at $36 - 0.2 \times 90 = $18$ , and it costs $6 + 0.1 \times 90 = $15$ to make. It is worth making. But the buyer values it at only $30 - 18 = $12$ , which is less than the $15 cost, so the buyer does not buy it. Every unit from 80 to 100 is lost in the same way. This is underproduction.

A positive externality. MSB lies $6 above MPB. Buyers count only their own benefit, so the market stops at Q* = 80, where MPB meets MPC. Society's best quantity is Q_opt = 100, where MSB meets MPC. The hatched triangle is the deadweight loss: the net value of the 20 units that are never made.

The value lost is the triangle between $Q^{*}$ and $Q_{opt}$ , above MPC and below MSB. Its base is 20 units. Its height at $Q^{*} = 80$ is the gap between MSB and MPC there, which is $20 - 14 = $6$ . So the deadweight loss is $\frac{1}{2} \times 20 \times 6 = $60$ a week. It is the same size as the loss in the pollution market, because the gap is the same $6, only on the other side.

Stop and think

A student says: "A positive externality means the good is good for society, so the market makes too much of it." What is wrong with this?

Show the answer

The direction is backwards. Buyers decide using only their own benefit, MPB. The benefit to others is left out of the decision, so buyers value each unit less than society does. They stop buying too early. A positive externality leads to too little, not too much. "Positive" describes the effect on other people, not the size of the market.

The subsidy that puts the missing benefit back

The problem is a benefit with no price on it. The fix is to attach one. If the government pays $6 for each unit, the person who produces or buys the unit now receives a reward for the benefit it gives to others.

The subsidies lecture showed how this works. A subsidy of $s$ per unit paid to sellers lowers the supply curve by $s$ . With $s = $6$ , supply becomes $P = 0.1 Q$ . Setting it equal to demand:

30 - 0.2 Q Q = 0.1 Q = 100

The market now reaches $Q_{opt} = 100$ on its own. Buyers pay $P_{c} = 30 - 0.2 \times 100 = $10$ . Sellers receive $P_{p} = 10 + 6 = $16$ . The government pays $6 \times 100 = $600$ a week.

A $6 subsidy lowers supply from MPC to MPC − $6. The market moves from Q* = 80 to Q_opt = 100. Buyers pay Pc = $10 and sellers receive Pp = $16. The tinted rectangle is the cost of the subsidy, $6 × 100 = $600 a week.

The subsidies lecture ended with a question it did not fully answer. In a market where each unit gave $4 of benefit to outsiders, a $4 subsidy did slightly better than a $5 one. It promised that this lecture would show why the best subsidy equals the external benefit per unit. Here is the reason.

With a subsidy of $s$ , a buyer and seller together act as if each unit were worth $MPB + s$ . They keep trading as long as $MPB + s$ is above MPC. If $s$ equals the external benefit, then $MPB + s$ is exactly MSB. The market now counts every benefit that society counts, so it stops exactly where society would. A smaller subsidy counts only part of the external benefit, so the market still stops too early. A larger subsidy counts more benefit than really exists, so the market goes too far.

That is it. That is the whole idea. The subsidy does not need to be clever. It only needs to pay each buyer for the benefit their purchase gives to others.

Is the $600 wasted?

The subsidies lecture showed that, in an ordinary market, a subsidy costs taxpayers more than it gives buyers and sellers. The gap was a deadweight loss. Will the same happen here? Decide before you read on.

Predict, then reveal — what does the $6 subsidy do to society?

In this market, each unit gives $6 of benefit to people outside it. The government pays a $6 subsidy on every unit, which costs $600 a week, and the market moves from 80 to 100 units. Counting buyers, sellers, taxpayers and the people who receive the external benefit, what happens to society's total welfare?

Society gains $60 a week. Before the subsidy: buyers and sellers gain $960 (the area between MPB and MPC up to 80 units) and outsiders receive 80 × $6 = $480, a total of $1440. With the subsidy: buyers gain $1000 and sellers $500, outsiders receive 100 × $6 = $600, and taxpayers pay $600. The total is 1000 + 500 + 600 − 600 = $1500. That is $60 more, exactly the deadweight loss the externality was causing.

Same tool, same diagram, opposite verdict. In an ordinary market, a subsidy pays for units that are not worth making. Here, it pays for units that are worth making, once the benefit to others is counted. This mirrors the tax in the last lecture exactly. A tax was harmful in an ordinary market and helpful against a negative externality; a subsidy is harmful in an ordinary market and helpful for a positive one.

The size still matters, just as it did for the pollution tax. A $3 subsidy, half the right size, removes three quarters of the loss. A $12 subsidy, twice the right size, leaves a $60 loss, as bad as doing nothing.

Vaccination: when the benefit has a threshold

Vaccination is the clearest positive externality, but it has a special feature. The external benefit is not the same for every dose. It depends on how many other people are already vaccinated.

Each infected person passes a disease on to some number of others. In a population where nobody is immune, that average number is called $R_{0}$ ("R nought"). If each case leads to more than one new case, an outbreak grows. If it leads to fewer than one, chains of infection die out. Vaccination lowers the number of people each case can reach. Once enough people are immune, each case leads to fewer than one new case, and outbreaks fade even among people who are not vaccinated. This is called herd immunity.

In the simplest textbook model, the share of people who must be immune is $1 - \frac{1}{R _{0}}$ . Measles is one of the most contagious diseases known: $R_{0}$ is usually put at about 12 to 18, and about 92% to 95% of a population needs to be vaccinated for herd immunity. Explore what this means for one person's choice.

Real-world model — herd immunity and the free rider

A simplified model, invented for teaching. Choose how contagious the disease is, then move the share of people vaccinated. The curve shows the chance that an unvaccinated person catches the disease if an outbreak arrives. Watch what happens to that chance as more other people are vaccinated.

How contagious:

Share vaccinated: 50%

A simplified model, invented for teaching: everyone mixes equally, and the vaccine always works. For a disease with R₀ = 5, the herd immunity threshold is 1 − 1/5 = 80%. The table shows the chance that an unvaccinated person catches the disease if an outbreak arrives. From 80% upwards, outbreaks fade out.

Vaccinated	Each case infects	Risk if unvaccinated
0%	5	≈ 99%
50%	2.5	≈ 89%
70%	1.5	≈ 58%
75%	1.25	≈ 37%
80% +	1 or fewer	≈ 0%

For R₀ = 2 the threshold is 50%; for R₀ = 15 it is about 93%.

Look at what the model shows. When few people are vaccinated, vaccination protects you well, and your private benefit is large. As more other people are vaccinated, your own risk falls, so your private benefit from vaccination falls too. Above the threshold, your private benefit is almost zero, because everyone else is protecting you. This is a free rider: someone who enjoys a benefit without paying for it.

Here is the trap. If many people reason like this, coverage slips below the threshold, and outbreaks return. The private benefit is smallest exactly where the social benefit matters most. The model leaves a great deal out: real vaccines are not perfect, people do not mix equally, and new babies are born without immunity. But the main idea survives all of these.

This is why governments rarely leave vaccination to the market. Common tools are free vaccination paid for from taxes, campaigns that explain the benefits, and, in some countries, a requirement for certain vaccinations before children start school.

Stop and think

A government runs an information campaign about vaccination. Can a campaign alone bring the market to $Q_{opt}$ ?

Show the answer

Only partly. A campaign can correct mistakes people make about their own benefit, for example if they underestimate how bad the disease is for them. That moves MPB up to where it should be. But even a perfectly informed person still gains nothing personally from protecting others. So the gap between MPB and MSB, the external benefit, is still there after the campaign. Information fixes an information problem. It does not put a price on a benefit to others.

A real case: free bed nets in Kenya

Worked Example — Free or cheap? Malaria bed nets (Kenya, 2006–2009)

The policy. Nets treated with insecticide protect sleepers from the mosquitoes that carry malaria. They also protect neighbours: when many people in an area sleep under treated nets, fewer mosquitoes survive to bite and infect anyone, including people without nets. In 2006–2009, the economists Jessica Cohen and Pascaline Dupas ran an experiment in 20 prenatal clinics in western Kenya. Pregnant women were offered nets at different, randomly chosen prices: free, or heavily subsidised at up to about $0.60. The market price of a net was about $5 to $7.

The prediction. The positive-externality model says that a large subsidy should raise the number of nets close to the social optimum. Some people argued the opposite about the last step: if nets are completely free, people who do not value them will take them and not use them, so a small charge would target nets better.

What happened. Demand was very sensitive to price. Raising the price from zero to $0.60 reduced the share of women taking a net by about 60 percentage points. Women who paid were no more likely to use their nets than women who received them free. The study was published in 2010 in the Quarterly Journal of Economics, and summarised by the research network J-PAL.

The verdict. In this setting, even a small price kept many people from the nets, and charging did not improve how the nets were used. Because the protection spreads to the whole community, the authors concluded that free distribution was both more efficient and more cost-effective than charging. This is one study, among pregnant women in one region, so it does not prove that "free" is always best. But it shows that a subsidy's size matters most when the external benefit is large and people are poor.

Education and research

Education gives large private benefits, such as higher earnings. It is also widely believed to bring external benefits: a better-educated population tends to be healthier, more productive and better informed as citizens, although these effects are hard to measure precisely. Governments therefore provide schooling free, subsidise further study, and in most countries make school compulsory up to a certain age.

Health and education are among the largest areas of government spending. But they are not the two largest in most rich countries. Across OECD countries in 2019, the largest area was social protection, such as pensions and unemployment benefits, at 13.3% of national income (GDP) on average. Health came next at 7.9%, and education took 5.1% (OECD, Government at a Glance 2021). Positive externalities are one reason for spending on health and education; fairness, giving everyone a chance whatever their family's income, is another.

Research is a positive externality of production. When a firm makes a discovery, other firms learn from it through published work, workers who change jobs, and copying. For these, many textbooks draw MSC below MPC. Each unit of research costs society less than its private cost, once the benefit to others is subtracted. The result is the same: too little research. One large study of US firms by Nicholas Bloom, Mark Schankerman and John Van Reenen (2013) estimated that the social returns to research are at least twice the private returns.

Governments respond with research subsidies, tax credits and public funding for universities. Another response is the patent: a legal right to be the only seller of a new invention for a limited time. A patent lets the inventor capture more of the value of the idea, so the reward for research rises. The cost is a temporary monopoly, with higher prices while the patent lasts. It is a trade-off, not a free fix.

Tool	How it raises quantity	Main weakness
Subsidy	Pays for the external benefit, so MPB + s = MSB	Needs the size of the benefit; costs taxpayers
Free provision	The government supplies the good at no charge	Can overshoot; quality may suffer
Information	Corrects mistakes about private benefit	Cannot add the benefit to others
Rules (compulsory schooling, vaccination requirements)	Sets the quantity directly	Removes choice; hard to enforce
Patents (research)	Lets inventors capture more of the value	Temporary monopoly and higher prices

Three classic mistakes

Trap 1 — Getting the direction wrong

A positive externality leads to underproduction: $Q^{*} < Q_{opt}$ . Buyers ignore the benefit to others, so they stop too early.

Trap 2 — "Every subsidy creates a deadweight loss"

That is true only when there is no externality. With a positive externality, a subsidy close to the external benefit removes a deadweight loss. Always ask whether the market was efficient before the subsidy.

Trap 3 — The wrong triangle

The deadweight loss lies between $Q^{*}$ and $Q_{opt}$ , below MSB and above MPC. It is not the triangle between MPB and MPC: that area is gained by buyers and sellers, not lost.

Try it yourself

Questions 1 and 2 use MPB: P = 30 − 0.2Q and MPC: P = 6 + 0.1Q.

Each unit gives $3 of benefit to outsiders. Find $Q_{opt}$ , the deadweight loss without intervention, the best subsidy, the prices buyers pay and sellers receive, and the cost of the subsidy.

Hint 1
MSB is $P = 33 - 0.2 Q$ . Set it equal to MPC.

Hint 2
The best subsidy equals the external benefit, so supply becomes $P = 3 + 0.1 Q$ .

Full answer
$33 - 0.2 Q = 6 + 0.1 Q$ , so $27 = 0.3 Q$ and $Q_{opt} = 90$ . Without intervention, the market makes 80, so the deadweight loss is $\frac{1}{2} \times 10 \times 3 = $15$ a week. With a $3 subsidy, $30 - 0.2 Q = 3 + 0.1 Q$ gives $Q = 90$ . Buyers pay $30 - 18 = $12$ , sellers receive $12 + 3 = $15$ , and the subsidy costs $3 \times 90 = $270$ a week. Check: MPC at 90 is $6 + 9 = $15$ , equal to what sellers receive.
The external benefit is $6, but the government pays a $9 subsidy. Find the quantity and the deadweight loss that remains. Is the market now making too much or too little?

Hint 1
Supply becomes $P = - 3 + 0.1 Q$ . Set it equal to demand.

Hint 2
The triangle has a base of $Q - 100$ and a height of $9 - 6$ .

Full answer
$30 - 0.2 Q = - 3 + 0.1 Q$ , so $33 = 0.3 Q$ and $Q = 110$ . That is 10 units above the optimum, so the market now makes too much. Deadweight loss $= \frac{1}{2} \times 10 \times 3 = $15$ a week, a quarter of the $60 loss with no subsidy. The subsidy costs $9 \times 110 = $990$ , against $600 for the right-sized one.
Return to the subsidies lecture's market: $Q_{d} = 100 - 2 P$ and $Q_{s} = 20 + 3 P$ , with $Q^{*} = 68$ . Each unit gives $4 of benefit to outsiders. Find $Q_{opt}$ , and show that a $4 subsidy reaches it.

Hint 1
Write both curves as prices: MPB is $P = 50 - \frac{1}{2} Q$ and MPC is $P = \frac{Q - 20}{3}$ . Then MSB is $P = 54 - \frac{1}{2} Q$ .

Hint 2
That lecture found that a subsidy of $s$ raises the quantity by $1.2 s$ .

Full answer
Set MSB equal to MPC: $54 - \frac{1}{2} Q = \frac{Q - 20}{3}$ . Multiply by 6: $324 - 3 Q = 2 Q - 40$ , so $5 Q = 364$ and $Q_{opt} = 72.8$ . A $4 subsidy raises the quantity by $1.2 \times 4 = 4.8$ , from 68 to 72.8. That is exactly $Q_{opt}$ , which is why the $4 subsidy beat the $5 one in that lecture.
A disease has $R_{0} = 4$ . In the simple model, what share of people must be immune for herd immunity? The vaccine protects only 90% of the people who receive it. What share must be vaccinated now? Explain why a government might pay for vaccination near this level, rather than leave it to individual choice.

Hint 1
The threshold for immunity is $1 - \frac{1}{R _{0}}$ .

Hint 2
If a share $v$ is vaccinated, the share immune is $0.9 v$ . Set $0.9 v = 0.75$ .

Full answer
The share immune must be $1 - \frac{1}{4} = 0.75$ , or 75%. With a 90% effective vaccine, $0.9 v = 0.75$ , so $v \approx 0.833$ : about 83% must be vaccinated. Near this level, an unvaccinated person is mostly protected by others, so their private benefit from vaccination is small, and many may choose not to bother. But each person who drops out pushes coverage towards the point where outbreaks can grow again. The social benefit of each vaccination is far larger than the private benefit, so the market (individual choice) would give too little. Free vaccination removes the private cost, so more people choose it.

The bigger idea

Negative and positive externalities are one idea seen from two sides. In both, a market leaves something out of the price. When it leaves out a cost, it makes too much; when it leaves out a benefit, it makes too little. The fix is the same in spirit: put the missing value back into the decision, with a tax in one case and a subsidy in the other, each equal to the external effect per unit.

Some goods take this to the extreme. Once a lighthouse shines, every ship benefits, and nobody can be stopped from benefiting. The benefit to others is almost the whole benefit, so markets may provide almost none. These are public goods, the next part of market failure.

Key concepts in this lesson

A positive externality makes MSB lie above MPB, so the market stops at Q* below Q_opt and underproduces.
The deadweight loss is the triangle between Q* and Q_opt, below MSB and above MPC: units worth more to society than they cost, which are never made.
The best subsidy equals the external benefit per unit, because then MPB plus the subsidy equals MSB and the market stops where society would.
With a positive externality, a well-sized subsidy removes a deadweight loss instead of creating one; one far too large can do more harm than none.
For vaccination, the private benefit falls as coverage rises, which creates a free-rider problem just where herd immunity matters most.