Consumer and producer surplus — the welfare diagram — Economics

Before this

The law of demand
Supply — how much sellers offer · coming soon
Equilibrium — where supply meets demand · coming soon

This leads to

The law of supply
Price controls
Negative externalities

A gain that no receipt shows

Ama buys a coffee every morning for $4. On a cold, tired morning, she would happily pay $9 for it. So each time she pays $4, she walks away $5 better off than if she had paid her limit.

That $5 appears on no receipt. The café's records show $4. Ama's bank shows $4. Yet the gain is real. It is the reason Ama is glad the café exists.

The café gains too. Suppose it would still sell that cup for $1, because $1 covers its costs. At $4, it is $3 better off than its limit.

So one cup, sold for $4, made two people better off by $5 and $3. Nothing was taken from anyone. A market is a machine for creating gains like these. This lecture measures them, for every buyer and every seller at once. Then it shows what destroys them.

Reading the demand curve upwards

The first lecture of this topic built a demand curve for coffee in a town: $Q_{d} = 500 - 50 P$ , with quantity in cups per week. It also promised a second way to read it. "At $4, buyers want 300 cups" also means "the 300th cup is worth exactly $4 to the buyer who values it least". Here is where that reading pays off.

Read the curve upwards, cup by cup. The first cups go to the buyers who value coffee most. They are worth almost $10. The 100th cup is worth $8, the 200th is worth $6, and the 300th is worth $4. The height of the demand curve above each cup is that cup's value to its buyer. Economists call this the buyer's willingness to pay.

Now set a price of $4. Every cup is sold for $4. The buyer of a cup worth $8 gains $4. The buyer of the 300th cup gains nothing, because the cup is worth exactly what it costs. Each buyer's gain is the gap between the curve and the price line.

Add up those gaps for every cup, and you get the total gain to buyers. Try it below.

Explore — adding up the buyers' gains

Move the price. Each bar is a group of cups, as tall as their value to the buyers. The shaded top of each bar is the buyers' gain. Then make the bars narrower.

Price: $4

At a price of $4, the six groups of cups worth more than $4 are bought. Their gains are shaded.

Each bar is 50 cups. Buyers' gains at $4: 50 × ($5.50 + $4.50 + $3.50 + $2.50 + $1.50 + $0.50) = 50 × $18 = $900. The triangle under D and above the price gives the same: ½ × 300 × $6 = $900.

Look at what happened. At every whole-dollar price, the bars add up to exactly the area of the triangle under the demand curve and above the price. When the bars are narrow, the staircase becomes the triangle. That is it. That is the whole idea: the buyers' total gain is an area on the diagram.

Consumer surplus

Consumer surplus is the difference between what buyers are willing to pay and what they actually pay, added up over every unit bought. On the diagram, it is the area below the demand curve and above the price.

With a straight demand curve, it is a triangle: $CS = \frac{1}{2} \times Q \times (P_{m a x} - P)$ , where $P_{m a x}$ is the price at which the demand curve meets the price axis.

For coffee at $4: $CS = \frac{1}{2} \times 300 \times (10 - 4) = $900$ per week. Notice the height of the triangle. It is $P_{m a x} - P$ , the gap above the price. It is not $P_{m a x}$ itself.

The sellers' side: the lowest acceptable price

Sellers have a limit too, and it runs the other way. For each cup, there is a lowest price at which some café is willing to sell it. Below that price, selling the cup is not worth the café's costs and effort.

The Supply lectures will explain why this lowest price rises as more cups are made. For now, take a supply curve for the town and read it the same way as demand, but from the sellers' side: $Q_{s} = 100 P - 100$ . Its height above each cup is the lowest price at which that cup would be sold. The first cups could be sold for about $1. The 200th needs at least $3. The 300th needs at least $4.

If the price is $4, the café selling the 100th cup gets $4 but would have accepted $2. It gains $2. Add these gains over every cup sold, and you get producer surplus: the area above the supply curve and below the price.

PS = \frac{1}{2} \times Q \times (P - P_{m i n})

Here $P_{m i n}$ is the price at which the supply curve meets the price axis. Producer surplus is not quite the same as profit. It is the sellers' gain over the lowest price they would have accepted for each unit.

Where the two sides meet

Which price will the town's coffee actually sell at? Buyers want $500 - 50 P$ cups. Sellers offer $100 P - 100$ cups. Only at one price do the two plans match:

500 - 50 P = 100 P - 100 \Rightarrow 600 = 150 P \Rightarrow P^{*} = 4, Q^{*} = 300

This is the equilibrium, marked $P^{*}$ and $Q^{*}$ . A later lecture shows how a market finds it. For now, look at what it does to the two surpluses.

The curves cross at P* = $4 and Q* = 300. Consumer surplus (darker) lies above the price and below D. Producer surplus (lighter) lies below the price and above S.

Consumer surplus is $\frac{1}{2} \times 300 \times (10 - 4) = $900$ . Producer surplus is $\frac{1}{2} \times 300 \times (4 - 1) = $450$ . Together they make the total surplus of $1 350 per week. This is the whole gain the coffee market creates for everyone in it.

Why the market quantity gives the most surplus

Here is the result that makes this diagram so important. At the equilibrium, total surplus is as large as it can possibly be. This is not a slogan. You can see why, one cup at a time.

Take the 250th cup. A buyer values it at $5, because $10 - 250 \div 50 = 5$ . A café would sell it for $3.50, because $1 + 250 \div 100 = 3.5$ . The cup is worth more than it costs to supply. So selling it adds $1.50 of surplus, whatever the price. The same is true of every cup up to the 300th.

Now take the 350th cup. A buyer values it at only $3. But a café would need at least $4.50 to make it. The cup costs more to supply than it is worth to anyone who wants it. Making it would destroy $1.50 of surplus.

So the best quantity is exactly where the value of the next cup equals its lowest acceptable price. That is where the two curves cross. Economists say that this quantity is allocatively efficient: every trade that creates a gain happens, and no trade that destroys one happens.

Stop and think

A government orders the town's cafés to sell 400 cups a week, more than the market would. More people get coffee. Does total surplus rise?

Show the answer

No, it falls. Every cup from 301 to 400 is worth less to its buyer than the lowest price at which a café would make it. At the 400th cup, buyers value it at $2 but it needs $5. Each of these extra cups destroys surplus. The loss is a triangle: $\frac{1}{2} \times 100 \times (5 - 2) = $150$ . So total surplus falls from $1 350 to $1 200. More is not always better. Efficient means the right quantity, not the largest one.

Deadweight loss: surplus that nobody gets

Now push the market the other way. Imagine an invented town that allows cafés to sell only 200 cups a week in total, and sells permits for them. What happens to the gains?

At 200 cups, the last cup is worth $6 to its buyer, but a café would sell it for $3. Buyers compete for the scarce cups, so the price rises to $6. Each permit is worth the $3 gap, and the town collects that. Use the graph below to change the limit and watch each area.

Graph builder — a limit on how many cups can be sold

Move the limit. Watch the four areas. Which one disappears completely, and which one only changes hands?

Limit: 200 cups per week

A limit of 200 cups, and how the surplus is shared at four different limits:

With a limit of 200 cups, buyers pay $6 and sellers keep $3 per cup. The hatched triangle is the deadweight loss.

Limit	CS ($)	Permits ($)	PS ($)	Lost ($)
300	900	0	450	0
250	625	375	312.50	37.50
200	400	600	200	150
100	100	600	50	600

"Permits" is the permit revenue and "Lost" is the deadweight loss. Consumer surplus, permit revenue and producer surplus add up to $1 350, $1 312.50, $1 200 and $750 for the four limits. The deadweight loss is what is missing from $1 350.

At a limit of 200 cups, buyers keep $400 of surplus and sellers keep $200. The town collects $600 from permits. That $600 is not lost. It has moved from buyers and sellers to the town, which can spend it. Something else has gone completely: the hatched triangle, worth $150.

That triangle is the 100 cups between 200 and 300. Each was worth more to a buyer than it would have cost to supply, but nobody was allowed to trade it. So its gain was never created. No one receives it. Economists call it deadweight loss: surplus that is destroyed, not moved.

The tighter the limit, the bigger the triangle. At a limit of 100 cups, the loss is $600, nearly half of everything the market could create.

Worked examples

Worked Example 1 — Surplus from two equations

In a market, $Q_{d} = 100 - 2 P$ and $Q_{s} = 3 P - 50$ , with prices in dollars. Find consumer surplus, producer surplus and total surplus at equilibrium.

Equilibrium. Set the plans equal: $100 - 2 P = 3 P - 50$ , so $150 = 5 P$ and $P^{*} = 30$ . Then $Q^{*} = 100 - 2 \times 30 = 40$ .

Where the curves meet the price axis. Demand: $Q_{d} = 0$ when $P = 50$ , so $P_{m a x} = 50$ . Supply: $Q_{s} = 0$ when $3 P = 50$ , so $P_{m i n} = 50 \div 3 \approx 16.67$ .

CS = \frac{1}{2} \times 40 \times (50 - 30) = 400

PS = \frac{1}{2} \times 40 \times (30 - \frac{50}{3}) = \frac{800}{3} \approx 266.67

Total surplus $= 400 + \frac{800}{3} = \frac{2000}{3} \approx $666.67$ . Keep $P_{m i n}$ as the exact fraction $\frac{50}{3}$ until the end. Rounding it to 16.67 too early gives a slightly wrong answer.

Check: $Q_{s} = 3 \times 30 - 50 = 40$ , the same as $Q_{d}$ . Notice also that buyers gain more than sellers here. This is because the demand curve is steeper above the price than the supply curve is below it. The triangle above the price is taller.

Worked Example 2 — A real case: Singapore's limit on cars

The policy. Since 1990, Singapore has limited how many vehicles can be on its roads. Each month, a fixed number of Certificates of Entitlement (COEs) is released, and buyers bid for them. A new car cannot be registered without one. From February 2018, the allowed growth in the number of cars was set to zero. The stated reason was land: about 12% of Singapore's land was already roads (Land Transport Authority, announced 2017).

The prediction. This is the coffee limit on a national scale. A fixed quantity creates a gap between the value of the last car to its buyer and the lowest price at which it would be sold. Buyers bid for the right to buy, so the COE premium should equal that gap. If demand for cars grows while the quantity stays fixed, the demand curve moves right but the limit does not. The gap, and the premium, must grow.

What happened. In the bidding on 18 October 2023, premiums reached records for that time: S$106 000 for the category of smaller cars and S$150 001 for larger cars (in Singapore dollars; Mothership, 18 October 2023). That is the price of the right to own a car, paid on top of the price of the car. Many forces moved the premium, but the diagram explains the key point: a fixed quantity turns rising demand into a rising premium, not into more cars.

The verdict. The premium is the permit-revenue rectangle, and it goes to the government. It is not lost. The real loss is the triangle: cars that buyers valued above their cost but could not buy. Singapore accepts that loss on purpose. The diagram on this page leaves something out: the cost that one more car places on everyone else, through congestion and land used for roads. A later lecture adds those costs to the diagram. Once they are counted, a limit can raise total welfare instead of lowering it.

Three classic mistakes

Trap 1 — Shading the spending instead of the surplus

The rectangle under the price line, $P \times Q$ , is what buyers spend. Consumer surplus is the triangle above the price and below the demand curve. Spending is money handed over. Surplus is the gain beyond that.

Trap 2 — The wrong height for the triangle

The height of the consumer surplus triangle is $P_{m a x} - P$ , not $P_{m a x}$ . For coffee at $4, the height is $10 - 4 = 6$ , not 10. Also keep the $\frac{1}{2}$ : a triangle is half the rectangle around it.

Trap 3 — Calling every lost surplus "deadweight loss"

When a limit or a policy moves surplus from buyers to the government, that part is a transfer, not a loss. Someone still has it. Deadweight loss is only the surplus that nobody gets, from trades that never happen.

Try it yourself

Work each one on paper first. Open a hint only when you are stuck.

In the coffee market ( $Q_{d} = 500 - 50 P$ ), find consumer surplus when the price is $6. Compare it with the $900 at a price of $4. Why does a $2 price rise more than halve the surplus?

Hint 1
Find the quantity demanded at $6 first. That is the base of the triangle.

Hint 2
The height is $10 - 6$ . Both the base and the height shrink.

Full answer
At $6, $Q_{d} = 500 - 300 = 200$ . $CS = \frac{1}{2} \times 200 \times (10 - 6) = $400$ . That is less than half of $900. The price rise shrinks the triangle in two ways at once. Each remaining buyer gains $2 less per cup (the height falls from 6 to 4), and 100 fewer cups are bought (the base falls from 300 to 200). Check: $400 \div 900 \approx 0.44$ .
In a market, $Q_{d} = 120 - 4 P$ and $Q_{s} = 2 P - 12$ . Find the equilibrium, consumer surplus, producer surplus and total surplus.

Hint 1
Set $120 - 4 P = 2 P - 12$ and solve for $P$ .

Hint 2
Demand meets the price axis where $120 - 4 P = 0$ . Supply meets it where $2 P - 12 = 0$ .

Full answer
$132 = 6 P$ , so $P^{*} = 22$ and $Q^{*} = 120 - 88 = 32$ . $P_{m a x} = 30$ and $P_{m i n} = 6$ . $CS = \frac{1}{2} \times 32 \times (30 - 22) = 128$ . $PS = \frac{1}{2} \times 32 \times (22 - 6) = 256$ . Total surplus $= $384$ . Check: $Q_{s} = 2 \times 22 - 12 = 32$ , the same as $Q_{d}$ .
In the market of question 2, the government allows only 20 units to be sold, and sells permits. Find the price buyers pay, the deadweight loss and the permit revenue.

Hint 1
At 20 units, find what the last unit is worth to buyers (from demand) and the lowest price at which it would be sold (from supply).

Hint 2
From $20 = 120 - 4 P$ , the value is $25. From $20 = 2 P - 12$ , the lowest price is $16. The gap is $9.

Full answer
Buyers pay $25. Each permit is worth $25 - 16 = $9$ , so permit revenue is $20 \times 9 = $180$ . The deadweight loss is the triangle between 20 and 32 units: $\frac{1}{2} \times 12 \times 9 = $54$ . Check: $CS = \frac{1}{2} \times 20 \times 5 = 50$ and $PS = \frac{1}{2} \times 20 \times 10 = 100$ . Then $50 + 180 + 100 = 330$ , and $330 + 54 = 384$ , the total from question 2.
A new coffee machine lowers every café's lowest acceptable price by $0.75 per cup, so supply becomes $Q_{s} = 100 P - 25$ . Demand is still $Q_{d} = 500 - 50 P$ . Find the new consumer and producer surplus. The price falls, yet producer surplus rises. Explain why.

Hint 1
Find the new equilibrium first. Then find where the new supply curve meets the price axis.

Hint 2
$500 - 50 P = 100 P - 25$ gives $P^{*} = 3.5$ . The new $P_{m i n}$ is $25 \div 100 = 0.25$ .

Full answer
$525 = 150 P$ , so $P^{*} = 3.5$ and $Q^{*} = 500 - 175 = 325$ . $CS = \frac{1}{2} \times 325 \times (10 - 3.5) = $1 056.25$ , up from $900. $PS = \frac{1}{2} \times 325 \times (3.5 - 0.25) = $528.125 \approx $528.13$ , up from $450. The price fell by $0.50, but each café's lowest acceptable price fell by $0.75. So on every cup it already sold, its gain grew by $0.25. It also sells 25 more cups. Both effects raise producer surplus. If the price had fallen by more than the costs, producers could have lost, so the result depends on the numbers.

The bigger idea

This diagram is the yardstick of welfare economics. Whenever economists judge a tax, a subsidy, a price control or a trade rule, they ask the same three questions. How does it change consumer surplus? How does it change producer surplus? How big is the deadweight loss?

Two warnings come with the yardstick. First, efficiency is not fairness. A market can be efficient while the gains go mostly to a few people. Second, the diagram counts only the buyers and sellers. When a trade affects other people, such as the drivers stuck behind one more car, the yardstick must be extended. Singapore's car limit shows both warnings at work.

Key concepts in this lesson

Consumer surplus is the gain to buyers from paying less than they were willing to pay, shown as the area below the demand curve and above the price.
Producer surplus is the gain to sellers from receiving more than their lowest acceptable price, shown as the area above the supply curve and below the price.
With straight-line curves, each surplus is a triangle with area $\frac{1}{2} \times base \times height$ , where the height is measured from the price.
At equilibrium, total surplus is as large as possible, because every unit worth more than its lowest acceptable price is traded and no other unit is.
A limit on quantity turns part of the surplus into a transfer to whoever holds the permits and destroys another part, the deadweight loss.