Subsidies — effect on equilibrium and welfare — Economics

Before this

This leads to

Everyone gains. So where is the catch?

A government pays sellers $5 for every unit they sell. Prices in the shops fall, so buyers are happier. Sellers receive more per unit than before, so they are happier too. More of the good is made and sold. It sounds like a policy with no losers.

But the $5 payments come from somewhere: from taxpayers. This lecture adds up the gains and the costs carefully. You will find that the gains to buyers and sellers are real, but they are smaller than the bill. The gap is a deadweight loss, just like the one a tax creates.

The last lecture ended with a promise: a tool that pushes prices the other way without forbidding any price. That tool is a subsidy, a payment from the government for each unit produced or bought. The lecture on tax incidence called it "a wedge the other way". Here is what that means.

What a subsidy does to the supply curve

Use the same market as the last three lectures. Demand is $Q_{d} = 100 - 2 P$ and supply is $Q_{s} = 20 + 3 P$ . Without any policy, the market clears at $P^{*} = $16$ and $Q^{*} = 68$ units per week.

Now the government pays sellers a subsidy of $s = $5$ per unit. A seller who charges the buyer $13 now receives $13 from the buyer plus $5 from the government, which is $18 in total. So there are two prices again, as with a tax:

$P_{c}$ is the price consumers pay in the shop.
$P_{p}$ is the price producers receive, including the subsidy: $P_{p} = P_{c} + s$ .

With a tax, the price buyers pay was above the price sellers keep. With a subsidy, it is the other way round. Sellers receive more than buyers pay, and the government fills the gap. That is the wedge the other way.

How do sellers respond? They decide how much to make by what they receive, $P_{p}$ . So supply depends on $P_{c} + 5$ :

Q_{s} = 20 + 3 (P_{c} + 5) = 35 + 3 P_{c}

At every shop price, sellers now offer 15 more units than before. On a diagram, the supply curve moves down by $5, which is the same as moving right by 15 units. Each unit that sellers would have offered at $18 is now offered at a shop price of $13, because the government adds the other $5.

Where the new equilibrium lands

Set demand equal to the new supply:

100 - 2 P_{c} 65 P_{c} = 35 + 3 P_{c} = 5 P_{c} = 13

So buyers pay $13, and sellers receive $P_{p} = 13 + 5 = $18$ . The quantity is $Q = 100 - 2 \times 13 = 74$ . Check with supply: $20 + 3 \times 18 = 74$ . Both curves agree.

A $5 subsidy moves supply down by $5 (dashed). Buyers now pay Pc = $13 and sellers receive Pp = $18, so 74 units are sold instead of 68. The tinted rectangle, $5 × 74 = $370, is what the government pays. The small grey triangle is the deadweight loss.

Compare with the old price of $16. Buyers pay $3 less. Sellers receive $2 more. The two parts add up to the subsidy: $3 + 2 = 5$ . This is the same 60–40 split as the $5 tax, for the same reason. The side that responds less to price takes the bigger share. With a tax that share was a burden. With a subsidy it is a benefit. Buyers here respond less (2 units per $1, against sellers' 3), so buyers get 60% of the subsidy.

Notice what this means for a common claim. "A $5 subsidy will cut prices by $5" is true only in a special case, when supply responds without limit. Usually part of the subsidy stays with sellers.

The bill, and who it really pays

The government pays $5 on every unit sold, not only on the extra ones. So the cost to taxpayers is:

s \times Q = 5 \times 74 = $370 per week

Now find the gains, using the welfare diagram from the lecture on consumer and producer surplus. Buyers gain the area between the old price $16 and the new price $13, left of the demand curve. This is a trapezium with parallel sides 68 and 74 and height 3:

Buyers’ gain = \frac{1}{2} \times (68 + 74) \times 3 = $213

Sellers gain the area between $16 and $18, left of the supply curve. The same trapezium shape, with height 2:

Sellers’ gain = \frac{1}{2} \times (68 + 74) \times 2 = $142

Together, buyers and sellers gain $213 + 142 = $355$ . The government pays $370. The missing $15 is the deadweight loss, the grey triangle:

DWL = \frac{1}{2} \times 6 \times 5 = $15

The welfare effects of a subsidy

A subsidy $s$ per unit moves supply down by $s$ . Buyers pay less ( $P_{c} < P^{*}$ ), sellers receive more ( $P_{p} > P^{*}$ ), and the quantity rises above $Q^{*}$ .

Government cost $= s \times Q$ = buyers' gain + sellers' gain + deadweight loss. The deadweight loss is $\frac{1}{2} \times (extra units) \times s$ .

Why the extra units waste money

Where does the $15 go? It is spent on the 6 extra units, from 68 to 74. Look at what each of those units is worth, and what it costs.

The demand curve tells you the most a buyer will pay for each unit. For the 69th unit, that is about $15.50. For the 74th unit, it is $13. The supply curve tells you the lowest price a seller will accept, which reflects the cost of making it. For the 69th unit, that is about $16.33. For the 74th, it is $18.

So every extra unit costs more to make than it is worth to the person who buys it. Without the subsidy, nobody would trade it, and rightly so. The subsidy pays people to make things that are worth less than the resources used. For the 74th unit, the loss is $18 - 13 = $5$ . For the 69th, it is less than $1. Add up all those losses and you get the triangle, $15.

This is the mirror image of a tax. A tax stops 6 units that were worth making, so the market makes too little. A subsidy adds 6 units that were not worth making, so the market makes too much. Either way, the quantity moves away from $Q^{*} = 68$ , and that move is where the deadweight loss comes from.

Stop and think

A student says: "Buyers gain and sellers gain, so society must be better off with the subsidy." What is missing from this argument?

Show the answer

The taxpayers. The $370 has to be raised from somebody, usually through taxes, and those people are part of society too. Count all three groups: buyers gain $213, sellers gain $142, and taxpayers lose $370. The total is $213 + 142 - 370 = - $15$ . So, in this simple market, society is $15 a week worse off. A policy can make the two sides of a market happier and still make society poorer, because the bill falls on people who are not in the market.

How the loss grows

Suppose the government doubles the subsidy. Before you look at the graph below, make a prediction.

Predict, then reveal — doubling the subsidy

In this market, a $5 subsidy causes a deadweight loss of $15 a week. The government raises the subsidy to $10. What happens to the deadweight loss?

With a $10 subsidy, supply becomes Qs = 20 + 3(Pc + 10) = 50 + 3Pc. Setting 100 − 2Pc = 50 + 3Pc gives Pc = $10, Pp = $20 and Q = 80, which is 12 more units than without the subsidy.

The deadweight loss is ½ × 12 × 10 = $60, which is four times the $15 from a $5 subsidy. Doubling the subsidy doubles both the base and the height of the triangle.

So small subsidies waste little, and large ones waste a lot. The same is true of taxes. Now try every size of subsidy from $0 to $10.

Graph builder — the size of the subsidy

Demand is Qd = 100 − 2P and supply is Qs = 20 + 3P. Choose the subsidy per unit. Watch the two prices, the bill and the deadweight loss. Compare s = 5 with s = 10.

Subsidy per unit s: $5

The effects of four sizes of subsidy in this market (all amounts per week):

s	Pc	Pp	Q	Bill	DWL
$2	$14.80	$16.80	70.4	$140.80	$2.40
$5	$13	$18	74	$370	$15
$8	$11.20	$19.20	77.6	$620.80	$38.40
$10	$10	$20	80	$800	$60

Does it matter who receives the money?

The tax lecture showed that it does not matter whether buyers or sellers send the tax to the government. The same is true of a subsidy. Suppose the government pays the $5 to buyers instead, as a refund for each unit they buy. A buyer who pays the shop price $P$ now spends only $P - 5$ , so demand becomes $Q_{d} = 100 - 2 (P - 5) = 110 - 2 P$ . Setting this equal to $20 + 3 P$ gives a shop price of $18. Sellers receive $18, buyers pay $18 - 5 = $13$ after the refund, and 74 units are sold. Exactly the same result. Only the label on the cheque changes.

Worked examples

Worked Example 1 — A subsidy in a new market

Demand is $Q_{d} = 120 - 4 P$ and supply is $Q_{s} = 3 P - 20$ . The government pays sellers $s = $7$ per unit. Find the two prices, the quantity, the bill and the deadweight loss.

Without the subsidy: $120 - 4 P = 3 P - 20$ , so $140 = 7 P$ , $P^{*} = $20$ and $Q^{*} = 40$ .

With the subsidy, supply is $3 (P_{c} + 7) - 20 = 3 P_{c} + 1$ . Then $120 - 4 P_{c} = 3 P_{c} + 1$ , so $119 = 7 P_{c}$ and $P_{c} = $17$ . Sellers receive $P_{p} = 17 + 7 = $24$ . The quantity is $120 - 4 \times 17 = 52$ .

The bill is $7 \times 52 = $364$ . Buyers gain $\frac{1}{2} \times (40 + 52) \times 3 = $138$ . Sellers gain $\frac{1}{2} \times (40 + 52) \times 4 = $184$ . The deadweight loss is $\frac{1}{2} \times 12 \times 7 = $42$ . Check: $138 + 184 + 42 = 364$ .

Here sellers get the bigger share, $4 of the $7. That fits the rule: buyers respond more (4 units per $1, against sellers' 3), so sellers, the less responsive side, take more of the subsidy.

A real case: Cash for Clunkers

Worked Example 2 — A short subsidy for new cars (United States, 2009)

The policy. In July and August 2009, during a deep recession, the US government ran the Car Allowance Rebate System, known as "Cash for Clunkers". A buyer who traded in an old car that used a lot of fuel received a credit of $3,500 or $4,500 towards a new, more fuel-efficient car. The aims were to support the car industry and to improve fuel economy. The budget was $3 billion. The US Government Accountability Office reports that 677,842 credits were paid, worth about $2.85 billion, an average of about $4,200 each.

The prediction. This is a subsidy paid to buyers, so the model above applies. The price buyers pay falls, the price sellers receive rises, and more cars are sold. The model also says the government pays on every subsidised car, including cars that people would have bought anyway.

What happened. Sales jumped. Atif Mian and Amir Sufi (NBER Working Paper 16351, 2010) estimated that the programme added about 360,000 car sales in July and August 2009. On that estimate, roughly half of the 677,842 credits went to purchases that were not extra. Then came a twist that the one-period model cannot show. The extra purchases were mostly pulled forward: people bought a car a few months earlier than they had planned. Mian and Sufi found that the boost was almost completely reversed by March 2010, only seven months after the programme ended. A later review by Ted Gayer and Emily Parker (Brookings, 2013) reached a similar view on timing. They put the cost at about $1.4 million per job created and would not recommend repeating the programme.

The verdict. The model predicted the direction correctly: lower prices for buyers and a rise in sales. It also predicted the hidden cost: the bill covered many cars that would have been sold anyway, just like the rectangle in our diagram covers all 74 units, not only the 6 extra ones. On $2.85 billion and 360,000 extra sales, each extra sale cost about $7,900, nearly twice the average credit. The timing effect shows a limit of the simple model, which compares one period with another and ignores what buyers expect next. Supporters reply that in a deep recession, moving spending earlier can itself be the goal, and that the swap also replaced some of the least fuel-efficient cars on the road. Economists still disagree about how much that was worth.

When a subsidy can make a market better

Everything so far assumed one thing: the demand curve shows the full value of each unit. Sometimes it does not. A vaccination protects the person who gets it, but it also protects the people around them, who pay nothing. When a good brings benefits to people outside the market, buyers ignore those benefits, and the market makes too little of it. Economists call this a positive externality.

In that case, $Q^{*}$ is not the best quantity for society. A subsidy that pushes the quantity up can move the market towards the best quantity, not away from it. Many countries subsidise vaccinations, education and, more recently, electric cars partly on this argument. The UK, for example, gave grants towards new plug-in cars from 2011 to 2022. Whether each of these subsidies is the right size is a separate question, and the lecture on positive externalities answers it with the same diagram.

Without such a reason, a subsidy needs a careful look. Farm subsidies in the European Union and the United States are long-running examples. Critics argue that they encourage overproduction and send most of the money to large farms. Supporters point to steadier food supplies and rural incomes. The welfare diagram does not settle that argument on its own, but it tells you exactly which numbers to look for.

Three classic mistakes

Trap 1 — The bill covers only the extra units

The government pays $s$ on every unit sold, not only on the units the subsidy adds. Here the bill is $5 \times 74 = $370$ , not $5 \times 6 = $30$ . Most of the money goes on units that would have been traded anyway.

Trap 2 — Prices fall by the whole subsidy

A $5 subsidy cut the shop price by $3, not $5. Sellers kept the other $2. The split depends on how strongly each side responds to price, exactly as with a tax.

Trap 3 — Shifting supply up

A tax moves supply up, because sellers need more from buyers. A subsidy moves supply down by $s$ , or right, because sellers can accept less from buyers. Check the direction by asking: does the policy make selling more or less attractive?

Try it yourself

Work each one on paper first. Questions 1 and 2 use the market Qd = 100 − 2P, Qs = 20 + 3P.

The government pays sellers a subsidy of $2.50 per unit. Find $P_{c}$ , $P_{p}$ , the quantity, the bill and the deadweight loss.

Hint 1
Supply becomes $20 + 3 (P_{c} + 2.5) = 27.5 + 3 P_{c}$ .

Hint 2
$100 - 2 P_{c} = 27.5 + 3 P_{c}$ gives $72.5 = 5 P_{c}$ .

Full answer
$P_{c} = $14.50$ and $P_{p} = $17$ . The quantity is $100 - 29 = 71$ , which is 3 more than 68. The bill is $2.5 \times 71 = $177.50$ . The deadweight loss is $\frac{1}{2} \times 3 \times 2.5 = $3.75$ . Half the $5 subsidy gives a quarter of its $15 loss, because the loss grows with the square of the subsidy.
With the $5 subsidy, a newspaper says: "The subsidy costs taxpayers $370 a week, and buyers and sellers gain $370 between them, so nothing is lost." Find the mistake.

Hint 1
Work out the buyers' gain and the sellers' gain separately, as trapezia.

Hint 2
Each trapezium has parallel sides 68 and 74.

Full answer
Buyers gain $\frac{1}{2} \times (68 + 74) \times 3 = $213$ and sellers gain $\frac{1}{2} \times (68 + 74) \times 2 = $142$ , a total of $355, not $370. The newspaper has assumed that every dollar of the bill becomes a gain. But $15 is spent on the 6 extra units, which cost more to make than they are worth to buyers. That $15 is the deadweight loss.
A city offers a $300 subsidy on new bicycles for one month only, then stops. Bicycle sales triple during that month. Explain why the monthly jump probably overstates the effect of the subsidy on how many bicycles people own.

Hint 1
Think about people who were already planning to buy a bicycle in the next few months.

Hint 2
What happened to car sales after Cash for Clunkers ended?

Full answer
Many buyers would have bought a bicycle anyway, a little later. A one-month subsidy gives them a reason to buy now instead. So sales jump during the month and then fall below normal afterwards, as with Cash for Clunkers. To measure the real effect, compare total sales over a longer period, say a year, with what would have happened without the subsidy. Also, the city pays $300 on every bicycle sold that month, including those that would have been bought anyway.
Return to the $5 subsidy. Suppose each unit of this good also gives $4 of benefit to people outside the market, who do not pay for it. Is society better or worse off with the subsidy than without it? Would a $4 subsidy be even better?

Hint 1
The 6 extra units now bring $4 each of outside benefit, which the $15 deadweight loss did not count.

Hint 2
With a $4 subsidy, the quantity rises by $1.2 \times 4 = 4.8$ units, and the deadweight loss is $\frac{1}{2} \times 4.8 \times 4$ .

Full answer
With $5: the 6 extra units bring $6 \times 4 = $24$ of outside benefit, and the deadweight loss counted against buyers and sellers is $15. So society gains $24 - 15 = $9$ a week. With $4: the extra units are 4.8, bringing $4.8 \times 4 = $19.20$ of outside benefit, and the loss is $\frac{1}{2} \times 4.8 \times 4 = $9.60$ . So society gains $19.20 - 9.60 = $9.60$ , slightly more. The best subsidy equals the outside benefit per unit. The lecture on positive externalities shows why.

The bigger idea

Taxes and subsidies are two sides of one picture. Each drives a wedge between the price buyers pay and the price sellers receive. Each splits its effect between the two sides in the same way, with the less responsive side taking more. And each creates a deadweight loss by moving the quantity away from $Q^{*}$ : a tax gives too little, a subsidy gives too much. The loss grows with the square of the wedge.

That is the whole story when the market's own prices measure every cost and benefit. When they do not, $Q^{*}$ is no longer the best quantity, and the same tools can make a market better instead of worse. The unit on market failure starts there.

Key concepts in this lesson

A subsidy of s per unit moves supply down by s: buyers pay Pc, sellers receive Pp = Pc + s, and the quantity rises above Q*.
The subsidy is shared between buyers and sellers in the same way as a tax: the less responsive side gets the larger share.
The government's bill is s × Q, paid on every unit sold, and it equals the buyers' gain plus the sellers' gain plus the deadweight loss.
The deadweight loss is ½ × (extra units) × s. It comes from units that cost more to make than they are worth to buyers, and it grows with the square of the subsidy.
When a good brings benefits to people outside the market, a subsidy can move the quantity towards the best level for society.