A per-unit subsidy is a tax in reverse (Per-Unit Taxes): the government pays for each unit, so the market grows. Buyers and sellers both gain, but there is a bill.
Key idea
A subsidy of $s$ per unit makes the price sellers receive exceed the price buyers pay: $P_s - P_b = s$. Quantity rises above $Q^*$. The cost to taxpayers is $s \times Q_s$. Buyers' and sellers' gains together are less than that cost; the difference is the deadweight loss, $\tfrac{1}{2} \times s \times (Q_s - Q^*)$.
Why it works
Paid to sellers, the subsidy lowers the price each one needs by $s$: supply shifts down. Buyers pay less, sellers receive more, and more is traded. But the extra units between $Q^*$ and $Q_s$ cost more to produce than buyers value them, which is why the market didn't produce them before. The government pays for the whole rectangle $s \times Q_s$, while buyers and sellers gain only part of it; the leftover triangle is the deadweight loss.
Worked example: A \$2 subsidy
$Q_d = 20 - 2P$ and $Q_s = -4 + 2P$. The government pays sellers \$2 per unit. Find $P_b$, $P_s$, the quantity, the cost to taxpayers and the deadweight loss.
- Sellers receive $P_b + 2$: $20 - 2P_b = -4 + 2(P_b + 2)$, so $20 = 4P_b$ and $P_b = 5$; $P_s = 7$.
- $Q_s = 20 - 2(5) = 10$ (up from 8).
- Cost to taxpayers: $\$2 \times 10 = \$20$.
- $DWL = \tfrac{1}{2} \times 2 \times (10 - 8) = \$2$.
Answer: $P_b = \$5$, $P_s = \$7$, $Q = 10$, cost \$20, DWL \$2.
Worked example: Where did the \$20 go?
In the same market, how much do consumer and producer surplus rise?
- Before: $CS = PS = \$16$ (from Consumer and Producer Surplus).
- After: $CS = \tfrac{1}{2} \times 10 \times (\$10 - \$5) = \$25$ and $PS = \tfrac{1}{2} \times 10 \times (\$7 - \$2) = \$25$.
- Gains: \$9 each, \$18 in all, against a \$20 cost. The missing \$2 is the deadweight loss.
Answer: Each rises by \$9; together \$18, less than the \$20 cost.
Common mistake
Counting the gains to buyers and sellers but forgetting the cost to taxpayers, and so concluding that a subsidy must raise total surplus. In a market with no spillovers it lowers total surplus. Check: always subtract the government's cost, $s \times Q_s$.
When a good benefits people outside the market, such as vaccines, the case changes: a subsidy can move the market toward the efficient amount. That is the subject of Positive Externalities.