How to Calculate a Per-Unit Subsidy
A corrective per-unit subsidy equals the marginal external benefit at the socially optimal quantity, the vertical gap between MSB and MPB at Q optimal.
The Per-Unit Subsidy formula
Calculator
Enter private demand, social demand and marginal social cost to size the corrective subsidy and its cost.
Private demand is P = 60 − 2Q, what buyers alone will pay.
How far private demand falls per extra unit. Enter 2 for P = 60 − 2Q.
MSB is MPB plus the external benefit, so 60 + 15 = 75.
Normally the same slope as MPB, since the spillover is a constant per unit.
Supply is P = 10 + 3Q with no cost spillover in this example.
How far cost rises per extra unit. Enter 3 for P = 10 + 3Q.
Pay $15 per unit, the gap between MSB and MPB at 13 units, and buyers act on the full social benefit.
- Free-market quantity
- 10
- Socially optimal quantity
- 13
- Price sellers receive
- $49
- Price buyers pay
- $34
- Total cost to government
- $195
- Market verdict
- Underproduction
Where private demand meets supply, so buyers act on their own benefit only and take 10 units.
Where MSB meets MSC. The free market falls 3 units short of it.
Sellers receive $49 while buyers pay only $34, and the government covers the difference.
The seller price minus the subsidy, which is exactly what private demand will bear at the optimal quantity.
Subsidy per unit times the quantity traded after the subsidy, which is the optimal quantity.
How to calculate Per-Unit Subsidy, step by step
- 1Find the socially optimal quantity. Q optimal is where marginal social benefit equals marginal social cost, which sits to the right of the free-market quantity for a positive externality.
- 2Measure the external benefit there. At Q optimal, subtract marginal private benefit from marginal social benefit; that vertical gap is the marginal external benefit.
- 3Set the subsidy equal to that gap. The per-unit subsidy equals the marginal external benefit, which lifts private demand until buyers act on the full social benefit.
- 4Split the price. Sellers receive the price on the supply curve at Q optimal, buyers pay that price minus the subsidy, and the difference is the subsidy per unit.
- 5Compute total government spending. Multiply the per-unit subsidy by the quantity traded after the subsidy, which is Q optimal.
Worked example: Per-Unit Subsidy
Flu shots have marginal private benefit P = 60 − 2Q and marginal social cost P = 10 + 3Q. Each shot gives $15 of protection to other people, so MSB = 75 − 2Q. Free market: 60 − 2Q = 10 + 3Q gives Q = 10 at a price of $40. Social optimum: 75 − 2Q = 10 + 3Q gives 5Q = 65, so Q = 13, where MSC = 10 + 39 = $49 and MSB = 75 − 26 = $49. The per-unit subsidy is $15. Sellers receive $49, buyers pay 49 − 15 = $34, which matches private demand at Q = 13 (60 − 26 = $34). Total cost to the government = $15 × 13 = $195, and the old deadweight loss of ½ × 3 × 15 = $22.50 is gone.
Per-Unit Subsidy questions
How is a per-unit subsidy different from a lump-sum subsidy?
A per-unit subsidy pays a fixed amount on every unit sold, so it shifts the curve and changes the quantity traded. A lump-sum subsidy is one fixed payment that leaves marginal cost and marginal benefit unchanged, so quantity does not move.
How much does the subsidy cost the government?
Total cost equals the per-unit subsidy times the quantity traded after the subsidy. In the example above that is $15 × 13 = $195.
Can a subsidy create deadweight loss?
Yes, if it is larger than the marginal external benefit it pushes output past the socially optimal quantity and creates a new deadweight loss. Set exactly equal to the external benefit, it removes the loss from underproduction instead.
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