How to Find the Socially Optimal Quantity
The socially optimal quantity is the output where marginal social benefit equals marginal social cost: set MSB = MSC and solve for Q.
The Socially Optimal Quantity formula
Calculator
Enter linear demand and supply plus the spillover per unit to solve MSB = MSC and compare with the market.
The a in demand P = a − bQ, which is marginal private benefit.
The b in P = a − bQ. Enter it as a positive number.
The c in supply P = c + dQ, which is marginal private cost.
The d in P = c + dQ.
Spillover cost of each unit. Enter a negative number for an external benefit.
At 16 units the last unit's value to society equals its full cost to society, both $68.
- Market quantity
- 20
- Market price
- $60
- Price at the optimum
- $68
- Quantity gap
- 4
- Deadweight loss
- $32
- Market outcome
- Overproduces
Left alone the market settles where private benefit meets private cost, at 20 units.
The price buyers pay at the market quantity.
Marginal social benefit, which equals marginal social cost, at the efficient quantity.
How far the market quantity sits from the socially optimal one.
Half the $16 spillover times a quantity gap of 4 comes to $32.
The spillover cost never lands on the producer, so the market quantity sits 4 above the optimum. A per-unit tax of $16 closes the gap.
How to calculate Socially Optimal Quantity, step by step
- 1Write the private curves. MPB is demand and MPC is supply; setting MPB = MPC gives the quantity the market produces on its own.
- 2Add the external effect. Add the marginal external cost to MPC for a negative externality, or the marginal external benefit to MPB for a positive one.
- 3Set MSB equal to MSC. Solve MSB = MSC for quantity, the output where the last unit's value to society equals its full cost to society.
- 4Compare with the market quantity. A negative externality makes the optimal quantity smaller than the market quantity; a positive externality makes it larger.
- 5Name the correction. Close the gap with a per-unit tax equal to the external cost, or a per-unit subsidy equal to the external benefit, measured at the optimal quantity.
Worked example: Socially Optimal Quantity
Demand (MPB) is P = 100 − 2Q and private supply (MPC) is P = 20 + 2Q. The market clears where 100 − 2Q = 20 + 2Q, so 4Q = 80 and Q = 20 at a price of 100 − 2(20) = $60. Now suppose each unit dumps $16 of pollution on third parties, so MSC = 20 + 2Q + 16 = 36 + 2Q. Setting MSB = MSC gives 100 − 2Q = 36 + 2Q, so 4Q = 64 and the socially optimal quantity is Q = 16. Check it: MSB = 100 − 2(16) = $68 and MSC = 36 + 2(16) = $68. The market overproduces by 20 − 16 = 4 units, and the deadweight loss is 0.5 × $16 × 4 = $32.
Socially Optimal Quantity questions
Is the socially optimal quantity the same as the market quantity?
Only when there is no externality, because MSB then equals MPB and MSC equals MPC, so both conditions pick the same output. Any spillover cost or benefit drives the two quantities apart.
How do you find deadweight loss from the socially optimal quantity?
Measure the triangle between MSB and MSC across the gap between the market quantity and the optimal quantity: DWL = 0.5 × the per-unit external effect × the difference in quantities.
Which quantity does a Pigouvian tax target?
The socially optimal quantity. A per-unit tax equal to the marginal external cost raises private cost up to MSC, so profit-seeking output falls to where MSB = MSC.
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