Pigouvian Tax Explained: Graph, Formula, and Worked Example

A Pigouvian tax is a per-unit tax on an activity that creates a negative externality, set equal to the marginal external cost at the socially optimal quantity. In formula form, tax per unit = MSC - MPC measured at Q, where Q is the quantity at which marginal social benefit equals marginal social cost. The tax makes producers pay for the damage they impose on bystanders, so the market cuts output to the efficient level on its own and the deadweight loss disappears.

The idea comes from the economist Arthur Pigou, and it is the standard answer to pollution-type externalities in AP Microeconomics Unit 6 and in intro college courses. Below: the graph, a full numeric example, the subsidy version, and the real limits.

Why a market with an externality needs a tax at all

A market weighs only private costs. A steel mill pays for coal, labor, and machines, its marginal private cost (MPC), but not for the smoke that harms people downwind. That unpaid damage per unit is the marginal external cost (MEC), and adding it to private cost gives marginal social cost (MSC):

• MSC = MPC + MEC
• The market quantity solves MPB = MPC (demand meets private supply).
• The socially optimal quantity solves MSB = MSC.

Because MSC sits above MPC, the market overproduces: every unit between Q* and the market quantity costs society more than it is worth to buyers, and the lost value is a deadweight loss. For the full background on spillovers, read externalities explained, and for the AP unit treatment, see the public goods and externalities module.

The Pigouvian tax graph: MSC vs MPC vs MSB

Draw price on the vertical axis and quantity on the horizontal axis, then add three curves:

1. MSB, downward sloping. With a production externality, buyers capture all the benefit, so demand (MPB) and MSB are the same curve.

2. MPC, upward sloping. This is the market supply curve.

3. MSC, upward sloping and above MPC. The vertical gap between MSC and MPC is the marginal external cost.

The market settles where MSB crosses MPC, at Q market. Society wants Q*, where MSB crosses MSC, further left. The deadweight loss triangle has its tip at the MSB = MSC crossing and its tall side running from demand up to MSC above Q market.

Now add the tax. A per-unit tax of t shifts the private supply curve up by exactly t, because sellers now need t more per unit to be willing to supply it. If t equals the gap between MSC and MPC at Q, the taxed supply curve passes through the MSB = MSC point. The market now clears at Q, buyers pay the price on MSC, sellers keep the price on MPC, and the gap between those two prices is the tax. You can drag these curves yourself on the externality sandbox, then switch the tax on to see the market move to Q* and the deadweight loss triangle disappear.

See it move

Make it move

A negative externality: the market produces more than the socially optimal quantity. Tap it to drag the curves, open the full version, or put it on your own site free, or turn it into a five-minute class activity.

How the tax removes the deadweight loss

The tax does not ban anything or set a quantity. It changes the price signal so the private decision matches the social one. After the tax:

• A buyer only buys a unit if it is worth at least the taxed price, which equals MSC at the margin.
• A seller only sells a unit if the after-tax price covers MPC.
• So every unit that gets traded is worth at least its full social cost, and every unit worth more than its social cost still gets traded.

That is the efficiency condition MSB = MSC. On a market with no externality, a per-unit tax creates deadweight loss by pushing output below the efficient level. On a market with a negative externality, the same tax moves output toward the efficient level and removes the loss already there.

Worked example: finding the optimal tax, revenue, and welfare gain

Suppose a chemical plant sells to a market with these curves (P in dollars, Q in tons):

• Demand (MPB = MSB): P = 120 - 2Q
• Private supply (MPC): P = 20 + 2Q
• Each ton causes $20 of damage to nearby residents, so MEC = 20 at every quantity.
• Marginal social cost: MSC = 20 + 2Q + 20 = 40 + 2Q

### Step 1: The free-market outcome

Set MPB = MPC: 120 - 2Q = 20 + 2Q, so 100 = 4Q and Q market = 25 tons.

Price: P = 120 - 2(25) = 120 - 50 = $70. Check with supply: 20 + 2(25) = 70. Both curves give $70.

### Step 2: The socially optimal quantity

Set MSB = MSC: 120 - 2Q = 40 + 2Q, so 80 = 4Q and Q* = 20 tons.

Check: MSB = 120 - 2(20) = 80 and MSC = 40 + 2(20) = 80. They match at $80. (The socially optimal quantity calculator walks through this same step.)

### Step 3: The Pigouvian tax

Measure the external cost at Q: MSC - MPC = 80 - (20 + 2 × 20) = 80 - 60 = $20 per ton*.

With the tax, private supply becomes P = 20 + 2Q + 20 = 40 + 2Q, which is identical to MSC. Setting it equal to demand gives Q = 20, so the tax lands the market exactly on Q*.

### Step 4: Prices after the tax

• Buyers pay $80 (read off demand at Q = 20).
• Sellers keep 80 - 20 = $60 (read off MPC at Q = 20: 20 + 40 = 60).
• The buyer price rose by $10 and the seller price fell by $10, so the burden is split evenly here because the two curves have equal slopes. See tax incidence for why the less elastic side bears more of the tax.

### Step 5: Tax revenue

Revenue = tax × quantity after the tax = 20 × 20 = $400.

### Step 6: The welfare gain

The deadweight loss before the tax is a triangle with a base of 25 - 20 = 5 tons. Its height is the gap between MSC and MSB at Q = 25: MSC = 40 + 50 = 90 and MSB = 70, so the height is 90 - 70 = 20. Area = 0.5 × 5 × 20 = $50. The tax removes the whole triangle, so the welfare gain is $50.

Check it a second way by adding up every piece of surplus:

Taxed consumer surplus uses 120 - 80 = 40 as its height; producer surplus uses 60 - 20 = 40. Total surplus rises from $750 to $800, a gain of exactly $50, matching the triangle. Buyers and sellers each lose surplus, but the government collects $400 and residents suffer $100 less damage, so society is better off. The Pigouvian tax calculator runs a second example if you want more practice.

When the external cost changes with output

Often the damage per unit grows with output (the tenth ton of smoke on a polluted day hurts more than the first). Then the rule "measure the gap at Q*" matters.

Keep the same demand and MPC, but let MEC = Q, so MSC = 20 + 3Q.

• Optimum: 120 - 2Q = 20 + 3Q, so 100 = 5Q and Q* = 20. Check: MSB = 80 and MSC = 20 + 60 = 80.
• MEC at Q* is 20 (since MEC = Q), so the correct tax is $20.
• MEC at the market quantity of 25 would be $25. A $25 tax gives 45 + 2Q = 120 - 2Q, so Q = 18.75, which overshoots below Q*.

The tax is the damage of the last efficient unit, not the damage at the old market quantity.

Pigouvian subsidies for positive externalities

The same logic runs in reverse when a good creates benefits for bystanders, such as vaccinations or education. Now the marginal external benefit (MEB) opens a gap between MSB and MPB, and the market underproduces. The fix is a per-unit subsidy equal to MEB at the socially optimal quantity.

Example: demand (MPB) is P = 90 - Q, supply (MPC = MSC) is P = 10 + Q, and each unit gives $20 of benefit to others, so MSB = 110 - Q.

• Market: 90 - Q = 10 + Q gives Q = 40 at P = 50.
• Optimum: 110 - Q = 10 + Q gives Q* = 50, where MSB = 60 and MSC = 60.
• Subsidy = MSB - MPB at Q* = 60 - 40 = $20. Sellers receive $60 and buyers pay 60 - 20 = $40, which is exactly MPB at Q = 50.
• Government cost = 20 × 50 = $1,000.
• Deadweight loss removed = 0.5 × (50 - 40) × 20 = $100.

The per-unit subsidy calculator covers this case step by step.

Real limits of Pigouvian taxes

The graph makes the policy look exact. In practice, several problems get in the way.

• Measuring the external cost is hard. Nobody sees the MSC curve. Economists estimate damage from health studies, property values, and climate models, and reasonable estimates can differ widely. Estimates of the social cost of carbon are the best-known case. A tax set too low leaves some deadweight loss; a tax set too high causes underproduction and a new loss.
• The damage varies by place. A ton of emissions in a dense city does more harm than the same ton in an empty area, but one tax rate treats them the same.
• Quantity is uncertain. A tax fixes the price of pollution but not the amount. If the goal is a firm cap, a cap-and-trade system fixes the quantity and lets the permit price adjust instead.
• Fairness and politics. Taxes on energy can take a bigger share of income from poorer households, which is why proposals often pair a carbon tax with rebates.

Private bargaining is the other main alternative when only a few parties are involved. The Coase theorem explained shows when that works without any tax.

Common mistakes

• Measuring the tax at the market quantity. The tax equals MEC at Q*. With a constant MEC the two readings agree, but on an exam always say "at the socially optimal quantity."
• Shifting the wrong curve. A tax on producers shifts supply (MPC) up by the amount of the tax, so the taxed supply curve passes through the MSB = MSC point at Q* (with a constant MEC it lies exactly on MSC). It does not move demand or MSC.
• Saying the tax creates deadweight loss. A correctly sized Pigouvian tax removes deadweight loss. Only an oversized tax creates a new one.
• Using the market quantity for revenue. Revenue = tax × the quantity sold after the tax, which is Q*.
• Forgetting the external damage in total surplus. Social surplus = consumer surplus + producer surplus + tax revenue - external damage.