Government intervention in a market: formula sheet

Key formulas

Efficiency condition

Allocative efficiency when MSB = MSC

MSB is marginal social benefit and MSC is marginal social cost. Intervention aims to move output toward this point.

Marginal social cost with a negative externality

MSC = MPC + MEC

MPC is marginal private cost and MEC is marginal external cost. The market produces where MPB = MPC, which is too much.

Marginal social benefit with a positive externality

MSB = MPB + MEB

MEB is marginal external benefit. The market produces where MPB = MPC, which is too little.

Deadweight loss from over- or under-provision

DWL = ½ × (difference in price at the margin) × (Q difference)

Valid only when the curves are straight lines, so the loss is a triangle. Use area of a triangle.

Public good test

Non-excludable and non-rivalrous

Both features must hold for a pure public good. The free-rider problem follows.

Equilibrium condition

Qd(P*) = Qs(P*)

Solve this first. P* is the market price without controls.

Price ceiling rule

Binding if Pmax < P*; non-binding if Pmax ≥ P*

A binding ceiling gives a shortage.

Price floor rule

Binding if Pmin > P*; non-binding if Pmin ≤ P*

A binding floor gives a surplus.

Shortage

Shortage = Qd(Pmax) − Qs(Pmax)

Valid only when Pmax < P*.

Surplus

Surplus = Qs(Pmin) − Qd(Pmin)

Valid only when Pmin > P*.

Quantity traded under a binding control

Q traded = min(Qd, Qs) at the controlled price

Assumes no government purchase of the excess and no black market.

Per-unit tax wedge

Pc − Pp = t

Pc is the price buyers pay, Pp is the price sellers keep, t is the tax per unit. Holds for a specific tax.

Ad valorem tax per unit

t = r × Pp (rate on the pre-tax seller price, equivalent to r ÷ (1 + r) × Pc), or t = r × Pc (if the rate is applied to the tax-inclusive price the buyer pays)

Check how the question defines the rate. Per-unit tax changes with price.

Consumer share of tax burden

Consumer burden = (Pc_new − P_old) × Q_new

Use the new quantity traded.

Producer share of tax burden

Producer burden = (P_old − Pp_new) × Q_new

Consumer burden + producer burden = tax revenue = t × Q_new.

Tax revenue

Revenue = t × Q_new

Q_new is the quantity traded after the tax.

Deadweight loss of a tax (linear curves)

DWL = ½ × t × (Q_old − Q_new)

Area of the triangle between the curves, from the new quantity to the old one.

Subsidy per unit

Pc = Pp − s

Buyers pay Pc, sellers receive Pp, s is the subsidy per unit. Cost to government = s × Q_new.

Incidence and elasticity

Consumer share ÷ Producer share = PES ÷ |PED| (approximately, for small taxes)

The more elastic side bears less. This is a guide, not an exact rule for large taxes on non-linear curves.

Marginal social cost

MSC = MPC + MEC

Used for negative externalities in production. MEC is the extra cost borne by third parties per unit.

Marginal social benefit

MSB = MPB + MEB

Used for positive externalities in consumption. MEB is the extra benefit to third parties per unit.

Socially efficient output

MSB = MSC

The market outcome is where MPB = MPC. With an externality these two points differ.

Pigouvian tax

Tax per unit = MEC at the socially efficient output

Shifts MPC up to MSC. If MEC is constant, the tax equals that constant.

Pigouvian subsidy

Subsidy per unit = MEB at the socially efficient output

Shifts MPB up to MSB for a positive externality.

Public good test

Public good = non-excludable and non-rival

Both conditions must hold. A good with only one is a quasi-public good, a club good or a common resource.

Effect of a binding quota

Quota < free-market quantity ⇒ supply falls, price rises, quantity traded = quota

If the quota is at or above the free-market quantity, it does not bind and nothing changes.

Quota rent per unit

Quota rent = market price with quota − price at which the licence holder can obtain the good

For an import quota, this is the domestic price minus the world price. It goes to licence holders, not the government, unless licences are auctioned.

Net judgement of intervention

Net gain = benefits of correcting the failure − (administration costs + distortions + unintended costs)

If net gain is negative, the intervention is a government failure. Use this as a way of thinking, not a number to compute unless data are given.

Market failure vs government failure

Market failure: free market outcome is inefficient. Government failure: intervention leaves outcome worse or too costly.

Keep the two ideas apart in every answer.

Quick revision

• Market failure includes externalities, public goods, imperfect information and market power. Equity is a separate reason to intervene.
• A price ceiling is a legal maximum price. It only binds if set below the equilibrium price, and it causes a shortage.
• A price floor is a legal minimum price. It only binds if set above the equilibrium price, and it causes a surplus.
• Price controls reduce the traded quantity, which creates a welfare loss and can lead to black markets or queues.
• A tax on sellers shifts supply left. A subsidy shifts supply right.
• Tax incidence depends on relative elasticity. The less elastic side bears more of the tax.
• A tax usually creates a deadweight loss. The loss grows as demand and supply become more elastic.
• A negative externality means social cost exceeds private cost, so the market produces too much.
• A positive externality means social benefit exceeds private benefit, so the market produces too little.
• Public goods are non-excludable and non-rival, so the free-rider problem means private markets under-provide them.
• Quotas limit quantity. Regulation sets rules on behaviour, prices or standards.
• Government failure means a policy leaves the outcome worse or costlier than the problem it aimed to fix.

Common mistakes

• Saying any good the government provides is a public good. Fix: Use the two tests: non-excludable and non-rivalrous. Education is publicly provided but is a merit good, not a pure public good.
• Confusing merit goods with goods that have positive externalities. Fix: A merit good is defined by society's judgement and information problems. Externalities can be one reason, but not the only one.
• Saying every price ceiling causes a shortage. Fix: Always compare the ceiling with P* first. A ceiling at or above P* has no effect.
• Calculating a shortage or surplus using the wrong price. Fix: Substitute the controlled price into both demand and supply, then subtract.
• Assuming the party who pays the tax to the government bears it. Fix: Always find the new Pc and Pp and compare with the old price. The burden depends on elasticity, not on who sends the cheque.
• Shifting the wrong curve, or shifting demand for a tax on sellers. Fix: A tax on sellers shifts supply. A tax on buyers shifts demand down by the tax. The result for Pc and Pp is the same either way.
• Calling any good that the government provides a public good. Fix: Use the definition only. Education and healthcare are often state-provided but are excludable and rival, so they are not pure public goods.
• Setting the Pigouvian tax equal to the market price or to total external cost. Fix: The tax per unit equals the marginal external cost at the efficient output. Check it by seeing that MPC plus tax equals MSC there.
• Saying a quota always raises prices. Fix: State that a quota raises price only if it is set below the free-market quantity. Otherwise it is non-binding.
• Mixing up market failure and government failure. Fix: Market failure is the free market failing. Government failure is the intervention failing. Name the one you are discussing.

Exam tips

• Define the term first. Examiners often award a mark for a precise definition of market failure, public good or merit good.
• In calculation questions, find the market and social optimum separately, then compute the triangle area. Show each equation.
• Give a concrete example for each type of failure. Keep examples short and standard.
• End every discussion question with a judgement that weighs market failure against government failure.
• In multiple-choice questions, read the wording for non-excludable and non-rivalrous carefully. Many options differ by one word.
• Write P* first in every answer. It shows the examiner you know when a control is binding.
• Label the axes, P*, the controlled price, Qd, Qs and the shortage or surplus on every diagram. Marks go to labels.
• In MCQs, check the position of the control price against P* before reading the options. Non-binding cases are a common trap.