Market failure: the smoke and the wedge
Introduced| English | Meaning |
|---|---|
| externality/ekˈstɜːnælɪti/ | A cost or benefit that falls on people outside the transaction. |
| marginal social cost/ˈmɑːdʒɪnl ˈsəʊʃl kɒst/ | The private supply cost plus the £1. |
| tax wedge | tax wedge |
| incidence/ˈɪnsɪdəns/ | incidence |
The smoke nobody's bill shows
- The flour mill upstream burns cheap coal; the smoke drifts over the city. Every loaf's flour carries a cost that appears on no invoice — £1.20 of damage per loaf's worth of grinding.
- The £4 market from lesson 0302 quietly over-produces bread. This lesson measures the gap, prices the repair, and states honestly where repairs fail.
The externality
- An externality 外部性 is a cost or benefit that falls on people outside the transaction. The buyers and bakers both gain from loaf 780 — the neighbours pay for it.
- The marginal social cost 社会边际成本 is the private supply cost plus the £1.20 damage: $MSC = MPC + 1.2$. The demand curve, meanwhile, is the true marginal social benefit — the smoke never touched the buyers' side.
Private marginal cost at Q = 740 is £3.40 and the external cost is £1.20 a loaf. What is the marginal social cost, in £?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$MSC = MPC + 1.2 = 3.40 + 1.20 = £4.60$ — exactly the price buyers pay under the corrective tax, because at the optimum willingness-to-pay equals full social cost.
A buyer values loaf 780 at £4.20 and its private cost is £3.80. Why is baking it still wrong?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The baker's cost omits £1.20 of smoke: the full cost is $3.80 + 1.20 = £5.00$ against a £4.20 benefit. Loaves 741–800 all fail this test.
Two quantities, one wedge apart
- The private market clears where demand meets supply: 800 loaves at £4. The SOCIAL optimum sets MSB = MSC: 740 loaves.
- Every loaf from 741 to 800 is worth more than its private cost but less than its full social cost — check loaf 780: a buyer values it at £4.20, the baker's cost is £3.80, but the city pays £5.00 all told.
The optimum sets demand (1200 − 100P) against supply plus wedge (400 + 100(P − 1.2)). What is the optimal quantity, in loaves?
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Solving $1200 - 100P_c = 400 + 100(P_c - 1.2)$ gives $P_c = £4.60$ and $Q = 740$ — sixty loaves fewer than the private market's 800.
Price the smoke: the Pigouvian wedge
- A tax of exactly £1.20 a loaf makes sellers' private cost tell the social truth. Buyers and sellers now solve the wedge equation: $1200 - 100P_c = 400 + 100(P_c - 1.2)$.
- The answer lands ON the optimum: buyers pay $P_c = £4.60$, bakers keep $P_p = £3.40$, quantity 740. The tax wedge 税收楔子 collects $1.2 \times 740 = £888$ for the clean-air fund, and the 60 loaves of damage never happen.
The £1.20 tax is levied on 740 loaves. What does the clean-air fund receive, in £?
Difficulty 2/5 · 140 coins + a small bonus on first completion today
$740 \times 1.2 = £888$ — a transfer, not a gain; the welfare gain itself is the small over-production triangle, about £36 a day.
The £888 of tax revenue is the welfare gain from the policy.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The revenue is a transfer from buyers and sellers to the fund. The GAIN is the avoided damage net of lost value — the small triangle of about £36 a day.
Who really pays: incidence
- The price rose £0.60 for buyers and fell £0.60 for bakers — an even incidence 归宿, and no accident: the curves' slopes are ±100, so their elasticities at equilibrium match.
- The lesson generalises: the STATUTORY payer is paperwork; ELASTICITIES divide the burden. The side that can escape (more elastic) pays less, whoever remits the tax.
The £1.20 tax split £0.60 to buyers and £0.60 to sellers. Why so even?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Incidence follows elasticities. With slopes ±100 the two sides respond equally, so the wedge halves; a stiffer side would pay more regardless of who remits.
The wider family of failures
- Public goods — street lighting is non-rival and non-excludable, so no price can fund it; the market supplies too little without taxes or provision.
- Information failures — the seller of a stale loaf knows more than the buyer; such gaps can unpick a market entirely. Each failure has its own repair, and none is "the price is wrong" by itself.
Name the failure before choosing the repair
Each market failure has its own shape — an external cost, a non-excludable good, or an information gap. Sort the city's problems; one market is healthy.
Match each failure to its natural repair.
Difficulty 2/5 · 140 coins + a small bonus on first completion today
Repairs match failures: price the externality, provide the public good, mend the information gap — and leave honest prices alone.
The limits, stated honestly
- The £1.20 is a measurement. No invoice prints the true damage curve; most real policy fights are fights about this number.
- Enforcement costs. The pure gain here is a small triangle (about £36 a day); if collection and metering cost more, the repair loses to the disease.
- Wrong-target taxes. The same £1.20 levied as plain revenue, with the mill still unpriced, moves the quantity but never touches the smoke.
The council levies the same £1.20 as a plain revenue tax and leaves the mill unpriced. What happens?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
A wedge moves the quantity, but only pricing the EXTERNAL COST changes the cost the transaction imposes. Same size, wrong target.
Recap
- MSC = MPC + external cost; the optimum sets MSB = MSC.
- A tax equal to the external cost lands exactly on the optimum; elasticities, not paperwork, divide the burden.
- Every intervention carries a measurement and its limits.