Market Failure and Externalities: Question 2

Worked solution

Part (a): Why this is a negative externality of production

Rockford Cement Works decides how much cement to produce by weighing up its own private costs against the price it can charge. It has no financial reason to take account of costs falling on people who are not part of the transaction. But the dust released as a by-product of production increases respiratory illness in Millbrook, imposing a real cost on residents that the firm does not pay.

This means: $MSC = MPC + MEC$

where $MEC > 0$ is the marginal external cost of the dust. Because the firm’s production decisions are guided only by $MPC$, not the higher $MSC$, it produces more than is socially desirable. This uncompensated third-party cost is exactly what defines a negative externality of production.

Part (b): Free-market output and socially optimum output

Free-market output: firms maximise profit where the price they receive (equal to $MPB$, since consumers pay based on their own private benefit) equals their marginal private cost, $MPB = MPC$. Reading down the table, at an output of 4 thousand bags, $MPB = MSB = 60$ and $MPC = 60$, both equal $60 per bag. So the free-market output is 4 thousand bags per week.

Socially optimum output: allocative efficiency requires marginal social benefit to equal marginal social cost, $MSB = MSC$. At an output of 3 thousand bags, $MSB = 70$ and $MSC = 70$, both equal $70 per bag. So the socially optimum output is 3 thousand bags per week.

Because the free-market output (4 thousand bags) exceeds the socially optimum output (3 thousand bags), the negative externality causes the market to over-produce cement relative to what is socially efficient.

Part (c): External cost and deadweight welfare loss

At the free-market output of 4 thousand bags per week: $MEC = MSC - MPC = 80 - 60 = 20$

So the marginal external cost is $20 per bag at this output (matching the constant MEC column in the table).

Between the socially optimum output (3 thousand bags) and the free-market output (4 thousand bags), the gap between $MSC$ and $MSB$ widens from $0 (at 3 thousand bags, where $MSB = MSC = 70$) up to $20 per bag (at 4 thousand bags, where $MSC - MSB = 80 - 60 = 20$). Since both $MSC$ and $MSB$ change at a constant rate with output, this gap grows in a straight line, so the deadweight welfare loss is the area of a triangle: $DWL = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times (4-3) \times 20 = 10$

Because output is measured in thousand bags, this deadweight welfare loss of $10$ represents $10 thousand ($10,000) per week, the value of resources wasted producing the units between the socially optimum and the free-market output, where the extra social cost of production exceeds the extra social benefit.

Part (d): A corrective tax

A specific (per-unit) tax of $20 per bag, set equal to the constant marginal external cost, would add $20 to Rockford Cement Works’ cost for every bag produced, so its effective marginal cost becomes: $MPC + tax = MPC + MEC = MSC$

Facing this higher effective marginal cost, the profit-maximising firm would now choose its output where $MPB = MPC + tax$, which, because $MPC + tax = MSC$, is exactly where $MPB = MSC$, i.e. at the socially optimum output of 3 thousand bags per week. The tax “internalises” the externality by making Rockford Cement Works pay the full social cost of its production.

Final answers

• (a) MSC exceeds MPC because of the uncompensated marginal external cost (the dust/health cost) imposed on Millbrook residents
• (b) Free-market output = 4 thousand bags/week; socially optimum output = 3 thousand bags/week
• (c) MEC at the free-market output = $20 per bag; deadweight welfare loss = $10 thousand ($10,000) per week
• (d) A specific tax of $20 per bag (equal to the MEC) would move output to the socially optimum level