Public Economics — Externalities, Public Goods & Taxation
Public economics examines when private markets fail to produce socially optimal outcomes and what government can do about it. Externalities — costs or benefits imposed on third parties — drive a wedge between private and social optima. Pigouvian taxes and subsidies can correct externalities without distorting incentives beyond the correction. Public goods (non-rival and non-excludable) suffer from the free-rider problem and require public provision or regulation. Taxation inevitably creates deadweight loss; the Ramsey rule and Diamond-Mirrlees results guide optimal tax design. The incidence of any tax depends on relative supply and demand elasticities, not on who legally pays it.
Externalities
A negative externality means the social cost exceeds private cost. The wedge at any output $Q$:
$$MSC = MPC + MEC$$
Private equilibrium: $MPC = MB$. Social optimum: $MSC = MSB$. The Pigouvian tax $t^* = MEC$ evaluated at the socially optimal quantity corrects the market without distorting other margins.
When property rights are clearly defined and transaction costs are zero, private bargaining produces the efficient outcome regardless of who holds the initial rights — without government intervention. The practical limit: high transaction costs make bargaining infeasible in large-number externality cases (e.g., climate change).
A good is a pure public good if it is both non-rival (one person's use does not reduce availability for others) and non-excludable (non-payers cannot be excluded). This creates the free-rider problem — private markets underprovide public goods.
Samuelson rule for optimal provision:
$$\sum_{i=1}^{n} MRS_i = MRT$$
The sum of each person's marginal willingness to pay equals the marginal cost of production.
A specific tax $t$ creates deadweight loss:
$$DWL = \frac{1}{2} \cdot t \cdot |\Delta Q|$$
Tax burden on consumers: $\frac{|E_s|}{|E_d|+|E_s|}\cdot t$. Burden falls entirely on consumers if supply is perfectly elastic, entirely on producers if supply is perfectly inelastic.
To raise a given amount of revenue with minimum deadweight loss, tax rates should be set inversely proportional to demand elasticities:
$$\frac{t_i}{P_i} \propto \frac{1}{|E_i|}$$
This means goods with inelastic demand should face higher tax rates — equating the marginal deadweight loss per dollar of revenue across all taxed goods.
Demand: $P = 200 - Q$. Private supply: $P = 20 + Q$. Marginal external cost $= \$30$ per unit. Find the Pigouvian tax and compare private and social equilibria.
- Private eq.: $200-Q=20+Q \Rightarrow Q_{\ priv}=90$, $P=110$.
- Social supply: $P=50+Q$.
- Social eq.: $200-Q=50+Q \Rightarrow Q^*=75$, $P^*=125$.
- Tax $t^*=\$30$ shifts supply up, reducing output from 90 to 75.
$|E_d|=1.5$, $|E_s|=2.5$, $P^*=\$40$, $Q^*=1{,}000$, tax $t=\$5$. Estimate DWL.
- $\Delta Q \approx \frac{|E_d||E_s|}{|E_d|+|E_s|} \cdot \frac{Q}{P} \cdot t = \frac{1.5\times 2.5}{4} \cdot \frac{1000}{40} \cdot 5 \approx \frac{3.75}{4}\cdot 125 \approx 117$.
- $DWL \approx \frac{1}{2}(5)(117) = \$292.50$.
Practice Problems
Show Answer Key
1. Private: $100-Q=10+Q \Rightarrow Q=45$, $P=55$. Social supply $P=30+Q$; social eq: $100-Q=30+Q \Rightarrow Q^*=35$, $P^*=65$. Pigouvian tax $= \$20$, reducing output from 45 to 35.
2. Lighthouse: non-rival (many ships use it) and non-excludable (can't easily deny access) — public good. Cable TV: non-rival but excludable (can be scrambled) — club good, not pure public good.
3. $\sum MRS = 60+45+30=\$135 > MC=\$120$. Samuelson condition satisfied — efficient to provide.
4. Consumer share $= |E_s|/(|E_d|+|E_s|) = 1.5/2.0 = 75\%$. Consumers bear $\$3$ of the $\$4$ tax.
5. $DWL = \frac{1}{2}(10)(200)=\$1{,}000$.
6. Carbon tax: sets the price of emissions; quantity adjusts. Cap-and-trade: sets the quantity of emissions; price adjusts via permit market. Both can achieve the same efficient outcome under certainty; under uncertainty they differ in how they distribute cost risk.
7. Education generates positive externalities (more productive workforce, lower crime, better civic participation) — private demand undervalues it, so the market undersupplies it. A subsidy raises private demand toward the social optimum.
8. Ramsey: higher tax on less elastic goods. Tax good A more (inelastic demand causes less distortion per dollar of revenue).
9. When transaction costs are high, property rights are unclear, many parties are affected (holdout/free-rider problems), or information is asymmetric — Coase bargaining breaks down.
10. The social marginal cost curve (MSC) lies above the private marginal cost curve (MPC) by the amount of the external cost per unit. The private equilibrium (where demand crosses MPC) produces too much output at too low a price relative to the social optimum (where demand crosses MSC).