Demonstration 1 of 4
Trading toward the Pareto frontier
Which reallocations of bread and cheese make someone better off without hurting anyone?
A values bread twice as much as cheese and B the reverse, so each swap raises both utilities. The corner where A has all the bread and B all the cheese is efficient: any further move hurts someone.
Scroll sideways for the whole equation
b is loaves of bread and c wedges of cheese; there are 8 of each. Both start with (4, 4). One swap moves one cheese wedge from A to B and one loaf from B to A.
Predict first. After 2 swaps, does an unpaid loaf grab still fail the Pareto test?
Choose an example
Scroll sideways for the whole figure
Constructed example: the chapter's hypothetical bread and cheese economy (0, 1 and 4 swaps and the unpaid loaf); 2 swaps and grabs after swaps are added for comparison.
Calculated values
- A holds (bread, cheese)
- (5, 3)
- B holds (bread, cheese)
- (3, 5)
- u_A
- 13
- u_B
- 13
- Pareto improvement on (4, 4)
- yes
After 1 swap of one cheese wedge for one loaf. A holds (5, 3) so u_A = 2 x 5 + 3 = 13; B holds (3, 5) so u_B = 3 + 2 x 5 = 13. Against 12 and 12 at the start this is a Pareto improvement: both gain.
Worked steps
- A: 2 x 5 + 3 = 13 (change +1)
- B: 3 + 2 x 5 = 13 (change +1)
- Pareto improvement: yes
Use the idea
A Pareto test compares each person with their own starting point; it never adds one person's gain to another's loss.
Where the conclusion applies
Linear utilities, fixed totals and a comparison against the starting allocation only. Pareto efficiency says nothing about fairness.
Check your understanding: After 2 swaps without a grab, what are both utilities?
Chapter 6 source: section "Pareto criterion".
Demonstration 2 of 4
Tax wedge and the surplus triangle
How much surplus does a parking tax destroy, and when does avoided congestion make up for it?
The tax cuts trades whose value exceeded their cost; the lost surplus on those trades is the triangle. Revenue is a transfer, not a loss. If each visit also harms others, the avoided harm can outweigh the triangle.
Scroll sideways for the whole equation
P is dollars per visit and Q hundreds of daily visits. The tax drives a wedge between the buyer price and the seller price. Damage is the congestion cost each visit imposes on others.
Predict first. With $3 of damage per visit, is a $6 tax better or worse than a $4 tax?
Choose an example
Scroll sideways for the whole figure
Constructed example: the chapter's hypothetical parking market ($4 tax, $3 damage); taxes of $2 and $6 are added for comparison.
Calculated values
- Visits
- 6,000
- Buyer price
- $14
- Seller price
- $10
- Tax revenue
- $24,000
- Consumer surplus
- $18,000
- Producer surplus
- $18,000
- Deadweight loss
- $4,000
- Avoided damage
- $0
- Net gain over no tax
- -$4,000
With a tax of 4, 20 - 0.10Q - (4 + 0.10Q) = 4 gives Q = 60 hundred, 6,000 visits. Revenue is 4 x 6,000 = $24,000. The triangle is 0.5 x 4 x 2,000 = $4,000. Damage of 0 per visit avoided on 2,000 visits is $0, so against no tax the result is a net social loss of $4,000.
Worked steps
- Q = (16 - 4) / 0.20 = 60 hundred
- Buyer price = 20 - 0.10 x 60 = 14; seller price = 10
- Revenue = 4 x 6,000 = 24,000
- DWL = 0.5 x 4 x 2,000 = 4,000
- Avoided damage = 0 x 2,000 = 0
- Net = 0 - 4,000 = -4,000
Use the idea
Judge a tax by the triangle plus any external cost it removes, not by revenue.
Where the conclusion applies
Linear schedules, a constant damage per visit and no costs of collecting the tax.
Check your understanding: With a $6 tax and $3 damage, what is the net social gain over no tax?
Chapter 6 source: section "Deadweight loss".
Demonstration 3 of 4
Subsidizing the train when roads cannot be priced
When roads cannot be priced, does a distorting transit subsidy raise welfare?
When one distortion cannot be removed, adding another can raise welfare. The subsidy pays off through the road externality it reduces, but financing it has its own cost.
Scroll sideways for the whole equation
A car trip imposes $12 of congestion on others. Moving it to the train costs the operator $6 and the traveler $2 of convenience. An $8 subsidy induces the switch; each subsidy dollar costs lambda dollars of surplus to raise.
Predict first. At what cost per subsidy dollar does the subsidy break even?
Choose an example
Scroll sideways for the whole figure
Constructed example: the chapter's hypothetical commuter (cost of funds 0 and 0.75, the $12 toll); 0.25 is added for comparison.
Calculated values
- Cost of funds
- $0
- Net welfare per trip
- $4
- Break-even cost per subsidy dollar
- 0.50
- Road toll
- no
Net welfare = 12 - 6 - 2 - 0 x 8 = 4, so society gains $4 per trip. The subsidy breaks even when the cost per dollar is 4 / 8 = 0.50.
Worked steps
- Resource gain = 12 - 6 - 2 = 4
- Cost of funds = 0 x 8 = 0
- Net = 4 - 0 = 4
Use the idea
Before copying a first-best rule such as marginal cost pricing, ask which other distortions are fixed and what the policy's financing costs.
Where the conclusion applies
One trip, constant per-trip values and a subsidy that is a pure transfer apart from its financing cost.
Check your understanding: With a cost of funds of 0.25, what is net welfare from the subsidy?
Chapter 6 source: section "General theory of second best".
Demonstration 4 of 4
A safety mandate that markets will not supply
Why does no insured driver pay for safety alone, and when does a mandate fix it?
Because the insurer cannot see the action, a driver who acts shares the saving with 99 others and keeps only 0.06. A rule that everyone must act, or a way to observe action, captures the full saving.
Scroll sideways for the whole equation
100 fully insured drivers face a $100 loss. A hidden $3 action cuts accident probability from 0.10 to 0.04. Insurers charge the pool's expected claim per driver.
Predict first. With an enforcement cost of $2 per driver, is the mandate still worthwhile?
Choose an example
Scroll sideways for the whole figure
Constructed example: the chapter's hypothetical insurance pool (no policy, mandate, telematics, enforcement 0 and 4); enforcement of 2 is added for comparison.
Calculated values
- Pool real cost
- $700
- Premium per driver
- $4
- Gain over no policy
- $300
- Premium cut from acting alone
- $0.06
Real cost = 100 x 3 + 100 x 0.04 x 100 + 100 x 0 = 700, against 1,000 with no policy: a gain of $300.
Worked steps
- Prevention = 300
- Accident loss = 100 x 0.04 x 100 = 400
- Enforcement = 0
- Total = 300 + 400 + 0 = 700
- Gain = 1,000 - 700 = 300
Use the idea
When a market's prices cannot reflect a hidden action, compare a rule's resource saving with its enforcement cost.
Where the conclusion applies
Identical drivers, full insurance, competitive premiums and a standard that works exactly like the hidden action.
Check your understanding: What is the pool's real cost under the mandate with enforcement of $2?
Chapter 6 source: section "Greenwald-Stiglitz constrained inefficiency".