Demonstration 1 of 4
Vote trading in a three-member council
When does a vote trade create surplus, and when does it only shift costs onto the excluded member?
A logroll lets members trade support so that projects pass that would fail separately. The traders accept when their own gains exceed the taxes they pay for the partner's project; the excluded member pays without benefit. Whether the council gains depends on total benefit against total cost.
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Districts A, B and C share the tax cost of every project equally. Each project gives its host the gross benefit. Delta V is a district's payoff from passing both projects and Delta W their sum.
Predict first. At tax cost 3, does the package still create aggregate surplus?
Choose an example
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Constructed example: the chapter's hypothetical council (benefit 9, tax cost 2 and 4); a tax cost of 3 and host benefits 6 and 12 are added.
Calculated values
- Host's own project
- 7
- Delta V_A
- 5
- Delta V_B
- 5
- Delta V_C
- -4
- Delta W
- 6
- Trade accepted
- Yes
Separately each project fails two to one, since only its host gains 9 - 2 = 7. In the package A and B each get 7 - 2 = 5, so both traders gain, so the trade is accepted; C pays for both and gets -4. In total the package creates aggregate surplus: Delta W = 6.
Worked steps
- Each project: host 9 - 2 = 7, others -2
- Delta V_A = 7 - 2 = 5; Delta V_B = -2 + 7 = 5
- Delta V_C = -2 - 2 = -4
- Delta W = 5 + 5 + (-4) = 6
Use the idea
Judge a package deal by its total benefit against total cost, not by whether the members who negotiated it gain.
Where the conclusion applies
Three equal tax shares, sincere separate voting, binding trades and benefits only to the host.
Check your understanding: With host benefit 9 and tax cost 3, what are coalition and total payoffs?
Chapter 71 source: section "Logrolling".
Demonstration 2 of 4
Expected bailouts change the plan
How does an expected rescue in bad times change which project an organization picks?
A rescue that arrives only after losses truncates the downside the organization bears. The perceived value of risky expansion rises while its true value does not, so plans the public would reject get chosen.
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The basic plan nets 10. The large plan costs 140 and yields 180 with strong demand or 80 with weak demand. An expected bailout covers part of the weak-state loss of 60; exposure is its expected cost to the treasury.
Predict first. What bailout size makes the operator indifferent at 50/50 odds?
Choose an example
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Constructed example: the chapter's hypothetical transit operator (basic 10, cost 140, outcomes 180 and 80 at 0.5, bailout 0 or 50); bailouts 30 and 60 and probabilities 0.3 and 0.7 are added.
Calculated values
- True large-plan value
- -10.00
- Perceived large-plan value
- 15.00
- Basic plan
- 10.00
- Chosen plan
- Large
- Treasury exposure
- 25.00
Under a hard budget the large plan is worth 0.5 x 40 + 0.5 x (-60) = -10.00. With the expected bailout the operator sees 0.5 x 40 + 0.5 x (-10) = 15.00 against the basic plan's 10, so it chooses the large plan. Expected treasury exposure is 25.00.
Worked steps
- True: 0.5 x 40 + 0.5 x (-60) = -10.00
- Weak state with bailout 50: 80 - 140 + 50 = -10
- Perceived: 0.5 x 40 + 0.5 x (-10) = 15.00
- Exposure: 0.5 x 50 = 25.00
Use the idea
When a body expects support after losses it helped cause, evaluate its plans with the loss it would bear without support.
Where the conclusion applies
Two demand states, risk neutrality, a bailout only in the weak state and capped at the loss.
Check your understanding: With bailout 30 and weak probability 0.5, which plan is chosen?
Chapter 71 source: section "Soft budget constraint".
Demonstration 3 of 4
When is studying the ballot worth it?
When does a voter find it worth paying to become informed?
The instrumental value of information is the stake times the chance one's vote matters. In a large electorate that product is tiny, so ignorance is rational unless decisiveness rises or the knowledge has other uses.
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c is the cost of study (3 hours at $18), B = $600 the household's stake in the better outcome, p the chance the ballot is decisive and d any reusable private value of the research.
Predict first. With a 1 percent pivot chance and no reusable value, does the resident study?
Choose an example
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Constructed example: the chapter's hypothetical resident (3 hours at $18, stake $600, pivot 1/200,000 or 0.10, reusable value 0 or $70); a pivot chance of 0.01 and reusable value $30 are added.
Calculated values
- Cost of study
- $54.00
- Expected pivotal benefit pB
- $0.003
- Reusable value d
- $0
- Net return
- -$53.997
- Decision
- Stay uninformed
Study costs 3 x 18 = 54 dollars. With a 1/200,000 chance of deciding, the expected pivotal benefit is 1/200,000 x 600 = $0.003; adding reusable value $0 gives $0.003, so the net return is -$53.997 and the resident stays uninformed.
Worked steps
- c = 3 x 18 = 54
- pB = 1/200,000 x 600 = 0.003
- Net = 0.003 + 0 - 54 = -53.997
- Net return is negative, so stay uninformed
Use the idea
To predict how informed voters will be, compare the cost of learning with stake times decisiveness plus any private use of the knowledge.
Where the conclusion applies
Risk neutrality, a fixed stake, a known pivot probability and no enjoyment from being informed.
Check your understanding: With pivot 0.01 and reusable value $30, what is the net return?
Chapter 71 source: section "Rational ignorance".
Demonstration 4 of 4
Everyone's project, no one's bill
Why do districts sharing a common budget approve projects that cost more than they are worth?
Shared finance lets each district count the full local benefit but only its share of the cost. When every district does the same, each pays for everyone's projects and all end up worse off.
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n districts pay equal shares of national taxes. Each project costs $10 million and benefits only its sponsoring district. A sponsor bears 10/n of its own project's cost.
Predict first. With 2 districts sharing finance, does a $6M project still get proposed?
Choose an example
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Constructed example: the chapter's hypothetical federation (5 districts, benefit $6M, cost $10M, and the district-levy case, shown as 1 district); 2 and 10 districts and benefits $4M and $12M are added.
Calculated values
- Own cost share ($M)
- 2.00
- Perceived gain ($M)
- 4.00
- Projects passed
- 5
- District final position ($M)
- -4.00
- Aggregate surplus ($M)
- -20.00
With 5 districts sharing finance, a sponsor pays 10 / 5 = 2.00 million of its own project and sees a gain of 6 - 2.00 = 4.00 million, so every representative proposes and all 5 projects pass. Each district gets 6 but pays 5 x 2.00 = 10.00, ending at -4.00. In truth each project costs more than it is worth: 6 - 10 = -4 million per project, so aggregate surplus is -20 million.
Worked steps
- Cost share = 10 / 5 = 2.00
- Perceived gain = 6 - 2.00 = 4.00
- True net = 6 - 10 = -4
- Aggregate = 5 x (6 - 10) = -20
Use the idea
Check the cost share decision makers face at the proposal margin; tying costs to the beneficiary removes the bias.
Where the conclusion applies
Equal tax shares, purely local benefits, identical districts and approval of every proposal.
Check your understanding: With 10 districts and benefit $4M, what is the perceived gain and aggregate surplus if all pass?
Chapter 71 source: section "Fiscal common-pool problem".