Demonstration 1 of 4
Free entry burns the fishery rent
Where does free entry leave the fish stock, and what happens to the rent?
Each entrant compares its own catch with its own cost and ignores the stock it depletes for everyone else. Entry continues until revenue per unit of effort equals its cost, so the stock falls until the rent is gone. Limiting access keeps a larger, cheaper-to-fish stock and a rent.
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X is the fish stock, G(X) its natural growth, which equals the sustainable harvest in a steady state. Harvest is H = 0.01 E X for effort E. p is the dockside price per unit of harvest and c the full cost of a unit of effort. X_OA is the open-access stock where effort earns zero profit.
Predict first. If the price doubles from $10 to $20 under free entry, what happens to rent?
Choose an example
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Constructed example: the chapter's hypothetical fishery (growth 0.5X(1 - X/100), price $10, effort cost $5, controlled stock 75, and the $20 price case); prices $7.50 and $15 and costs $2.50 and $7.50 are added for comparison.
Calculated values
- Open-access stock X_OA
- 50.00
- Sustainable harvest
- 12.50
- Effort E
- 25.00
- Revenue
- $125.00
- Effort cost
- $125.00
- Open-access rent
- $0.00
- Controlled harvest (X = 75)
- 9.3750
- Controlled effort
- 12.50
- Controlled rent
- $31.25
Free entry pushes the stock to where a unit of effort just pays: 10 x 0.01 x X = 5, so X_OA = 50.00. Growth there is 12.50, which needs effort 12.50 / (0.01 x 50.00) = 25.00. Revenue $125.00 equals effort cost $125.00, so the rent is $0.00. At the controlled stock of 75, growth is 9.375 and effort 12.5, giving revenue $93.75 against cost $62.50, so holding the stock at 75 earns a rent of $31.25 a year that free entry would burn.
Worked steps
- X_OA = 5 / (0.01 x 10) = 50.00
- H = G(X_OA) = 0.5 x 50.00 x 0.50 = 12.50
- E = 12.50 / (0.01 x 50.00) = 25.00
- Revenue = 10 x 12.50 = $125.00; cost = 5 x 25.00 = $125.00; rent = $0.00
- Controlled X = 75: revenue = 10 x 9.375 = $93.75; cost = 5 x 12.5 = $62.50; rent = $31.25
Use the idea
To see whether a shared resource is losing its rent, compare total revenue with the full cost of the effort applied; under open access they converge whatever the price.
Where the conclusion applies
Logistic growth, harvest proportional to effort and stock, constant price and cost, identical fishers and a steady state. Real fisheries add uncertainty, adjustment lags and enforcement costs.
Check your understanding: With price $10 and effort cost $7.50, what is the open-access stock and the rent?
Chapter 69 source: section "Gordon open-access rent dissipation".
Demonstration 2 of 4
Setting the corrective tax rate
What tax moves a polluting market to the efficient quantity, and what if the rate is miscalibrated?
The market stops where buyers' value equals private cost and ignores the damage, so it overproduces. A tax equal to the marginal damage makes producers face the social cost. A tax above the damage removes units worth more than their full cost; one below leaves harmful units in place.
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Q is output in thousands of units. MB is buyers' marginal benefit, MPC the marginal private cost and MSC = MPC plus the external damage per unit. A per-unit tax shifts MPC up by the tax.
Predict first. If damage is really $5 but the $12 tax is kept, is output too high or too low?
Choose an example
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Constructed example: the chapter's hypothetical market (MB = 100 - Q, MPC = 20 + Q, damage $12 and $5, tax $12); damages $0 and $20, tax rates $0 and $5 and the welfare-loss figure are added.
Calculated values
- Untaxed output Q^p (000)
- 40.00
- MSC at Q^p ($)
- 72.00
- Efficient Q* (000)
- 34.00
- Taxed output (000)
- 34.00
- Taxed vs efficient
- Efficient
- Welfare loss, derived ($000)
- 0.00
With damage $12 per unit, the efficient quantity solves 100 - Q = 20 + Q + 12, so Q* = (80 - 12) / 2 = 34.00 thousand. A $12 tax gives (80 - 12) / 2 = 34.00 thousand, so output is efficient: the tax matches the damage. The shaded triangle between MB and MSC is the derived welfare loss, 0.00 thousand dollars; the book does not print this figure.
Worked steps
- Untaxed: 100 - Q = 20 + Q, so Q^p = 80 / 2 = 40.00
- MSC at Q^p = 20 + 40 + 12 = 72.00
- Efficient: 100 - Q = 20 + Q + 12, so Q* = 68 / 2 = 34.00
- Taxed: 100 - Q = 20 + Q + 12, so Q_t = 68 / 2 = 34.00
- Loss = 0.5 x 0.00 x (2 x 0.00) = 0.00 thousand dollars (derived)
Use the idea
Before calling a charge Pigouvian, compare its rate with an estimate of marginal damage at the efficient quantity, not with the damage at today's output.
Where the conclusion applies
Linear schedules, constant damage per unit, competitive pricing and no other distortions. The welfare-loss triangle is derived from these schedules; the chapter does not print it.
Check your understanding: With damage $5 and tax $12, how far is output from efficient and what is the welfare loss?
Chapter 69 source: section "Pigouvian correction".
Demonstration 3 of 4
How many prize teams is too many?
When does adding prize contestants raise expected social value, and when do teams still enter?
More teams raise the chance of success but with diminishing returns, while cost grows in proportion. Each entrant counts its expected share of the prize, not the duplication it adds, so private entry can continue past the number that maximizes social value.
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V = $1,500,000 is the social value of success, P = $1,000,000 the prize, and each team spends $80,000 and succeeds independently with probability p. q_n is the chance at least one of n teams succeeds and W_n = q_n V - 80,000 n the expected social value. W_D is the direct contract's value.
Predict first. Does moving from 4 to 12 teams raise or lower expected social value, and do teams still want to enter?
Choose an example
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Constructed example: the chapter's hypothetical contest (V $1,500,000, prize $1,000,000, cost $80,000, p 0.25, 4 and 12 teams, direct contract 0.40 and $300,000); 1 and 8 teams and success probabilities 0.15 and 0.35 are added. The book's W_12 = 492,485.475 uses a rounded q_12 and is approximate; the exact value 492,485.472 is shown.
Calculated values
- Success chance q_n
- 0.68359375
- Total team cost
- $320,000
- Expected social value W_n
- $705,390.625
- Expected prize per team
- $170,898.44
- Direct contract W_D
- $300,000
- Teams enter
- Yes
With 4 teams each succeeding with probability 0.25, the chance that at least one succeeds is 1 - 0.75^4 = 0.68359375, so expected social value is $705,390.625, above the direct contract's $300,000. Each team's expected prize, $170,898.44, covers its $80,000 cost, so teams want to enter.
Worked steps
- q = 1 - 0.75^4 = 0.6835937500
- W = 0.6835937500 x 1,500,000 - 4 x 80,000 = 1,025,390.625 - 320,000 = 705,390.625
- Prize per team = 0.6835937500 x 1,000,000 / 4 = 170,898.44
- W_D = 0.40 x 1,500,000 - 300,000 = 300,000
Use the idea
When designing a prize, compare the marginal gain in success probability from one more team with that team's cost, and use entry limits or staged rounds if private entry overshoots.
Where the conclusion applies
Independent success across teams, a prize split at random among winners, no administration cost and risk-neutral teams. Correlated approaches overstate the gain from adding teams.
Check your understanding: With 8 teams at p = 0.25, what are W_8 and each team's expected prize?
Chapter 69 source: section "Prize Incentives, Contest Design, and Subsidized Experimentation".
Demonstration 4 of 4
Four users, one basin
How far does private use of a shared basin overshoot the coordinated level, and what does it cost?
A user who draws one more unit raises everyone's depletion cost but bears only its own share. The private first-order condition therefore ignores most of the marginal cost, and the gap grows with the number of users.
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Each of N users draws x and gets benefit 20x - x^2. Total use is X = N x and the depletion cost k X^2 (k = 0.1 in the book) is shared equally. x^N is the noncooperative choice, x* the coordinated one, and W(x) total welfare.
Predict first. As users rise from 4 to 8 (same k), does the private choice x^N change?
Choose an example
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Constructed example: the chapter's hypothetical basin (4 users, B = 20x - x^2, D = 0.1X^2); 2 and 8 users and coefficients 0.05 and 0.2 are added for comparison.
Calculated values
- Private use per user x^N
- 9.09
- Total private use X^N
- 36.36
- Coordinated use per user x*
- 7.14
- Total coordinated use X*
- 28.57
- Welfare W^N
- 264.46
- Welfare W*
- 285.71
- Welfare gain
- 21.25
- Withdrawal cut
- 7.79
Each of the 4 users bears only 1/4 of the depletion cost, so privately each draws x^N = 20 / 2.20 = 9.09, 36.36 in total. Coordination counts the full cost and sets x* = 20 / 2.80 = 7.14, 28.57 in total. Cutting withdrawal by 7.79 raises welfare from 264.46 to 285.71, a gain of 21.25.
Worked steps
- Private: 20 - 2x - 2 x 0.10 x = 0, so x^N = 20 / 2.20 = 9.0909
- Coordinated: 20 - 2x - 2 x 0.10 x 4 x = 0, so x* = 20 / 2.80 = 7.1429
- W^N = 4 x 9.090909 x (20 - 1.40 x 9.090909) = 264.46
- W* = 4 x 7.142857 x (20 - 1.40 x 7.142857) = 285.71
- Gain = 285.7143 - 264.4628 = 21.25; cut = 36.3636 - 28.5714 = 7.79
Use the idea
Compare the marginal cost each user actually faces with the full marginal depletion cost; rules, quotas or charges that close the gap align private and coordinated use.
Where the conclusion applies
Identical users, quadratic benefits and depletion cost, equal cost sharing and one period. Real basins add heterogeneity, dynamics and monitoring costs.
Check your understanding: With 8 users and k = 0.1, what are x^N, x* and the welfare gain from coordination?
Chapter 69 source: section "Tragedy of the Commons".