Classwork 4
Optimal Harvesting in a Common-pool Fishery
Consider a fishery where the total benefit (TB) and total cost (TC) of fishing effort \(E\) are:
- \(TB(E)=1.5E - E^{2}\) (in $)
- \(TC(E)=0.5E\) (in $)
where \(E\) is the level of fishing effort (e.g., number of boat-days).
Key Concepts
Before beginning, recall these definitions:
Average Benefit (AB):
\(AB(E) = \dfrac{TB(E)}{E}\).
In economics, AB measures benefit per unit of input; here, it shows the revenue per boat-day.
For this fishery: \(AB(E) = \dfrac{1.5E - E^2}{E} = 1.5 - E\).Average Cost (AC):
\(AC(E) = \dfrac{TC(E)}{E}\).
In economics, AC measures cost per unit of input; here, it is the cost per boat-day.
For this fishery: \(AC(E) = \dfrac{0.5E}{E} = 0.5\).Marginal Benefit (MB):
\(MB(E) = \dfrac{dTB}{dE}\).
In economics, MB is the extra benefit from one more unit of input; here, the additional revenue from one more boat-day.
For this fishery: \(MB(E) = 1.5 - 2E\).Marginal Cost (MC):
\(MC(E) = \dfrac{dTC}{dE}\).
In economics, MC is the extra cost from one more unit of input; here, the additional cost of adding one boat-day.
For this fishery: \(MC(E) = 0.5\).
Benefit–Cost Table
Benefit–Cost Table
| Effort \(E\) |
TB \(1.5E - E^2\) |
MB \(1.5 - 2E\) |
AB \(1.5 - E\) |
TC \(0.5E\) |
MC \(0.5\) |
AC \(0.5\) |
|---|---|---|---|---|---|---|
| 0.05 | 0.073 | 1.400 | 1.450 | 0.025 | 0.500 | 0.500 |
| 0.10 | 0.140 | 1.300 | 1.400 | 0.050 | 0.500 | 0.500 |
| 0.15 | 0.202 | 1.200 | 1.350 | 0.075 | 0.500 | 0.500 |
| 0.20 | 0.260 | 1.100 | 1.300 | 0.100 | 0.500 | 0.500 |
| 0.25 | 0.312 | 1.000 | 1.250 | 0.125 | 0.500 | 0.500 |
| 0.30 | 0.360 | 0.900 | 1.200 | 0.150 | 0.500 | 0.500 |
| 0.35 | 0.402 | 0.800 | 1.150 | 0.175 | 0.500 | 0.500 |
| 0.40 | 0.440 | 0.700 | 1.100 | 0.200 | 0.500 | 0.500 |
| 0.45 | 0.473 | 0.600 | 1.050 | 0.225 | 0.500 | 0.500 |
| 0.50 | 0.500 | 0.500 | 1.000 | 0.250 | 0.500 | 0.500 |
| 0.55 | 0.522 | 0.400 | 0.950 | 0.275 | 0.500 | 0.500 |
| 0.60 | 0.540 | 0.300 | 0.900 | 0.300 | 0.500 | 0.500 |
| 0.65 | 0.552 | 0.200 | 0.850 | 0.325 | 0.500 | 0.500 |
| 0.70 | 0.560 | 0.100 | 0.800 | 0.350 | 0.500 | 0.500 |
| 0.75 | 0.562 | 0.000 | 0.750 | 0.375 | 0.500 | 0.500 |
| 0.80 | 0.560 | -0.100 | 0.700 | 0.400 | 0.500 | 0.500 |
| 0.85 | 0.552 | -0.200 | 0.650 | 0.425 | 0.500 | 0.500 |
| 0.90 | 0.540 | -0.300 | 0.600 | 0.450 | 0.500 | 0.500 |
| 0.95 | 0.522 | -0.400 | 0.550 | 0.475 | 0.500 | 0.500 |
| 1.00 | 0.500 | -0.500 | 0.500 | 0.500 | 0.500 | 0.500 |
| 1.05 | 0.473 | -0.600 | 0.450 | 0.525 | 0.500 | 0.500 |
| 1.10 | 0.440 | -0.700 | 0.400 | 0.550 | 0.500 | 0.500 |
| 1.15 | 0.402 | -0.800 | 0.350 | 0.575 | 0.500 | 0.500 |
| 1.20 | 0.360 | -0.900 | 0.300 | 0.600 | 0.500 | 0.500 |
| 1.25 | 0.312 | -1.000 | 0.250 | 0.625 | 0.500 | 0.500 |
| 1.30 | 0.260 | -1.100 | 0.200 | 0.650 | 0.500 | 0.500 |
| 1.35 | 0.202 | -1.200 | 0.150 | 0.675 | 0.500 | 0.500 |
| 1.40 | 0.140 | -1.300 | 0.100 | 0.700 | 0.500 | 0.500 |
| 1.45 | 0.073 | -1.400 | 0.050 | 0.725 | 0.500 | 0.500 |
| 1.50 | 0.000 | -1.500 | 0.000 | 0.750 | 0.500 | 0.500 |
Part A — Total Curves
Draw the Total Benefit (TB) and Total Cost (TC) curves as functions of fishing effort \(E\). Place these curves in the top panel of a two-panel figure.
Part B — Marginal Curves
On the bottom panel, draw the Marginal Benefit (MB) and Marginal Cost (MC) curves as functions of fishing effort \(E\).
Part C — Efficient and Open-Access Effort Levels
Make sure the top and bottom panels share the same horizontal scale for effort \(E\), so the relationship between total and marginal spaces is consistent.
- Recall that:
- \(MB(E)=\dfrac{d\,TB}{dE}=1.5 - 2E\)
- \(MC(E)=\dfrac{d\,TC}{dE}=0.5\)
On both panels, mark and label the two benchmark effort levels using vertical guide lines that align across panels:
The efficient effort \(E^*\): Draw a vertical line at \(E^*\) on both the top (TB/TC) and bottom (AB/AC/MB/MC) panels and label it “\(E^*\)”.
The open-access effort \(E_{OA}\): Draw a vertical line at \(E_{OA}\) on both panels and label it “\(E_{OA}\)”.
Part D — Intuition Behind \(E^*\) and \(E_{OA}\)
Explain the intuition behind these two effort levels:
- Why does \(E^*\) arise?
- Why does \(E_{OA}\) arise under open access?
Part E — Policy Intervention with a License Fee
- License fee: Explain, both analytically and intuitively, how charging a license fee (e.g., license fee per boat-day) can reduce fishing effort from the open-access level \(E_{OA}\) down to the efficient level \(E^*\).
- Explain why it is important to align individual fishers’ incentives with the socially optimal level of effort in a fishery. In your answer, discuss:
- Incentive problem: Why do individual fishers’ private decisions differ from what is best for the group?
- Economic rent: What happens to economic rents under open access when incentives are not aligned?
- Sustainability: Why does aligning incentives matter for the long-run health of both the fish stock and the fishing economy?