Midterm Exam: Questions and Answers

Grading summary and worked examples

Review the October 19 midterm, the grading criteria and assigned points, and example answers for every question.

Published

October 19, 2026

Midterm exam grading summary

The table summarizes fractional credit earned for each grading criterion across the 30 graded exams. Assigned points gives each criterion’s nominal maximum; these weights sum to 100 points. A fractional score of 1 represents full credit, 0.5 represents half credit, and 0 represents no credit. Recorded credit above the nominal maximum is preserved, so some values can exceed 1.

The mean, sample standard deviation (SD), and histogram are computed below. The mean is the average fraction of credit earned, rather than the share of students receiving full credit.

Try each question before expanding its example answer. The original exam allowed 50 minutes for 100 points.

Time: 50 minutes. Total: 100 points. Select one answer per multiple-choice item. Use the supplied information and show brief reasoning. Only simple arithmetic is needed. Suggested section times leave 4 minutes for review.

I. Multiple choice (15 points; 3 minutes)

1. [3 points] Maintained fishing effort is already above the effort that produces MSY. On the sustainable catch and effort curve, moving still farther beyond that effort tends to:

A. increase sustainable catch without limit.
B. leave the maintained fish stock unchanged.
C. eliminate all real fishing costs.
D. reduce sustainable catch as the maintained stock falls.

Example answer: I.1

1. D. Beyond the peak of the sustainable catch and effort curve, extra maintained effort leaves a smaller stock that supports less natural growth and therefore less sustainable catch.

2. [3 points] Under Hotelling’s benchmark, the expected growth rate of net resource price exceeds the market interest rate. A forward-looking private owner has an incentive to:

A. extract more today and invest the proceeds.
B. delay extraction.
C. ignore future net prices.
D. remain indifferent between extracting now and later.

Example answer: I.2

2. B. Leaving the resource underground offers a higher expected return than extracting now and investing the net proceeds. The owner has an incentive to postpone extraction until the arbitrage opportunity disappears.

3. [3 points] A pollution loss occurs only in the distant future. A higher discount rate, with the future loss unchanged, gives that loss:

A. a larger present value.
B. the same present value.
C. a smaller present value.
D. a negative physical size.

Example answer: I.3

3. C. Discounting gives the same future monetary loss a smaller present value when the discount rate is higher.

4. [3 points] Under Hartwick’s rule and its substitution assumptions, which action helps maintain consumption as a nonrenewable resource is depleted?

A. Invest all scarcity rent in other forms of productive capital.
B. Consume all scarcity rent immediately.
C. Invest only the interest earned on scarcity rent.
D. Preserve every unit of the resource underground.

Example answer: I.4

4. A. Hartwick’s rule calls for investing all scarcity rent in other productive capital. Under its substitution assumptions, this can offset the loss of resource wealth and maintain consumption.

5. [3 points] A cheaper substitute permanently reduces a mineral’s value below its extraction cost. Which outcome can be economically optimal?

A. Extract the entire deposit because leaving ore unused always wastes resources.
B. Stop extraction while some ore remains underground.
C. Increase extraction because the mineral is now less valuable.
D. Ignore extraction costs when deciding whether to continue mining.

Example answer: I.5

5. B. Physical availability does not guarantee that extraction produces a positive net benefit. When the substitute provides the service for less than the extraction cost, leaving some ore unextracted can be economically optimal.

II. Fill in the blanks (19 points; 5 minutes)

Write a word or phrase in each blank.

1. [4 points] The property-right characteristic that makes the owner receive the benefits and bear the costs of resource use is \underline{\hspace{4cm}}; the characteristic that allows rights to be voluntarily transferred from one owner to another is \underline{\hspace{4cm}}.

Example answer: II.1

1. Exclusivity; transferability. Exclusivity assigns the benefits and costs of resource use to the owner; transferability allows voluntary exchange of rights between owners.

2. [3 points] The largest fish stock the environment can support without fishing, where net natural growth is zero, is the \underline{\hspace{5cm}}.

Example answer: II.2

2. Carrying capacity. The upper zero-growth stock in the model; not the stock that produces MSY.

3. [3 points] A defined community excludes outsiders and enforces rules for its members’ resource use. This property regime is \underline{\hspace{5cm}}.

Example answer: II.3

3. Common property. Also called a common-property regime. The community regulates access and use, so this is not open access.

4. [3 points] Benefiting from a public good while relying on others to pay for its provision is called \underline{\hspace{5cm}}.

Example answer: II.4

4. Free riding. A person benefits from public-good provision while relying on others to contribute.

5. [6 points] Maximizing the sum of current and discounted future net benefits concerns \underline{\hspace{4cm}}; deciding how benefits and costs should be shared across generations concerns \underline{\hspace{4cm}}; leaving future generations no worse off than the present generation concerns \underline{\hspace{4cm}}.

Example answer: II.5

5. Efficiency; fairness; sustainability. Efficiency concerns maximizing net benefits; fairness concerns how benefits and costs are shared across generations; sustainability requires leaving future generations no worse off.

III. Short answers (20 points; 8 minutes)

Aim for 2 to 4 sentences per answer.

1. [10 points] A myopic private owner extracts a scarce nonrenewable resource, ignoring both extraction pollution and future net benefits. Assume the socially efficient allocation includes both current and future extraction. Would a per-unit tax equal to marginal external pollution damage achieve efficiency? Explain your answer.

Example answer: III.1

1. No. A pollution-only tax does not account for the future benefits that this owner ignores. Current extraction causes pollution and reduces the stock available for future use, so both costs must enter the owner’s decision. A pollution-only tax leaves the myopic owner extracting too much today because the owner still ignores the future social net benefit forgone.

A Pigouvian pollution tax should equal marginal external pollution damage. A resource-depletion tax should equal marginal user cost: the present value of future social net benefit forgone by extracting one additional unit today. The depletion component should reflect future net benefits after future extraction costs and pollution damages.

Use either one combined per-unit tax equal to marginal external pollution damage plus marginal user cost, or two separate per-unit taxes with those respective rates. Evaluate the components at the socially efficient allocation. This myopic owner internalizes neither cost without the correction.

2. [10 points] An efficient allocation gives the current generation $120 in net benefits and the future generation $80, with no saving or transfers. These amounts are undiscounted. Does it also meet the sustainability criterion? Explain how the two criteria differ.

Example answer: III.2

2. No. Future net benefit of 80 dollars is below current net benefit of 120 dollars. Efficiency maximizes the sum of net benefit today and discounted net benefit in the future. The sustainability criterion compares generations’ own well-being without discounting and requires future generations to be no worse off. Efficiency therefore does not automatically imply sustainability. Actual saving and transfers might change the generational comparison, but none occur in the stated allocation.

IV. Analytical question: a fishery growth shock (18 points; 9 minutes)

Slower natural growth changes sustainable revenue as shown. Fish price is $2 per ton, real effort costs are unchanged, and there are no other external damages. For Questions 1 and 2, compare steady states. AR and MR are average and marginal revenue from effort; AC and MC are average and marginal cost.

Curves are distinguished by line type, as shown in the legend below.

Black-and-white panels show initial and post-shock total revenue with total cost, and the corresponding average revenue, marginal revenue, and constant average and marginal cost curves for each case. Numbered effort axes and per-effort dollar units support reading efficient, MSY, and open-access effort.

1. [9 points] Identify each initial effort and briefly state its graph criterion.

Target	Initial effort	Graph criterion
Static efficient	\underline{\hspace{1.2cm}}	\underline{\hspace{5cm}}
MSY	\underline{\hspace{1.2cm}}	\underline{\hspace{5cm}}
Open access (OA)	\underline{\hspace{1.2cm}}	\underline{\hspace{5cm}}

Example answer: IV.1

1. The initial effort levels are:

Target	Effort	Graph criterion
Static efficient	4	MR=MC; equivalently, the largest gap between total revenue and total cost
MSY	6	Peak sustainable revenue at a fixed fish price; equivalently, MR=0
Open access	8	AR=AC; equivalently, positive-effort equality of total revenue and total cost

The zero-effort intersection is not the positive open-access outcome.

2. [6 points] After the shock, what effort corresponds to each target?

Static efficient: \underline{\hspace{1.4cm}} MSY: \underline{\hspace{1.4cm}} OA: \underline{\hspace{1.4cm}}

Example answer: IV.2

2. After the shock: efficient effort 2, MSY effort 3, OA effort 4.

The supplied fishery panels with the efficient, MSY, and open-access points marked for the initial and post-shock cases.

3. [3 points] The graph below shows the same fishery at its old efficient effort, E=4, and starting stock of 80 tons. Immediately after growth slows, effort and stock are unchanged. Does the stock begin to rise or fall? Use the growth and harvest values to explain.

Curves are distinguished by line type, as shown in the legend below.

Natural growth before and after the shock and the harvest line for effort four are shown against fish stock. At the marked starting stock of eighty tons, initial growth and harvest are sixteen tons per year, while growth after the shock is eight. The graph corresponds to the revenue and effort panels.

Example answer: IV.3

3. The stock begins to fall. At stock 80, harvest remains 16 tons while natural growth falls from 16 to 8 tons per year. Harvest exceeds growth by 8 tons. The revenue curves compare steady states, whereas this question concerns the initial adjustment from the old stock.

V. Analytical question: allocation over time (10 points; 7 minutes)

Allocate a 60-ton deposit between today (Q_1) and ten years from now (Q_2), with Q_1+Q_2=60. There is no pollution. Private marginal net benefits, in dollars per ton, are

MNB_1=90-Q_1,\qquad MNB_2=120-Q_2\quad\text{(undiscounted)}.

For an amount X_t in t years, PV(X_t)=X_t/(1+r)^t. Let r_p be the owner’s market interest rate and r_s the social discount rate. Use these ten-year discounting factors:

\frac{1}{(1+r_p)^{10}}=\frac12,\qquad \frac{1}{(1+r_s)^{10}}=\frac23.

The forward-looking owner uses r_p; society uses r_s. Each maximizes the sum of net benefit today and discounted net benefit in ten years at the respective rate. The top axis shows Q_2=60-Q_1; both discounted future MNB curves are given.

Curves are distinguished by line type, as shown in the legend below.

Four black curves labeled A to D. Solid curve A falls from 90 to 30; long-dashed curve B rises from 60 to 120; dotted curve C rises from 30 to 60; dash-dot curve D rises from 40 to 80. Current extraction runs from zero to sixty tons along the bottom axis, while future extraction runs from sixty to zero along the top.

1. [4 points] Identify curves A–D. Write a name or formula in each blank.

A: \underline{\hspace{5cm}}	B: \underline{\hspace{5cm}}
C: \underline{\hspace{5cm}}	D: \underline{\hspace{5cm}}

Example answer: V.1

1. The four curves are:

Curve	Name	Formula
A (solid)	Today’s marginal net benefit	MNB_1=90-Q_1
B (long-dashed)	Future marginal net benefit, undiscounted	MNB_2=120-Q_2
C (dotted)	Owner’s discounted future marginal net benefit, using r_p	MNB_2/(1+r_p)^{10}=\tfrac12(120-Q_2)
D (dash-dot)	Discounted future marginal net benefit using the social discount rate r_s	MNB_2/(1+r_s)^{10}=\tfrac23(120-Q_2)

2. [6 points] Shade society’s loss from the owner’s allocation. Explain why shifting extraction to the future improves the outcome and where this shift should stop. Do not calculate the size of the area.

Example answer: V.2

2. The owner’s allocation is 40 tons today and 20 later; society’s optimum is 30 tons today and 30 later. Shade the region between today’s MNB and the socially discounted future MNB, from Q_1=30 to Q_1=40. For those extra current tons, the discounted future net benefit forgone exceeds the current marginal net benefit. Moving these tons to the future raises the sum of current and discounted future net benefits until the two marginal values match.

The provided discounted future curves are shown with the welfare loss shaded. The owner's curve crosses today's marginal net benefit at forty tons today, while the social curve crosses at thirty. The welfare loss is shaded between today's marginal net benefit and the social discounted future curve from thirty to forty tons of current extraction.

VI. Analytical question: extraction and pollution (18 points; 14 minutes)

Use Part V’s deposit, marginal net benefits, and discounting factors. Extraction causes $20 of external damage per ton today and $5 per ton in ten years, at the time of extraction. In Questions 1 and 2, the owner is forward-looking as in Part V and does not pay for this damage. Present values use r_p for private MNB and r_s for social MNB.

Curves are distinguished by line type, as shown in the legend below.

Four black curves show private and social marginal net benefit today and the present value of each future marginal net benefit. Private future values use the owner's market interest rate; social future values use the social discount rate. Extraction today rises along the bottom axis while extraction in ten years falls along the top axis.

1. [4 points] Mark the allocation optimal for society and label Q_1^{*} and Q_2^{*}. Shade the welfare loss if the owner instead extracts 40 tons today. Do not calculate the size of the area.

Example answer: VI.1

1. Society’s optimum is Q_1^{*}=20, Q_2^{*}=40, where the two present-value social marginal net benefits both equal 50. Shade between the two social curves over Q_1=20 to Q_1=40. The shaded gap uses social marginal net benefits after accounting for damage.

The four private and social curves are shown, with the pollution-adjusted social optimum marked at twenty tons today and forty later. The loss from forty tons of current extraction is shaded only between the two social marginal net benefit curves.

2. [3 points] Compare society’s allocation between the two periods with Part V. Would pollution change that allocation? Why?

Example answer: VI.2

2. Yes. Society shifts from 30 tons today and 30 later to 20 today and 40 later. Today’s damage of 20 dollars per ton exceeds the present value of the 5 dollars of damage from a future ton. Pollution therefore lowers today’s social marginal net benefit more than it lowers the discounted future marginal net benefit, favoring later extraction in this example. This is a result for these damages, not a general rule that every externality favors delay.

3. [7 points] A myopic private owner considers only today’s private net benefits, ignoring future benefits and pollution. Before any tax, how many tons would this owner extract today? What per-ton corrective tax on extraction today would induce society’s optimal allocation? Draw the tax-adjusted current private MNB curve on the graph below and mark the owner’s extraction after tax.

Curves are distinguished by line type, as shown in the legend below.

The original four-curve pollution graph is repeated for the tax question. It shows private and social marginal net benefits today and their discounted future curves. Students add the tax-adjusted current private marginal net benefit curve and mark extraction after tax; no tax solution is supplied.

Example answer: VI.3

3. Before tax, the myopic owner extracts all 60 tons today, leaving none for the future. Today’s private MNB=90-Q_1 is still positive at the stock limit: 90-60=30. Its unconstrained zero at 90 tons is infeasible because the deposit contains only 60 tons.

Impose $70 per ton of extraction today. The social optimum is Q_1=20 and Q_2=40. With this tax, the myopic owner stops where today’s private MNB after tax is zero:

90-Q_1-\tau=0.\qquad \text{At }Q_1=20,\quad \tau=90-20=\boxed{70}.

Shift the current private MNB curve down by 70: the tax-adjusted curve is 20-Q_1, with a vertical intercept of 20 dollars per ton and a horizontal intercept at Q_1=20. Mark the owner’s after-tax extraction at 20 tons today, leaving 40 tons for the future. Only the nonnegative portion of the tax-adjusted curve is shown on the supplied axes.

The original four curves remain. A fifth solid curve with open square markers shows private marginal net benefit after a seventy-dollar tax, running from twenty dollars per ton at zero current extraction to zero at twenty tons. The graph marks after-tax extraction of twenty tons and untaxed myopic extraction of sixty tons, the deposit limit, where private marginal net benefit remains thirty dollars per ton.

For this myopic owner, the tax accounts for both the $20 current external damage and the $50 marginal user cost, the present value of future social marginal net benefit forgone at the optimum. A pollution-only tax of $20 does not account for the future benefits that this owner ignores.

4. [4 points] An innovation reduces today’s external marginal cost from $20 to $5, with no change in extraction costs or future damage. Add the new current social MNB curve to the graph below. Compared with Question 1 in Part VI, does society’s optimal current extraction rise or fall? Do not calculate the extraction quantity.

Curves are distinguished by line type, as shown in the legend below.

The same four-curve pollution graph is repeated for adding the new current social marginal net benefit curve. It shows current private and social marginal net benefits, future private marginal net benefit discounted at the owner's rate, and future social marginal net benefit discounted at the social rate. The innovation curve is not supplied.

Example answer: VI.4

4. The new current social curve is 85-Q_1: shift the existing curve up by 15 dollars per ton. The optimal amount extracted today rises because the current marginal damage is smaller; future extraction falls.