Classwork 5

Allocating Gravel Across Two Generations

NoteThe case

We will allocate construction gravel from a single fixed deposit between the current generation and the next generation, 35 years from now. The exercise connects an ordinary supply-and-demand model to marginal net benefit, present value, user cost, and a resource depletion tax.

Model and units

For the current generation,

Q_D=200-5P \qquad\Longleftrightarrow\qquad P=40-0.2Q_D,

and

Q_S=5P \qquad\Longleftrightarrow\qquad P=0.2Q_S.

• Q is measured in millions of tons.

• P is measured in dollars per ton.

• The next generation has the same marginal-benefit and marginal-cost functions.

• The deposit contains 100 million tons of extractable gravel for the two generations.

• The annual discount rate is 4%. Over 35 years, use the approximation

  (1.04)^{35}\approx 4.

Task 1 · Find the static market outcome

Ignore the next generation for now.

1. Draw the current supply and demand curves.
2. Solve algebraically for the current-generation market equilibrium, (Q_M,P_M).
3. Briefly explain what this outcome leaves out.

Task 2 · Construct marginal net benefit

1. Interpret inverse demand as marginal benefit, MB(Q), and inverse supply as marginal cost, MC(Q).

2. Derive

   MNB(Q)=MB(Q)-MC(Q).

3. Draw the current generation’s MNB(Q) curve for quantities from zero through the static market quantity.

4. What does a positive value of MNB(Q) mean?

Task 3 · Allocate the gravel stock across generations

Let Q_1 be gravel used by the current generation and Q_2 be gravel used by the next generation.

1. Write the stock constraint connecting Q_1 and Q_2.
2. Write the present value of the next generation’s marginal net benefit. Use the 35-year discount factor of 4.
3. Find the efficient allocation by setting the current marginal net benefit equal to the present value of the next generation’s marginal net benefit.
4. Report Q_1^* and Q_2^* and verify the equality numerically.
5. Draw a graph like the two-period allocation graph from lecture. Put Q_1 on the horizontal axis, so moving right means more gravel for the current generation and less for the next generation.

Task 4 · Price the user cost

1. At the efficient allocation, calculate the marginal user cost of one more million tons used by the current generation.
2. What per-ton resource depletion tax would make the current market account for that user cost?
3. Add the following to a current-generation market graph:
   • demand;
   • the original supply curve;
   • supply plus marginal user cost; and
   • supply plus the resource depletion tax.
4. Calculate the new current-generation market price with the tax. Also report the net price received by sellers after paying the tax.
5. Explain why supply plus marginal user cost and supply plus the constant tax meet at the efficient quantity but need not be the same curve everywhere.

Task 5 · Change the discount rate

Without doing new algebra or drawing another graph, explain how each outcome would change if the annual discount rate were:

1. higher than 4%; and
2. lower than 4%.

For each case, discuss the present value of future marginal net benefit, current gravel use, gravel left for the next generation, and the current resource depletion tax.

Exit sentence

Complete this sentence in one or two lines:

Using one more ton today has a user cost when __________ because __________.

ImportantUnits matter

Quantities are in millions of tons, while prices, marginal net benefits, user costs, and taxes are in dollars per ton.

Adapted from Exercise 1 in Chapter 5 of Jonathan M. Harris and Brian Roach, Environmental and Natural Resource Economics: A Contemporary Approach, 5th ed.