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Efficient Supply of Renewable and Non-Renewable Resources

Econ 1661 Review SectionFebruary 24th, 2012

Rich Sweeney(Based on slides from Robyn Meeks and Matt Ranson)

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Lots to catch up on today

• Part 1: Non-renewable resources– Review two-period model– Go through practice problem

• Part 2: Review of open-access resources– Explanation of fishery biology graphs and efficient

yeilds– No problem yet, as Prof. Stavins is going to finish

this topic on Monday

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Part 1: Non-renewable resources outline

• Static versus dynamic efficiency• Two-period model – Mathematical solution – Graphical Solution

• Resource prices and Hotelling Rule

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Static vs. dynamic efficiency• Static efficiency

– Economically efficient allocation maximizes net benefits (TB-TC)– At this point MB=MC – Key: incremental benefits associated with the last unit preserved are exactly

equal to the incremental opportunity cost of preserving the last unit • Dynamic efficiency

– Economically efficient allocation maximizes the present value of net benefits– At this allocation, PV (Marginal Net Benefits) are equal across time periods

• Discounting: present value = future value / (1+r)t

– Trade off between marginal net benefits in different time periods because of budget constraint (total stock of resource)

– Dynamically efficient allocation: requires that present value of the marginal net benefit from the last unit in Period 1 equals the present value of the marginal net benefit in Period 2 (for 2 period model)

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Non-renewable resources: dynamic efficiency

• Dynamically efficient allocation:PDV [MNB1 (q1)] = PDV [MNB2 (q2)]

• This allocation maximizes the present value of net benefits

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• Price = MUC + MEC • Marginal extraction cost (MEC) • Marginal user cost (MUC):• “scarcity rent”• is the opportunity cost of

forgone future consumption • is the additional marginal value

of a resource due to its scarcity • If MUC = 0 and P = MEC, then the resource is not economically scarce• Marginal Opportunity Cost

= MUC + MEC

Non-renewable resources: dynamic efficiency

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Example: 2-period non-renewable resource model

• Basic setup:– There are two periods.– There is a total stock of 20 units of oil in the ground.– Consumer demand for oil is:

• q1 (p1) = 20 – 2.5p1 in period 1 p1 (q1) = 8 - .4q1

• q2 (p2) = 20 – 2.5p2 in period 2 p2 (q2) = 8 - .4q2

– The marginal cost of extracting the resource is:• MEC1 = 2 in period 1

• MEC2 = 1 in period 2 (because of improved technology)

– The interest rate is r = .1

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• First, what quantities would be supplied in each period if producers were myopic?

• Real Question: What are the socially optimal (ie dynamically efficient) quantities of resource extraction in the two periods?

• Answer: Dynamic efficiency implies that we want to choose q1

and q2 to maximize the present discounted value of the net benefits of oil extraction in the two periods.

• So, we want to maximize:PDV[NB1 (q1 )]+PDV[NB2 (q2)]

• ...while taking into account the stock constraint:q1 +q2 = 20.

Two-period non-renewable resource model

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Mathematical solution • Two equations define the optimal extraction in the two periods:

– Condition #1 (Maximization):

PDV [MNB1 (q1)] = PDV [MNB2 (q2)]

• This equation says that when the PDV of net benefits is maximized, the PDV of the marginal net benefits in period 1 must equal the PDV of the marginal net benefits in period 2.

• If this weren't true, then we could increase the discounted net benefits by switching a unit of extraction from period 1 to 2 (or vice versa), and so we wouldn't be at a maximum.

– Condition #2 (Resource Constraint): q1+q2 = 20

• This is just the constraint that we can't extract more oil than we have.

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• Six steps to find the optimal q*1 and q*2 :– Step 1: Write down marginal extraction costs in each period.

• MEC1 = 2 in period 1

• MEC2 = 1 in period 2

– Step 2: Write down marginal benefits in each period. Remember that marginal benefits are given by the inverse demand function.• MB1 (q1) = p1 (q1) = 8 - .4q1

• MB2 (q2) = p2 (q2) = 8 - .4q2

– Step 3: Calculate marginal net benefits in each period. Marginal net benefits are just equal to marginal benefits minus marginal extraction costs.• MNB1 (q1) = MB1 (q1) - MEC1 (q1) = 8 - 4q1 - 2 = 6 - .4q1

• MNB2 (q2) = MB2(q2) - MEC2 (q2) = 8 - 4q2 - 1 = 7 - .4q2

Mathematical solution

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– Step 4: Write down the present discounted value of marginal net benefits in each period. Remember that the PDV of x dollars t years from now at interest rate r is PDV = x / (1+r )t

• PDV [MNB1 (q1)] = (1+.1)0 * (6 - .4q1) = 6 - .4q1

• PDV [MNB2 (q2 )] = (1+.1)-1 * (7 - .4q2 ) = 6.363 - .363q2

– Step 5: Write down conditions #1 and #2.• Condition #1: Maximization

PDV[MNB1 (q1)] = PDV[MNB2 (q2 )], which we can rewrite as:

6 - .4q1 = 6.363 - .363q2

• Condition #2: Resource Constraint– q1+q2 = 20

Mathematical solution

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• Step #6: We finally have two equations (Conditions #1 and #2) and two unknown variables (q1 and q2 ). We can now use algebra to solve for the socially optimal q*1 and q*2 :

6 - .4(20 - q 2) = 6.363 - .363q2 q* 2 = 10.952

q1 + 10.952 = 20 q*1 = 9.048

• So, in this example, the socially optimal quantity of extraction is higher in period 2. Even though net benefits in period 2 are discounted, the marginal cost of extraction is lower in period 2. Thus, it makes sense to extract more in the second period, when extraction is cheaper.

Mathematical solution

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• First 4 steps are the same as in the mathematical solution: – Step 1: Write down marginal extraction costs in each period.

• MEC1 = 2 in period 1

• MEC2 = 1 in period 2

– Step 2: Write down marginal benefits in each period. Remember that marginal benefits are given by the inverse demand function.• MB1 (q1) = p1 (q1) = 8 - .4q1

• MB2 (q2) = p2 (q2) = 8 - .4q2

– Step 3: Calculate marginal net benefits in each period. Marginal net benefits are just equal to marginal benefits minus marginal extraction costs.• MNB1 (q1) = MB1 (q1) - MEC1 (q1) = 8 - 4q1 - 2 = 6 - .4q1

• MNB2 (q2) = MB2(q2) - MEC2 (q2) = 8 - 4q2 - 1 = 7 - .4q2

– Step 4: Write down the present discounted value of marginal net benefits in each period. • PDV [MNB1 (q1)] = (1+.1)0 * (6 - .4q1) = 6 - .4q1

• PDV [MNB2 (q2 )] = (1+.1)-1 * (7 - .4q2 ) = 6.363 - .363q2

Graphical solution

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• Step 5: Draw the PDV of marginal net benefits in each period on the same graph (see following slides).– This graph is a little complicated, so be careful.– We'll put PDVs on the vertical axes.– We'll put extraction quantities (q1 and q2) on the horizontal

axis, running in opposite directions. Since the total stock of the resource is 20, each of the axes will end at 20.• By drawing the quantities in opposite directions on the same

axis, we are building in the resource constraint (Condition #2).

– We'll then plot the PDV of MNB in period 1 on the left side, and the PDV of MNB in period 2 on the on the right side.

Graphical solution

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• Step 6: Find the optimal extraction quantities where the two lines intersect (see following slides).– This is just a graphical version of Condition #1

(maximization).– If we picked some other point, then we could

increase the sum of discounted net benefits in the two periods by switching a unit of extraction from period 1 to 2 (or vice versa). Thus, we wouldn't be at a maximum.

Graphical solution

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Graphical solution

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Graphical solution

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Graphical solution

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Graphical solution

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Graphical solution

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Graphical solution

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Resource prices • To calculate the optimal price in the two periods, we can just plug the

optimal quantities into the inverse demand functions: p 1(q 1) = 8 - .4q 1 = 8 - .4 * 9.048 = 4.38 p 2(q 2) = 8 - .4q 2 = 8 - .4 * 10.952 = 3.62

– NOTE: should be optimal real (current ) prices.

• Why aren't these prices equal to the marginal extraction costs?The answer is because the prices also include a marginal user cost (also known as scarcity rent) that accounts for the fact that once we extract a unit of oil, it is gone forever. To calculate marginal user cost, we just subtract the marginal extraction cost from the optimal price in each period: MUC 1 = p 1 - MEC 1 = 4.38 - 2 = 2.38 MUC 2 = p 2 - MEC 2 = 3.62 - 1 = 2.62

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Hotelling Rule • At the dynamically efficient allocation of a non-renewable resource with

constant MEC, the MUC rises over time at the rate of interest (the opportunity cost of capital)– This is the “no arbitrage” condition.

• We check the Hotelling Rule with for this example. The interest rate was r = .1, so

MUC 1 * 1.10 = MUC 2

2.380952 * 1.10 = 2.619048• So Hotelling Rule does hold in this example. • Question: What would happen if there were no property rights?

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Part 2: Renewable, open-access resources outline

• Biological aspects of fisheries• Efficient allocations: fisheries • Issues of open-access in fisheries

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Biological dimension of fisheries

• natural equilibrium (Scc)– the population size that would

persist in the absence of outside influences

– stable– “carrying capacity”

• minimum viable population (SMVP)– level of the population below which

growth is negative – unstable – below this population the species

could become extinct • critical depensation

Logistic Growth Curve with Harvest

Population growth increases as population increases

Population increases lead to eventual decline in growth

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Biological dimension of fisheries

• sustainable yield – Population for which

catch level = growth rate of population

– This can be maintained • maximum sustainable yield (SM)

• population size that yields the largest catch that can be perpetually sustained

• maximum sustainable yield = maximum growth

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Logistic Growth Curve with Harvest

Population growth increases as population increases

Population increases lead to eventual decline in growth

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Few additional points on biological dimension

• Very small and very large populations generate small rates of growth

• One population level will have the greatest possible annual growth rate (maximum sustainable yield)

• Except for at the stock maximum, equilibrium at any desired yield can be achieved through 2 different levels of fishing effort (high and low)

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Recall : Static Efficiency• To achieve static efficiency (single time period),

undertake policy to the point at which marginal benefits equal marginal costs

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0 1 2 3 4 5 6 7 8 9 10

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Quantity of Pollution Control

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Total Benefits

Total Costs

Marginal Benefits

Marginal CostsNet Benefits

Q*

Total Benefits and Total Costs Marginal Benefits and Marginal Costs

Q*

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Efficient sustainable yield

• We use three assumptions to simplify analysis/graphs: – price/fish constant over all catch

levels– Constant MC of fishing effort – Quantity of fish caught

proportional to existing stock of fish

• Static vs. dynamic – For details on impact of

discount rate, see Tietenberg.

Efficient level of effort

Slope=MB

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Efficient sustainable yield

• Efficient sustainable yield occurs where – MB=MC– Net benefits are

maximized

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Efficient level of effort

Slope=MB

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Market exploitation under open access

• In an open-access fishery, the rent is a stimulus for new fishermen to enter – As long as TB > TC

competitors will enter the market

• Under open-access, there will be an inefficient level of effort: – Effort will increase until

TB=TC– Overexploitation,

overcapitalization, depleted stocks

– Rents dissipate entirely

Total benefits (total revenues)

Total cost

Under open access

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Reasons for Ec:OPEN ACCESS

• Contemporaneous external cost: one fisher affects another (if I catch, you can’t and vice versa)

• Intertemporal external costs: take today, less for tomorrow

• Does not take into account future value MUC (this is why Hotelling rule for non-renewables required secure property rights)

• Market on its own cannot achieve efficient result.

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Additional information

• Prof. Stavins will be taking questions at the end of class Monday.

• PRE-MIDTERM OFFICE HOURS: Rich and Liz will hold office hours together on Monday (2/27) from 5 to 7 pm, and on Tuesday (2/28) from 6 to 8 pm. Both sets of office hours will occur in room 401 in Taubman. We will post review notes and problems in advance, which we encourage you to review prior to coming to office hours.

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Summer Undergrad RA Opportunity

“The Economic Returns to Investments in Clean Energy Technologies in the American Recovery

and Reinvestment Act of 2009”

Faculty: Joe AldyMore info at: http://environment.harvard.edu/student-resources/undergraduate-summer-research-fund/opportunities