Dynamic Efficiency: Hotelling’s Rule

Environmental Economics II Spring 2014Lecture based in part on Harris and Roach 2013 and Field 2008

Present Value Calculations

Vo = Vn/(1 + r)n

$1000 next year is worth today:

Vo = $1000/(1.05) = $953

$1000 in seven years is worth today:

Vo = $1000/(1.05)7 = $710

Net Present Value

• NPV is the present value of revenues minus present value of costs.

NPV = R0 + R1/(1+r) + R2/(1+r)2 + R3/(1+r)3 ... Rn/(1+r)n - C0 - C1/(1+r) - C2/(1+r)2 –

C3/(1+r)3 ... Cn/(1+r)n

• NPV = ∑(Rn/(1+r)n - Cn/(1+r)n )

Present Value of Periodic Payments:

V0 = p {1- (1 + r)-n}/r

where p is payment and n is number of years

START

One

Series

Annual

Periodic

NumberOf

Payments

Terminating

Terminating

Perpetual

Terminating

Perpetual

Future

Present

Future

Present

Future

Present

Future

Present

Future

Present

Vn=Vo(1 + r)n

V0=Vn/(1 + r)n

Vn= Infinity

Vn= Infinity

TimeBetweenPayments

EvaluationPeriod

Time ofValue

Formula FormulaName

Legend

Future value* of an amount

Present value of an amount

Future value* terminating annual series

Present value of a terminating annual series

Present value of a perpetual annual series

Future value* of a terminating periodic series

Present value of a terminating periodic series

Present value of a perpetual periodic series

r = Annual interest rate/100. (If payments are fixed in real terms, r is real; if payments are fixed in nominal terms, r is nominal.

V0 = Present value (or initial value)

Vn = Future value after n years (including interest)

n = Number of years of compounding or discounting

p = Amount of fixed payment each time in a series (occurring annually or every t years)

t = Number of years between periodic occurrences of p

* The future value of any terminating serires is its present value formula time (1 + r)n. Adapted from Gunter and Haney (1978), by permission.

Decision tree from: Klemperer, W. (1996) Forest Resource Economics & Finance. McGraw Hill.

PV of string of payments using Excel

Present Value Calculations and Resources

• Renewable Resource Problem: Harvest such that resource grows at same rate as money in the bank.

• Non-renewable Resource Problem: Extract such that price of resource grows at same rate as money in the bank. – Hotelling’s Rule

Hotelling’s Rule

• Net benefits (CS + PS) over time are maximized (dynamic efficiency) when net price increases at the discount rate.

• Managers will extract at this rate.• Therefore, resource extraction will be

“socially efficient”.

• Let’s test with a simple two-period example.

Dynamic Efficiency Model - Disclaimer

This analysis of dynamic efficiency for non-renewable resource extraction is based on a highly simplified modeling framework, in order to provide an accessible introduction to the topic, along with important insights, without complex mathematics.

Simplifying Assumptions

1. Marginal extraction cost is constant2. Demand is constant3. There is a competitive market with no market

irregularities such as cartels4. There is perfect information. i.e. Market

participants are fully informed of current and future demand, marginal extraction costs, the discount rate, available stocks, and market price

5. No externalities!

We will look at the most basic case with just two time periods: today (period 1) and next year (period 2)

Dynamic Efficiency Example: Model

Demand curve: P = $300 – 0.25Q

Supply curve: P = $20

Total stock = 1,000 barrels

r = .05

Dynamic Efficiency ExampleDemand curve: P = $300 – 0.25QSupply curve: P = $20Total stock = 1,000 barrelsr = .05

Sell all the first year: Q = 1,000• P = $300 – 0.25(1,000) = $50• Net benefit = CS + PS• CS = (300 – 50) x 1000/2 = $125,000• PS = (50 – 20) x 1000 = $30,000

• Net benefit = $125k + $30K = $155k

Net Benefits = CS + PS

300

20

1200

50

1000Stock

P

Q

CS

PS

Demand curve

Supply

Marginal Net Benefits

280

11201000Stock

P

Q

Marginal Net Benefits

Total Net Benefits

Sell half each year. Q1 = Q2 = 500Demand curve: P = $300 – 0.25QSupply curve: P = $20Total stock = 1,000 barrelsr = .05

• The first year:• P = $300 – 0.25(500) = $175• CS = (300 – 175) x 500/2 = 31,250• PS = (175 – 20) x 500 = 77,500• CS + PS = $108,750

• Year 2: $108,750/(1.05) =$103,571• Total = 108,750 + 103,571 =

$212,321

Dynamic Efficiency: (P2– MC)/(P1– MC) = 1+ r

Demand curve (MWTP): P = $300 – 0.25QSupply curve (MC): P = $20Total stock = 1,000 barrelsr = .05

• Maximize net benefits by selling quantities such that marginal net benefits (MWTP – MC) increases at the rate of interest: 5%

(P1 - 20) = (P2 – 20)/1.05

$300 – 0.25Q1 – $20 = ($300 – 0.25Q2– $20 )/1.05

Dynamic Efficiency Example Cont.

$280 – 0.25Q1 = ($280 – 0.25Q2)/1.05

1.05 ($280 – 0.25Q1)= ($280 – 0.25Q2)

294 - 0.2625Q1 = 280 – 0.25Q2

Q2 = 1000 – Q1

294 - 0.2625Q1 = 280 – 0.25(1000 – Q1)

294 - 0.2625Q1 = 280 – 250 + 0.25Q1

264 = 0.5125Q1

Q1 = 515.1

Q2 = 1000 – Q1 = 484.9

Q1 = 515.1Q2 = 484.9

• Net Price (price – cost) should increase by 5%

• Find price for each period by filling in the demand equation P = 300 - .25Q

• P1 = $300 – 0.25(515.1) = $171.225

• P2 = $300 – 0.25(484.9) = $178.775

• (178.8 – 20)/(171.2 – 20) = 1.05 TA DA!

Q1 = 515.1 P1 = $171.2 Q2 = 484.9 P2 = $178.8

• Calculate present value of net benefits.CS period 1= (300 – 171.2) x 515.1 / 2 = 33,172PS period 1 = (171.2 – 20) x 515.1 = 77,883

CS period 2= (300 – 178.8) x 484.9/2 = 29,385PS period 2 = (178.8 – 20) x 484.9 = 77,002

• 33,172 + 77,883 + (29,385 + 77.002)/1.05 = $212,376

Graphic Dynamic EfficiencyDemand curve: P = $300 – 0.25QSupply curve: P = $20Total stock = 1,000 barrelsr = .05280

P

Q1

267 = (280/1.05)

1000Q2 515484

500500

P/1.05

Dynamically Efficient Equilibrium and Discount Rate

How will the dynamically efficient allocation of the fixed resource stock change if the discount rate (r) becomes larger?

Intuition?

Graphic Dynamic Efficiency r= 10%

280

P

Q1

267 = (280/1.05)

1000Q2 515484

255 = (280/1.10)

More extraction in period 1

Dynamically Efficient Equilibrium Intuition: why this model is amazing

If the net price increases at the interest rate, then the present value of marginal profit is equal across time periods (Hotelling’s rule).

Resource managers have no incentive to change this production path over time, ceteris paribus. This solution also generates the largest PV of total net benefits (CS + PS) over time.

Dynamically Efficient Equilibrium, Profits, and Size of Stock

• When a resource is abundant, then consumption today does not involve an opportunity cost of lost profit in the future, since there is plenty available for both today and the future. In a perfectly competitive market, P = MC and marginal profit would be zero.

• As the resource becomes increasingly scarce, consumption today involves an increasingly high opportunity cost of foregone profit in the future. As resources become increasingly scarce, P increases relative to MC and profits grow.

Hotelling Rent or Scarcity Rent

• The profit due to resource scarcity in competitive markets.

• Economic profit that can persist in certain natural resource cases due to the fixed stock of the resource.

• Due to fixed stock, consumption of a resource unit today has an opportunity cost equal to the present value of the marginal profit from selling the resource in the future.

User Costs

• Value of the resource in its natural state, such as oil in the ground.

• Equal to the opportunity costs associated with using the resource now such that it will not be available in the future.

• The marginal user costs (MUC) are the opportunity cost associated with using one more unit today instead of saving it for the future.

• In theory, the user cost should be the cost an extractor pays to the owner of the resource: royalty or rent.

• Royalty or rent is money derived, not from having special

skills or timing or insights, but simply from owning or having access to a resource.

MEX = marginal extraction costMEC = marginal external costsMUC= marginal user costsp=private e=external s=social

MEX (i.e. marginal private cost)

MEX + MEC

MEX + MEC + MUC (= MTC in book)

QpQs Qe Q oil

P

Demand

Oil Extraction and Externality Costs and User Costs

MEX = marginal extraction costMEC = marginal external costsMUC= marginal user costsp=private e=external s=social

MEX (i.e. marginal private cost)

MEX + MEC

MEX + MEC + MUC (= MTC in book)

QpQs Qe Q oil

P

Demand

Oil Extraction and Externality Costs and User Costs

Dead weight loss due to external costs

Dead weight loss due to EC and UC

Finding the optimal extraction using Excel

Price-cost ratio is where PV of net benefits is maximized

Q1

Another example

Comparison• Both stock of 1000 and r = .05 and • Supply: P = 20

Example 1– Demand P = 300 - .25Q – Q1 = 515, P1 = $171

Example 2– Demand P= 100 - .01Q – Q1 = 685, P1 = $93

Why are the extraction rates so different?

Calculate the demand elasticity

• ΔQ/ΔP x P/Q

• Example 1: 171/515 x 1/.25 = 1.33

• Example 2: 93/685 x 1/.01 = 13.58

Effect of Elasticity on Quantity Extracted – Less Elastic Demand

P

Q

Effect of Elasticity on Quantity Extracted – Very Elastic Demand

P

Q

Graphic Dynamic Efficiency: Demand P = 100 - .01Q

Supply: P = 20Stock = 1000

100

P

Q1

95.3 = (100/1.05)

1000Q2 680320

P/1.05

Impact of Elasticity on Extraction

• More elastic the demand, more resources extracted in the present.

Intuition?

Why might prices not follow Hotelling’s Rule?

1. Stock increases due to new discoveries increased extraction now

2. Technological change – More new discoveries– MC of extraction decreasing in future less

extraction now– MC decreases in all periods slightly more

extraction now– New substitutes more extraction now

3. Demand increase over time– Unpredicted no change in extraction rate– Predicted Increased extraction now

Why might prices not follow Hotelling’s Rule?

4. Awareness of scarcity increases extract less now

5. r increases extract more now6. Government regulation coming extract

more now (like MC increasing in future)7. Market irregularities extraction and price

irregularities8. Expectations:

Increase scarcity decrease extraction nowGovernment regulation increase extraction

now

Why might prices not follow Hotelling’s Rule?

9. Backstop technology extract more now

10. Imperfect information erratic pricing and extraction

Graphic Dynamic Efficiency:Demand curve: P = $300 – 0.25QSupply curve: P = $20Total stock = 1,500 barrelsr = .05

100

P

Q1

95.2

1500Q2 515484 More extraction in period 1 (and 2)

Graphic Dynamic Efficiency: MC in period 2 falls: Net benefits

increases in period 2.

280

P

Q1

267 = (280/1.05)

1000Q2 515484Less extraction in period 1

276 = (280 – 10)/1.05

Graphic Dynamic Efficiency: MC in both periods fall: Net benefits increases both

periods.

280

P

Q1

267 = (280/1.05)

1000Q2 515484Less extraction in period 1

276 = (290 – 10)/1.05

290

Graphic Dynamic Efficiency: Demand is predicted to increase in

period 2

280

P

Q1

267 = (280/1.05)

1000Q2 515484Less extraction in period 1

Second ExampleDemand: P = 100 - .01Q

MC in period 2 falls. Less extraction now.

MC in both periods fall. Slightly more extraction now.

Demand in period 2 increase - predicted. Less extraction now.

r increases – More extraction now

In-Class Exercises

For your math pleasure: Hotelling’s rule for multiple periods (totally

optional!)Demand: Pi = a – bqi

Supply is fixed: P = c

i (aqi – bqi2/2 – cqi)/(1+r)i + [Qtot - i qi],

where i = 0, 1, 2, …, n.

If Qtot is constraining, then the dynamically efficient solution satisfies:

(a – bqi – c)/(1+r)i - = 0, i = 0, 1, …, n.

[Qtot - i qi] = 0