Net Present Value Analysis

Andrew Foss(andrew_foss@ksg09.harvard.edu)

Economics 1661 / API-135Environmental and Resource Economics and PolicyHarvard University

February 13, 2009Review Section

Agenda

Fundamental Theories of Welfare Economics

Static Efficiency

Dynamic Efficiency

Cost-Effectiveness Analysis and Benefit / Cost Ratios

Internal Rate of Return

Equivalent Annual Net Benefits

Readings on Benefit-Cost Analysis

Private Goods and Public Goods

Excel Workbook Embedded Here: 2

Microsoft Office Excel 97-2003 Worksheet

Fundamental Theories of Welfare Economics:Pareto Criterion and Pareto Optimality

Pareto Criterion: A policy change is an improvement if at least some people are made better off and no one is made worse off

Pareto Optimality: No other feasible policy could make at least one person better off without making anyone else worse off

3Adam’s Payment

Beth’s Payment

StatusQuo

Policy A

Policy B

Policy C

Policy D

Feasibility Frontier

Possible Payments to Adam and Beth Which satisfy Pareto Criterion?

‒ Policy A does ‒ Policy B does not‒ Policy C does not‒ Policy D does‒ All policies in light gray triangle

Which satisfy Pareto Optimality?

‒ Policy A does not ‒ Policy B does not‒ Policy C does‒ Policy D does‒ All policies on feasibility frontier (because nothing “better” from there)

$25 $100

$25

$100

Fundamental Theories of Welfare Economics:Kaldor-Hicks Criterion

Kaldor-Hicks Criterion: A policy change is an

improvement if the “winners” could fully compensate the

“losers” and still be better off themselves

– Also known as Potential Pareto Improvement Criterion

Kaldor-Hicks Criterion rules out policies with total

benefits smaller than total costs (that is, policies with

negative net benefits, where NB = TB - TC)

When the Kaldor-Hicks Criterion is used to compare all

feasible policy options, the best is that which maximizes

net benefits

– If all policies have negative net benefits, keep the status quo4

0

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0 1 2 3 4 5 6 7 8 9 10

Ma

rgin

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arg

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Quantity of Pollution Control

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tal B

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Quantity of Pollution Control

Static Efficiency

To achieve static efficiency (single time period),

undertake policy to the point at which marginal

benefits equal marginal costs

5

Total Benefits

Total Costs

Marginal Benefits

Marginal CostsNet Benefits

Q*

Total Benefits and Total Costs Marginal Benefits and Marginal Costs

Q*

Dynamic Efficiency:Overview

To achieve dynamic efficiency (multiple time periods),

undertake policy with highest net present value

If all policies have negative NPV, keep the status quo

Discount rate should reflect social opportunity cost

U.S. Office of Management and Budget (OMB)

published guidance on discount rate and benefit-cost

analysis in Circular A-4 (September 2003):http://www.whitehouse.gov/omb/assets/regulatory_matters_pdf/a-4.pdf

6

T

tttt

r

CB

0 )1(NPV

Dynamic Efficiency:Discounting

Benefits and costs far in the future are more sensitive

to discount rate than near-term benefits and costs

– Run discounting program in Excel workbook embedded on p. 1

7

r = 3% → NPV = $29M r = 10% → NPV = -$10M

-$250

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$0

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0 1 2 3 4 5 NPV

Net

Ben

efit

s (m

illi

on

s)

Undiscounted Net Benefits Discounted Net Benefits

-$250

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0 1 2 3 4 5 NPV

Net

Ben

efit

s (m

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s)

Undiscounted Net Benefits Discounted Net Benefits

Dynamic Efficiency:Discounting

When costs are incurred up front and benefits occur in

the future, low discount rates result in higher NPVs than

high discount rates

8

Relationship between Discount Rate and NPVwith Upfront Costs and Future Benefits

-$50

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$0

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0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Net

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Dynamic Efficiency:Power Plant Example

You are a special assistant to Gov. Schwarzenegger of

California. He wants to shut down a coal-fired power plant

and replace it with either a hydropower plant or a natural gas-

fired plant. He asks you to analyze the options.

Assumptions (unrealistic…)

– Both plants can be built in 1 year and operate for 5 years

– Both plants yield annual benefits of $50M relative to coal

– Hydropower plant has upfront fixed costs of $100M and

annual operating costs of $5M

– Natural gas plant has upfront fixed costs of $40M and

annual operating costs of $20M

– Discount rate is 7 percent, but also try 3 and 10 percent

9

Dynamic Efficiency:Power Plant Example

Hydropower has a slightly higher NPV than natural gas

at 7 percent discount rate, but lower at 10 percent

10

$0

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$120

r = 3% r = 7% r = 10%

Ne

t Pre

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Va

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(m

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)

Hydropower Natural Gas

Cost-Effectiveness Analysis andBenefit / Cost Ratios

Cost-effectiveness analysis answers the question,

“Does the policy achieve its purpose at least cost?”

Benefits and costs over time should be discounted

When benefits are not monetized, undertake projects

in increasing order of cost per unit of benefit

When benefits are monetized, calculate benefit / cost

ratios and undertake projects in decreasing order of

benefit / cost ratios, provided they are greater than 1

There are problems with both these rules, however11

Qq)(qCi

ii

iiqi

s.t. min

Cost-Effectiveness Analysis andBenefit / Cost Ratios

Cost-effectiveness analysis is less robust than NPV

– Insensitive to scale

– Sensitive to impact definitions (e.g., costs as negative benefits)

12

$0 $20,000

Engine Modifications

Humid Air Motor (HAM)

Direct Water Injection (DWI)

Selective Catalytic Reduction (SCR)

Low-Sulfur Fuel

Shore-side Power

Engine Modifications

Engine Modifications

Engine Modifications

Engine Modifications

Diesel Particulate Filter (DPF) and SCR

Cost-Effectiveness ($/ton NOx)

Oceangoing Vessels

Harbor Craft

Cargo Handling Equipment

Heavy-Duty Vehicles

Locomotives

$11-$50

$280-$350

$450

$540-$790

$3,200

$14,500-$18,500

$1,800

$2,000

$12k-$16k

$1k-$5k

$4k-$14k

Reducing Nitrogen Oxide (NOx) Emissions at Ports‒ Unclear what scale of benefits could result from each measure

‒Unclear to what degree policies should be undertaken

‒ Unclear whether one or several policies should be undertaken

‒ Unclear what probability distributions underlie uncertainty bars

Internal Rate of Return:Overview

Internal rate of return answers the question, “What

discount rate would make NPV zero?”

When costs are incurred up front and benefits occur in

the future, undertake project if IRR > r

In the first discounting example (p. 7), IRR ≈ 8 percent

Internal rate of return is less robust than NPV, and it

should not be used to rank projects when constraints

make it impossible to undertake them all

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0)1(

NPV0

T

tt

tt

IRR

CB

Internal Rate of Return:Power Plant Example

A nuclear power plant can be built in 1 year for $100M, can operate

for 5 years, yields annual benefits of $55M relative to coal, has

annual operating costs of $5M, and has decommissioning costs of

$155M in Year 6

14-$6

-$5

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$0

$1

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0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Net

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IRR??

IRR is not useful in this case because there are costs in the future

Equivalent Annual Net Benefits

Suppose the hydropower plant replacing the coal plant

in California can operate for 10 years and the natural

gas plant can still only operate for 5 years

– At r = 7 percent, NPVhydro = $216M and NPVgas = $83M

– At r = 32* percent, NPVhydro = $32M and NPVgas = $30M

* This is an unusually high discount rate, but it illustrates the point for the example numbers

Calculate equivalent annual net benefits to compare

these projects of different duration

– At r = 7 percent, EANBhydro = $27M and EANBgas = $16M

– At r = 32 percent, EANBhydro = $8M and EANBgas = $9M 15

Trr

rNPVEANB

)1(1

Readings on Benefit-Cost Analysis:Arrow et al. (1996)

Benefit-cost analysis is a important framework for

making regulatory decisions

– Careful consideration of benefits and costs

– Common unit of measurement for disparate impacts (dollars)

– Useful tool for improving effectiveness of regulation

– Techniques for incorporating uncertainty

But benefit-cost analysis should not be the sole basis

for making regulatory decisions

– Consideration of distributional impacts as well

– Perhaps not necessary to perform benefit-cost analysis for

minor regulations

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Readings on Benefit-Cost Analysis:Goulder and Stavins (2002)

Discounting does not shortchange the future, so long as

an appropriate discount rate is used

– It simply puts current values and future values of benefits and

costs in equivalent monetary terms; apples-to-apples

comparison

– It accounts for time value of money (interest) and not inflation:

rnominal ≈ inflation + rreal

When the “winners” of a policy do not actually

compensate the “losers,” the Kaldor-Hicks criterion

carries less weight

Lowering the discount rate to increase NPV is

problematic because it mixes efficiency and equity 17

Private Goods and Public Goods:Beekeeper and Farmer Example

A beekeeper and a farmer are neighbors. The bee-

keeper’s bees help pollinate the farmer’s orchard.

The beekeeper’s marginal benefit from Q beehives is

MBbeekeeper(Q) = 10 – Q

The beekeeper’s marginal cost is constant at

MCbeekeeper(Q) = 7

The farmer’s marginal benefit from Q beehives is

MBfarmer(Q) = 5 – Q

If the beekeeper ignores impacts on the orchard, how

many beehives will the beekeeper have? What if the

beekeeper takes impacts on the orchard into account? 18

Private Goods and Public Goods:Beekeeper and Farmer Example

If the beekeeper treats the beehives as a private good,

the beekeeper will have 3 beehives

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0

5

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0 5 10 15

MB

or

MC

Quantity of Beehives

MB_beekeeper MC_beekeeper

Q*

MCbeekeeper

MBbeekeper = MCbeekeeper

10 – Q = 7

Q* = 3 beehives

MBbeekeeper

0

5

10

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0 5 10 15

MB

or

MC

Quantity of Beehives

MB_beekeeper MC_beekeeper MB_farmer MB_society

Private Goods and Public Goods:Beekeeper and Farmer Example

If the beekeeper treats the beehives as a public good,

the beekeeper will have 4 beehives

– Public goods are underprovided by private decision-making

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Q*

MBbeekeeper

MCbeekeeper

MBsociety = MBbeekeeper + MBfarmer

(vertical sum of MBs for public goods)

For 0 ≤ Q ≤ 5 (where both MBs ≥ 0),

MBsociety = (10 – Q) + (5 – Q) = 15 – 2Q

MCsociety = MCbeekeeper

MBsociety = MCbeekeeper

15 – 2Q = 7

Q* = 4 beehives

MBsociety

MBfarmer

Private Goods and Public Goods:Beekeeper and Farmer Example

If the beekeeper and farmer can negotiate without

transaction costs, what outcome would we expect?

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Increasing the number of beehives from 3 to 4 gives the

beekeeper extra benefits of $6.50 (area under MBbeekeeper)

but extra costs of $7 (area under MCbeekeeper), so the

beekeeper’s profit decreases by $0.50

Increasing the number of beehives from 3 to 4 gives the

farmer extra benefits of $1.50 (area under MBfarmer) and

does not impose extra costs on the farmer

By the Coase Theorem, the farmer could give the bee-

keeper between $0.50 and $1.50 to have 4 beehives

Private Goods and Public Goods:Steel Mill and Laundry Example

A steel mill generates $1000 in profits and can install

technology to eliminate its emissions at a cost of $400

A laundry can operate either upwind or downwind of the

steel mill, with different building sizes at the locations

– If the laundry operates upwind of the steel mill,

• It can generate $400* in profits if mill releases emissions

• It can generate $600* in profits if mill releases no emissions* Suppose profits differ even when upwind because emissions depress local economy

– If the laundry operates downwind of the steel mill,

• It can generate $200 in profits if mill releases emissions

• It can generate $1000 in profits if mill releases no emissions

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Private Goods and Public Goods:Steel Mill and Laundry Example

Laundry Upwind Laundry Downwind

Steel Mill Emissions

Steel Mill $1000 $1000

Laundry $400 $200

Joint $1400 (= $1000 + $400) $1200 (= $1000 + $200)

No Steel Mill Emissions

Steel Mill $600 (= $1000 - $400) $600 (= $1000 - $400)

Laundry $600 $1000

Joint $1200 (= $600 + $600) $1600 (= $600 + $1000)

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Steel Mill and Laundry Profits

Private Goods and Public Goods:Steel Mill and Laundry Example

What is the socially efficient arrangement?

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– The socially efficient arrangement has highest joint profits for

the steel mill and laundry

– Joint profits are highest ($1600) when the steel mill has no

emissions and the laundry operates downwind

Private Goods and Public Goods:Steel Mill and Laundry Example

Suppose the steel mill and laundry cannot bargain What arrangement will occur if the steel mill has the right

to release emissions?

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– The steel mill will release emissions, because its profits are higher if it releases

emissions ($1000) than if it does not ($600)

– The laundry will operate upwind, because its profits are higher if it operates

upwind ($400) than if it operates downwind ($200)

What arrangement will occur if the laundry has the

right to clean air?– The steel mill will have to install the technology to eliminate its emissions, leaving

it with $600 in profits

– The laundry will operate downwind, because its profits are higher if it operates

downwind ($1000) than if it operates upwind ($600)

Private Goods and Public Goods:Steel Mill and Laundry Example

Suppose the steel mill and laundry can bargain costlessly What arrangement will occur if the steel mill has the right

to release emissions?

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– The laundry can increase its profits from $400 operating upwind with emissions to

$1000 operating downwind without emissions. The laundry is willing to pay the

steel mill up to the difference, $600, to install the technology. The steel mill is

willing to accept anything more than $400, so they make some deal in this range.

What arrangement will occur if the laundry has the

right to clean air?– The steel mill is willing to pay up to $400 to avoid installing the technology, but

the laundry is only willing to accept $600 or more to allow emissions and operate

upwind, so there is no deal.

With bargaining, steel mill installs technology and

laundry operates downwind (the efficient arrangement)

regardless of allocation of rights (Coase theorem)