Cost benefit analysis

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Chapter 11 Cost benefit analysis

11.1 Intertemporal welfare economics

 find out how to do project appraisal

 learn about cost–benefit analysis and its application

to the environment

 be introduced to some alternatives to cost–benefit

analysis

Cost-benefit analysis

Cost-benefit analysis, CBA , is the social appraisal of marginal

investment projects, and policies, which have consequences over time

It uses criteria derived from welfare economics , rather than commercial criteria

CBA seeks to correct project appraisal for market failure

Environmental impacts of projects/policies are frequently externalities , both negative and positive

CBA seeks to attach monetary values to external effects so that they can be taken account of along with the effects on ordinary inputs and outputs to the project/policy

CBA is the same as BCA – Benefit-cost analysis

Intertemporal efficiency

) C , (C U

U

) C , (C U

U

B 1

B 0

B B

A 1

A 0

A A

Intertemporal efficiency requires the satisfaction of 3 conditions

Equality of individuals’ consumption discount rates

Equality of rates of return to investment across firms Equality of the common consumption discount rate with the common rate of return

Given that CBA is concerned with consequences over time, and based in welfare economics, a key idea is that of intertemporal efficiency

Discount rate equality

MRUS C c A0, 1 = MRUSB C c0, 1 otherwise one could be made better off without making the other

Then the first intertemporal efficiency condition is stated as

rA=rB = r (11.2)

Note: consumption discount rates are not constants

Shifting consumption over time

ΔCC ΔCC

ΔCC ΔCC

ΔCC

) ΔCC (

ΔCC δ

0 1 0

0 1

0

0 1

where ΔCC1 is the second period increase in consumption, Ca

1Cb

1, resulting from the first period increase in investment ΔCI0, Cb

0Ca

0 For ΔCI0 = ΔCC0, this is

which is the negative of the slope of the transformation frontier minus 1, which can

be written

where s is the slope of the frontier

Rate of return equality

If each firm were investing as indicated by C01b and C02b, then period 1

consumption could be increased, without loss of period 0 consumption, by having firm 1, where the rate of return is higher, increase investment by the amount firm 2, where the rate of return is lower, reduced its investment.Only where rates of return are equal is this kind of period 1 gain

impossible For N firms, the second intertemporal efficiency condition is

(11.3)

Equality of discount rate and rate of return

If the first two conditions are satisfied, can consider representative individual and firm.

Point a corresponds to intertemporal efficiency, b and c do not as from either could reallocate consumption as between periods so as to move on to a higher consumption indifference curve.

At a the slopes of the consumption indifference curve and the consumption ttansformation frontier are equal The third condition is

r

Intertemporal optimality

As in the single period situation, the intertemporal efficiency

conditions do not fix a unique intertemporal allocation.

That requires a social welfare function with utilities as arguments Will consider this under ‘Optimal growth models’

Markets and intertemporal efficiency –futures markets

Futures Markets

X at t is treated as a different commodity from X and t+1

For N commodities and M periods there are MN dated commodities Contracts are written at the start of the first period for trades an all commodities at all future

dates.

Then, it is effectively the static case.

Given that all ideal circumstances apply in all MN markets, the conditions for

intertemporal efficiency will be satisfied.

Futures markets are, in fact, rare – standardised raw materials, financial instruments

Markets and intertemporal efficiency – loanable funds market

Loanable Funds Market

x is the bond coupon paid on the first day of period 1

Pb is the price the bond trades for on the first day of period 0

i is the interest rate

A seller is a borrower

A buyer is a lender

Individuals - utility maximisation

UU is a consumption indifference curve, slope –(1+r)

Line C1max C0max is the budget constraint, slope –(1+i)

Optimum is at C*0 in period 0 and C*1 in period 1, where

r = iwhich will hold for all individuals, satisfying the first

intertemporal efficency condition

Firms – present value maximisation

Owners of firms can shift consumption by

investing in firmdealing in the bond market

AB shows C0 C1 combinations on account of varying investment, slope –(1+ δ), δ is the rate of return to investment

RS with slope -(1+i) shows how consumption can be shifted via bond market dealing

The optimum level of investment is at a, where the present value of the firm is

maximised, and where

i = δ

In the second stage the owner maximises utility, by bond market dealing, at b where

i = rAll owners so act, and the three conditions for intertemporal efficiency are satisfied

Optimal growth modelling; discrete time

A representative individual model for two periods

Maximise

1 1

So with diminishing marginal utility, C1 is greater than C0 for

ρ, the utility discount rate, less then δ, the rate of return on

investment For ρ = δ consumption is constant

Optimal growth modelling: continuous time

dt )e

U(C

0 t

ρt t

(11.8)

where ρ is a parameter, the utility discount rate, and δ is a variable,

the rate of return to capital accumulation For δ > ρ, given

diminishing marginal utility, C is growing Consumption growth

ceases when δ = ρ

Optimal growth in the basic model

A model with resource input to production

dt )e

U(C

W t

0 t

ρt t

In this model intertemporal efficiency is not trivial.

There are two forms of investment, in capital and in the resource stock

Efficiency requires that the rates of return on the two are equal.

See chapter 14 especially

Utility and consumption discount rates 1

Figure 11.8 Indifference curves in utility and consumption space

 0   1 

1 1

1 1

1

1/ 1

C C

C C

U U

r

U U

Utility and consumption discount rates 2

In continuous time

where

r is the consumption discount rate

ρ is the utility discount rate

η is is the elasticity of marginal utility for the instantaneous utility function

g is the growth rateFor g>0 r> ρ and r would be positive for ρ = 0For g = 0 r = ρ

E R

N N N N

N i

The project should go ahead iff NPV≥0

The present value of expenditures E is

The present value of receipts R is

The present value of the project is

(11.16) Which for N = R - E is

(11.17)

Private project appraisal – the Net Present Value test 2

The NPV of a project is the amount by which it increases the

firm’s net worth It is the present value of the surplus, after

financing the project, at the end of the project lifetime.

Private project appraisal - risk

Year Net cash flow 1

–6.81

Table 11.4 One project, two possible cash flows Table 11.5 Calculation of expected NPV

Where the firm is prepared to assign probabilities, the criterion for going ahead with the project is the expected NPV – the probability weighted sum of the mutually exclusive cash flow outcomes.

This assumes that the decision maker is risk-neutral

Chapter 13 on decision making in the face of imperfect knowledge of the future.

Social project appraisal

CBA is the social appraisal of projects

CBA uses the NPV test

CBA can be approached in two ways

As an extension of private appraisal where externalities are taken into account

In terms of social welfare enhancementThe first stages of CBA are

proper project/policy identificationforecasting all of the consequences of the project/policy for all of the affected individuals in each year of the project/policy lifetime

Then

expressing consequences in terms of monetary gains/losses for aggregation to an NPV number

Social appraisal: an illustrative project

Table 11.6 Net benefit (NB) impacts consequent upon an illustrative project

r)(1

NB

Generally, go ahead if

CBA as a potential pareto improvement test

A positive NPV indicates that, with due allowance for the dating of costs and benefits, the project delivers a surplus of benefit over cost The consumption gains involved are greater than the consumption losses, taking account of the timing of gains and losses The existence of a surplus means that those who gain from the project could compensate those who lose and still be better off

Finance by taxation – two periods

The initial investment is ΔCI0, equal to -ΔCC0, and the consumption increment on account of going ahead with the project is ΔCC1 First period consumers lose an amount ΔCC0, equal to ΔCI0, and second period consumers gain ΔCC1, and the question is whether the gain exceeds the loss From the viewpoint of the first period, the second period gain is worth ΔCC1/(1+r), so the question is whether

is true, which is the NPV test discounting at r.

Finance by borrowing – two periods

The government funds the project by borrowing and the project displaces the marginal private sector project with rate of return δ In this case the cost of the public sector project is ΔCI0 in the first period plus δΔCI0 in the second, this being the extra consumption that the private sector project would have generated in the second period In this case, from the viewpoint of the first period, the gain exceeds the loss if:

δΔCI ΔCI

r 1

0 1









CBA as welfare increase test 1

Time period

A ΔUUA,0 ΔUUA,1 ΔUUA,2 ΔUUA,3 ΔUUA

B ΔUUB,0 ΔUUB,1 ΔUUB,2 ΔUUB,3 ΔUUB

C ΔUUC,0 ΔUUC,1 ΔUUC,2 ΔUUC,3 ΔUUC

Table 11.7 Changes in utility (∆U) consequent on an illustrative project

W = W(UA,0, , UC,3) positive – the project should go ahead

Or with an intratemporal social welfare function mapping individual utilities into a social aggregate Ut

W = W(U0, U1, U2, U3) positive – the project should go ahead

where a widely entertained particular form is

3 3 2

2 1

0

ρ) (1

ΔCU ρ)

(1

ΔCU ρ

1

ΔCU ΔCU

CBA as welfare increase test 2

Utility variations consequent on going ahead with the project cannot be estimated But, using the methods of chapter 12, monetary equivalent gains and losses can be estimated

t

C U ρ 1

1 W

t

C r 1

1 W

1 U

ρ)U (1

r

Ct

1 Ct

CBA as welfare increase test 3

1

1 W

The NPV test is interpreted as a test that identifies projects that yield welfare improvements - positive and negative consumption changes, net benefits that is, are added over time after

discounting, so that

(11.24)and for ΔCW > 0 the project is welfare enhancing and should go ahead

Finance by taxation two periods

1

r 1

1 ΔCI

where the rhs is NPV, positive for ΔCC1/(1+r) > ΔCI0

Finance by borrowing, two periods

If the project crowds out the marginal private sector project

δΔCI r

1

ΔCC ΔCI

where the rhs is NPV, positive if

r 1

δΔCI ΔCI

r

1

0 1









Choice of discount rate 1

There is disagreement about the discount rate that should be used in CBA

This matters because the result of the NPV test can be very sensitive to the number used for the discount rate

This is especially true where the project lifetime is long, as it often is with projects

with environmental consequences – the lifetime is when the longest lasting

consequence ceases, not when the project stops yielding the benefits which were its purpose – nuclear electricity generation and waste products

Time Horizon Years Discount rate % 25 50 100 200

Choice of discount rate 2

With no market failure r = i = δ

Given market failure which to use?

Generally agreed that whichever way looking at CBA – potential pareto improvement

or welfare enhancing – should use r, the consumption discount rate

Shadow pricing

While it is agreed that r should be used to discount, δ appears in the NPV criterion as

r 1

δΔCI ΔCI

r

1

0 1

δΔCI r

1

ΔCC ΔCI

which apply with finance by borrowing when there is crowding out – δ is there to

adjust the initial cost, shadow price it, for the displacement of the marginal private

sector project.

At one time it was thought that, on account of crowding out, proper shadow pricing

of all inputs and outputs was important in CBA And difficult.

Now the dominant view is that given international capital mobility, crowding out is

not a problem – that the supply of capita for private sector projects could be treated

as perfectly elastic.

Shadow pricing is not now seen as necessary in CBA

Choice of discount rate 3 – descriptive versus prescriptive

Regarding CBA as about potential pareto improvement aligns with the descriptive approach

to determining a number for r – it should be the post tax return on risk free lending reflecting the rate at which people are willing to exchange current for future consumption.

Those who regard CBA as a test for welfare enhancement tend to adopt the prescriptive

approach to a number for r, according to

r = ρ + ηg (11.23)

where

ρ is the utility discount rate

η is the elasticity of the marginal utility of consumption

g is the growth rate

Some economists want to get values for ρ and η from observed behaviour, some from

ethical considerations.

Much of the controversy among economists over the Stern Review of the climate change problem focussed on the numbers used in (11.23) – Stern took an ethical prescriptive

position

Box 11.2 Discount rate choices in practice

US Office of Management and Budget

7% as an estimate of pre-tax return on capital

US Environmental Protection Agency

For intragenerational – descriptive, r as 2-3%

For intergenerational - r = ρ + ηg with ρ = 0 on ethical grounds, gives r 0.5% to 3%

For lifetimes greater than 30 years

because of ‘uncertainty about the future’

The Stern Review

Implicit r of 2.1% from r = ρ + ηg with ρ = 0.1%, η = 1, and g = 2%

Environmental cost-benefit analysis

Look at wilderness area development

With Bd for development benefits and Cd for development costs, and ignoring

where EC is the present value of the stream of the net value of the project’s

environmental impacts over the lifetime of the project

EC stands for Environmental Cost

From (11.27) the project should go ahead if

NPV’ = Bd – Cd > EC (11.28)

As can consideration of EC*/N, where N is the size of the relevant affected

population, which is not necessarily restricted to visitors, and may include people from a wider area than the host country, as with an internationally recognised

wilderness/ conservation area inscribed as ‘world heritage’

Box 11.3 Mining at Coronation Hill?

In 1990 there emerged a proposal to develop a mine at Coronation Hill in the Kakadu national park, which is listed as a World Heritage Area The Australian federal government referred the matter to the Resource Assessment Commission, which undertook a very thorough exercise in environmental valuation using the Contingent valuation Method, implemented via a survey of a sample of the whole Australian population This exercise produced a range of estimates for the median willingness to pay, WTP, to preserve Coronation Hill from the proposed development, the smallest

of which was $53 per year If it is assumed, conservatively, that the $53 figure is WTP per household, and this annual environmental damage cost is converted to a present value capital sum

in the same way as the commercial NPV for the mine was calculated, the EC to be compared with the mine NPV' is, in round numbers, $1500 million This 'back of the envelope' calculation assumes

4 million Australian households, and a discount rate of 7.5%.

It was pointed out that given the small size of the actual area directly affected, the implied per hectare value of Coronation Hill greatly exceeded real estate prices in Manhattan, whereas it was 'clapped out buffalo country' of little recreational or biological value In fact, leaving aside environmental considerations and proceeding on a purely commercial basis gave the NPV' for the mine as $80 million, so that the threshold per Australian household WTP required to reject the mining project was, in round numbers, $3 per year, less than one-tenth of the low end of the range

of estimated household WTP on the part of Australians Given that Kakadu is internationally famous for its geological formations, biodiversity and indigenous culture, a case could be made for extending the existence value relevant population, at least, to North America and Europe

In the event, the Australian federal government did not allow the mining project to go ahead It is not clear that the CVM application actually played any part in that decision