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Lecture Note 15: Social Cost Benefit Analysis - University of ...

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<strong>Lecture</strong> <strong>Note</strong> <strong>15</strong>: <strong>Social</strong> <strong>Cost</strong> Bene…t <strong>Analysis</strong>:<br />

Distribution<br />

Part 2B: Paper 1.<br />

1 Introduction<br />

Dr. T.S. Aidt<br />

<strong>University</strong> <strong>of</strong> Cambridge<br />

Michaelmas 2011 (corrected April 2012)<br />

The discussion <strong>of</strong> SCBA in lecture note 13-14 ignored the distributional impact<br />

<strong>of</strong> the project or policy under consideration and focussed on deriving e¢ ciency<br />

based rules to help guide public policy. For some projects, say, an evaluation<br />

<strong>of</strong> whether the MOD should buy one type <strong>of</strong> …ghter jet rather than another,<br />

this may not be too far o¤ the mark, but for others policies, say, an evaluation<br />

<strong>of</strong> the costs and bene…ts <strong>of</strong> an in-work bene…t programme or a public health<br />

intervention, distribution is at the heart <strong>of</strong> the matter. So, an important question<br />

is: if distributional issues should be taken into account when conducting<br />

a SCBA and if so, how. Both <strong>of</strong> these questions are somewhat controversial.<br />

The purpose <strong>of</strong> this lecture note is to discuss the theoretical issues underpinning<br />

these questions.<br />

The note is organized as follows. We begin by asking under which sorts <strong>of</strong><br />

conditions it is appropriate to ignore distribution when undertaking a SCBA.<br />

As we shall see it hard to make a clean welfare theoretical case for not taking<br />

distribution into account. Next, we consider two alternative ways <strong>of</strong> incorporating<br />

distribution into social cost bene…t analysis. The …rst approach is the<br />

"adjusted social weights" approach. This is the method used by the UK government.<br />

The second approach is the "marginal cost <strong>of</strong> funds" approach. This<br />

approach is widely used in the US and many other countries.<br />

2 <strong>Social</strong> cost bene…t analysis and distribution<br />

Before getting to the "how to" question, we need to understand if we really<br />

need to bother with distribution when undertaking SCBA, i.e., can we make a<br />

Disclaimer: This note may contain mistakes. If you spot any, please bring them to my<br />

attention.<br />

1


convincing case for leaving distribution to one side and focus on e¢ ciency?<br />

To set the stage, let us return to the framework from section ?? but instead<br />

<strong>of</strong> assuming that all consumers are identical, let us assume that they di¤er<br />

in their tastes (i.e., have di¤erent utility function), that they have di¤erent<br />

levels <strong>of</strong> income (e.g., because they got di¤erent types <strong>of</strong> human capital that<br />

allow them to hold di¤erent types <strong>of</strong> jobs or got other sources <strong>of</strong> income than<br />

wage income), and that the social welfare function is individualistic, but not<br />

necessarily utilitarian. More speci…cally, replace assumptions 1, 3 and 7 from<br />

lecture note 13-14 by 1’, 3, and 7’:<br />

1’ The economy is populated by many di¤erent consumers, h = 1; ::; H.<br />

1. There are two goods x1 and x2 with consumer prices q = (q1; q2) and<br />

producer prices p = (p1; p2). The two goods are private goods and they<br />

are sold in markets, i.e., a market price exists for both <strong>of</strong> them.<br />

3’ The direct utility function <strong>of</strong> consumer h is Uh(x h 1; x h 2) and the budget<br />

constraint is q1x h 1 + q2x h 2 = m h where m h is the given income <strong>of</strong> consumer<br />

h.<br />

2. Each consumer maximizes utility subject to the budget constraint. This<br />

involves buying each <strong>of</strong> the two goods up to the point where<br />

@Uh<br />

@x1<br />

@Uh<br />

@x2<br />

= hq1 (1)<br />

= hq2; (2)<br />

where h is the marginal utility <strong>of</strong> income (the Lagrange multiplier on the<br />

budget constraint) for consumer h.<br />

3. The two goods are produced with capital and labour, l0 and k. The total<br />

supply <strong>of</strong> these are assumed to be …xed. The producer prices <strong>of</strong> labour<br />

and capital, respectively, are p0 and pk.<br />

4. The production functions are x1 = F 1 (l 1 0; k 1 ) and x2 = F 2 (l 2 0; k 2 ) where<br />

l j<br />

0 and kj is the amount <strong>of</strong> labour and capital, respectively employed in<br />

the production <strong>of</strong> the two goods.<br />

7’ <strong>Social</strong> welfare is individualistic, i.e.,<br />

with<br />

@SW F<br />

@Uh<br />

SW F (U1; :::; UH) (3)<br />

0. This includes as a special case the utilitarian social<br />

welfare function SW F = P H<br />

h=1 Uh but we shall also consider other speci-<br />

…cations.<br />

Suppose that we want to evaluate the net social value <strong>of</strong> a (small) project<br />

that increases the supply <strong>of</strong> x1 by x h 1 and, as a consequence, reduces the supply<br />

2


<strong>of</strong> x2 by x h 2 for consumer h. Under what conditions, we can judge the social<br />

value <strong>of</strong> this project without having to worry about distribution. The answer<br />

to this question will give us a good sense <strong>of</strong> whether distribution is a central or<br />

a marginal issue for social cost bene…t analysis.<br />

By de…nition the net bene…t (NB) <strong>of</strong> the project is the di¤erence between<br />

social welfare before and after, so we can write it as<br />

NB = SW F (U1(x 1 1 + x 1 1; x 1 2 x 1 2); :::; UH(x H 1 + x H 1 ; x H 2 x H 2 ))(4)<br />

This can be approximated by<br />

SW F (U1(x 1 1; x 1 2); :::; UH(x H 1 ; x H 2 )):<br />

NB = PH @SW F @Uh<br />

h=1 @Uh @xh 1<br />

x h 1<br />

PH @SW F @Uh<br />

h=1 @Uh @xh 2<br />

x h 2: (5)<br />

The …rst term on the right-hand side captures the social bene…t <strong>of</strong> the project.<br />

For a particular consumer h, the extra consumption <strong>of</strong> good x1 is x h 1. The<br />

extra utility <strong>of</strong> this is found by multiplying x h 1 by the marginal utility <strong>of</strong><br />

good x1 ( @Uh<br />

@x h 1<br />

). The social value <strong>of</strong> this increase in utility for consumer h is<br />

then found by multiplying by the marginal increase in social welfare per unit<br />

@SW F<br />

<strong>of</strong> utility enjoyed by consumer h ( ). Finally, the total social bene…t is<br />

@Uh<br />

found by summing over all consumers. The interpretation <strong>of</strong> the second term<br />

– representing the social opportunity cost – is similar, but you might just go<br />

through the details on your own to make sure you understand what it is.<br />

All consumers maximize their utility subject to given market prices and their<br />

given income, so before the project is introduced, the quantities consumed by<br />

each consumer h is determined by the following two conditions:<br />

@Uh<br />

@x h 1<br />

@Uh<br />

@x h 2<br />

= hq1 (6)<br />

= hq2; (7)<br />

where h is the marginal utility <strong>of</strong> income <strong>of</strong> consumer h. We note two things<br />

here. Consumers consume di¤erent amounts <strong>of</strong> the two goods partly because<br />

they have di¤erent tastes (i.e., their utility functions are di¤erent) and partly<br />

because they have di¤erent incomes (which contributes to making their private<br />

marginal utility <strong>of</strong> income di¤erent). They might also face di¤erent market<br />

prices, but we leave that complication aside here. We can substitute these<br />

optimality conditions into equation (5) to get:<br />

NB = PH @SW F<br />

h=1 @Uh<br />

hq1 x h 1<br />

PH @SW F<br />

h=1 @Uh<br />

hq2 x h 2: (8)<br />

Now, we are in a position to answer the question we set out to answer, i.e.,<br />

when can we ignore the distributional impact <strong>of</strong> a project in our calculation <strong>of</strong><br />

its social value? It is useful to start by specifying some su¢ cient conditions:<br />

3


1. The project has the same impact on the consumption <strong>of</strong> good x1 and x2<br />

for all consumers, i.e.,<br />

x h 1 = x1 and x h 2 = x2 for all h:<br />

2. All consumers have the same private marginal value <strong>of</strong> income, i.e., h =<br />

for all h. This, in turn, is satis…ed if (for example) all consumers have the<br />

same utility functions (tastes) and the same level <strong>of</strong> income. 1<br />

3. The social welfare function values utility increments <strong>of</strong> all consumers in<br />

the same way, i.e.,<br />

@SW F<br />

@Uh<br />

= @SW F<br />

@U<br />

for all h:<br />

This is, for example, satis…ed by the (unweighted) utilitarian social welfare<br />

function where = 1 for all h.<br />

@SW F<br />

@Uh<br />

Under these (su¢ cient) conditions, we can write equation (8) as<br />

@SW F<br />

NB = H<br />

@U<br />

[q1 x1 q2 x2] (9)<br />

and that takes us back to the scenario considered in <strong>Lecture</strong> note 13-14. The<br />

three conditions are su¢ cient, so we can get away with less. If, for example,<br />

we got conditions 2 and 3, but not condition 1 (i.e., the project might deliver<br />

di¤erent amounts to di¤erent individuals or cost them di¤erent amounts in<br />

forgone consumption), we would get<br />

NB =<br />

@SW F<br />

@U<br />

PH [q1 h=1 xh PH 1 q2 h=1 xh2]; (10)<br />

which e¤ectively allow us to add the bene…t and cost increments up and then<br />

value them at market prices. Another possibility is that conditions 1 and 3<br />

holds but not condition 2. This case we get that<br />

NB =<br />

@SW F<br />

@U q1<br />

PH x1 h=1 h<br />

= [q1 x1 q2 x2]<br />

@SW F<br />

@U<br />

@SW F<br />

@U q2 x2<br />

P H<br />

h=1 h<br />

P H<br />

h=1 h: (11)<br />

Whichever <strong>of</strong> these we take as the benchmark, it should be clear now that<br />

we have to assume a lot for it to be valid to use the simply cost-bene…t rule<br />

that the project will increase social welfare if the market value <strong>of</strong> its output is<br />

greater than the market value <strong>of</strong> the inputs. 2 In fact, one could argue that these<br />

1 Other assumptions can also insure that everybody have the same marginal value <strong>of</strong> income.<br />

For example, if consumers have quasi-linear preferences, then they can have di¤erent<br />

preferences over some goods and di¤erent levels <strong>of</strong> income while preserving a common value<br />

<strong>of</strong> the marginal utility <strong>of</strong> income (typically assumed to be one).<br />

2 There is, <strong>of</strong> course, also the issue <strong>of</strong> whether market values exist and whether they represent<br />

the shadow value <strong>of</strong> the outputs and inputs.<br />

4


conditions are never satis…ed 100% and thus one should always take distribution<br />

into account in social cost bene…t analysis. On the other hand, it is also clear<br />

that it, from a practical point <strong>of</strong> view, is a lot more complicated and requires a lot<br />

more information about the impact <strong>of</strong> a project or policy to take distribution into<br />

account in a serious manner and some very eminent economists have argued that<br />

unless a project or a policy is explicitly designed to distribute income, one should<br />

leave distribution to one side and focus on selecting projects and policies that<br />

satisfy the simply e¢ ciency criteria embodied in equation (9). Redistribution<br />

should, then, be left to tax system or to expenditure programmes designed<br />

redistribute.<br />

Arnorld Harberger (<strong>of</strong> the <strong>University</strong> <strong>of</strong> Chicago) is perhaps the most famous<br />

supporter <strong>of</strong> this view. The “Harberger principle” can crudely be summarized<br />

as follows. Suppose that for a given project, we can divide the net bene…ts<br />

into those which have to do with e¢ ciency (i.e., the net bene…ts as they would<br />

be if we were to treat those a¤ected by the project as if there were identical)<br />

and the additional net bene…ts that have to do with distribution (i.e., the extra<br />

net bene…ts we would get when we treat those a¤ected by the project as being<br />

di¤erent in some fundamental way and give more weight to the bene…ts and<br />

costs <strong>of</strong> some than to others). Let the former be denoted NB E and the later be<br />

denoted NB D where superscript E refers to e¢ ciency and superscript D refers<br />

to distribution. The overall net bene…t <strong>of</strong> the project is<br />

NB = NB E + NB D :<br />

The “Harberger principle” (Harberger, 1978) is that one should not accept a<br />

project if NB E < 0 but NB > 0. Clearly, such a project is only justi…ed<br />

because <strong>of</strong> its distributional implications. Public sector projects are, the argument<br />

goes, an ine¢ cient way to achieve distributional ends. They should be<br />

dealt with through more direct tax-transfer programmes and scare resources for<br />

public projects should be concentrated on programmes which can pass a purely<br />

e¢ ciency-based cost-bene…t test. Implicit in this argument is the assumption<br />

that the focus is on public sector investment projects and that the tax-bene…t<br />

system is designed (approximately) to induce an optimal distribution <strong>of</strong> income<br />

prior to the project. For expenditure programmes, e.g. in the area <strong>of</strong> education,<br />

health and welfare, targeted more directly at distribution, the power <strong>of</strong><br />

the Harberger argument is clearly a lot weaker.<br />

3 How should distribution be integrated into<br />

the SCBA?<br />

On balance, there is a case for incorporating distributional considerations into<br />

social cost bene…t analysis, although the case is more compelling for some<br />

projects and policies than for others. How should it be done in practice? Equation<br />

(8) gives us a useful clue: we need to de…ne social welfare weights (or<br />

distributional weights) associated with each individual (or, in practice, with<br />

5


groups <strong>of</strong> individuals, say rich, middle income and poor; or men and women,<br />

etc.). This is, perhaps, most clearly seen by rewriting the equation as<br />

NB = P H<br />

h=1<br />

b<br />

h x h 1<br />

P H<br />

h=1<br />

c<br />

h x h 2; (12)<br />

where the welfare weight given to individual h in the calculation <strong>of</strong> the bene…ts<br />

(superscript b is for bene…t) is<br />

b @SW F @Uh<br />

h = =<br />

@Uh @x1<br />

@SW F<br />

@Uh<br />

hq1: (13)<br />

The weight given to individual h in the calculation <strong>of</strong> the opportunity cost<br />

(superscript c is for cost) is:<br />

c @SW F @Uh<br />

h = =<br />

@Uh @x2<br />

@SW F<br />

@Uh<br />

hq2: (14)<br />

The two weights might di¤er ins<strong>of</strong>ar as the marginal utility <strong>of</strong> the output delivered<br />

by the project is di¤erent from the marginal utility <strong>of</strong> consumption foregone<br />

as a consequence <strong>of</strong> the project. Nonetheless the principle is clear: we need to<br />

de…ne a sensible set <strong>of</strong> welfare weights and then weight the costs and bene…ts<br />

accordingly.<br />

This makes it more complicated to estimate the shadow prices. Recall in the<br />

simple world with one representative consumer, we de…ned the shadow price <strong>of</strong><br />

say the output x1 as the change in social welfare induced by a small change in<br />

project output relative to the marginal value <strong>of</strong> private consumption, i.e.,<br />

@SW F<br />

@x1<br />

1 = @SW F<br />

@U<br />

@U<br />

@x1<br />

1 : (<strong>15</strong>)<br />

When there are many di¤erent consumers, the change is social welfare induced<br />

by the project output is<br />

HX @SW F<br />

h=1<br />

@Uh<br />

@Uh<br />

@x h 1<br />

x h 1: (16)<br />

In the special case, where all consumers bene…t by the same amount from the<br />

project x h 1 = x1, the increase in social welfare induced by a small change<br />

in the (common) project output is P H<br />

PH h=1<br />

h=1<br />

@SW F<br />

@Uh<br />

@Uh<br />

@x h 1<br />

, which we see is equal to<br />

b<br />

h. But in the case <strong>of</strong> heterogenous consumers there is not a single<br />

marginal value <strong>of</strong> private income: there are, in principle, H di¤erent ones; one<br />

for each consumer. So how do we normalize the change in social value to get at<br />

the shadow price? One possibility is to use the average value = 1 PH H h=1 h<br />

and de…ne the shadow price <strong>of</strong> the (common) output as<br />

q SP<br />

1<br />

=<br />

P H<br />

h=1<br />

b<br />

h<br />

(17)<br />

but we could also choose, for example, to use the marginal value <strong>of</strong> money for<br />

a particular consumer, say, consumer 1, 1 as the unit <strong>of</strong> account.<br />

6


Exercise 1 Suppose an analyst decided to ignore distribution and simply use<br />

the market price <strong>of</strong> good 1, q1, as the shadow price and not qSP 1 . Would doing<br />

so over-estimate or under-estimate the social value <strong>of</strong> a unit <strong>of</strong> output from the<br />

project?<br />

As noted in <strong>Lecture</strong> note 13-14, we do not observe the opportunity cost<br />

<strong>of</strong> foregone consumption directly, so, in practice, it is, typically, estimated as<br />

the monetary cost <strong>of</strong> the project or policy programme. For this reason, the<br />

practical starting point for a social cost bene…t analysis that takes distribution<br />

into account is <strong>of</strong>ten an expression such as<br />

NB = P H<br />

h=1<br />

b<br />

h x h 1 C (18)<br />

where C is the monetary cost <strong>of</strong> the project. Given that, there are two broad<br />

approaches to incorporating distribution into social cost bene…t analysis that<br />

are widely used in practice: the "adjusted social weights approach" and the<br />

"marginal cost <strong>of</strong> public funds approach". We shall discuss the two in detail<br />

below, but …rst a little aside that you can jump if you are familiar with the<br />

concept <strong>of</strong> inequality aversion. It plays a key role in what follows, so make sure<br />

you know what it is.<br />

3.1 Aside: Inequality aversion<br />

An important consideration in calculating social welfare weights is the degree<br />

<strong>of</strong> inequality aversion embodied in the social welfare function. We make a<br />

distinction between inequality aversion de…ned over utility allocations (call this<br />

utility inequality aversion) and inequality aversion de…ned over consumption<br />

allocations (call this consumption inequality aversion). In practice, they <strong>of</strong>ten<br />

get confounded but conceptually they are di¤erent things. Let us begin by<br />

de…ne inequality aversion in general using a function g with constant relative<br />

inequality aversion:<br />

g(y) =<br />

a<br />

y1<br />

: (19)<br />

1 a<br />

We de…ne the degree <strong>of</strong> inequality aversion (with respect to the variable y) as<br />

a =<br />

@ 2 g<br />

@y2 y: (20)<br />

@g<br />

@y<br />

The parameter a controls the degree <strong>of</strong> inequality aversion exhibited by the<br />

function g. The higher is a, the greater the degree <strong>of</strong> aversion. To see the<br />

intuition, let us think <strong>of</strong> y as income and g as a utility function. If g is a<br />

linear function <strong>of</strong> income y, then a = 0 and the marginal utility is the same<br />

irrespective <strong>of</strong> the level <strong>of</strong> income, i.e., rich and poor will get the same amount<br />

<strong>of</strong> utility out <strong>of</strong> an extra unit <strong>of</strong> income. If a > 0, marginal utility is @g<br />

@y = y a .<br />

This is falling with income, so the marginal utility that a rich person gets from<br />

an extra unit <strong>of</strong> y is lower than the marginal utility that a poor person gets<br />

7


from an extra unit <strong>of</strong> income. How much lower depends on the curvature <strong>of</strong><br />

the function g, i.e., how fast marginal utility falls with income. It would seem<br />

reasonable to say that a function g exhibits more inequality aversion if marginal<br />

utility <strong>of</strong> income falls fast with income than when it falls slowly. After all, in<br />

the later case, it does not really matter at the margin if there are big income<br />

inequalities or not, while in the former case it does matter a great deal. The<br />

speed at which marginal utility falls is controlled by the second derivative <strong>of</strong> the<br />

function, i.e., by @2 g<br />

@y 2 = ay a 1 . We could, <strong>of</strong> course, use the (absolute) size <strong>of</strong><br />

the derivative as our measure <strong>of</strong> inequality aversion, but such a measure would<br />

not be invariant to a positive monotonic transformation <strong>of</strong> the utility function.<br />

So it is better to normalize by the marginal utility to get a measure <strong>of</strong> inequality<br />

aversion that is invariant to such transformations. This motivates the de…nition<br />

given in equation (20). 3 With a function such a g that got constant relative<br />

inequality aversion, inequality aversion is captured by one parameter a and the<br />

larger a is, the more inequality aversion there is.<br />

Now, let us get back to the distinction between consumption and utility<br />

inequality aversion. This is best illustrated by considering three di¤erent cases.<br />

1. Case 1: Utilitarian social welfare function and individual utility functions<br />

with constant relative inequality aversion. A utilitarian social welfare<br />

function is linear in the utility <strong>of</strong> each individual, so it exhibits no utility<br />

inequality aversion: the utility <strong>of</strong> one individual is a perfect substitute for<br />

that <strong>of</strong> another and the marginal social welfare <strong>of</strong> an increase in the utility<br />

<strong>of</strong> some individual is the same as for any other individual (and typically<br />

normalized to be equal to 1). If the individual utility function is <strong>of</strong> the<br />

type<br />

Uh = xh 1<br />

1<br />

1<br />

(21)<br />

then it exhibits consumption inequality aversion (and more so, the bigger<br />

is 0). This consumption inequality aversion is inherited by social<br />

welfare function, which in this case, then, exhibits consumption inequality<br />

aversion but not utility inequality aversion. We can write the social welfare<br />

function as<br />

HX HX<br />

SW F = Uh =<br />

xh 1<br />

1<br />

1<br />

: (22)<br />

h=1<br />

2. Case 2: Individual utility functions are linear in consumption (Uh = x h 1)<br />

and the social welfare function is <strong>of</strong> the following type:<br />

SW F =<br />

h=1<br />

HX (Uh) 1<br />

h=1<br />

1<br />

: (23)<br />

3 If you are familiar with the de…nition <strong>of</strong> risk aversion, you will see the similarity to that<br />

concept.<br />

8


In this case, the social welfare function exhibits utility inequality aversion.<br />

Although, individual utility functions do not exhibit consumption<br />

inequality aversion, the social welfare function does as we can write it as<br />

SW F =<br />

HX<br />

h=1<br />

x h 1<br />

1<br />

1<br />

: (24)<br />

This illustrates that if we pick = , the welfare weights generated by<br />

case 1 and 2 are going to be the same. That is, for case 1, we get<br />

and, for case 2, we get<br />

b @SW F @Uh<br />

h = = 1 x<br />

@Uh @x1<br />

h 1<br />

(25)<br />

b @SW F @Uh<br />

h =<br />

@Uh @x1<br />

= (Uh) 1 = x h 1 : (26)<br />

These are the same for = . Importantly, however, and are not conceptually<br />

the same object. The parameter is an attribute <strong>of</strong> individual<br />

preferences (and can, in principle, be estimated from observed consumption<br />

choices). Technically, is the elasticity <strong>of</strong> marginal utility with respect<br />

to consumption. The parameter represents an ethical choice that<br />

we make when we specify the social welfare function. Often we accept the<br />

utilitarian principle that we should just sum the utilities <strong>of</strong> individuals.<br />

When we do, this implies that we choose to set<br />

utility inequality aversion.<br />

= 1 and not allow for<br />

3. Case 3: Individual utility functions are<br />

and the social welfare function is<br />

Uh = xh 1<br />

SW F =<br />

1<br />

1<br />

HX (Uh) 1<br />

h=1<br />

1<br />

(27)<br />

: (28)<br />

So in this case, the individual utility functions exhibit consumption inequality<br />

aversion and the social welfare function exhibits utility inequality<br />

aversion. However, if we substitute the individual utility functions into<br />

the SWF, we can rewrite it as<br />

SW F = (1 )<br />

HX<br />

h=1<br />

x h 1<br />

1 "<br />

1 "<br />

; (29)<br />

where " = (1 ) + . This is very similar to the implied SWF from<br />

case 1 and 2. But as noted above, we now have to be careful with the<br />

choice <strong>of</strong> ": it is a combination <strong>of</strong> the characteristics <strong>of</strong> individual utility<br />

functions and the ethical choice embodied in the selection <strong>of</strong> the social<br />

welfare function.<br />

9


3.2 The adjusted social weights approach<br />

This approach is used in the UK and forms part <strong>of</strong> the Green Book 4 (the government’s<br />

manual for how to conduct social cost bene…t analysis). It uses a<br />

particular normalization to calculate the social welfare weights attached to the<br />

bene…ts <strong>of</strong> a project or a policy proposal. The approach, as well as the general<br />

pitfalls associated with the task <strong>of</strong> de…ning social welfare weights in a sensible<br />

way, can be illustrated through an example. 5 The example is based on the<br />

following assumptions:<br />

1. We consider a society with four individuals, h = 1; 2; 3; 4.<br />

2. The individuals’utility functions are Uh = log(x h 1). This is a special case<br />

<strong>of</strong> the utility function with constant inequality aversion considered above<br />

(with = 1).<br />

3. The pre-project consumption levels <strong>of</strong> the four individuals are 6<br />

x 1 1 = x 2 1 = $1000 per month<br />

x 3 1 = x 4 1 = $2000 per month<br />

4. The social welfare function is utilitarian, i.e., SW F + = P 4<br />

h=1 Uh, where<br />

we use the superscript + to indicate that this is net net <strong>of</strong> the monetary<br />

cost <strong>of</strong> the project.<br />

5. The (expected) impact <strong>of</strong> the project under consideration is to increase<br />

consumption <strong>of</strong> good x1 for each <strong>of</strong> the four individuals. We denote the<br />

extra consumption <strong>of</strong> individual h by x h 1 > 0.<br />

6. The monetary cost <strong>of</strong> the project is $C.<br />

Given this information, we want to evaluate if the project is socially desirable<br />

or not, i.e., we want to know if the net social bene…t<br />

NB =<br />

4X @SW F +<br />

h=1<br />

@Uh<br />

@Uh<br />

@x h 1<br />

is positive. Using the assumptions that individuals’utility functions are logarithmic<br />

and the SWF is utilitarian, we can write this as<br />

NB =<br />

4X<br />

x<br />

h=1<br />

h 1<br />

4 See Chapter 5 <strong>of</strong> the Green Book.<br />

5 I am greatful to Dr Tom Crossley for providing this example.<br />

6 We assume that the price <strong>of</strong> good 1 is una¤ected by the project. This allows us to measure<br />

pre- and post-project consumption levels in pounds, but we could as well have speci…ed the<br />

consumption levels in physical units.<br />

1<br />

10<br />

x h 1<br />

x h 1<br />

C;<br />

C


where the social weight attached to the bene…ts <strong>of</strong> individual h is simply h =<br />

. Since marginal utility is falling with the level <strong>of</strong> consumption, we put<br />

1<br />

x h 1<br />

less weight on individuals 3 and 4 (1=2000) who consume a lot <strong>of</strong> the project<br />

output to begin with and higher weight on individuals 1 and 2 (1=1000) who<br />

consume less <strong>of</strong> the project output initially. Table 1 shows what happens when<br />

we apply these weights to a project that takes $1 <strong>of</strong> uncommitted public funds<br />

and distribute it equally across the four individuals (through a non-distortionary<br />

lump sum transfer). The last column shows the welfare increments ( 1<br />

xh x<br />

1<br />

h 1) for<br />

each consumer. We see that the sum <strong>of</strong> these bene…ts adds up to a very small<br />

amount, namely 0:00075. This is a lot less than the $1 cost <strong>of</strong> the project, so it<br />

should be rejected. But this does not seem to make any sense. We got a pound<br />

<strong>of</strong> uncommitted funds, how can dividing it equally amongst individuals be such<br />

a bad idea?<br />

The issue is that we have (without saying it out load..) chosen to use public<br />

funds as the numeraire, i.e., as the unit in which we measure the cost <strong>of</strong> the<br />

project ($1 <strong>of</strong> public funds is worth $1). The bene…ts, however, 1<br />

xh x<br />

1<br />

h 1, are<br />

measured in units <strong>of</strong> utility. So the problem is that we are not comparing like<br />

with like. More fundamentally, the problem is that the choice <strong>of</strong> the utility functions<br />

must in practice be understood as the choice <strong>of</strong> ordinal utility functions.<br />

This means that we can subject it to any positive transformation and it will<br />

still represent the same underlying preferences over consumption. To see what<br />

this implies more clearly, let us transform the utility function and represent the<br />

individuals’preferences by the function e Uh = 10000 log(xh 1). Table 2 shows the<br />

re-calculated bene…ts <strong>of</strong> the divided-a-pound-equally project. Now, they add up<br />

to 7:5 and we conclude that the project is a good idea after all.<br />

Exercise 2 Does this logic strengthen or weaken Harberger’s case for not applying<br />

welfare weights at all?<br />

Clearly, to reach a sensible decision and avoid leaving the analysis wide open<br />

to manipulation by special interests, we need a principle that can tell us how<br />

to normalize or anchor the social welfare weights so that the costs and bene…ts<br />

become comparable.<br />

The Green Book suggests the following normalization. Begin by de…ning as<br />

the benchmark the social value <strong>of</strong> a uniform lump sum transfer, i.e., the social<br />

value <strong>of</strong> dividing $1 <strong>of</strong> uncommitted public funds equally amongst all citizens.<br />

This benchmark de…nes the opportunity cost <strong>of</strong> spending a pound on some other<br />

project or programme. The logic is that any project or programme would have<br />

to be better than simply spending the funds on a uniform lump sum transfer in<br />

order to be socially justi…ed.<br />

How does one implement this principle? What we need to do is to normalize<br />

the welfare weights <strong>of</strong> the four individuals in such a way that a project that<br />

divides $1 equally amongst them breaks even, i.e., yields a social bene…t equal<br />

to the cost <strong>of</strong> $1. Formally, we want the normalized social weights –denoted<br />

11


y b h –to satisfy the following equation:<br />

4X<br />

h=1<br />

b<br />

1<br />

h<br />

4<br />

1 = 0 (30)<br />

where " 1<br />

4 " is the extra consumption given to each consumer and " 1" is the<br />

monetary cost <strong>of</strong> this policy. This equation implies that P 4<br />

h=1 b h = 4 (or more<br />

generally the number <strong>of</strong> individuals in society, H). The next step is to scale the<br />

to make them sum to 4. The scaling factor –<br />

sometime called the reference consumption level –can be de…ned as<br />

original social weights h = 1<br />

x h 1<br />

bc =<br />

4<br />

P4 =<br />

h=1 h<br />

4<br />

6=2000<br />

4000<br />

= ; (31)<br />

3<br />

where we use the original weights associated with Uh = log(x h 1). The normalized<br />

social weights (using this reference consumption level) are then calculated as<br />

b h = hbc: (32)<br />

Table 3 shows the normalized weights. Given these weights, we can now evaluate<br />

di¤erent projects. If a project yields positive net (social) bene…ts at these<br />

weights, the project dominates from a social point <strong>of</strong> view the next-best alternative<br />

<strong>of</strong> allocating the funds lump sum to individuals.<br />

Exercise 3 Verify that at the normalized social weights a project that divides<br />

1£ equally amongst all individuals breaks even.<br />

Exercise 4 Consider a project that allocates 1£ to consumers 1 and 2 and<br />

nothing to consumers 3 and 4. Does that pass the cost bene…t test at the normalized<br />

social weights? What about a project that allocates £ 1 to consumers 3<br />

and 4 and nothing to the other two? Would you reach the same conclusion if<br />

the assumed utility function was Uh = 20 log(x h 1) + 30 instead <strong>of</strong> Uh = log(x h 1)?<br />

The proposed normalization – the "Green Book" normalization – is obviously<br />

just one possibility. Many others could be considered. However, it is<br />

based on a fairly strong rationale. Clearly, one way to dispose <strong>of</strong> a 1£ <strong>of</strong> uncommitted<br />

funds is to give it back to individuals in a lump sum fashion. So in<br />

that sense, it does de…ne a feasible benchmark, but one could also give the 1£<br />

back as cut in the income tax or spend it on policing or something else. The<br />

reason for using "divided-a-pound-equally" as the benchmark to normalize the<br />

distributional weights is an argument that many actual tax systems, including<br />

the UK system, can be approximated fairly well by a system where individuals<br />

get the same universal bene…ts and pay into the system roughly in proportion to<br />

their incomes, i.e., that the tax liabilities <strong>of</strong> an individual can be approximated<br />

by T h = B + tm h where m h is the total income <strong>of</strong> individual h, B0 is the<br />

(monetary) value <strong>of</strong> the universal bene…t, and t is the (average) tax rate per<br />

unit <strong>of</strong> income.<br />

12


Exercise 5 Illustrate graphically what determines the amount <strong>of</strong> redistribution<br />

in a tax system like this.<br />

Against this background, one can then argue that most redistribution takes<br />

place through public expenditure programmes (the "B" in the equation for the<br />

tax payment) rather than through the tax system as such. "B" includes spending<br />

on welfare, health, education etc. Given that, one can go on to argue that<br />

the next-best use <strong>of</strong> uncommitted public funds is to spend them on a universal<br />

bene…t since this is how most redistribution happens in practice. Accordingly, if<br />

one wants to make a case for spending uncommitted funds di¤erently, one should<br />

demonstrate that the alternative use is better than this alternative. While this<br />

argument got merit, it can be challenged.<br />

As noted above the need for the normalization arose because the monetary<br />

value <strong>of</strong> uncommitted public funds were chosen as the numeraire, i.e., as the<br />

unit in which we measure the cost <strong>of</strong> the project ($1 <strong>of</strong> public funds is worth<br />

$1). Recall from our previous discussion that the theoretically correct way to<br />

measure the opportunity cost <strong>of</strong> a project or policy is to record the social value<br />

<strong>of</strong> all the goods that are being displaced by the fact that resources are drawn<br />

into the project. In equation (5) this is represented by the term<br />

PH @SW F @Uh<br />

h=1 @Uh @xh 2<br />

x h 2: (33)<br />

Working out what x h 2 is in practise is <strong>of</strong>ten impossible and so one might<br />

wonder if the direct inputs to the project could not be used instead as a proxy<br />

–along the lines <strong>of</strong> what we did in <strong>Lecture</strong> note 13-14. For many public sector<br />

programmes aimed at redistribution or insurance, it is very di¢ cult to work<br />

out exactly how much more labour, capital and so is attributed to the project<br />

directly and in any case this would hardly capture the full impact <strong>of</strong> the project<br />

on the wider economy and it would not tell us not to add up these costs (i.e.,<br />

what the distributional weights on the cost should be). Adding all this up, it is<br />

clear that for a large class <strong>of</strong> public policies the only practical way to measure<br />

the "opportunity cost" <strong>of</strong> the policy is to start with an estimate <strong>of</strong> its monetary<br />

cost and the analyst is therefore <strong>of</strong>ten forced by practical considerations to use<br />

public funds as the numeraire. In those cases, a normalization is required to<br />

make the bene…ts and costs comparable.<br />

3.3 The marginal cost <strong>of</strong> public funds approach<br />

The "adjusted social weights approach" is the approach adopted in the UK. In<br />

the USA, where social cost bene…t is more widely applied than here, it is common<br />

to use an alternative approach that we might call "the marginal cost <strong>of</strong> public<br />

funds approach". This approach can be justi…ed in two di¤erent ways. First, it is<br />

<strong>of</strong>ten the case that the public funds that may be devoted to a particular project<br />

or policy must be raised through distortionary taxation. Thus, the next-best<br />

alternative use <strong>of</strong> the funds is not a uniform lump sum transfer rather the nextbest<br />

alternative is not to levy the distortionary taxes needed to raise the funds<br />

13


in the …rst place. This suggests that the …scal implications <strong>of</strong> a project (both<br />

on the cost and bene…t side) should be adjusted for the marginal cost <strong>of</strong> public<br />

funds in order to re‡ect the true social cost or bene…ts <strong>of</strong> the project. This line<br />

<strong>of</strong> reasoning is based on e¢ ciency considerations. Second, the approach is also<br />

occasionally justi…ed by noting that many projects and government programmes<br />

distribute income or resources from the general taxpayer to the speci…c groups<br />

that bene…t from the project or programme. This in undoubtedly true in many<br />

cases: a new school will bene…t those who use it, but may be …nanced from<br />

general revenues; a welfare bene…t programme will bene…t those how qualify<br />

for it, but may be …nanced from general revenues, etc. Given that, one can<br />

argue that di¤erent weights should be given to the bene…ts (the social surplus<br />

created by the programme) than to the net revenue cost <strong>of</strong> the programme or<br />

project, but without going the all the way and applying di¤erent weights to<br />

each individual. That is, all "bene…ciaries" are treated as if they were the same<br />

and all "contributors" are treated as if they were the same, but the two groups<br />

are treated di¤erently. This is the distributional justi…cation for the approach.<br />

To see the logic behind this approach more clearly and to make it clear under<br />

which conditions it is valid, let us return to the example from section 3.2, but<br />

instead <strong>of</strong> specifying the cost <strong>of</strong> the project in monetary terms at the outset, we<br />

now assume that the opportunity cost is foregone consumption <strong>of</strong> some other<br />

good x2 and that the utility function <strong>of</strong> individual h is<br />

Uh = log x h 1 + log x h 2:<br />

The bene…t <strong>of</strong> the project is as before x h 1 for individual h (which we assume<br />

is provided free <strong>of</strong> charge) while the cost is a reduction in consumption <strong>of</strong> good<br />

x2 in the order <strong>of</strong> x h 2. Under these assumptions, we can evaluate the project<br />

by estimating:<br />

NB = SW F +<br />

SW F ;<br />

where (recall that we assume that the social welfare function is utilitarian in<br />

the example)<br />

SW F + =<br />

SW F + =<br />

4X @SW F<br />

h=1<br />

@Uh<br />

4X @SW F<br />

h=1<br />

@Uh<br />

@Uh<br />

@x h 1<br />

@Uh<br />

@x h 2<br />

x h 1 =<br />

x h 2 =<br />

4X<br />

h=1<br />

4X<br />

h=1<br />

@Uh<br />

@x h 1<br />

@Uh<br />

@x h 2<br />

So far, this is invariant to the choice <strong>of</strong> utility function because we are specifying<br />

the costs and bene…ts in the same units. Let us rewrite the bene…t di¤erential<br />

= hq1 to get<br />

( SW F + ) by exploring the utility maximization condition @Uh<br />

@x h 1<br />

SW F + = q1<br />

4X<br />

h=1<br />

x h 1<br />

x h 2:<br />

h x h 1: (34)<br />

The opportunity cost <strong>of</strong> the programme in terms <strong>of</strong> the units <strong>of</strong> good x2 that<br />

must be scari…ed can be linked to the …nancial cost <strong>of</strong> the project, C. Let us<br />

14


suppose that the cost is covered by increasing some tax. 7 For the purpose at<br />

hand, let us, however, assume that the funds are raised through the tax t2 levied<br />

on good x2. The revenue collected per unit increase in this tax rate is denoted<br />

R= t2 and the …nance requirement is that R = C, i.e., the extra revenue<br />

R covers the monetary cost C. Faced with this tax increase, consumer h will<br />

reduce his demand for good x2 by<br />

x h 2 = @xh 2<br />

@t2<br />

t2<br />

(35)<br />

where @xh<br />

2<br />

@t2 is the slope <strong>of</strong> his demand function for good x2. We can use that to<br />

rewrite the cost di¤erential ( SW F ) as follows:<br />

SW F =<br />

4X<br />

h=1<br />

@Uh<br />

@xh @x<br />

2<br />

h 2<br />

@t2<br />

t2: (36)<br />

From the de…nition <strong>of</strong> the …nance requirement, we can write t2 = C<br />

R= t2 and<br />

substitute that into the cost di¤erential:<br />

The term @Uh<br />

@x h 2<br />

SW F =<br />

C<br />

4X<br />

@Uh<br />

R= t2 @x<br />

h=1<br />

h 2<br />

@x h 2<br />

:<br />

@t2<br />

@x h<br />

2 represents, in units <strong>of</strong> utils, the welfare cost for consumer h<br />

@t2<br />

<strong>of</strong> the necessary tax increase. We can de…ne the associated welfare cost in units<br />

<strong>of</strong> income by dividing through with the private marginal cost <strong>of</strong> income h, i.e.,<br />

Uh<br />

t2<br />

1<br />

h<br />

@Uh<br />

@x h 2<br />

@x h 2<br />

;<br />

@t2<br />

Uh where represents the welfare cost <strong>of</strong> the tax for consumer h in units <strong>of</strong><br />

t2<br />

private income. Now, let us substitute this into the cost di¤erential:<br />

SW F =<br />

C<br />

4X<br />

R= t2<br />

h=1<br />

The term P4 h=1 h Uh represents the social (utility cost) <strong>of</strong> the tax increase<br />

t2<br />

needed to …nance the project and we could write this more compactly be denoting<br />

it simply by SW F= t2 = P4 h=1 h Uh . This allows us to write the cost<br />

t2<br />

di¤erential as<br />

SW F =<br />

h<br />

Uh<br />

t2<br />

:<br />

SW F= t2<br />

C: (37)<br />

R= t2<br />

7 It could be many taxes, in fact, if the tax-expenditure system were designed optimally<br />

and in a joint-up way, then we know from the discussion <strong>of</strong> the Ramsey tax problem, that it<br />

would be optimal to adjust all taxes in order to raise the funds at the lowest possible social<br />

cost.<br />

<strong>15</strong>


This is simply telling us that the utility cost <strong>of</strong> the project is that C = R<br />

units <strong>of</strong> private income have to be given up. Since this happens by levying<br />

the distortionary tax t2, the social welfare cost per unit <strong>of</strong> revenue collected is<br />

SW F= t2 SW F<br />

R= = t2 R . Let us now link this to the marginal cost <strong>of</strong> public funds<br />

(or what Boardman et al. calls the marginal excess burden <strong>of</strong> taxation). The<br />

(social) marginal cost <strong>of</strong> public funds (SMCF) is de…ned as<br />

SMCF =<br />

1 SW F= t2<br />

R= t2<br />

(38)<br />

where = 1 P4 4 h=1 h is the average marginal utility <strong>of</strong> income for the (entire)<br />

population (in our example the four individuals). We divide by this parameter in<br />

the de…nition in order to convert the welfare cost measured in utils into a welfare<br />

cost in units <strong>of</strong> income. With this in mind, we can do the …nal substitution into<br />

the cost di¤erential:<br />

SW F = (SMCF ) C =<br />

(SMCF ) C<br />

4<br />

Let us now make the critical assumption: assume that the marginal utility <strong>of</strong><br />

income is the same for all consumers (i.e., h = for all h). This allows us to<br />

write the cost di¤erential as<br />

and the bene…t di¤erential as<br />

The net bene…t is<br />

NB = q1<br />

4X<br />

h=1<br />

SW F = (SMCF ) C (39)<br />

SW F + =<br />

4X<br />

h=1<br />

4X<br />

q1 x h 1: (40)<br />

h=1<br />

x h 1 (SMCF ) C: (41)<br />

We can link the social marginal cost <strong>of</strong> public funds (SMCF ) to the marginal<br />

excess tax burden as follows:<br />

SMCF = 1 + MET B; (42)<br />

which then gives us the formula that Boardman et al. recommends in chapter 4.<br />

In short, this tells us that we should adjust the total …nancial cost <strong>of</strong> the project<br />

using an estimate <strong>of</strong> the marginal cost <strong>of</strong> public funds and, more generally, that<br />

we should adjust all project e¤ects on government revenues and costs by the<br />

factor 1 + MET B. This method is widely used in the US. It has intuitive<br />

appeal and is easy to apply in practice. However, the main point <strong>of</strong> going<br />

through the derivations above was to make it clear that it is based on two<br />

critical assumptions. The …rst <strong>of</strong> these we already discussed: namely that the<br />

16<br />

h:


marginal value <strong>of</strong> income is the same for everyone. The second, which we did not<br />

stress in calculations because we assumed a utilitarian social welfare function<br />

from the beginning, is that all individuals must have the same weight in social<br />

welfare function (which is true for the utilitarian social welfare function).<br />

Both <strong>of</strong> these assumptions seriously undermine the case for redistribution in<br />

general. If all individuals have the same private marginal value <strong>of</strong> income, then<br />

were the case, then the main reason for any government to get involved with<br />

distribution is void (remember that one key reason a social planner would want<br />

to distributive income is precisely that the marginal value <strong>of</strong> income is di¤erent<br />

for di¤erent people). If on top <strong>of</strong> that the social welfare function puts equal<br />

weight on everyone, then there is not case for distribution left. If is as if the<br />

government thinks that everyone is the same. This makes it clear, on the one<br />

hand, that the marginal cost <strong>of</strong> public funds approach is really about e¢ ciency:<br />

it provides a systematic way to integrate into the SCBA the fact that public<br />

funds must be raised through distortionary taxation and in situations where<br />

the control area <strong>of</strong> the analyst is such that the broader tax system must simply<br />

be taken as given, it makes sense to adjust the …scal implications <strong>of</strong> a project<br />

or programme for the e¢ ciency cost <strong>of</strong> raising the extra revenue with the preexisting<br />

tax instruments (or if the project generates public revenue to take into<br />

account that this will then reduce the pressure on other sources <strong>of</strong> revenue). On<br />

the other hand, the two assumption required to make the approach theoretically<br />

valid clearly make it hard to justify the approach as a way to take distribution<br />

into account. At the surface is may seem intuitive that projects that distribute<br />

from the general taxpayer who must pay higher taxes as a consequence <strong>of</strong> a<br />

project to the groups <strong>of</strong> individuals that bene…ts from the project (who typically<br />

constitute a small subset <strong>of</strong> all taxpayers). But if we think (assume) that the<br />

marginal value <strong>of</strong> income is the same for everyone, then we should not be very<br />

concerned with distribution in the …rst place and we could as well raise the<br />

required funds using a poll tax. This would avoid any distortions but then<br />

there is no need to make any adjustment in the …rst place.<br />

In practice, however, it is clear that poll taxes are not being levied on a<br />

large scale (and the experience in the UK in the 1980s suggests that they may<br />

not be entirely non-distortionary if we count the social cost <strong>of</strong> the tax rebellion<br />

that was triggered). Accordingly, many cost-bene…t analysts take the view that<br />

adjusting the e¤ects on the governments budget <strong>of</strong> projects for the marginal<br />

cost <strong>of</strong> public funds is a sensible way to proceed in particular when the analyst<br />

is not able to re-design the entire tax system and thus must take whatever taxes<br />

are levied as given. So, if we take that line, we are only left with one question:<br />

which value should be using for the adjustment. Boardman suggests that the<br />

Marginal Excess Tax Burden per unit <strong>of</strong> revenue raised is between 0:33 and 0:46<br />

for all taxes taking together, from 0:11 to 0:39 for the sales tax and from 0:31 to<br />

0:65 for the income tax. These estimates apply to the USA; for other countries,<br />

see Klever and Kreiner (2006). They estimate that the marginal cost <strong>of</strong> public<br />

funds for the UK is about 1:26.<br />

Exercise 6 Can the fact that the marginal excess tax burden di¤ers by tax base<br />

17


e consistent with an optimally designed tax system?<br />

Exercise 7 This exercise shows that the marginal cost <strong>of</strong> funds approach can<br />

sometimes work even if the marginal value <strong>of</strong> private income is not the same<br />

for everyone (that assumption is su¢ cient for it to work). Suppose that h<br />

is di¤erent for the four consumers, but that (1) they all contribute to the cost<br />

<strong>of</strong> the project and (2) they all bene…t equally from the project ( xh 1 = x1).<br />

Show that under those conditions, it is true that P4 h=1 q1 x1 (SMCF ) C > 0<br />

implies that social welfare is increasing. Would this also be true if the project<br />

only bene…ts consumer 1, i.e., x1 1 > 0 and xh 1 = 0 for h = 2; 3; 4?<br />

4 Conclusion<br />

As a matter <strong>of</strong> principle, public projects and policy programme do have distributional<br />

impacts because they a¤ect individuals di¤erently and because di¤erent<br />

individuals value the bene…ts and costs di¤erently. From a practical point<br />

<strong>of</strong> view, short-cut <strong>of</strong>ten have to be taken and the distinction between project<br />

which got limited distributional impact and projects with a clear and signi…cant<br />

distributional impact is a useful starting point. For projects with a clear distributional<br />

impact, say an in-work bene…t programme, the adjusted social weights<br />

approach provides a transparent set <strong>of</strong> rules to apply to reach a sensible decision.<br />

For projects without a clear distributional impact, say procurement <strong>of</strong> a new<br />

…ghter jet for the army, it may be reasonable to proceed as if everyone is just<br />

the same and focus on selecting projects that pass the e¢ ciency test proposed<br />

by Harberger and it may also make sense in doing so to take into account that<br />

the marginal cost <strong>of</strong> public funds is greater than one and adjust the …scal ‡ows<br />

accordingly. In doing so, however, one is not really dealing with distributional<br />

issues in a serious way.<br />

5 <strong>Note</strong>s on the literature<br />

There is a vast literature on social cost bene…t analysis. To navigate it, I<br />

recommend the following core readings. It is useful to stick to one textbook<br />

and I recommend Boardman et al. (2006, 2011). Boardman (2011) chapter 4<br />

and 19 (not 18 as indicated in the course outline) contain an introductionary<br />

discussion <strong>of</strong> the marginal cost <strong>of</strong> public funds approach and a little on how to<br />

use distributional weights mostly from a practical point <strong>of</strong> view. Pearce and<br />

Nash (1981) got some useful material in chapter 3 on distributional weights.<br />

Chapter 5 <strong>of</strong> the Green Book explains how the UK government deals with<br />

distributional weighting. You may …nd it useful to read the lecture note before<br />

engaging with these readings. The note is intended to provide you with the<br />

theoretical background needed to understand and evaluate the exposition by<br />

Boardman, Pearce and Nash and the material in the Green Book.<br />

Further readings, for those who want a deeper understanding <strong>of</strong> the material,<br />

the collection <strong>of</strong> articles, including the introduction chapter, in Layard and<br />

18


Glaister (1994) are highly recommended, but they assume that you are familiar<br />

with the basic ideas.<br />

References<br />

[1] Boardman, et al. (2011). <strong>Cost</strong>-Bene…t <strong>Analysis</strong>, concepts and practice. Pearson<br />

International Edition, (4th edition).<br />

[2] HMT 2003, The Green Book: Appraisal and Evaluation in Central Government<br />

("The Green Book") available for downloading at http://www.hmtreasury.gov.uk/media/3/F/green_book_260907.pdf<br />

[3] Pearce, D.W., and C.A. Nash, 1981. The <strong>Social</strong> Appraisal <strong>of</strong> Projects: A<br />

Text in <strong>Cost</strong>-Bene…t <strong>Analysis</strong>. MacMillan.<br />

[4] Kleven, Henrik Jacobsen and Claus Thustrup Kreiner, 2006. The marginal<br />

cost <strong>of</strong> public funds: Hours <strong>of</strong> work versus labor force participation. Journal<br />

<strong>of</strong> Public Economics 90 (10-11), 1955-1973.<br />

[5] Layard, R. and S. Glaister, 1994. <strong>Cost</strong>-Bene…t <strong>Analysis</strong>, Cambridge. 2nd<br />

edition.<br />

[6] Harberger, A.C. 1978. On the use <strong>of</strong> distributional weights in social costbene…t<br />

analysis. Journal <strong>of</strong> Political Economy 86(2), S87-S120.<br />

19


Table 1: Dividing a Pound between the four consumers: Take one.<br />

Individual<br />

1<br />

2<br />

3<br />

4<br />

Individual<br />

1<br />

2<br />

3<br />

4<br />

Extra consumption<br />

0.25<br />

0.25<br />

0.25<br />

0.25<br />

Extra consumption<br />

0.25<br />

0.25<br />

0.25<br />

0.25<br />

Welfare weight<br />

1/1000<br />

1/1000<br />

1/2000<br />

1/2000<br />

Sum: 6/2000<br />

Welfare weight<br />

10<br />

10<br />

5<br />

5<br />

Sum: 30<br />

Incremental Welfare<br />

1/4000<br />

1/4000<br />

1/8000<br />

1/8000<br />

Sum: 6/8000<br />

=0.00075<br />

Table 2: Dividing a Pound between the four consumers: Take two.<br />

Incremental Welfare<br />

2.5<br />

2.5<br />

1.25<br />

1.25<br />

Sum: 7.5


Table 3: The re-scaled social welfare weights<br />

Individual<br />

1<br />

2<br />

3<br />

4<br />

Consumption<br />

1000<br />

1000<br />

2000<br />

2000<br />

Original weight<br />

1/1000<br />

1/1000<br />

1/2000<br />

1/2000<br />

Scale factor<br />

4000/3<br />

4000/3<br />

4000/3<br />

4000/3<br />

Scaled weight<br />

4/3<br />

4/3<br />

2/3<br />

2/3

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