Lecture Note 15: Social Cost Benefit Analysis - University of ...
Lecture Note 15: Social Cost Benefit Analysis - University of ...
Lecture Note 15: Social Cost Benefit Analysis - University of ...
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<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