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Social discount rates
Journal
of Public Economics
1 (1972)
359-378.
0 North-Holland
SOCIAL DISCOUNT
Publishing
Company
RATES*
J.A. KAY St. John’s College, Oxford, England First version
received
February
1972, revised version received
July 1972
1. Introduction We live in an economy which rnay or may not be on an optimum growth path, but probably is not. In that economy, it is necessary for government agencies to undertake the evaluation of proposed investment projects, and to choose some discount rate for the purpose. Asked for guidance, the government indicates that it discounts future consumption at a rate p, and that its empirical studies indicate that the return obtained on capital invested in the public sector is Y, with r > p. The object of this paper is to consider what advice an economist might give on the basis of this rather inadequate and unsatisfactory information. If a(t) represents the stream of costs and benefits which the project generates over time, we seek some rate q such that the project should be adopted if and only if li a(t)e- 4 t dt 2 0. The information is sufficient for the analysis of two polar cases. If a project displaces only private consumption, it should be required to return p, the social time preference (STP) rate: q is equal to p. If it affects,only private investment, it may be expected to provide a return at least as great as r, the social opportunity cost (SOC) rate: q is equal to Y. In general, however, the truth will lie somewhere between these extremes: and it is natural to argue that an appropriate criterion will similarly be somewhere be-
* Earlier versions of this paper were given to a seminar in Oxford and to the European Winter Symposium of the Econometric Society in 1972. I am grateful for comments received, particularly those of A.B. Atkinson, J.H. D&e, J.S. Flemming, J.A. Mirrlees, G.B. Richardson and the referees.
360
J.A. Kay, Social discount
rates
tween the two extremes of the restrictive SOC rate and the permissive STP rate. It has been suggested (Sandmo and D&e, 1971) that a weighted average of the two is appropriate. This paper takes issue with that view. It follows an argument of Arrow’s (1966) which suggests that even when the effects on private investment are taken into account, the appropriate discount rate remains p. Arrow’s argument may be paraphrased as follows. If a project requires & 1 at time T, and this & 1 is withdrawn from private disposable income, the ultimate result of the repercussions, immediate and long term, on consumption and on investment, will be a stream of changes in consumption. Discounted to T, , this stream is worth w say. The project returns & 1 at some other time T2. This sets up a similar stream of changes in the opposite direction. Discounted to T2, this stream is also worth w. Hence the effect of the complications of investment and reinvestment is to multiply the value of the costs and benefits of the project by w. These complications therefore alter the magnitude of the net benefits of the project, but do not affect their sign. Thus ji ceePf dt is simply w Jomae- P tdt. If the project has positive present value at p, we should adopt it. If w is constant, then, 4 is equal to p and the STP rate is the appropriate discount rate for public investment. This is demonstrated in section 4, in which the assumptions implied by Arrow’s supposition of constant w are made explicit. Divergence between 4 and p can occur if, and only if, these assumptions are violated. Section 5 therefore considers the possibility that the multiplier w is variable because the costs and benefits of projects enter private income in different ways. Section 6 deals with the case where no finite w exists. Section 7 explores the impact of variations in w due to changes in savings rates or rates of return on private investment. These possibilities do suggest that minor deviations from the path that Arrow blazed through the tangled jungle of complex general criteria may sometimes be necessary. But they do not, I think, seriously weaken the proposition that this is the path which economists, and following them governments, should take. The STP rate p remains a central estimate of 4: the SOC rate r is a parameter of interest only for the refined analysis of marginal cases. This conclusion differs sharply from that reached by other writers in this field, and sections 8- 10 give some consideration to the reasons. But first sections 2 and 3 outline the model and some preliminary argument.
J.A. Kay, Social discount rates
361
2. The model I assume a closed economy in which there is a homogeneous private capital stock, with a single instantaneous rate of return r(t). The costs and benefits of public projects are independent of the size of the private capital stock. (This assumption is unnecessary if benefit streams are valued on the basis of the private capital stock which would actually exist if the project were undertaken.) All potential output is fully utilised in consumption, private investment or government activity. As a consequence of the project, private sector income at time I is increased by v(t): private consumption by c(t): and the private capital stock by k(t). Let a(t) be the value of the benefits (project expenses are regarded throughout as negative benefits) accruing at time t. Hence y(t) =a(t) + r(t) k(t) = c(t) + k(t). Define s(r) = g From national
.
income identities,
v(t) = c(f) + S&t) + S,(t) , where S,(t) is private saving and s,(t) government saving. At this stage, no hypothesis is made or implied concerning the determinants of savings. s(t) is a parameter, defined for all t, which may be observed ex post. The approach meshes most easily with a Keynesian theory of savings behaviour, where the marginal propensity to save is a familiar parameter: but its applicability is not confined to this theory, and the effect on the model of alternative assumptions about savings can be viewed in terms of the effect of these assumptions on s(t). If the government adjusts its spending and receipts so that whatever government expenditures the project requires are tax financed, presumably using monetary policy if necessary to maintain full employment, then new government saving is zero and s(t) can be regarded as the private sector savings rate. If, however, alternative policy mixes which include unbalanced changes in the budget are employed, the savings which necessarily match changes in private investment may be done by the government as well as by the private sector, and s(t) is then a rate which includes government as well as private saving. Whether this s(t) is
362
J.A. Kay, Social discount rates
greater or less than s,(t), the private sector savings rate, depends on whether an expansionary fiscal policy does more to stimulate private investment than private savings or vice versa. I shall not make judgement on this point. In what follows, it will be tempting to regard s(t) as referring to purely private saving; the caveat entered here should be borne in mind.
3. The nature of the benefits
of public sector projects
Three kinds of benefits from public projects can be distinguished. They are: (i) returns to the public sector: cost savings, sales receipts, initial expenses; (ii) pecuniary returns to the private sector: uncompensated costs and cost savings resulting from the project, e.g. fuel savings and savings in working time as a result of a road improvement; (iii) non-pecuniary returns to the private sector - leisure time savings, noise, pollution, etc. The distinction between (i) and (ii) is natural but unimportant. Resources generated in the public sector are of value only to the extent that they are reinvested in the public sector or transferred to the private sector. For the moment, exclude the possibility of public sector reinvestment: I shall later show that introducing this makes no difference to the results. Then the value of type (i) benefits is, in effect, the value of the type (ii) benefits which they induce. This implies that if resources are generated in the public sector as a result of the returns from public sector projects, their value should be reduced (increased) by the costs (benefits) involved in transferring these resources to the private sector, but that subject to this qualification they can be treated identically to returns accruing directly to the private sector. The distinction between (ii) and (iii), though less natural, is more important. The objective is to separate monetisable from non-monetisable benefits, and the reason is that a higher savings rate can be expected from money income of type (ii) than from psychic income of type (iii). As Feldstein (1970) points out, (iii) is not normally included in econometric studies of savings behaviour - though the reasons may be practical rather than conceptual. But it is plausible to expect different savings rates. Suppose an individual lives for two periods, and has an
J.A. Kay, Social discount
additively
separable utility
rates
363
function
and gl and g2 government where x1, x2 are his money expenditures expenditures in periods 1 and 2 respectively. This he maximises subject to a constraint of form x1 + kx2 = yl , where y1 is his income in period 1. Then an increase in y Will generally prompt him to save more in period 1, while an increase ingl sufficient to induce an equal change in u will not do so. However, if the utility function is not separable changes in g will generally have some effect on savings. There is a range within which the distinction becomes fuzzy, particularly where a discrete price change to final consumers is involved. The effect of a price reduction which represents the lower cost of units previously purchased at a higher price seems clearly a pecuniary benefit, not differing from an equivalent cash bonus. But it is not clear how one would wish to allocate the consumers’ surplus triangle: and since the distinction sought is intuitive rather than rigorous there is no real recourse where intuition fails. But it seems unlikely that for many projects the decision would hinge criticially on the classification chosen.
4. The Arrow theorem It is now possible to pursue more formally -the intuitive argument outlined in section 1. I shall prove the Arrow theorem under a set of rather restrictive assumptions similar to those made or implied by Arrow. The consequences of relaxing these assumptions will be considered in subsequent sections. Theorem I. If (a) a project and r(t) are constant, (c) sr proceeds of public projects, only if it has positive present Proof.
From the definitions
k = sa
+ srk .
generates
only pecuniary returns, (b) s(t) is no reinvestment of the the project should be undertaken if and value when discounted at the.STP rate p. < p, (d) there
J.A. Kay, Social discount
364
rates
Hence k(T) =J
a exp
[ --ST(X
- T)]
dx
0
y=a+rk.
Hence Jyexp(-pf)dl=
7 a exp(-pt)dt+rs
7 exp(-pt)
0
0
0
=
7~exp (-pt)dt
ja
exp [-sr(x-t)]dxdt
0
Iaexp
+ TS 7 exp ]-(p-sr)t
0
0
Integrating
(--Sr.x)dxdt
.
0
this last term by parts gives
i
Y
exp (-pt)dt
=
0
9:a exp (-pt)dt
+6
7 a exp (-pt)dt
0
0
and so ./ 0
c w (-pt)dt = (1 --s)
syexp(-pt)dt 0
p(1
= Fs
-s)
m
a exp (-pt)dt
.
0
In the terminology of section 1, then, w is equal to (p-ps)/(p--KS), and hence is constant over time. The effect of investment and reinvestment is to magnify the costs and benefits of the project, while leaving their sign unchanged. 4, the appropriate discount rate for public projects, is equal to the SOC rate p. Sections 5, 6 and 7 indicate the effects of relaxing assumptions (a), (c) and (b) respectively. Generalisation of (d), however, is immediate. Let part of the returns of the project be invested in subsequent projects, with the possibility of subsequent reinvestment of these surpluses. Then so long as none of these further projects fail to meet the criterion of positive present value at the STP rate, the overall effect is the generation of a new project over a longer time period, comprising only com-
J.A. Kay, Social discount rates
365
ponent projects meeting the present value criterion and hence itself meeting this criterion. Since the theorem implies this overall project has a positive effect on the welfare integral, the effect of the initial project is positive.
5. Non-pecuniary
benefits
Feldstein ( 1970) implies that the existence of non-pecuniary benefits reduces Arrow’s result to an exceptional case, but that is not necessarily correct. It is true that if there are non-pecuniary benefits, then a different savings rate can be expected from them. Let is be s’. Let the stream of non-pecuniary benefits be b(t). Then a precisely analogous argument shows that the value of the consequent consumption stream is
W’ =
(’p -“)’ m b exp --s’r s
(-pt)dt
.
0
It is no longer possible to provide a completely general decision rule without knowing the values of the parameters s and s’. However, a substantial range of cases can be covered. If both pecuniary and nonpecuniary benefit streams have positive present value at the STP rate, the project should be undertaken: if they are both negative then the project should be rejected. Suppose one is positive, the other negative. Then if, as we may assume, s > s’, (1 -s)p p-sr
> Cl-S’)P p--s)T’
Let A be the present value of the pecuniary non-pecuniary benefits. Then if A,B>O,
benefit
w>o
A > 0,B < 0,
W>
0
ifA+B>
0
A < 0,B > 0,
W’< 0
ifA+B<
0
A,B
WC0
stream, B that for
366
J.A. Kay, Social discount
rates
The typical case is probably one in which the non pecuniary benefit stream is positive. If the pecuniary benefit stream is also positive, the project can be recommended. If it is negative, and the present value of the whole benefit stream is also negative, the project should be rejected. However if the project has net pecuniary costs and positive non pecuniary benefits, it is not possible to make a decision without further analysis.
6. Convergence
criteria and their significance
In theorem 1 it was assumed that sr < p, a condition required to ensure convergence of the welfare integral. Such convergence appears either to have been taken for granted or assured by the introduction of a finite time horizon in most previous treatments. Since sr > p cannot be ruled out a priori, it is worth considering its implications. I shall employ an “overtaking” criterion, i.e. a path c* is to be preferred to c if for some T, > 0 and all T > TO, J,‘c* exp (-pt)dt > J,‘c exp (-pt)dt. Then we have: Theorem 2 If rs > p, a project
yielding only pecuniary benefits and terminating at time x is acceptable if and only if J,” a exp (-rst)dt > 0: i.e. a project must have positive present value when discounted at the rate YS. Proof
If the project
k = rsk
terminates
at x, then for t > x
k = k(x) exp [rs(t-x)]
y exp (-pt)
= rk(x)
and so y = rk(x) exp [rs(t-x)]
exp (-rsx).exp
[t(rs-p)].
Hence T s
y
exp (-pt)dt
= rk(x~lpp(-rsX){exp
[(rs-p)
Tl - exp [(rs-p)x]}
x
Thus for all sufficiently large T, ,,’ y exp(-pt)dt has the sign of k(x): but from a result of theorem 1, k(x) has the sign of J,” a exp (-rst)dt:
.
J.A. Kay, Social discount
rates
367
and so J,‘c exp(-pt)dt is positive (negative) for all T > T, as Ji a exp (-srt)dt is greater than (less than) zero. Non convergence problems of this kind introduce the possibility that no optimum policy exists for the economy. Full investigation of this would require a more complete specification of the constraints on public and private investment than is attempted here. But the concern of this paper - and in particular of this section - is not with the characterisation of an optimum, but rather with the comparison of two nonoptimal paths. What is shown here is that a path in which a public project yieldings YS is undertaken will overtake an otherwise identical one in which it is not. I find this a compelling argument in favour of such projects. Thus more generally the appropriate discount rate for public investment is the higher of p and TS. My own view is that this will usually mean p. First note that if YS> p, the integral evaluating private investment does not converge. Theorem 3. consumption,
v, the value of private investment in terms of current is finite if and only if rs < p, and is then equal to
(r-rs)/(p-rsj. Proof: Consider
the stream of income change in private investment. Then y = rk
Ii = sy
W=(~-S)~Y
changes consequent
k = exp (rst)
on a unit
y = r exp (rst)
exp(--pf)dr=(l--S)ryexp[(srp)f]dl 0
0
= (r-rs)/(p-KS)
if p > rs .
It is not, I suppose, impossible that the value of private investment in the British economy today is infinite: but it would be a somewhat disturbing conclusion, and one which does not appear to be reflected in current policy. Nor is it the conclusion to which armchair speculation as to plausible values of r, s and p is likely to lead. On a Keynesian savings hypothesis, 25% is a high value for s, the proportion of benefits saved, and 15% a high value for r: while 4% for p is a rather modest figure.
J.A. Kay, Social discount rates
368
of s and r
7. Variability
Suppose r and s can vary through time. r(t) is now the rate of return earned at time t by the private capital stock k(t) in existence at time t. s(t) is the proportion of y(t) which is invested. Clearly the simple result of theorem 1 cannot survive this generalisation. If a project has its returns mainly in a period when the savings rate is low then even if it earns the STP rate it may be inadvisable to undertake it: similar problems may apply if withdrawals from private income occur in a period when the return on private investment is especially high. To consider this problem it is necessary to make more specific assumptions about the nature of the time pattern of the returns generated by the project. I shall suppose that what is involved is in general postponement: a project withdraws income from the private sector now and returns it later. Clearly this is true of the vast majority of public sector projects. It is desirable to postpone private sector income if the rate of return on the postponement exceeds the rate at which the value of private sector income is declining, i.e. if rp
>
_Lw/Lw a~
ay ’
where rp is the public sector rate of return. A project earning p while postponing disposable income will thus be acceptable if
Define u(t) as the value of investment per unit of consumption at time t, and u’(t) as the value u(t) would take if the current savings rate s(t) and rate of return on private capital r(t) were to be maintained indefinitely: thus from theorem 3, u’(t) =
r(t) -r(t>s(t) p -r(tMt) ’
Theorem 4. The rate at which the value of private disposable declines is not less than p if the savings rate s(t) is not declining
income and the
J.A. Kay, Social discount rates
369
value of private investment is not less than it would be if current rates and private rates of return persisted. I.e., if
and
i(t) 2 0
u(t) 2 v’(t) .
Proof W = 7 c(t) exp (-pt)dt 0
i!$={( 1 -
s(t)) + s(t) u(t)} exp (-pt)
aw +
&= aY
-p ay
aW av>
0,
{i(t)(u(t)
- 1) + s(t)ti(t)}
exp(-pt)
Since and
u(t)>
1,
ifalso
s(t)>O,
Now y=rk
and
k=sy,
so that
k(x) = k, exp j rsdt 0
hence y(x) = r(x) k, exp j rsdt 0
u(t) =
7 {1 -
s(x)} r(x) exp
exp {-p(x-t))b
t = 7 t
11 -r(x))r(x)exp(S(rrp)di)
dx
,
savings
370
J.A. Kay, Social discount
rates
(1 - s(t)} r(t) .
C(t) = {p - r(t)s(t>} u(t) Thus u(t) > 0
if
r(t) (1 -s(t)) dt)
a
= u’(t)
p -r(t)s(t)
Theorem 5. If u’(x) > u’(t) for all x > t, then u(t) 2 u’(t). Proof u(t) = 7 (1 - S(Y)} I t
exp
j (Y(x)s(x)-_~)dx
dy .
t
If u’(x)
a
u’(t)
= _
)
r(x)s(x)
- p2
r(x){1 - s(x))
r;I(‘“z;;;s;t)
r(t) (1-s(t)) Wdt)-P
= u’(t)
.
Thus if u’, the value of investment corresponding to maintenance of the current levels of Y and S, never falls below its value at t as r and s vary in future, the actual value of investment now, u(t), is at least as great as u’(t): and this ensures that postponement of private sector income is desirable. We require that if the economy’s savings and investment behaviour were to be frozen, now should be the worst possible time to do it. This formulation is rather complicated: it is perhaps easier to examine the future values of s and Y which will ensure the result, and this is done in fig. 1. The use of p as a basis for accepting projects is justified if the savings rate is not now declining and s and r are expected to remain
J.A. Kay, Social discount rates
371
t-
V’P
v’-
I-- r-5 p- rs
I I
Fig. 1.
within the hatched area in future. If neither initial level, the satisfaction of these conditions
s nor Y fall below their is assured.
8. Private sector optimisation Preceding sections have just allowed saving to happen; no assumptions have been made about its motivation. Clearly any particular set of such assumptions will generate some particular temporal pattern of savings rates. Diamond ( 1968) has considered, in a framework similar to that used here, the effect of optimising behaviour by the private sector. If this approach is to be adopted, it is necessary to set up a model of private sector behaviour which allows a divergence between STP and SOC rates to persist. The simplest such assumption is that individuals also discount at the STP rate: but taxes: appropriation of capital’s product by labour, other distortions and externalities imply a social rate of return to private investment in excess of the private rate: and the social rate implied by a private rate of return equal to the STP rate is the SOC rate. In such a model, consider a tax-financed public sector project which requires investment of 1 this period and returns (1 + p) next period. An
372
J.A. Kay, Social discount rates
individual faced with such a change in his disposable income will increase his consumption by 1 this period and reduce it by (1 + p) in the next. The spillover of (r-p) will be lost: there will be minor additional adjustments of essentially second order. Such an argument suggests that only a project returning r, the SOC rate, is acceptable. The reason for this result is that private sector behaviour of this kind falls into a category which has already been rejected as unrealistic. If the private sector regards its time pattern of consumption as optimal, it will undertake transactions which more or less exactly offset the effects of any government project on it. In consequence the savings rate is close to one: the convergence criterion is not satisfied and in accordance with theorem 2 the return required from public sector investment is sr and hence approaches r. But for reasons discussed earlier it is hard to take seriously a situation in which the convergence criterion fails to be met, and a model which generates such behaviour is difficult to justify. This is not unreasonable. It does not seem realistic to visualise current dissaving in anticipation of the future benefits of the electricity generation programme: though it is plausible to anticipate such dissaving in the light of its current burdens. And it is suggested below that a model may incorporate essential features of the life cycle savings hypothesis without leading to unacceptable implications. A related approach is that of Sandmo and Dreze (1971), who consider a two period general equilibrium model in which resources are released for public investment by manipulation of the interest rate. They show that this requires that a weighted average of STP and SOC rates be employed in appraisal. It is natural to attribute this result to their restriction to a two period case, with the consequent exclusion of reinvestment complications. Further analysis by Dreze (1972), however, suggests that a modified version of their result holds for a longer but still finite - time period. But it seems that restriction of the analysis to any finite time horizon introduces a possible asymmetry between the effects of costs and benefits. If N periods are considered, the effect of a cost incurred in period 1 is spread over the succeeding (IV - 1) periods: while the impact of a benefit in period N is spread over (N- 1) preceding periods. For reasons similar to those outlined in criticism of the Diamond approach, this seems an unrealistic description of actual behaviour, and the model below incorporates related considerations in a more plausible way.
J.A. Kay, Social discount rates
373
9. Savings from permanent income If instead of dividing current income into current consumption and additions to homogeneous savings, consumers plan to spread an increment to current income over a number of periods, the results are not affected in any substantial way. Suppose we define some function g(x) over an interval [O,T] such that consumption at time t, c(t), is Jtg(x)y(t-x)dx. Then
and JJ -h: = r/i(t). Assume T s 0
Lemma
g(x)d.x
1
.
1. The convergence
p>
Proof
<
r
(
condition
1 -jg(x)exp(-px)dx 0
in this model is
>
.
Write T C’ =
s
T g(x)
exp (-px)dx
0
After the termination
of the project, T
ry-$=r
s 0
g(x)Y(t--xW
c” =
s 0
g(x)dx
.
374
J.A. Kay, Social discount rates M
M
r s Y exp(-Pt)df 0
- s j, 0
MT
= r s Jg(x)y(t-x)
exp(-pt)dt
exp(-pt)dxdt
0 0
M
M
r s Y 0
exP (-@)dt
- Y(M) exp (-PM)
+ y. - p s
y exp (-pt)dt
0 MT
=r
g(x).!J(t-x)
J-s
exp [-p(t-x)1
exp (-px)dxdt
0 0 T M-x =r
g(x)Y(z)
ss
exp (-pz)
exp (-px)dzdx
0 --x M
=r
T
s 0
Y(z)
exp (-pz)
s g(x) 0
exp (-px)dxdz
TM -r
g(x) ss 0 M-x
.
exp (-pz)dzdx
exp (-px)y(z)
Write -J for this last term. Then we have M s 0
yo-~W) y
exp (-pt)dt
=
ev-pW+J
p-r(1
-c’)
’
Note that since JOTg(x)dx < 1, if y. > 0, y is always positive decreasing, and conversely: suppose y. > 0. Then
J< v4W
1g(x) 0
=
exp (Gpx) 7
exp (-pz)
and non-
dzdx
M-X
(r/p)_dW j k?(x)exp (-px)
[ exp [-p(M-x)]
- exp (-@I)]
dx
0
=
(r/p)yGW exp (-pM) Cc”-c’) y(M) exp (--pM) - J > i y(M) exp (--pM){p - r(c”-ccl)}
.
J.A. Kay, Social discount rates
375
Hence M s
y exp (-pt)
dt <
0
YO
p-r(l-c’)
.
But y is always positive: hence (0” y exp (-Pt)dt similar argument applies if y. < 0. Lemma 2. If the convergence u(c) = c’uo/) )
criterion
is satisfied,
certainly
converges.
then
where u(c) = l c(x) exp (-px)dx 0
and
d_Y)=
J 0
y(x) exp (-px)dx
.
Proot By a similar argument to that above, but now assuming convergence, - T U(C) = s s g(x)y(t-x) exp (-pt)dxdt 0 0 = 7 g(x) exp (-px) 0 =
7 y(z) exp (-pz)dz 0
c’u(y) .
Theorem 6 If consumption at any time t is&r g(x) y(t-x)dx, and p > r (1 - JOTexp (-px)g(x)dx}, then a project ent value at the STP rate is acceptable. Proof T s 0
&)y(t--x)dx
with Jl g(x)dx < 1 with positive pres-
J.A.Kay,Social discount rates
376
j
ev(-pf)df
Y
- (l/r)
0
7O;-;l> exp (-pt)dt 0
-T = ss
&)A-x>
exp (-pt)dxdt
0 0
uti>- (p/r) j
0, -a) exp (-pt)dt
= u(c)
from lemma
1.
0
Hence from lemma 2, u(c) 5
(P-r(1
I
= 9: a
-c')) I
Hence if the convergence criterion value of the project is positive.
10. Inefficient
exp (-pr)dt
.
0
is satisfied,
V(C) > 0 iff the present
production
The argument of this paper implies that the economy should be characterised by productive inefficiency: that intertemporal marginal rates of transformation in the public and private sectors should diverge. The public sector will undertake investment which would be rejected by the private sector because its returns are inadequate. Diamond and Mirrlees ( 197 1) have shown that productive efficiency is desirable under rather weak assumptions, and have suggested that this result should imply equality of public sector discount rate and SOC rate. Their conclusion differs from that of this paper because the situation analysed here implies some constraint on the ability or willingness of the government to vary commodity taxes. A slightly different set of producer prices would imply a higher level of private investment, which the welfare function employed here would judge desirable. Hence commodity taxes cannot be optimal, and since the Diamond-Mirrlees result is valid only as one aspect of a scheme of optimal taxes their result is inapplicable. In this context, “commodity taxes” refer to taxes on the same com-
J.A. Kay, Social discount rates
377
modity at different points in time. It is not, perhaps, surprising to find that such taxes are not optimal in the Diamond-Mirrlees sense. In part, the explanation is that optimality in this field involves considerations which the Diamond-Mirrlees model does not adequately capture; in particular the macroeconomic implications of investment and intertemporal tax variations and the impact on managerial responsibility of excessive investment subsidy. In part, the cause may simply be governmental incompetence. It is more important to identify and remove constraints which prevent the attainment of full optimality than to devise second best policies for operation under them; but given the apparent universality of such constraints the second best analysis remains a useful exercise.
11. Conclusion Many economists - certainly the author of this paper - have believed that positive present value at the STP rate is too weak a criterion to apply to public investment projects, and that the higher rate of return earned in the private sector should imply a more rigorous selection of public projects. This view is not generally correct; it is conceivable - if, for example, savings rates and rates of return increase over the life of the project - that the reverse is the case, and that some projects whose present value at the STP rate is negative are desirable. For these, and for projects whose benefits largely take the form of surpluses, complex general rules are required. But for most cases, and in particular for the commercial projects of nationalised industries, accept projects which earn the STP rate is an appropriate criterion. Employment of the SOC rate is defensible only under implausible assumptions with disturbing implications. In the U.K., effort should be diverted from the easy but irrelevant task of finding the SOC rate to the harder but necessary determination of the STP rate.
References Arrow, K.J., 1966, Discounting and public investment criteria, in: Kneese and Smith (eds.), Water research, resources for the future (Baltimore). Diamond, P.A., 1968. The opportunity costs of public investment: comment, Quarterly Journal of Economics 82.682-8.
318
J.A. Kay, Social discount
rates
Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production, American Economic Review 61, 8-27, 261-278. Dreze, J.H., 1972, Discount rates and public investment: a post-scriptum (mimeo, C.O.R.E.). Feldstein, M.S., 1970, Choice of technique in the public sector; a simplification, Economic Journal 80,985-990. Sandmo, A. and J.H. Dreze, 1971, Discount rates for public investment in closed and open economies, Economica 38,395-412.
of Public Economics
1 (1972)
359-378.
0 North-Holland
SOCIAL DISCOUNT
Publishing
Company
RATES*
J.A. KAY St. John’s College, Oxford, England First version
received
February
1972, revised version received
July 1972
1. Introduction We live in an economy which rnay or may not be on an optimum growth path, but probably is not. In that economy, it is necessary for government agencies to undertake the evaluation of proposed investment projects, and to choose some discount rate for the purpose. Asked for guidance, the government indicates that it discounts future consumption at a rate p, and that its empirical studies indicate that the return obtained on capital invested in the public sector is Y, with r > p. The object of this paper is to consider what advice an economist might give on the basis of this rather inadequate and unsatisfactory information. If a(t) represents the stream of costs and benefits which the project generates over time, we seek some rate q such that the project should be adopted if and only if li a(t)e- 4 t dt 2 0. The information is sufficient for the analysis of two polar cases. If a project displaces only private consumption, it should be required to return p, the social time preference (STP) rate: q is equal to p. If it affects,only private investment, it may be expected to provide a return at least as great as r, the social opportunity cost (SOC) rate: q is equal to Y. In general, however, the truth will lie somewhere between these extremes: and it is natural to argue that an appropriate criterion will similarly be somewhere be-
* Earlier versions of this paper were given to a seminar in Oxford and to the European Winter Symposium of the Econometric Society in 1972. I am grateful for comments received, particularly those of A.B. Atkinson, J.H. D&e, J.S. Flemming, J.A. Mirrlees, G.B. Richardson and the referees.
360
J.A. Kay, Social discount
rates
tween the two extremes of the restrictive SOC rate and the permissive STP rate. It has been suggested (Sandmo and D&e, 1971) that a weighted average of the two is appropriate. This paper takes issue with that view. It follows an argument of Arrow’s (1966) which suggests that even when the effects on private investment are taken into account, the appropriate discount rate remains p. Arrow’s argument may be paraphrased as follows. If a project requires & 1 at time T, and this & 1 is withdrawn from private disposable income, the ultimate result of the repercussions, immediate and long term, on consumption and on investment, will be a stream of changes in consumption. Discounted to T, , this stream is worth w say. The project returns & 1 at some other time T2. This sets up a similar stream of changes in the opposite direction. Discounted to T2, this stream is also worth w. Hence the effect of the complications of investment and reinvestment is to multiply the value of the costs and benefits of the project by w. These complications therefore alter the magnitude of the net benefits of the project, but do not affect their sign. Thus ji ceePf dt is simply w Jomae- P tdt. If the project has positive present value at p, we should adopt it. If w is constant, then, 4 is equal to p and the STP rate is the appropriate discount rate for public investment. This is demonstrated in section 4, in which the assumptions implied by Arrow’s supposition of constant w are made explicit. Divergence between 4 and p can occur if, and only if, these assumptions are violated. Section 5 therefore considers the possibility that the multiplier w is variable because the costs and benefits of projects enter private income in different ways. Section 6 deals with the case where no finite w exists. Section 7 explores the impact of variations in w due to changes in savings rates or rates of return on private investment. These possibilities do suggest that minor deviations from the path that Arrow blazed through the tangled jungle of complex general criteria may sometimes be necessary. But they do not, I think, seriously weaken the proposition that this is the path which economists, and following them governments, should take. The STP rate p remains a central estimate of 4: the SOC rate r is a parameter of interest only for the refined analysis of marginal cases. This conclusion differs sharply from that reached by other writers in this field, and sections 8- 10 give some consideration to the reasons. But first sections 2 and 3 outline the model and some preliminary argument.
J.A. Kay, Social discount rates
361
2. The model I assume a closed economy in which there is a homogeneous private capital stock, with a single instantaneous rate of return r(t). The costs and benefits of public projects are independent of the size of the private capital stock. (This assumption is unnecessary if benefit streams are valued on the basis of the private capital stock which would actually exist if the project were undertaken.) All potential output is fully utilised in consumption, private investment or government activity. As a consequence of the project, private sector income at time I is increased by v(t): private consumption by c(t): and the private capital stock by k(t). Let a(t) be the value of the benefits (project expenses are regarded throughout as negative benefits) accruing at time t. Hence y(t) =a(t) + r(t) k(t) = c(t) + k(t). Define s(r) = g From national
.
income identities,
v(t) = c(f) + S&t) + S,(t) , where S,(t) is private saving and s,(t) government saving. At this stage, no hypothesis is made or implied concerning the determinants of savings. s(t) is a parameter, defined for all t, which may be observed ex post. The approach meshes most easily with a Keynesian theory of savings behaviour, where the marginal propensity to save is a familiar parameter: but its applicability is not confined to this theory, and the effect on the model of alternative assumptions about savings can be viewed in terms of the effect of these assumptions on s(t). If the government adjusts its spending and receipts so that whatever government expenditures the project requires are tax financed, presumably using monetary policy if necessary to maintain full employment, then new government saving is zero and s(t) can be regarded as the private sector savings rate. If, however, alternative policy mixes which include unbalanced changes in the budget are employed, the savings which necessarily match changes in private investment may be done by the government as well as by the private sector, and s(t) is then a rate which includes government as well as private saving. Whether this s(t) is
362
J.A. Kay, Social discount rates
greater or less than s,(t), the private sector savings rate, depends on whether an expansionary fiscal policy does more to stimulate private investment than private savings or vice versa. I shall not make judgement on this point. In what follows, it will be tempting to regard s(t) as referring to purely private saving; the caveat entered here should be borne in mind.
3. The nature of the benefits
of public sector projects
Three kinds of benefits from public projects can be distinguished. They are: (i) returns to the public sector: cost savings, sales receipts, initial expenses; (ii) pecuniary returns to the private sector: uncompensated costs and cost savings resulting from the project, e.g. fuel savings and savings in working time as a result of a road improvement; (iii) non-pecuniary returns to the private sector - leisure time savings, noise, pollution, etc. The distinction between (i) and (ii) is natural but unimportant. Resources generated in the public sector are of value only to the extent that they are reinvested in the public sector or transferred to the private sector. For the moment, exclude the possibility of public sector reinvestment: I shall later show that introducing this makes no difference to the results. Then the value of type (i) benefits is, in effect, the value of the type (ii) benefits which they induce. This implies that if resources are generated in the public sector as a result of the returns from public sector projects, their value should be reduced (increased) by the costs (benefits) involved in transferring these resources to the private sector, but that subject to this qualification they can be treated identically to returns accruing directly to the private sector. The distinction between (ii) and (iii), though less natural, is more important. The objective is to separate monetisable from non-monetisable benefits, and the reason is that a higher savings rate can be expected from money income of type (ii) than from psychic income of type (iii). As Feldstein (1970) points out, (iii) is not normally included in econometric studies of savings behaviour - though the reasons may be practical rather than conceptual. But it is plausible to expect different savings rates. Suppose an individual lives for two periods, and has an
J.A. Kay, Social discount
additively
separable utility
rates
363
function
and gl and g2 government where x1, x2 are his money expenditures expenditures in periods 1 and 2 respectively. This he maximises subject to a constraint of form x1 + kx2 = yl , where y1 is his income in period 1. Then an increase in y Will generally prompt him to save more in period 1, while an increase ingl sufficient to induce an equal change in u will not do so. However, if the utility function is not separable changes in g will generally have some effect on savings. There is a range within which the distinction becomes fuzzy, particularly where a discrete price change to final consumers is involved. The effect of a price reduction which represents the lower cost of units previously purchased at a higher price seems clearly a pecuniary benefit, not differing from an equivalent cash bonus. But it is not clear how one would wish to allocate the consumers’ surplus triangle: and since the distinction sought is intuitive rather than rigorous there is no real recourse where intuition fails. But it seems unlikely that for many projects the decision would hinge criticially on the classification chosen.
4. The Arrow theorem It is now possible to pursue more formally -the intuitive argument outlined in section 1. I shall prove the Arrow theorem under a set of rather restrictive assumptions similar to those made or implied by Arrow. The consequences of relaxing these assumptions will be considered in subsequent sections. Theorem I. If (a) a project and r(t) are constant, (c) sr proceeds of public projects, only if it has positive present Proof.
From the definitions
k = sa
+ srk .
generates
only pecuniary returns, (b) s(t) is no reinvestment of the the project should be undertaken if and value when discounted at the.STP rate p. < p, (d) there
J.A. Kay, Social discount
364
rates
Hence k(T) =J
a exp
[ --ST(X
- T)]
dx
0
y=a+rk.
Hence Jyexp(-pf)dl=
7 a exp(-pt)dt+rs
7 exp(-pt)
0
0
0
=
7~exp (-pt)dt
ja
exp [-sr(x-t)]dxdt
0
Iaexp
+ TS 7 exp ]-(p-sr)t
0
0
Integrating
(--Sr.x)dxdt
.
0
this last term by parts gives
i
Y
exp (-pt)dt
=
0
9:a exp (-pt)dt
+6
7 a exp (-pt)dt
0
0
and so ./ 0
c w (-pt)dt = (1 --s)
syexp(-pt)dt 0
p(1
= Fs
-s)
m
a exp (-pt)dt
.
0
In the terminology of section 1, then, w is equal to (p-ps)/(p--KS), and hence is constant over time. The effect of investment and reinvestment is to magnify the costs and benefits of the project, while leaving their sign unchanged. 4, the appropriate discount rate for public projects, is equal to the SOC rate p. Sections 5, 6 and 7 indicate the effects of relaxing assumptions (a), (c) and (b) respectively. Generalisation of (d), however, is immediate. Let part of the returns of the project be invested in subsequent projects, with the possibility of subsequent reinvestment of these surpluses. Then so long as none of these further projects fail to meet the criterion of positive present value at the STP rate, the overall effect is the generation of a new project over a longer time period, comprising only com-
J.A. Kay, Social discount rates
365
ponent projects meeting the present value criterion and hence itself meeting this criterion. Since the theorem implies this overall project has a positive effect on the welfare integral, the effect of the initial project is positive.
5. Non-pecuniary
benefits
Feldstein ( 1970) implies that the existence of non-pecuniary benefits reduces Arrow’s result to an exceptional case, but that is not necessarily correct. It is true that if there are non-pecuniary benefits, then a different savings rate can be expected from them. Let is be s’. Let the stream of non-pecuniary benefits be b(t). Then a precisely analogous argument shows that the value of the consequent consumption stream is
W’ =
(’p -“)’ m b exp --s’r s
(-pt)dt
.
0
It is no longer possible to provide a completely general decision rule without knowing the values of the parameters s and s’. However, a substantial range of cases can be covered. If both pecuniary and nonpecuniary benefit streams have positive present value at the STP rate, the project should be undertaken: if they are both negative then the project should be rejected. Suppose one is positive, the other negative. Then if, as we may assume, s > s’, (1 -s)p p-sr
> Cl-S’)P p--s)T’
Let A be the present value of the pecuniary non-pecuniary benefits. Then if A,B>O,
benefit
w>o
A > 0,B < 0,
W>
0
ifA+B>
0
A < 0,B > 0,
W’< 0
ifA+B<
0
A,B
WC0
stream, B that for
366
J.A. Kay, Social discount
rates
The typical case is probably one in which the non pecuniary benefit stream is positive. If the pecuniary benefit stream is also positive, the project can be recommended. If it is negative, and the present value of the whole benefit stream is also negative, the project should be rejected. However if the project has net pecuniary costs and positive non pecuniary benefits, it is not possible to make a decision without further analysis.
6. Convergence
criteria and their significance
In theorem 1 it was assumed that sr < p, a condition required to ensure convergence of the welfare integral. Such convergence appears either to have been taken for granted or assured by the introduction of a finite time horizon in most previous treatments. Since sr > p cannot be ruled out a priori, it is worth considering its implications. I shall employ an “overtaking” criterion, i.e. a path c* is to be preferred to c if for some T, > 0 and all T > TO, J,‘c* exp (-pt)dt > J,‘c exp (-pt)dt. Then we have: Theorem 2 If rs > p, a project
yielding only pecuniary benefits and terminating at time x is acceptable if and only if J,” a exp (-rst)dt > 0: i.e. a project must have positive present value when discounted at the rate YS. Proof
If the project
k = rsk
terminates
at x, then for t > x
k = k(x) exp [rs(t-x)]
y exp (-pt)
= rk(x)
and so y = rk(x) exp [rs(t-x)]
exp (-rsx).exp
[t(rs-p)].
Hence T s
y
exp (-pt)dt
= rk(x~lpp(-rsX){exp
[(rs-p)
Tl - exp [(rs-p)x]}
x
Thus for all sufficiently large T, ,,’ y exp(-pt)dt has the sign of k(x): but from a result of theorem 1, k(x) has the sign of J,” a exp (-rst)dt:
.
J.A. Kay, Social discount
rates
367
and so J,‘c exp(-pt)dt is positive (negative) for all T > T, as Ji a exp (-srt)dt is greater than (less than) zero. Non convergence problems of this kind introduce the possibility that no optimum policy exists for the economy. Full investigation of this would require a more complete specification of the constraints on public and private investment than is attempted here. But the concern of this paper - and in particular of this section - is not with the characterisation of an optimum, but rather with the comparison of two nonoptimal paths. What is shown here is that a path in which a public project yieldings YS is undertaken will overtake an otherwise identical one in which it is not. I find this a compelling argument in favour of such projects. Thus more generally the appropriate discount rate for public investment is the higher of p and TS. My own view is that this will usually mean p. First note that if YS> p, the integral evaluating private investment does not converge. Theorem 3. consumption,
v, the value of private investment in terms of current is finite if and only if rs < p, and is then equal to
(r-rs)/(p-rsj. Proof: Consider
the stream of income change in private investment. Then y = rk
Ii = sy
W=(~-S)~Y
changes consequent
k = exp (rst)
on a unit
y = r exp (rst)
exp(--pf)dr=(l--S)ryexp[(srp)f]dl 0
0
= (r-rs)/(p-KS)
if p > rs .
It is not, I suppose, impossible that the value of private investment in the British economy today is infinite: but it would be a somewhat disturbing conclusion, and one which does not appear to be reflected in current policy. Nor is it the conclusion to which armchair speculation as to plausible values of r, s and p is likely to lead. On a Keynesian savings hypothesis, 25% is a high value for s, the proportion of benefits saved, and 15% a high value for r: while 4% for p is a rather modest figure.
J.A. Kay, Social discount rates
368
of s and r
7. Variability
Suppose r and s can vary through time. r(t) is now the rate of return earned at time t by the private capital stock k(t) in existence at time t. s(t) is the proportion of y(t) which is invested. Clearly the simple result of theorem 1 cannot survive this generalisation. If a project has its returns mainly in a period when the savings rate is low then even if it earns the STP rate it may be inadvisable to undertake it: similar problems may apply if withdrawals from private income occur in a period when the return on private investment is especially high. To consider this problem it is necessary to make more specific assumptions about the nature of the time pattern of the returns generated by the project. I shall suppose that what is involved is in general postponement: a project withdraws income from the private sector now and returns it later. Clearly this is true of the vast majority of public sector projects. It is desirable to postpone private sector income if the rate of return on the postponement exceeds the rate at which the value of private sector income is declining, i.e. if rp
>
_Lw/Lw a~
ay ’
where rp is the public sector rate of return. A project earning p while postponing disposable income will thus be acceptable if
Define u(t) as the value of investment per unit of consumption at time t, and u’(t) as the value u(t) would take if the current savings rate s(t) and rate of return on private capital r(t) were to be maintained indefinitely: thus from theorem 3, u’(t) =
r(t) -r(t>s(t) p -r(tMt) ’
Theorem 4. The rate at which the value of private disposable declines is not less than p if the savings rate s(t) is not declining
income and the
J.A. Kay, Social discount rates
369
value of private investment is not less than it would be if current rates and private rates of return persisted. I.e., if
and
i(t) 2 0
u(t) 2 v’(t) .
Proof W = 7 c(t) exp (-pt)dt 0
i!$={( 1 -
s(t)) + s(t) u(t)} exp (-pt)
aw +
&= aY
-p ay
aW av>
0,
{i(t)(u(t)
- 1) + s(t)ti(t)}
exp(-pt)
Since and
u(t)>
1,
ifalso
s(t)>O,
Now y=rk
and
k=sy,
so that
k(x) = k, exp j rsdt 0
hence y(x) = r(x) k, exp j rsdt 0
u(t) =
7 {1 -
s(x)} r(x) exp
exp {-p(x-t))b
t = 7 t
11 -r(x))r(x)exp(S(rrp)di)
dx
,
savings
370
J.A. Kay, Social discount
rates
(1 - s(t)} r(t) .
C(t) = {p - r(t)s(t>} u(t) Thus u(t) > 0
if
r(t) (1 -s(t)) dt)
a
= u’(t)
p -r(t)s(t)
Theorem 5. If u’(x) > u’(t) for all x > t, then u(t) 2 u’(t). Proof u(t) = 7 (1 - S(Y)} I t
exp
j (Y(x)s(x)-_~)dx
dy .
t
If u’(x)
a
u’(t)
= _
)
r(x)s(x)
- p2
r(x){1 - s(x))
r;I(‘“z;;;s;t)
r(t) (1-s(t)) Wdt)-P
= u’(t)
.
Thus if u’, the value of investment corresponding to maintenance of the current levels of Y and S, never falls below its value at t as r and s vary in future, the actual value of investment now, u(t), is at least as great as u’(t): and this ensures that postponement of private sector income is desirable. We require that if the economy’s savings and investment behaviour were to be frozen, now should be the worst possible time to do it. This formulation is rather complicated: it is perhaps easier to examine the future values of s and Y which will ensure the result, and this is done in fig. 1. The use of p as a basis for accepting projects is justified if the savings rate is not now declining and s and r are expected to remain
J.A. Kay, Social discount rates
371
t-
V’P
v’-
I-- r-5 p- rs
I I
Fig. 1.
within the hatched area in future. If neither initial level, the satisfaction of these conditions
s nor Y fall below their is assured.
8. Private sector optimisation Preceding sections have just allowed saving to happen; no assumptions have been made about its motivation. Clearly any particular set of such assumptions will generate some particular temporal pattern of savings rates. Diamond ( 1968) has considered, in a framework similar to that used here, the effect of optimising behaviour by the private sector. If this approach is to be adopted, it is necessary to set up a model of private sector behaviour which allows a divergence between STP and SOC rates to persist. The simplest such assumption is that individuals also discount at the STP rate: but taxes: appropriation of capital’s product by labour, other distortions and externalities imply a social rate of return to private investment in excess of the private rate: and the social rate implied by a private rate of return equal to the STP rate is the SOC rate. In such a model, consider a tax-financed public sector project which requires investment of 1 this period and returns (1 + p) next period. An
372
J.A. Kay, Social discount rates
individual faced with such a change in his disposable income will increase his consumption by 1 this period and reduce it by (1 + p) in the next. The spillover of (r-p) will be lost: there will be minor additional adjustments of essentially second order. Such an argument suggests that only a project returning r, the SOC rate, is acceptable. The reason for this result is that private sector behaviour of this kind falls into a category which has already been rejected as unrealistic. If the private sector regards its time pattern of consumption as optimal, it will undertake transactions which more or less exactly offset the effects of any government project on it. In consequence the savings rate is close to one: the convergence criterion is not satisfied and in accordance with theorem 2 the return required from public sector investment is sr and hence approaches r. But for reasons discussed earlier it is hard to take seriously a situation in which the convergence criterion fails to be met, and a model which generates such behaviour is difficult to justify. This is not unreasonable. It does not seem realistic to visualise current dissaving in anticipation of the future benefits of the electricity generation programme: though it is plausible to anticipate such dissaving in the light of its current burdens. And it is suggested below that a model may incorporate essential features of the life cycle savings hypothesis without leading to unacceptable implications. A related approach is that of Sandmo and Dreze (1971), who consider a two period general equilibrium model in which resources are released for public investment by manipulation of the interest rate. They show that this requires that a weighted average of STP and SOC rates be employed in appraisal. It is natural to attribute this result to their restriction to a two period case, with the consequent exclusion of reinvestment complications. Further analysis by Dreze (1972), however, suggests that a modified version of their result holds for a longer but still finite - time period. But it seems that restriction of the analysis to any finite time horizon introduces a possible asymmetry between the effects of costs and benefits. If N periods are considered, the effect of a cost incurred in period 1 is spread over the succeeding (IV - 1) periods: while the impact of a benefit in period N is spread over (N- 1) preceding periods. For reasons similar to those outlined in criticism of the Diamond approach, this seems an unrealistic description of actual behaviour, and the model below incorporates related considerations in a more plausible way.
J.A. Kay, Social discount rates
373
9. Savings from permanent income If instead of dividing current income into current consumption and additions to homogeneous savings, consumers plan to spread an increment to current income over a number of periods, the results are not affected in any substantial way. Suppose we define some function g(x) over an interval [O,T] such that consumption at time t, c(t), is Jtg(x)y(t-x)dx. Then
and JJ -h: = r/i(t). Assume T s 0
Lemma
g(x)d.x
1
.
1. The convergence
p>
Proof
<
r
(
condition
1 -jg(x)exp(-px)dx 0
in this model is
>
.
Write T C’ =
s
T g(x)
exp (-px)dx
0
After the termination
of the project, T
ry-$=r
s 0
g(x)Y(t--xW
c” =
s 0
g(x)dx
.
374
J.A. Kay, Social discount rates M
M
r s Y exp(-Pt)df 0
- s j, 0
MT
= r s Jg(x)y(t-x)
exp(-pt)dt
exp(-pt)dxdt
0 0
M
M
r s Y 0
exP (-@)dt
- Y(M) exp (-PM)
+ y. - p s
y exp (-pt)dt
0 MT
=r
g(x).!J(t-x)
J-s
exp [-p(t-x)1
exp (-px)dxdt
0 0 T M-x =r
g(x)Y(z)
ss
exp (-pz)
exp (-px)dzdx
0 --x M
=r
T
s 0
Y(z)
exp (-pz)
s g(x) 0
exp (-px)dxdz
TM -r
g(x) ss 0 M-x
.
exp (-pz)dzdx
exp (-px)y(z)
Write -J for this last term. Then we have M s 0
yo-~W) y
exp (-pt)dt
=
ev-pW+J
p-r(1
-c’)
’
Note that since JOTg(x)dx < 1, if y. > 0, y is always positive decreasing, and conversely: suppose y. > 0. Then
J< v4W
1g(x) 0
=
exp (Gpx) 7
exp (-pz)
and non-
dzdx
M-X
(r/p)_dW j k?(x)exp (-px)
[ exp [-p(M-x)]
- exp (-@I)]
dx
0
=
(r/p)yGW exp (-pM) Cc”-c’) y(M) exp (--pM) - J > i y(M) exp (--pM){p - r(c”-ccl)}
.
J.A. Kay, Social discount rates
375
Hence M s
y exp (-pt)
dt <
0
YO
p-r(l-c’)
.
But y is always positive: hence (0” y exp (-Pt)dt similar argument applies if y. < 0. Lemma 2. If the convergence u(c) = c’uo/) )
criterion
is satisfied,
certainly
converges.
then
where u(c) = l c(x) exp (-px)dx 0
and
d_Y)=
J 0
y(x) exp (-px)dx
.
Proot By a similar argument to that above, but now assuming convergence, - T U(C) = s s g(x)y(t-x) exp (-pt)dxdt 0 0 = 7 g(x) exp (-px) 0 =
7 y(z) exp (-pz)dz 0
c’u(y) .
Theorem 6 If consumption at any time t is&r g(x) y(t-x)dx, and p > r (1 - JOTexp (-px)g(x)dx}, then a project ent value at the STP rate is acceptable. Proof T s 0
&)y(t--x)dx
with Jl g(x)dx < 1 with positive pres-
J.A.Kay,Social discount rates
376
j
ev(-pf)df
Y
- (l/r)
0
7O;-;l> exp (-pt)dt 0
-T = ss
&)A-x>
exp (-pt)dxdt
0 0
uti>- (p/r) j
0, -a) exp (-pt)dt
= u(c)
from lemma
1.
0
Hence from lemma 2, u(c) 5
(P-r(1
I
= 9: a
-c')) I
Hence if the convergence criterion value of the project is positive.
10. Inefficient
exp (-pr)dt
.
0
is satisfied,
V(C) > 0 iff the present
production
The argument of this paper implies that the economy should be characterised by productive inefficiency: that intertemporal marginal rates of transformation in the public and private sectors should diverge. The public sector will undertake investment which would be rejected by the private sector because its returns are inadequate. Diamond and Mirrlees ( 197 1) have shown that productive efficiency is desirable under rather weak assumptions, and have suggested that this result should imply equality of public sector discount rate and SOC rate. Their conclusion differs from that of this paper because the situation analysed here implies some constraint on the ability or willingness of the government to vary commodity taxes. A slightly different set of producer prices would imply a higher level of private investment, which the welfare function employed here would judge desirable. Hence commodity taxes cannot be optimal, and since the Diamond-Mirrlees result is valid only as one aspect of a scheme of optimal taxes their result is inapplicable. In this context, “commodity taxes” refer to taxes on the same com-
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modity at different points in time. It is not, perhaps, surprising to find that such taxes are not optimal in the Diamond-Mirrlees sense. In part, the explanation is that optimality in this field involves considerations which the Diamond-Mirrlees model does not adequately capture; in particular the macroeconomic implications of investment and intertemporal tax variations and the impact on managerial responsibility of excessive investment subsidy. In part, the cause may simply be governmental incompetence. It is more important to identify and remove constraints which prevent the attainment of full optimality than to devise second best policies for operation under them; but given the apparent universality of such constraints the second best analysis remains a useful exercise.
11. Conclusion Many economists - certainly the author of this paper - have believed that positive present value at the STP rate is too weak a criterion to apply to public investment projects, and that the higher rate of return earned in the private sector should imply a more rigorous selection of public projects. This view is not generally correct; it is conceivable - if, for example, savings rates and rates of return increase over the life of the project - that the reverse is the case, and that some projects whose present value at the STP rate is negative are desirable. For these, and for projects whose benefits largely take the form of surpluses, complex general rules are required. But for most cases, and in particular for the commercial projects of nationalised industries, accept projects which earn the STP rate is an appropriate criterion. Employment of the SOC rate is defensible only under implausible assumptions with disturbing implications. In the U.K., effort should be diverted from the easy but irrelevant task of finding the SOC rate to the harder but necessary determination of the STP rate.
References Arrow, K.J., 1966, Discounting and public investment criteria, in: Kneese and Smith (eds.), Water research, resources for the future (Baltimore). Diamond, P.A., 1968. The opportunity costs of public investment: comment, Quarterly Journal of Economics 82.682-8.
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Diamond, P.A. and J.A. Mirrlees, 1971, Optimal taxation and public production, American Economic Review 61, 8-27, 261-278. Dreze, J.H., 1972, Discount rates and public investment: a post-scriptum (mimeo, C.O.R.E.). Feldstein, M.S., 1970, Choice of technique in the public sector; a simplification, Economic Journal 80,985-990. Sandmo, A. and J.H. Dreze, 1971, Discount rates for public investment in closed and open economies, Economica 38,395-412.