Pa ke , Ch is
Wo king Pape
De i ing alues o he social a e o ime p e e ence
New Zealand T easu y Wo king Pape , No. 25/01
P o ided in Coope a ion wi h:
The T easu y, New Zealand Go e nmen
Sugges ed Ci a ion: Pa ke , Ch is (2025) : De i ing alues o he social a e o ime p e e ence, New
Zealand T easu y Wo king Pape , No. 25/01, New Zealand Go e nmen , The T easu y, Welling on
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WORKING PAPER
De i ing alues o he social a e
o ime p e e ence
Ch is Pa ke
New Zealand T easu y Wo king Pape 25/01
Feb ua y 2025
DISCLAIMER
: The iews, opinions, indings, and conclusions o ecommenda ions exp essed in his wo king pape a e
s ic ly hose o he au ho (s). They do no necessa ily e lec he iews o he New Zealand T easu y o he New
Zealand
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iew o in o m and s imula e wide deba e.
NZ TREASURY
WORKING PAPER 25/01
De i ing alues o he social a e o ime p e e ence
MONTH/YEAR
Feb ua y 2025
AUTHOR
Ch is Pa ke
The T easu y
1 The Te ace
Welling on 6011
New Zealand
Email ch is.pa ke @ easu y.go .nz
URL
T easu y websi e a Feb ua y 2025:
h ps://www. easu y.go .nz/publica ions/wp/wp-25-01
ACKNOWLEDGEMENTS
Co y Da is, o assis ing wi h Py hon; o Luke Symes, Tim Ng,
D Zacha y Tu k, D G aeme Gu h ie, and P o esso Ben
G oom o quali y assu ance/pee e iew; Ki s en Jensen, D
Ch is Thompson, Shane Domican, and Dominick S ephens o
commen . All e o s emain hose o he au ho .
NZ
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WP 25/01 | De i ing alues o he social a e o ime p e e ence
i
Abs ac
This epo es ima es he ‘social a e o ime p e e ence’ (SRTP) o New Zealand
public policy app aisal using a Ramsey equa ion. I uses he Wei zman (1998) me hod
o de i e a declining discoun a e schedule o 2% ( eal) o yea s 1–30, 1.5% o yea s
31–100, and 1% o yea s 101 on. The epo does no compa e and con as social
discoun a e me hods and how o use hem in p ac ice.
JEL CLASSIFICATION
H43, H50
KEYWORDS
Discoun ing; social discoun a e; ime p e e ence, Go e nmen
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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Execu i e summa y
Cons an social a e o ime p e e ence
We de i ed a Ramsey equa ion based on a social wel a e unc ion o he consump ion
o goods and se ices o e all u u e ime. We equi ed ha he social a e o ime
p e e ence (SRTP) is no less han he g ow h a e o consump ion o ensu e ini eness
o ne p esen alues. To cha ac e ise ou unce ain y abou SRTP pa ame e s we use
iangula p obabili y dis ibu ions o plausible alues o he pu e a e o ime
p e e ence, annihila ion isk, he elas ici y o ma ginal social wel a e, and he g ow h
a e o eal pe capi a consump ion. These a e se ou below:
𝒓𝒓= 𝝆𝝆+𝜶𝜶+𝝁𝝁𝝁𝝁 >𝝁𝝁
Pa ame e s
T iangula dis ibu ion inpu s Commen
Minimum Mode Max
Pu e a e ime p e e ence 𝝆𝝆 0.00% 0.25% 2.0% Values used in majo wel a e
economics s udies
Annihila ion isk 𝜶𝜶 0.00% 0.05% 0.20% Ex inc ion isk pe cen u y
Elas ici y ma ginal social
wel a e wi h espec o
consump ion 𝝁𝝁
0.25 0.75 1.5 Reasonably ole able
consump ion loss om
ans e s
G ow h a e o eal
consump ion pe capi a 𝝁𝝁
0.75% 1.25% 2.0% His o ical pe capi a eal
consump ion and p ojec ions
Resul ing ou pu
SRTP 𝒓𝒓 0.8%
(Mean)
2.1% * 4.8%
Range o es ic ed alues o
SRTP. Rounded o one d.p.
* Be ween 1.2%–3.3%, wi h 95 pe cen con idence.
We use Mon e Ca lo simula ion o de i e a ange o SRTP alues cen ing a ound 2%,
and be ween 1.2%–3.3% wi h 95 pe cen con idence. We disca ded abou 15% o he
d aws because hey esul ed in a SRTP less han he g ow h a e o consump ion.
Declining SRTP
Using he Wei zman (1998) me hod we de i ed he ollowing declining ce ain y
equi alen SRTP schedule (using hal pe cen age poin inc emen s):
Schedule o SRTPs Yea s
2% 1-30
1.5% 31-100
1% 101+
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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Con en s
Execu i e summa y ............................................................................................ i
1. In oduc ion .................................................................................................... 1
2. De e mining he anges o pa ame e s ....................................................... 2
2.1 Pu e a e o ime p e e ence ................................................................. 2
2.2 Annihila ion isk ..................................................................................... 3
2.3 The elas ici y o ma ginal social alue o consump ion .......................... 3
2.4 G ow h a e o consump ion .................................................................. 6
3. Mon e Ca lo simula ion o pa ame e s ......................................................... 8
4. Declining ‘ce ain y equi alen ’ SRTP ........................................................ 10
Re e ences ....................................................................................................... 15
Appendix 1 – Ma hema ical basis o SRTP ................................................... 18
The concep o he ‘social a e o ime p e e ence’ .................................... 19
Appendix 2 – Py hon code o Mon e Ca lo analysis .................................... 22
Lis o igu es
Figu e 1: Annual g ow h o pe capi a eal p i a e consump ion (GDP) ................ 6
Figu e 2: Annual g ow h in eal GDP pe capi a (pe cen age) .............................. 7
Figu e 3: Mon e Ca lo simula ion ou pu ( es ic ed se ) ....................................... 9
Figu e 4: Discoun ac o s .................................................................................. 11
Figu e 5: A e age discoun ac o ...................................................................... 11
Figu e 6: Ma ginal and a e age ce ain y equi alen SRTPs .............................. 13
Figu e 7: Compa ing discoun ac o s (DFs) ....................................................... 13
Figu e 8: Discoun ac o s wi h and wi hou declining SRTP............................... 14
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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De i ing alues o he social a e o
ime p e e ence
1. In oduc ion
As pa o he T easu y e iew o he Public Sec o Discoun Ra e (T easu y Ci cula
2024/15, T easu y 2024a), his no e p o ides an es ima e o he ange o plausible
alues o he social a e o ime p e e ence (SRTP) based on he ollowing o m o
Ramsey equa ion:
𝑟𝑟= 𝜌𝜌+𝛼𝛼+𝜇𝜇𝜇𝜇 (1)
The SRTP (𝑟𝑟) es ima e based on a Ramsey equa ion uses he pa ame e s:
• pu e a e ime p e e ence 𝜌𝜌
• annihila ion isk 𝛼𝛼
• elas ici y o ma ginal social wel a e wi h espec o consump ion 𝜇𝜇
• g ow h a e o eal consump ion pe capi a 𝜇𝜇.
Appendix 1 p o ides a de ailed ma hema ical de i a ion o he equa ion and
pa ame e s abo e. We e iew he li e a u e and e iew al e na i e o mula ions o
he Ramsey equa ion.
This no e p o ides a ange o an es ima ed cons an SRTP, as well as a declining
SRTP schedule based on a ce ain y-equi alen app oach. This documen does no
can ass o he app oaches o se ing he public sec o discoun a e.1
C eedy and Passi (2017) and C eedy (2007) explain ha a SRTP can be de i ed
based on a mix o empi ical es ima es and alue judgemen s o decision make s ac ing
on behal o socie y using a ‘social wel a e unc ion’. They would assign ‘wel a e
sco es’ (analogous o u ili y ha an indi idual migh assign o hei own p e e ences)
o di e en ou comes o de e mine which policy op ions a e p e e able o socie y, based
on a se o e hical conside a ions.
1 Mo e can be ound in G imes (2023), Abelson and Dal on (2023), NZIER (2024a), Tu k (2024),
G oom e al (2022), and C eedy and Passi (2017).
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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2. De e mining he anges o pa ame e s
Fo he gene al social wel a e unc ion o all u u e wel a e we used o ou analysis
o be ini e, he discoun a e needs o be g ea e han he g ow h a e in pe pe ui y,
ie, 𝑟𝑟>𝑔𝑔, o ha :2
𝜌𝜌+𝛼𝛼+𝜇𝜇𝑔𝑔 >𝑔𝑔 (2)
We do no place es ic ions di ec ly on each pa ame e as a unc ion o he o he
pa ame e s, bu a he on he o al esul ing SRTP alue. We implici ly assume each
pa ame e alue is no independen by disca ding ce ain combina ions o pa ame e
alues. Fo example, i he pu e a e o ime p e e ence 𝜌𝜌 and he annihila ion isk is
hough o be e y low, hen we would equi e a somewha highe alue o ma ginal
elas ici y o ma ginal wel a e 𝜇𝜇.
Applying his es ic ion is no common in he li e a u e, and no is using Mon e Ca lo
simula ion o iden i y a ange o possible ou comes whe eby he possibili y o 𝑟𝑟<𝑔𝑔
occu ing is mo e appa en . While we a e con iden we should exclude combina ions o
alues whe e 𝑟𝑟<𝑔𝑔, we conside esul s in he analysis bo h wi h and wi hou excluding
d aws whe e 𝑟𝑟<𝑔𝑔.
2.1 Pu e a e o ime p e e ence
The pu e a e o ime p e e ence is highly con en ious and no ma i e; eg, some a gue
i should be ze o (eg, Ramsey 1928) on he basis o “in e gene a ional equi y”. C eedy
and Passi (2017) documen anges be ween 0.1% o 1.5% used by policy making
au ho i ies globally o by well-known wel a e economics s udies. Some o he li e a u e
conside s alues o 0% (eg, HM T easu y 2024, G oom e al 2022). Sense Pa ne s
(2022) p o ides a Māo i economic wo ld iew om he iwi ( ibe) Ngāi Tūhoe,
sugges ing he u u e should no be discoun ed.
D upp e al (2018) su eyed 262 global expe s and 38 pe cen hough i should be
ze o, bu he esponses we e subs an ially igh -skewed wi h a mode o ze o, median
o 0.5%, and a mean o 1.10%. Tu k eplica ed he su ey o New Zealand economis s
and wi hin a g oup o 33 sel -iden i ied expe s ound a mode o 0.1%, median o 1.0%,
and a mean o 1.7%. Based on hese s udies, we assume i lies in he ange 0% o 2%,
and assume a con inuous iangula dis ibu ion o i wi h a mode o 0.25% (implying
a subs an ial igh -skew); his esul s in a mean es ima e o 0.75%.
2 Tha he discoun a e needs o exceed he g ow h a e in pe pe ui y is desc ibed as an ‘economically
na u al assump ion’ in Dixi (1993 p13) and Dixi and Pindyck (1994 p138).
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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Figu e 3: Mon e Ca lo simula ion ou pu ( es ic ed se )
Abou 15% o he d aws we e disca ded om he es ic ed se because he SRTP
was less han he g ow h a e d awn, some imes by as much as 1 pe cen age poin .
Disca ding hese led o he ange o SRTP being highe han wha one migh suppose
by conside ing he anges o alues o he indi idual a iables in isola ion. This is seen
by compa ing he wo ows in Table 4 abo e.
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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4. Declining ‘ce ain y equi alen ’ SRTP
The wo T easu y-commissioned epo s o G imes (2023) and Abelson and Dal on
(2023) e iewed di e en app oaches o declining discoun a es. These expe s
ad ised us (in hese epo s and in wo kshops) ha we use he Wei zman (1998)
app oach o de i e a ce ain y equi alen declining discoun a e o e a ce ain u u e
cash low, which is he app oach aken he e.11 Wei zman’s insigh was i one is
unce ain which discoun a e o use, a he han deduce some a e age o he discoun
a es one should ins ead a e age he p esen alues o a (ce ain) dolla o impac
in he u u e (ie, he discoun ac o s) om each po en ial discoun a e. The equi alen
discoun a e ha gi es he same expec ed ne p esen alue is no he simple a e age
o he po en ial a es bu is weigh ed owa ds he lowe o he po en ial a es. The
u he in ime he discoun ed cash low, he close he ce ain y equi alen discoun
a e is o he lowes po en ial one conside ed. One in ui ion is he la ge a es annhila e
hemsel es quicke , lea ing he lowes o ha e an inc easingly la ge in luence.
Thus he ce ain y equi alen discoun a e declines o e ime.
This app oach is sui able when he e is pe manen disag eemen o unce ain y abou
e hical judgemen s ha lead o he e ogenei y in he app op ia e SRTP o use and
u u e bene i s eams a e ce ain. This, all else equal, se es o lowe he hu dle a e
o long-li ed now o ne e p ojec s.
Gu h ie (2021) conside s how he Wei zman esul applies o in es men s ha a e
i e e sible, ha e iming lexibili y, and whose bene i s eams a e unce ain. He inds
a ia ion in discoun a es also inc eases he op ion alue o delaying in es men ha
o se s — somewha o possibly mo e han — he dec ease in he hu dle a e o
long-li ed asse s. Fu he mo e, when he decision make is calcula ing he op imal ime
o exe cise an op ion hey need o conside wha hey a e likely o do in ha u u e poin
in ime o ensu e decisions a e ‘dynamically consis en ’. A e conside ing he ne e ec
o hese ma e s, he sugges s li le would be los in jus igno ing discoun a e
he e ogenei y o such in es men s. Bu i such he e ogenei y we e o be accoun ed o ,
he shows how o do i in an op ions alua ion model ha ensu es in es men policies
a e dynamically consis en .
The ange o possible ou comes o he SRTP illus a ed in Figu e 3 means ha
unce ain y abou he p esen alue o cons an expec ed bene i s and cos s inc eases
wi h dis ance in ime. Figu e 4 illus a es he discoun ac o 1/(1 + 𝑟𝑟)𝑡𝑡 o each
discoun a e 𝑟𝑟 om Table 4 as a unc ion o ime ho izon o e a 200 yea ho izon.
Lowe discoun a es esul in la e and highe discoun ac o cu es.
11 This app oach was u he cla i ied in Gollie and Wei zman (2010).
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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Figu e 4: Discoun ac o s
We ake he simple a e age o he wo se s o deciles ( om he es ic ed and
un es ic ed se s) o ind an a e age discoun ac o — he o ange cu e in Figu e 5.
Figu e 5: A e age discoun ac o
0
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0.76% 1.40% 1.58% 1.73% 1.87% 2.01%
2.16% 2.33% 2.54% 2.82%4.86% 0.30%
1.19% 1.40% 1.58% 1.74% 1.90% 2.06%
2.25% 2.46% 2.76% 4.86%
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A e age discoun ac o
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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The ‘ma ginal discoun a e’ is he one ha applies o he las yea o cash lows (also
known as he ins an aneous discoun a e, eg, C oppe e al 2014). Any declining
discoun a e (DDR) schedule should apply a he ma gin, a he han a he a e age,
so ha he discoun ac o cu e is smoo h and mono onically dec easing. Fo example,
he HM T easu y’s (2008) schedule o discoun a es is 3.5% o yea s 1-30 and 3% o
yea s 31-75. I a p ojec had, say, 31 yea li e, only he 31s yea is discoun ed a 3%.
Tha is, he HM T easu y’s ma ginal discoun a e be ween yea s 31-75 is 3%.
Figu e 6 illus a es bo h, wi h he (black) a e age ‘ce ain y equi alen ’ SRTP being he
single cons an alue ha would gi e he a e age discoun ac o as a ha yea . The
(blue) ma ginal ce ain y equi alen SRTP is calcula ed as he a e o change o he
a e age discoun ac o wi h espec o ime — i lies unde he a e age SRTP since
he a e age is educing.
Ra he han issue guidance on a con inuously declining public sec o discoun a e
we es ima e a simple schedule o changes, much like he HM T easu y in he Uni ed
Kingdom and inc easingly o he coun ies (D upp e al 2018). This schedule aims o
oughly app oxima e he blue ma ginal ce ain y equi alen SRTP, and is plo ed as he
g ey cu e, using ai ly ound numbe s o bo h he SRTP and he pe iod leng hs o
simplici y. I we s a wi h he a e age o abou 2% and keep ha cons an o 30 yea s
and hen educe he ma ginal SRTP o 1.5% o yea s 31 o 100, hen 1% he ea e ,
we ind he esul ing discoun ac o cu e ( he g ey cu e in Figu e 7) is close enough
o ha equi ed ( he o ange).
The declining discoun a e schedule leads o sligh unde -discoun ing a e yea 30.
Howe e , wo mino conside a ions we excluded om ou analysis migh ha e educed
he discoun a es in Figu e 6 by abou 0.15%–0.2% (inc easing inequali y in u u e
ollowing Emme ling e al (2017) discussed in oo no e 13 o maybe 0.16%, and he
adjus men by Gollie (2013) discussed in Appendix 1 o maybe up o 0.05%). I hese
wo adjus men s we e added hen he declining discoun a e schedule would be a e y
close app oxima ion.
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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Figu e 6: Ma ginal and a e age ce ain y equi alen SRTPs
Figu e 7: Compa ing discoun ac o s
0.80%
0.85%
0.90%
0.95%
1.00%
1.05%
1.10%
1.15%
1.20%
1.25%
1.30%
1.35%
1.40%
1.45%
1.50%
1.55%
1.60%
1.65%
1.70%
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1.80%
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1.95%
2.00%
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2.20%
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Discoun a e
Yea s
CE (a e age) discoun a e CE (ma ginal) discoun a e DDR schedule
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A e age discoun ac o Discoun ac o wi h DDR
WP 25/01 | De i ing alues o he social a e o ime p e e ence
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This declining SRTP schedule is summa ised in Table 5. No e, mos ou ine policy
ini ia i es in p ac ice may no ha e impac s a e 30 yea s; only highly signi ican and
ans o ma i e ini ia i es ha do no ha e oo much lexibili y in wai ing likely will be
ma e ially impac ed by a declining SRTP.
Table 5: Declining SRTP schedule
Schedule o declining SRTP Yea s
2% 1-30
1.5% 31-100
1% 101+
This declining SRTP leads o somewha signi ican di e ences in he discoun ac o s
in he e y long- e m, o help be e e lec in e gene a ional issues, so i is pe haps
wo h using o in e gene a ional issues. Figu e 8 augmen s Figu e 7 by including he
discoun ac o wi h a cons an SRTP o 2%. The e is an inc easing gap be ween he
g ey discoun ac o cu e wi h a declining SRTP and blue discoun ac o cu e wi h
a cons an SRTP; a discoun ac o o 0.1 is pushed ou 50 yea s, om abou yea 120
o yea 170. The sum o a uni o m cash low o $1 o e 200 yea s is abou $50 a a
cons an 2% discoun a e, and abou $59 wi h a declining discoun a e, which is a
19% inc ease. The impac o a declining SRTP would be p opo ionally much g ea e
i an ini ia i e had g owing bene i s o e he long- e m.
Figu e 8: Discoun ac o s wi h and wi hou declining SRTP
0.0
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105
110
115
120
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A e age discoun ac o DF wi h DDR No DDR
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Appendix 1 – Ma hema ical basis o
SRTP
The e a e wo key dimensions we will abs ac away om in he analysis ha ollows.
The i s is ha goods and se ices could a y be ween ma ke and non-ma ke i ems,
wi h a pa icula dis inc ion be ween na u al en i onmen al a ibu es and gene al
ma ke consumables. The second is indi idual ci izens can ha e as ly di e en
bundles o goods and se ices ha hey consume also, wi h inequali y be ween he ich
and poo . He e we suppose we can simply add oge he he alue o agg ega e pe
capi a consump ion 𝑐𝑐𝑡𝑡 in each pe iod 𝑡𝑡 o all kinds o goods and se ices o all people.
This means a e sion o he decision make o wi hin-pe iod inequali y is igno ed, and
we do no he e conside di e en in e empo al ea men s o na u al en i onmen
e sus o he commodi ies.
These wo simplica ions may no g ea ly a ec wha ollows. I a pa icula na u al
species we e o be inc easingly h ea ened wi h ex inc ion his could be ep esen ed
h ough an escala ing eal p ice assigned o each i em o ha species, and so be
implici ly cap u ed in 𝑐𝑐𝑡𝑡. Howe e , he e could be a heo e ical basis o conside ha ing
di e en discoun a es ac oss ca ego ies o consump ion (G imes 2023). I c oss-
sec ional ma e s o inequali y, inequi y, and dis ibu ional jus ice we e impo an
(as ou lined in Thompson 2022) hen he decision make may be able o weigh he
impac s o consump ion ac oss people di e en ly when de i ing he 𝑐𝑐𝑡𝑡 e m.12 Howe e ,
i ma e s o inequali y, inequi y, and dis ibu ional jus ice we e bo h c oss-sec ional and
occu ed o e ime (such as in e gene a ional po e y aps / p i ilege) hen mo e
gene al app oaches may be equi ed.13
A policy in e en ion may change he amoun o o al consump ion o e ime, and he
pu pose o he judgemen is whe he ha is p e e able o no in e ening. The decision
make may compa e he social wel a e in each s a e o de e mine which sco es highe .
12 This is easible o ‘wel a is ’ o consequen ionis heo ies o dis ibu i e jus ice, which assume ha
ou comes (and hei dis ibu ion) a e mo ally ele an . These heo ies include u ili a ianism (maximising
summed u ili ies), maxi-min (maximise he wel a e o he wo s -o pe son), p io i a ianism (weigh ing
he wo se-o highe ), and su icien a ianism (ensu ing all ha e enough u ili y). Weigh ing would no
add ess equi y acco ding o o he heo ies o jus ice ha ake ac o s o he han wel a e and i s
dis ibu ion o be mo ally ele an . These o he non-consequen ionis heo ies include luck
egali a ianism (equal oppo uni y a he han equal ou comes is impo an ), ela ional egali a ianism
(ensu ing he mo al equali y o people), and libe a ianism (jus and ai p ocesses and ac ions). These
la e h ee seem o i mos wi h he ways New Zealande s hink abou ai ness (NZIER 2024b).
13 Fo example, Emme ling e al (2017) adjus ed he SRTP o g owing in a-pe iod inequali y, which is
akin o educing he g ow h a e in consump ion. New Zealand was no modelled, bu o Aus alia hey
es ima e he SRTP would be 0.16% poin s lowe i 𝜇𝜇= 1 gi en he as e a e o g ow h o mean
household income han median household income.
