Working Paper
Notes on the Mfects of Cohort Size on Intergenerational Transfer
Robin Cowan
J a n u a r y 1986 WP-86-3
International Institute for Applied Systems Analysis
A-2361 Laxenburg, Austria
NOT FOR QUOT.4TION
WITHOUT THE PERMISSION OF THE AUTHOR
Notes on the Hfects of Cohort Size on Intergenerational Transfer
R o b i n C o w a n
J a n u a r y 1986 WP-86-3
This r e s e a r c h was conducted in conjunction with a summer r e s e a r c h s e m i n a r on h e t e r o g e n e i t y dynamics, u n d e r t h e d i r e c t i o n of James W.
Vaupel and Anatoli I. Yashin, in t h e Population P r o g r a m at IIASA led b y Nathan Keyfitz. .
W o r k i n g P a p e r s a r e interim r e p o r t s on work of t h e I n t e r n a t i o n a l I n s t i t u t e f o r -4pplied Systems Analysis a n d h a v e r e c e i v e d only limited review. Views o r opinions e x p r e s s e d h e r e i n d o not n e c e s s a r i l y r e p r e s e n t t h o s e of t h e I n s t i t u t e o r of i t s National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 L a x e n b u r g , Austria
Foreword
A group of eleven Ph.D. candidates from seven countries--Robin Cowan, An- drew Foster, Nedka Gateva, William Hodges, Arno Kitts, Eva Lelievre, Fernando Rajulton, Lucky Tedrow, Marc Tremblay, John Wilmoth, and Zeng Yi--worked togeth- e r a t IIASA from June 17 through September 6, 1985, in a seminar on population heterogeneity. The seminar w a s led by t h e two of us with t h e help of Nathan Key- fitz, leader of t h e Population Program, and Bradley Gambill, Dianne Goodwin, and Alan Bernstein, r e s e a r c h e r s in t h e Population Program, as well as t h e occasional participation of guest scholars at IIASA, including Michael Stoto, Sergei Scherbov, Joel Cohen, Frans Willekens, Vladimir Crechuha. and G e e r t Ridder. Susanne Stock, o u r s e c r e t a r y , and Margaret T r a b e r managed t h e seminar superbly.
Each of t h e eleven students in t h e seminar succeeded in writing a r e p o r t on t h e r e s e a r c h they had done. With only one exception, t h e students evaluated t h e seminar as "very productive"; t h e exception thought i t w a s "productive". The two of us agree: t h e quality of t h e r e s e a r c h produced exceeded o u r expectations and made t h e summer a thoroughly enjoyable experience. W e were particularly pleased by t h e i n t e r e s t and s p a r k l e displayed in o u r daily, hour-long colloquium, and by t h e s p i r i t of cooperation all t h e participants. both students and more senior r e s e a r c h e r s , displayed in generously sharing ideas and otherwise helping each other.
Robin Cowan succeeded in producing two p a p e r s o v e r t h e course of t h e sum- mer, t h e present p a p e r on how cohort size affects total lifetime consumption being one of them. In i t , Cowan develops a model, cleverly contrived t o shed Light on a darkly complex issue.
Anatoli I. Yashin James W. Vaupel
-
iii-
Acknowledgments
The author wishes t o acknowledge many useful conversations with Brian Arthur.
Notes on the Ettects of Cohort Size on Intergenerational Transfer
Robin Cowan Food Research Institute
Stanford University Stanford, California 94305
USA
Introduction
A s a member of t h e baby boom, I would like t o b e able t o blame my poverty on t h e bad luck of having been born in a l a r g e cohort, r a t h e r than or! pome inability I suffer in t h e money-making department. This excuse is only viable, though, if t h e r e is a systematic relationship between cohort size and p e r capita lifetime earnings. Arthur (1984) describes a world in which such a relationship does exist.
A s my l a r g e cohort t r i e s t o move up through t h e job hierarchy pyramid, t h e r e will b e a job market squeeze, and unless people before us r e t i r e early, many members of my cohort will find t h a t t h e i r advancement is slower than otherwise might b e ex- pected. My concern in this p a p e r i s not with t h e job market, however. I t is r a t h e r with t h e relationship between cohort size and intergenerational transfers. Arthur and McNicoll (1978) showed t h a t in a regime of stable populations. under typical mortality and fertility schedules, t h e higher t h e growth rate, t h e lower t h e individual's welfare. Analysis of stable populations cannot c a p t u r e t h e baby boom phenomenon though as a baby boom is very much a n example of a non-stable popu- lation. If t h e r e is a systematic relationship between cohort size and p e r capita in- come, one effect of phenomena like baby booms is t o c r e a t e inequalities in living standards between generations. Keyfitz (1985) t r i e s
to
devise ways in which t o el- iminate inequalities arising from intergenerational t r a n s f e r s . He discusses t h r e e t r a n s f e r mechanisms which equilibrate t h e quasi-interest rates which cohorts see as t h e r e t u r n on t h e money t h e pay into social security. His method involves pro- jections of both birth and death rates. My interest i s more general than this though. I wish to ask t h e question 'What would b e t h e effect on t h e net t r a n s f e r s of a cohort if i t were of a size different than i t actually is?" A s one might expect, t h eanswer depends o n t h e n a t u r e of t h e t r a n s f e r mechanism, and I will discuss a case in which t h e answer is, in f a c t . "Nothing." In t h i s c a s e , t h e r e i s equity between t h e generations, but t h i s equity i s not t h e focus of t h e e x e r c i s e .
In t h i s p a p e r I will d e s c r i b e a framework designed t o answer c e r t a i n questions about how c o h o r t size a f f e c t s total lifetime consumption. I will use a n overlapping generations model in which intergenerational t r a n s f e r s are t h e mechanism through which changes in c o h o r t size a f f e c t consumption levels. The model will seem r a t h e r contrived with r e s p e c t to many economies, in t h a t t h e state will intermediate a l l t r a n s f e r s . This assumption is not c r u c i a l to t h e r e s u l t s , as I will a r g u e later, but i t does facilitate discussion.
A Welfare State
Each individual b o r n in t h i s mythical economy h a s t h r e e distinct s t a g e s of life.
From b i r t h to a g e z -1 h e leads t h e life of a child, producing nothing and being s u p p o r t e d by his p a r e n t s who r e c e i v e baby-bonus cheques from t h e government which exactly o f f s e t ail child-rearing expenses. F o r purposes of exposition, I will s p e a k as if t h e child r e c e i v e s t h e s e cheques directly. From a g e z to a g e y -1 h e works and r e c e i v e s nothing f r o m t h e state e x c e p t t a x bills. Over t h i s time his pro- duction can b e d e s c r i b e d by a n age-earnings profile which will b e called f ( ~ ) where a is h i s age. During t h i s period of life h e also pays t a x e s which go solely to s u p p o r t t h o s e who are not in t h e working s t a g e of life. A t a g e y h e r e t i r e s and produces nothing. The government pays him a n old a g e pension on which h e lives until h i s dying day. The government h a s only one r o l e , namely to collect t r a n s f e r payments f r o m t h o s e working and d i s p e r s e t r a n s f e r r e c e i p t s to t h o s e not working.
This i t does costlessly, and o p e r a t e s only u n d e r t h e c o n s t r a i n t t h a t i t s budget must b e balanced at all times.
I now introduce a c o h o r t in whose welfare w e will b e interested. I t i s b o r n at time z e r o , and i s of size L . The c o h o r t b o r n just b e f o r e i t w a s b o r n at time -1 and i s of size L t h e c o h o r t b o r n just a f t e r i t will b e b o r n at time 1 and will b e of size L No one s a v e s in t h i s economy, so in e a c h period (i) of Its life a c o h o r t con- sumes e i t h e r i t s t r a n s f e r r e c e i p t s , Ri or i t s a f t e r - t a x production, t h a t is, produc- tion less t r a n s f e r payments, Qi - P i . W e should note t h a t t h e production of o u r c o h o r t in period i i s Qi
=
L.
f (i ).The t o t a l lifetime consumption of o u r c o h o r t c a n b e e x p r e s s e d a s a sum:
where o is t h e a g e a t d e a t h (which equals time of d e a t h f o r o u r cohort). Since t h e time when t h e c o h o r t is receiving is distinct from t h e time when i t is producing and paying, w e c a n divide i t s consumption into distinct p a r t s :
For simplicity, I will refer t o t h e time when i t is receiving a s A, and t h e time when i t is producing a s A C
.
S oThe government plays t h e c e n t r a l r o l e in determining t h e magnitude of t o t a l t r a n s f e r s e a c h period. E v e r y period, i t considers t h e t o t a l number of people who a r e receiving t r a n s f e r s t h a t period, L;, and t h e a g g r e g a t e output of t h e period, TQi, t o determine t h e t o t a l quantity t h a t should b e t r a n s f e r r e d ,
T q .
Clearly, t h e number of r e c i p i e n t s must b e considered, since if i t changes, t h e value of t r a n s f e r s n e c e s s a r y t o s u p p o r t them a t a given level a l s o changes, and in t h e same direction. T h e r e is nothing t o s a y t h a t t h e level of individual s u p p o r t must b e fixed, but I will assume i t bounded; below by subsistence, and above by some level of luxury. This is enough t o make L; a n argument. I will assume as well t h a t i t i s a"nice" argument. That is t o s a y t h a t
- am,
i s finite and continuous everywhere.a q
Total output is considered by t h e government since i t is t h e t a x b a s e . The l a r g e r t h e t a x base, t h e g r e a t e r t h e quantity (though not necessarily t h e proportion) t h a t c a n b e e x t r a c t e d . Each period then, t o t a l t r a n s f e r s c a n b e written as
mi =
TT(TQ~ ,L;).Having determined t o t a l t r a n s f e r s , t h e government must t h e n decide t h e s h a r e of t o t a l r e c e i p t s (payments) t h a t e a c h receiving (paying) c o h o r t i s t o b e assigned.
In making t h i s assignment, i t d o e s not d i f f e r e n t i a t e between people by a g e , and ig- n o r e s o t h e r potential differentiating c h a r a c t e r i s t i c s . A s a r e s u l t , r e c e i p t s are distributed t o a c o h o r t according t o t h e r a t i o of i t s size t o t h e t o t a l number of re- cipients. S o when o u r c o h o r t is receiving (during time A) in e a c h period i i t re- c e i v e s
-
L xq .
On t h e o t h e r hand, individuals paying can b e distinguished by t h e=I
q u a n t i t i e s t h e y p r o d u c e . The government u s e s t h i s information, a n d when o u r c o h o r t i s paying (on A C ) i t p a y s Pi
= -
Qi x e a c h p e r i o d , w h e r e TQi i s t h e ag-TQi
g r e g a t e p r o d u c t i o n of t h e economy at time i . Now t h e t o t a l consumption of o u r c o h o r t c a n b e w r i t t e n as
or m o r e explicitly,
Dividing t h r o u g h by t h e s i z e of o u r c o h o r t L , we g e t lifetime consumption p e r c a p i t a ,
To find t h e e f f e c t of c o h o r t s i z e o n t o t a l lifetime consumption, simply d i f f e r e n t i a t e with r e s p e c t t o L
.
A f t e r simplifying (and noting t h a t ?Ti
=
T T ( L ~ , T Q ~ ) , we g e tAs e x p e c t e d , c o h o r t s i z e a f f e c t s consumption in b o t h s t a g e s of life, a n d in b o t h cases t h e r e are positive a n d n e g a t i v e e f f e c t s . The f i r s t t e r m r e p r e s e n t s t h e e f f e c t of c o h o r t s i z e o n t r a n s f e r r e c e i p t s . When o u r c o h o r t i s r e c e i v i n g , t h e positive e f - f e c t may b e s e e n as a marginal e f f e c t : Adding a p e r s o n r a i s e s t h e t o t a l quantity of t r a n s f e r s . The n e g a t i v e e f f e c t i s a n a v e r a g e effec'~: T h a t t h e r e i s o n e m o r e p e r s o n means t h a t w h a t e v e r t r a n s f e r r e c e i p t s are a v a i l a b l e are s p r e a d among m o r e peo- ple. Clearly, t h e n e t e f f e c t o n t r a n s f e r r e c e i p t s d e p e n d s o n how t h e quantity of to-
t a l t r a n s f e r s r e s p o n d s t o c h a n g e s in t h e number of r e c i p i e n t s . If
-
i s a de-a ~ , t
c r e a s i n g function of L;, as seems likely t o b e t h e c a s e , t h e n t h e effect on p e r capi- ta t r a n s f e r r e c e i p t s , - o f increasing t h e c o h o r t , size i s negative. The second sum in
a(
C,)
aL r e p r e s e n t s t h e effect of c o h o r t size on t r a n s f e r payments. H e r e , e f f e c t s of population a r e transmitted t h r o u g h t h e i r e f f e c t s on production. The positive ef- f e c t i s t h e a v e r a g e e f f e c t : Individual payments a r e smaller as t h e r e are more peo- ple contributing t o t h e t r a n s f e r fund. The negative e f f e c t c a n b e s e e n a s marginal:
A l a r g e r c o h o r t means t h a t i t s production, and s o t o t a l production, will b e l a r g e r . This will h a v e a positive e f f e c t on t h e size of t o t a l t r a n s f e r s . H e r e , whether t h e n e t effect is positive o r negative depends on how t o t a l t r a n s f e r s responds t o t o t a l output.
The n e t r e s u l t of changing t h e size of a c o h o r t is, in g e n e r a l , indeterminate.
T h e r e a r e some s p e c i a l cases, however, where t h e t r a n s f e r mechanism i s of a t y p e f o r which determinate r e s u l t s c a n b e obtained.
In t h e f i r s t case, suppose t h a t in calculating t h e quantity of t o t a l t r a n s f e r s , t h e government maintains a fixed t a x rate and pays n o attention t o t h e t o t a l number of recipients. H e r e t r a n s f e r s are always a fixed p r o p o r t i o n , a , of t o t a l output, and so o u r c o h o r t , in e a c h period when i t i s paying, pays 1 0 0 . a p e r c e n t of i t s production into t h e t r a n s f e r fund. This i s t h e only qualification to t h e g e n e r a l scheme d e s c r i b e d above. H e r e , t h e r e c e i p t s in period i (on A) f o r o u r c o h o r t are Ri
=
- ( a . L T Q i ) . On A t production i s unchanged, and t h e payments o u r c o h o r tL5 makes are
S o f o r o u r c o h o r t , t o t a l lifetime consumption p e r c a p i t a becomes
C 1
- = x
a T Q i ~+ x
f ( i ) ( l - a ).
=
i d Li id'And, differentiating with r e s p e c t to t h e size of t h e c o h o r t
n
which is unambiguously negative. This r e s u l t c o r r e s p o n d s with o u r intuitions: If e a c h individual c o n t r i b u t e s a fixed p r o p o r t i o n of his income t o t h e t r a n s f e r fund, t h e n a l a r g e c o h o r t , when receiving, cannot a f f e c t t h e size of t h e fund, s o t h e same quantity is divided among more people. On t h e o t h e r hand, a f t e r t a x production i s constant, s o p e r c a p i t a consumption on A C i s unchanged. Notice also t h a t under t h i s scheme t h o s e receiving when t h e l a r g e r c o h o r t i s receiving are h u r t by i t s size, but those receiving when t h e l a r g e c o h o r t i s producing gain by i t s size. (For t h e c o h o r t s v e r y n e a r in b i r t h d a t e t o o u r c o h o r t t h e n e t e f f e c t will a l s o b e nega- tive though less s o t h a n f o r o u r c o h o r t . )
The second case with a n unambiguous r e s u l t i s t h e situation in which t h e government g u a r a n t e e s a c e r t a i n s t a n d a r d of living, call i t 8 , f o r t h o s e receiving.
In t h i s case t h e size of o u r c o h o r t i s positively r e l a t e d t o i t s welfare. H e r e , o v e r time A , when i t i s receiving, o u r c o h o r t r e c e i v e s Ri
=
BL. On A C production i s un- changed, but payments are Pi= -
QiTQi x BL;. Total lifetime consumption p e r c a p i t a c a n b e written as
Again, differentiating with r e s p e c t t o L yields a ( y ) C
-- - C
m i xJ
(i )2aL t a l c T Q ~ '
which i s positive. A l a r g e r c o h o r t means t h a t t o t a l production r i s e s . The number of r e c i p i e n t s d o e s not c h a n g e , however, s o e a c h p r o d u c e r i s r e q u i r e d t o pay l e s s in o r d e r t o provide s u p p o r t at t h e p r e s c r i b e d level t o those receiving.
The final unambiguous case which I will d e s c r i b e i s one in which c o h o r t size h a s no e f f e c t on lifetime consumption. Again, t h i s case c a n b e d e s c r i b e d simply by specifying t h e t o t a l t r a n s f e r s function. If w e specify t h a t t o t a l t r a n s f e r s are cal- culated by
=
K X L; X TQi, where K i s a constant, t h e nLikewise,
From equation (1) (page 4), w e see t h a t under this condition, c o h o r t size h a s no ef- f e c t on lifetime consumption. The major drawback of such a scheme, were i t t o b e implemented in o u r world, is t h a t t h e p a t t e r n of consumption o v e r a n individual's lifetime is determined ( a p a r t from his age-earnings profile) by t h e s i z e s of t h e c o h o r t s which are alive when h e is. This means t h a t , at l e a s t in principle, i t is pos- sible f o r someone t o spend p a r t of his life in t h e l a p of luxury, and p a r t of i t below subsistence. (That's a l l r i g h t I suppose, a s long as they come in t h a t o r d e r . )
Discussions and Extensions
I s t a t e d e a r l i e r t h a t t h e l a r g e r o l e which t h e state plays in t h i s economy i s not c r u c i a l t o t h e r e s u l t s of t h e model. That t h e state play some r o l e i s important, but not t h a t i t play t h e all-encompassing r o l e d e s c r i b e d in t h e model. I claim t h a t t h e t r a n s f e r mechanism d e s c r i b e d h e r e is not grotesquely d i f f e r e n t in effect from what w e might o b s e r v e in a n economy with l e s s government intervention. This i s c l e a r l y s o f o r t r a n s f e r s t o t h e young, which a r e generally p a r e n t t o child t r a n s f e r s . If a couple h a s more c h i l d r e n , t h e y are s u r e l y going t o spend more in t o t a l on t h e i r children. If p a r e n t s find themselves with more income, again i t seems v e r y likely t h a t t h e y will spend more on t h e i r children. One word of caution: The model d e s c r i b e s a world in which t h e production of t h e economy a s a whole a f f e c t s t h e size of t r a n s f e r s . I t is possible t h a t t h e economy could g e t r i c h e r without p a r e n t s of receiving children getting r i c h e r . I would a r g u e as follows: Most economies pro- vide many s e r v i c e s o u t of g e n e r a l t a x revenues: public education; defense; t r a n - s p o r t a t i o n system subsidies a n d s o on. If t h e s e s e r v i c e s a r e used by t h e non- producing p a r t of t h e population, then t h e y are a t r a n s f e r . The level of t h e s e ser- vices i n c r e a s e s as t h e production of t h e economy i n c r e a s e s . With r e g a r d t o t r a n s f e r s t o t h e r e t i r e d , f o r any economy with a n old a g e s e c u r i t y program, t h e model needs no additional explanation. But even s o , i t seems v e r y likely t h a t any intra-family t r a n s f e r s t o t h e r e t i r e d will b e a function both of t h e number of r e c i - pients (both p a r e n t s o r only one) and t h e income of t h o s e paying.
T h e r e a r e two obvious ways in which t h i s model c a n b e extended. The f i r s t h a s t o d o with t h e n a t u r e of production in t h i s economy. Until now w e h a v e assumed t h a t production i s l i n e a r in labour. That is, changing t h e s i z e of a c o h o r t does not c h a n g e i t s a v e r a g e p r o d u c t , j ' ( a > . However Finis Welch (1979) h a s shown t h a t t h e age-earnings profile of t h e baby-boom generation i s lower t h a n one would have ex- pected had t h a t g e n e r a t i o n not been s o l a r g e , and t h i s r e s u l t c a n b e easily incor-
p o r a t e d into t h e model. The second extension is t o t h e r e s u l t s r a t h e r t h a n t o t h e model. The model c a n b e used t o answer t h e question 'What would b e t h e effect on lifetime consumption of o u r c o h o r t if some o t h e r c o h o r t w e r e of a size o t h e r t h a n i t actually is?"
I will f i r s t a d d r e s s t h e case of non-linear production. The model changes only in t h a t c o h o r t size i s now included as a n argument in t h e age-earnings profile. In o r d e r t o make t h e problem t r a c t a b l e , I will assume t h a t t h e size of a c o h o r t h a s n o effect on t h e age-earnings profiles of o t h e r c o h o r t s . Now J" ( a ) becomes J" ( a ,L ).
aJ"
I t seems r e a s o n a b l e t o e x p e c t t h a t
-
i s negative in sign. If w e a c c e p t diminishing aLmarginal p r o d u c t , t h e n as c o h o r t size i n c r e a s e s , a v e r a g e p r o d u c t , which i s o n e way t h a t J"(.) c a n b e viewed, must fall. P e r c a p i t a lifetime consumption does not change from t h e simple model, b u t t h e f i r s t d e r i v a t i v e h a s added terms:
A s in t h e simple model, t h e sign is in g e n e r a l indeterminate, depending h e r e not only on t h e t r a n s f e r mechanism, but a l s o on t h e responsiveness of a v e r a g e p r o d u c t t o changes in c o h o r t size. Again, however, t h e r e are s p e c i a l cases in which t h e r e s u l t s are determinate.
In t h e f i r s t special case discussed in t h e simple model, t o t a l t r a n s f e r s were not a f f e c t e d by t h e number of r e c i p i e n t s , but were a l i n e a r function of a g g r e g a t e production. In t h i s c a s e ,
Under t h i s t r a n s f e r mechanism, t h e d e r i v a t i v e of lifetime consumption with r e s p e c t t o c o h o r t size h a s a n added t e r m in t h e extended model:
*
- - - C --
a TQiBL i d L;2 +
C
% , - a ) .i d ' aL
This r e s u l t , as in t h e simple model, i s unambiguously negative. If o u r c o h o r t i s l a r g e , t h e fixed quantity of available t r a n s f e r s i s divided among more people, s o
t h e c o h o r t i s h u r t . A s well, during t h e time when o u r c o h o r t is producing, t h e i r a v e r a g e product, of which t h e y e a c h consume t h e p r o p o r t i o n ( 1 -a), i s driven down. Under t h i s t r a n s f e r mechanism t h e n , t h e negative effect of being b o r n in a l a r g e c o h o r t i s more s e v e r e t h a n i t i s in t h e simple model.
The second s p e c i a l case specified t h a t t o t a l t r a n s f e r s w e r e not a f f e c t e d by t h e level of a g g r e g a t e production, but w e r e a l i n e a r function of t h e number of re- cipients:
Under t h i s t r a n s f e r scheme, c o h o r t size h a s n o effect on p e r c a p i t a t r a n s f e r r e c e i p t s , e i t h e r in t h e simple model o r in t h e extended model. Cohort size does af- fect both p e r c a p i t a production and t r a n s f e r payments in t h e extended model how- e v e r . H e r e t h e f i r s t d e r i v a t i v e i s
Noting t h a t (f(i .L)
+
L S ) i s equal t o t h e d e r i v a t i v e of a g g r e g a t e output with Cami a-
r e s p e c t t o c o h o r t size,
-
aL , w e see t h a t t h e sign of
-
i s ambiguous. (It is safe aLt o assume, I think, t h a t a g g r e g a t e output r i s e s if t h e size of a producing c o h o r t in- c r e a s e s . ) A s in t h e simple model, w e h a v e a positive effect--the second t e r m d e s c r i b e s t h e gain from having more p r o d u c e r s among whom t o divide a fixed quan- t i t y of t r a n s f e r payments. This effect is smaller t h a n in t h e simple model, though, s i n c e a v e r a g e p r o d u c t i s d r i v e n down by t h e e x t r a c o h o r t members. The f i r s t t e r m r e p r e s e n t s t h e p e r c a p i t a loss of after t a x production due t o t h e growth of t h e c o h o r t . Under t h e s e conditions, t h e h i g h e r i s t h e r a t i o of t o t a l t r a n s f e r s t o a g g r e - g a t e production, t h e l a r g e r t h e positive effect r e l a t i v e t o t h e negative effect.
In t h e original model t h e r e w a s a v e r y simple t r a n s f e r mechanism which would g u a r a n t e e t h a t n o c o h o r t gained o r lost solely due t o i t s size, viz. setting t o t a l t r a n s f e r s t o TTi
= K
xLtf x TQi. Under t h i s mechanism, at e a c h s t a g e of a c o h o r t ' s life, whether receiving o r paying, t h e n e t gain from changes in c o h o r t size i s zero. Indeed, t h i s i s t r u e at e v e r y period in i t s life. In t h e world where a v e r a g e product i s a function of c o h o r t size, t h i s can still b e a r r a n g e d without dif- ficulty f o r t h e p e r i o d s where t h e c o h o r t i s receiving ( i € A ) . Simply devise at r a n s f e r mechanism t h a t i s l i n e a r in L;. Unfortunately i t i s m o r e difficult f o r t h e p e r i o d s when t h e c o h o r t i s producing a n d paying
(i
E A t ) . The same a p p r o a c h would involve s e t t i n g t h e summand of t h e s e c o n d t e r m of equation (2) e q u a l t o z e r o :If w e assume t h a t
- am, a
TQc f a l l s as TQ r i s e s , a n d t h i s seems p e r f e c t l y r e a s o n a b l e ,t h e n in o r d e r f o r t h i s equation t o h a v e a solution, t h e n a t u r e of p r o d u c t i o n must b e s u c h t h a t f
(i .L) + L~ >
0. But w e h a v e o b s e r v e d t h a t f(i
.L)+
L- af i s equal t oa~ a~
t h e d e r i v a t i v e of a g g r e g a t e o u t p u t with r e s p e c t t o c o h o r t size. If w e c a n assume t h a t t h e marginal p r o d u c t of l a b o u r n e v e r g o e s t o z e r o , t h e n t h i s quantity i s always positive, a n d t h e equation i s , in p r i n c i p l e , solvable, a n d s o w e c a n find a t r a n s f e r scheme t h a t i s n e u t r a l t o c h a n g e s in c o h o r t size.
I t u r n now t o t h e question of o t h e r c o h o r t e f f e c t s . The model c a n b e used t o examine t h e e f f e c t o n o u r c o h o r t of c h a n g e s in s i z e of o t h e r c o h o r t s . F o r t h i s e x - e r c i s e I will retreat t o t h e simple model. I n c o r p o r a t i n g t h e e f f e c t of c o h o r t s i z e o n a v e r a g e p r o d u c t i s n o t a problem, b u t i t makes t h e p r e s e n t a t i o n c o n s i d e r a b l y l e s s t r a n s p a r e n t . The time of b i r t h of t h e o t h e r c o h o r t will e f f e c t i v e l y divide t h e life of o u r c o h o r t i n t o s e v e r a l s t a g e s . These s t a g e s c a n b e c h a r a c t e r i z e d by t h e activi- t i e s of t h e two c o h o r t s . In e a c h s t a g e t h e e f f e c t of t h e s i z e of t h e o t h e r c o h o r t o n o u r c o h o r t will t a k e o n a p a r t i c u l a r n a t u r e . I give d e t a i l e d examples of two dif- f e r e n t times of b i r t h f o r t h e o t h e r c o h o r t . T h e r e a r e , of c o u r s e , many t y p e s of
a-
Cb i r t h times, but f o r e a c h t y p e , t h e g e n e r a l p r o c e d u r e of determining
-
i s t h e 6Lsame.
S u p p o s e w e a r e i n t e r e s t e d in t h e e f f e c t on t h e t r a n s f e r s of o u r c o h o r t of a c h a n g e in s i z e of t h e c o h o r t b o r n s p e r i o d s after o u r s . These p e o p l e are s p e r i o d s y o u n g e r , a n d t h e i r c o h o r t i s of s i z e L,. The life of o u r c o h o r t c a n b e divided into s i x s t a g e s :
1)
when o u r c o h o r t i s a l i v e b u t c o h o r t C, i s not--from p e r i o d z e r o t o p e r i o d2) when both c o h o r t s are a l i v e a n d both a r e receiving--periods s t o x -1;
3) when o u r c o h o r t is producing and C, is receiving--periods x to x +s -1;
4) when both c o h o r t s are producing--periods x +s to y -1;
5) when o u r c o h o r t is receiving and C, i s producing--periods y t o y +s -1;
6 ) when both c o h o r t s are again receiving--periods y + s to w.
Consequently, p e r c a p i t a lifetime consumption c a n b e divided into t h e s e six s t a g e s , as can t h e e f f e c t s of size of c o h o r t C,. If w e t a k e t h i s d e r i v a t i v e w e get:
These s t a g e s only apply when s
<
x . Clearly, if s=
0, s t a g e s 3) and 5) d i s a p p e a r , and w e h a v e t h e original r e s u l t s . Also if s i s small, t h e c r o s s e f f e c t s will b e simi- lar to t h e own-cohort e f f e c t , p a r t i c u l a r l y if t h e age-earnings p r o f i l e i s relatively f l a t .A s opposed to examining t h e s e g e n e r a l r e s u l t s in detail, i t may b e more in- s t r u c t i v e t o look at t h e f i r s t two s p e c i a l cases. Recall t h a t in t h e f i r s t case t o t a l t r a n s f e r s were a l i n e a r function of a g g r e g a t e output
=
aTQi. Under t h i sa r r , a r r , rr, a-
Ct r a n s f e r mechanism
-
= 0 , and- =-
, and so many t e r m s in-
disap-a ~ g
aTQi TQi aLsp e a r . In s t a g e s 1 , 3 and 4 of t h e l i f e of o u r c o h o r t , t h e size of c o h o r t Cs h a s n o ef- f e c t on consumption. S o w e c a n write
The f i r s t and t h i r d sums a r e o v e r periods when both c o h o r t s are receiving. The middle sum i s o v e r p e r i o d s when o u r c o h o r t i s receiving but t h e o t h e r i s produc- ing. H e r e , a n added member of C, i n c r e a s e s t o t a l t r a n s f e r s by aJ ( i
-
s ) e a c h p e r i o d , of which e a c h member of o u r c o h o r t r e c e i v e s-.
1 This i s a positive e f f e c t ;=S
t h e o t h e r s are negative e f f e c t s . Clearly, e x a c t l y when C, i s b o r n (i.e. t h e size of s ) i s important in determining t h e r e l a t i v e sizes of t h e positive and negative ef- f e c t s . A s s a p p r o a c h e s z , t h e f i r s t sum d i s a p p e a r s , t h e second sum g e t s l a r g e , and t h e t h i r d sum g e t s small, v e r y possibly disappearing (if z
>
o-
y ).Let us now look at t h e o t h e r t r a n s f e r mechanism, where 7'Ti
= WS.
In t h i s c a s e , t h e size of c o h o r t C, h a s no e f f e c t on t h e consumption of o u r c o h o r t in s t a g e s one, two, five and six. P e r c a p i t a r e c e i p t s are g u a r a n t e e d , s o e f f e c t s on o u r c o h o r t can only o c c u r when i t is producing. W e c a n writeH e r e , when o u r c o h o r t is producing and t h e o t h e r i s receiving, a n e x t r a member
P
( i ) will r a i s e t o t a l t r a n s f e r s by6,
of which e a c h member of o u r c o h o r t must pay-.
TQi When both c o h o r t s are producing, a n e x t r a member of t h e o t h e r c o h o r t will r a i s e a g g r e g a t e output and so lower t h e p r o p o r t i o n of t h e t o t a l which o u r c o h o r t pro- duces, t h u s reducing i t s payments. Again, t h e value of s i s c r u c i a l in determining t h e r e l a t i v e sizes of t h e positive and negative e f f e c t s .
All of t h e a b o v e analysis h a s assumed t h a t t h e o t h e r c o h o r t w a s b o r n b e f o r e o u r s s t a r t e d producing, i.e., s
<
z. This need not b e t h e case. Suppose, f o r exam- ple, t h a t w e are i n t e r e s t e d in t h e r e l a t i o n between t h e number of c h i l d r e n o u r c o h o r t h a s and i t s lifetime consumption. In t h i s case t h e o t h e r c o h o r t will almost c e r t a i n l y b e b o r n after o u r s starts working. Suppose for simplicity t h a t in e a c h family a l l c h i l d r e n are b o r n in one, possibly multiple, b i r t h .Under t h e s e conditions t h e s t a g e s of life of o u r c o h o r t have d i f f e r e n t c h a r a c - t e r i s t i c s . I will number them 1' to 5'.
1 ' ) Our c o h o r t i s receiving and t h e o t h e r i s not yet alive--periods 0 t o z -1.
2') Our c o h o r t i s producing and t h e o t h e r is not y e t alive--periods z t o s -1.
3') O u r c o h o r t is producing and t h e o t h e r is receiving--periods s t o x +s -1.
4') Both c o h o r t s a r e producing--periods x +s t o y -1.
5 ' ) O u r c o h o r t i s receiving, and t h e o t h e r i s producing--periods y t o a.
In constructing t h e s e s t a g e s of life, I am thinking of t h e following 'typical' life plan. An individual begins t o work a t about a g e 20. Childbearing does not begin un- t i l a f t e r t h e individual begins work and i s completed by a g e 40. The individual re- t i r e s from t h e l a b o u r f o r c e a t a g e 65, a f t e r which h e lives a n o t h e r 15-20 y e a r s . This,
I
think, is not a n unreasonable approximation f o r t h e life plans of t h e a v e r - a g e North American, say. Under t h e s e circumstances, p a r e n t s and children will spend some of t h e same y e a r s m t h e l a b o u r f o r c e t o g e t h e r , but will not both b e re- t i r e d a t t h e same time. Consequently, only during t h r e e s t a g e s of i t s life does t h e number of children a f f e c t o u r c o h o r t :With qualifications t o c o v e r p a r t i c u l a r t r a n s f e r mechanisms, having more children will h u r t o u r c o h o r t , a s t h e r e a r e more people f o r i t t o s u p p o r t . On t h e o t h e r hand, when o u r c o h o r t i s r e t i r e d , t h e r e a r e more people t o s u p p o r t it, which i s c l e a r l y a benefit. When both p a r e n t s and children a r e working, t h e n e t e f f e c t will b e positive if t o t a l t r a n s f e r s i n c r e a s e with t o t a l product, but a t a d e c r e a s i n g r a t e . Once again, in g e n e r a l t h e r e s u l t i s ambiguous, but specific t r a n s f e r mechanisms will g e n e r a t e unambiguous r e s u l t s .
Conclusions
The main conclusion i s t h a t t h e size of one's c o h o r t can a f f e c t t h e level of o n e ' s t o t a l lifetime consumption through intergenerational t r a n s f e r s . Whether t h i s e f f e c t i s positively o r negatively r e l a t e d t o c o h o r t size depends on t h e specific t r a n s f e r mechanism employed however. And, indeed, t h e r e i s a t l e a s t one simple mechanism in which positive and negative e f f e c t s c a n c e l e a c h o t h e r , and s o con- sumption is unrelated t o c o h o r t size.
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Welch, F. (1979) E f f e c t s of C o h o r t Size o n E a r n i n g s : Baby Boom Babies' Financial Bust. J o u r n a l o f P o l i t i c a l Economy Vol. 8 7 , No. 5.
