We build an overlapping generations model with endogenous fertility choices as well as public and private expenditures on health. We find that the complementary effect of public health services on private health expenditures can provide an additional explanation behind a salient feature of demographic transition; that is, the fertility decline along the process of economic growth.Fertility; Economic growth; Health expenditures

Dimitrios Varvarigos

Intan Zanariah Zakaria

English

Research Papers in Economics

 
 
 
 
DEPARTMENT OF ECONOMICS 
   
 
 
 Endogenous Fertility in a Growth Model 
with Public and Private Health 
Expenditures  
 
 
 
 
 
 
Dimitrios Varvarigos, University of Leicester, UK 
 
Intan Zanariah Zakaria, University of Leicester, UK 
 
 
 
 
 
 
 
 
Working Paper No. 11/07 
November 2010 
   1 
Endogenous Fertility in a Growth Model with 
Public and Private Health Expenditures 
 
 
Dimitrios Varvarigos
a  Intan Zanariah Zakaria
b 
 
Department of Economics  
University of Leicester 
UK 
 
 
Abstract 
We build an overlapping generations model with endogenous fertility choices as 
well as public and private expenditures on health. We find that the complementary 
effect  of  public  health  services  on  private  health  expenditures  can  provide  an 
additional explanation behind a salient feature of demographic transition; that is, 
the fertility decline along the process of economic growth. 
 
 
 
JEL classification: J13, O41 
 
Keywords: Fertility, Economic growth, Health expenditures 
 
 
 
 
1   Introduction 
In a seminal  paper, Bhattacharya and Qiao (2007)  analyse a growth model  in  which 
public health spending is complementary to the expenditures that individuals incur for 
the improvement of their health status. They find that an increase in the public provision 
of  health  services  induces  individuals  to  reduce  their  saving  and,  correspondingly, 
                                                 
a Corresponding Author.  Address: Astley Clarke Building, University Road, Leicester LE1 7RH, UK. 
Email: dv33@le.ac.uk. Tel: ++44 (0) 116 252 2184 
b Intan Zanariah Zakaria acknowledges financial support from the Malaysian Ministry of Higher Education 
and the International Islamic University of Malaysia. The usual disclaimer applies. 
   2 
increase  the  resources  they  devote  towards  health  improvements.  As  a  result,  the 
dynamics  of  capital  intensity  become  non-monotonic  and  may  admit  periodic 
(endogenous limit cycles) or even aperiodic (i.e., chaotic) equilibria.      
     In this paper, we build an overlapping generations model in which individuals live for 
three periods – childhood, young adulthood and old adulthood. Our model utilises the 
main  idea  of  Bhattacharya  and  Qiao  (2007)  –  that  is,  public  health  spending  being 
complementary to private health spending – but modifies their set-up in the following 
manner. Firstly, we allow individuals to consume during both periods of their adulthood. 
Secondly,  individuals  incur  their  health  expenditure  during  the  final  period  of  their 
lifetime.  Finally,  we  assume  that  individuals  are  reproductive  during  their  young 
adulthood and they choose the number of their offspring in an optimal fashion.     
     Our results can be summarised as follows. We find that, rather than generating a 
trade-off between saving and private  health spending,  the  provision  of public health 
services induces an increase in both private health spending and saving; therefore, the 
dynamics of capital accumulation are monotonic and endogenous fluctuations cannot 
emerge. Furthermore, by motivating individuals to save more, the complementary nature 
of public and private  health expenditures  also  induces them to bear and raise fewer 
children  during  their  young  adulthood.  Hence,  this  complementarity  emerges  as  an 
additional explanatory factor behind a well-known stylised fact of demographic transition 
(e.g., Dyson and Murphy, 1985).
1    
 
2   The Economy 
Time is discrete and indexed by  0,1,... t . We consider an economy which produces a 
homogeneous commodity and is inhabited by reproductive individuals who live for three 
periods and belong to overlapping generations. Agents make decisions only after they 
reach  their  adulthood.  These  decisions  are  dictated  by  the  desire  to  maximise  their 
lifetime utility function 
  11 ln( ) ln( ) [ln( ) ln( )]
t
t t t t uc γ n β c h ,    0 γ ,  (0,1) β ,  (1) 
subject to the constraints  
  (1 )(1 ) t t t t c τ qn ω s ,    0 q ,  (0,1) τ ,  (2) 
  1 1 1 t t t t c r s x ,  (3) 
                                                 
1 For theoretical models of economic growth that explain various feautures of demographic transition, see 
Becker et al. (1990), Galor and Weil (2000) and Zhang and Zhang (2005) among others.      3 
 
1
11
t δε
tt h Hx ,     0 H ,  (0,1) δ .  (4)    
     In the previous expressions,  t c  denotes consumption during young adulthood,  1 t c  
denotes consumption during old adulthood,  t n  is the number of children raised by a 
young adult,  t ω  is the market wage per unit of labour,  t s  denotes saving,  1 t r  is the gross 
interest on saving,  1 t h  is the old adult’s health status and  1 t x  is the old adult’s spending 
towards health improvements. When young, each person is endowed with a unit of time 
which she allocates between raising children and providing labour services. Raising each 
child requires q units of time. Therefore, the young adult will use her remaining time to 
earn labour income – an income that is subject to a flat tax rate  τ . She divides her 
disposable income between consumption and saving. The latter is deposited to a financial 
intermediary with the purpose of providing the agent with retirement income when she 
becomes an old adult. When old, the agent can potentially face some health problems 
which she can tackle by using part of her retirement income for the improvement of her 
health  status.  The  remaining  part  of  retirement  income  is  used  so  as  to  satisfy  her 
consumption needs. 
     With respect to the link between public and private health expenditures, we follow 
Bhattacharya and Qiao (2007) in using the expression in (4) for which it is assumed that      
  11 () tt ε Z p ,  (5) 
where 
 
1
1
t
t
t
g
p
N
.  (6) 
The function  1 () t Zp  in (5) satisfies  (0) 1 Z ,  ( ) 1 Z ε ,  0 Z  and  0 Z .
2 In 
equation (6), the variable  1 t g  is the stock of public capital devoted to health services and 
t N  is the population of those agents who are young in period  t  and therefore will be 
old during  1 t . The presence of the variable  t N  is meant to capture a congestion-type 
effect. In particular, those who are young in period t  will access public health services 
when they become old, i.e., in period  1 t . We assume that, for given  1 t g , a larger 
population of agents mitigates the benefit accrued to each agent.      
     Given  the  above,  the  assumptions  illustrated  through  (4),  (5)  and  (6)  provide  a 
mechanism through which the provision of public health services is complementary to 
                                                 
2 We assume that  1 δε  in order to ensure the concavity of  1 t h  with respect to  1 t x .     4 
private health expenditures in that they promote their effectiveness in improving the 
agent’s health status during old adulthood.
3 This can be formally expressed through 
 
1 1 1
1
11
t t t
t
t t t
dh x g
δε δZ
dx h N
,  (7) 
i.e, the elasticity of health status with respect to private health spending is increasing to 
the stock of public capital that the government devotes towards health services.     
     In any period  t , there is a large number of competitive firms who combine labour 
from young adults,  t L , and capital from financial intermediaries,  t K , so as to produce 
t Y  units of output according to  
 
1 ()
αα
t t t t Y K A L ,   01 α   (8) 
The  variable  t A   indicates  some  type  of  labour-augmenting  technological  progress. 
Following Frankel (1962) and Romer (1986), we assume that is related to the average 
capital-labour ratio according to a learning-by-doing externality. That is 
  Ψ,
t
t
t
K
A
L
   Ψ0 .     (9) 
Firms who maximise profits will equate the marginal product of each input with the 
respective marginal cost. That is,  
 
1 (1 )
α α α
t t t t ω α K L A ,     (10) 
and  
 
11 ()
αα
t t t t r αK A L .  (11) 
 
3   Equilibrium 
After some straightforward algebra, the first order conditions associated with an agent’s 
optimal problem allow us to derive  the  following conditions for saving,  fertility  and 
private health expenditures:  
 
1
1
(1 )
(1 )
1 (1 )
t
tt
t
β δε
s τω
β δε γ
,  (12)  
                                                 
3 We can think of many examples that justify this assumption. The presence of qualified professionals – in 
the national health system – that offer support and advice on various difficulties that may emerge while 
people trying to quit smoking (e.g., cravings etc.) may provide an incentive for smokers to seek and buy 
treatments that support Nicotine Replacement Therapy (patches, gums etc.). Clinical depression can be 
combated more effectively if sufferers combine antidepressant medication with appropriate counselling by 
qualified psychiatrists – counselling that is sometimes offered by professionals employed in the national 
health system.        5 
 
1
/
1 (1 )
t
t
γq
n
β δε γ
,  (13) 
 
1
11
1
(1 )
1 (1 )
t
t t t
t
βδε
xr τω
β δε γ
.  (14) 
     Define  / t t t k K N  as capital per worker. Given  (1 ) t t t L qn N , we can combine 
(9), (10) and  tt KK  to write the equilibrium wage as  
 
(1 )
1
t
t
t
α Ak
ω
qn
,  (15) 
where 
1 Ψ
α A . Furthermore, assume that the government uses its collected revenues 
in period  t  so as to finance the formation of public capital that will be available next 
period, i.e., during  1 t , according to a balanced budget rule. That is,  
  1 (1 ) t t t t t t g τL ω τ qn N ω .  (16) 
Combining (16) with (5) and (6) leads to  
  1 ( (1 ) ) tt ε Z τ α Ak .  (17) 
Substituting (17) in (12) and (13) yields  
 
[1 ( (1 ) )]
(1 ) ( )
1 [1 ( (1 ) )]
t
t t t t
t
β δZ τ α Ak
s τ ω s k ω
β δZ τ α Ak γ
,  (18) 
 
/
()
1 [1 ( (1 ) )]
tt
t
γq
n n k
β δZ τ α Ak γ
.  (19) 
These solutions allow us to derive  
 
Proposition 1. The saving rate  () t sk  is increasing in the stock of capital per worker while the fertility 
rate  () t nk  is decreasing in the stock of capital per worker.  
 
     The intuition behind the result of Proposition 1 is the following. A higher capital 
stock  increases  the  government’s  revenues  and  allows  a  greater  provision  of  public 
capital  towards  health  services.  As  a  result,  the  effectiveness  of  private  health 
expenditures increases and old adults will find optimal to devote more resources towards 
them. Naturally, this implies that old adults would find desirable to have more resources 
available at the beginning of their old adulthood. Indeed, agents can achieve this by 
saving  a  larger  fraction  of  the  disposable  income  they  earn  when  young.  Therefore, 
agents decide to limit the resources they keep during their reproductive period – an 
outcome to which they respond by reducing the number of children they rear.      6 
     Notice that our result concerning the saving rate comes in stark contrast to the result 
in Bhattacharya and Qiao (2007). In their model, agents consume only during the final 
period of their lifetime whereas their health spending occurs during their youth because 
their motive is to increase their life expectancy. Therefore, their set-up generates a trade-
off  between  saving  and  private  health  expenditures  which,  combined  with  the 
complementary  effect  of  public  health  spending,  results  in  the  saving  rate  being 
decreasing in the stock of capital. This effect is responsible for non-monotonic capital 
dynamics  which  converge  to  limit  cycles  or,  possibly,  chaos.  Due  to  the  different 
mechanisms  in  our  paper,  such  non-monotonicities  do  not  emerge.  Moreover,  by 
allowing endogenous fertility choices we add another dimension to the implications from 
private-public health expenditures complementarities. To see these, we need to analyse 
the economy’s dynamics formally.
4  
 
4   Economic Growth and Endogenous Fertility 
The economy’s growth rate can be derived as follows. We begin by using the equilibrium 
condition in the financial market; that is,  1 t t t K s N . Combined with  1 t t t N n N , this 
condition becomes  
  1
t
t
t
s
k
n
.  (20) 
Substituting (15), (18) and (19) in (20), we can write the growth rate as 
1
1 1
t
t
t
k
θ
k
 
  1
[1 ( (1 ) )]{1 [1 ( (1 ) )] }
1 ( )
1 [1 ( (1 ) )]
tt
tt
t
ηβ δZ τ α Ak β δZ τ α Ak γ
θ θ k
β δZ τ α Ak
,  (21) 
where  (1 )(1 )/ η qA α τ γ . It is straightforward to check that  0 θ . Assuming that 
(0) 0 θ  holds, we have  1 tt kk  because the growth rate is positive. The long-run rate 
of growth is, thus, equal to  
 
(1 )[1 (1 ) ]
1
1 (1 )
ηβ δε β δε γ
θθ
β δε
.  (22) 
                                                 
4 Note that the fact that capital dynamics are monotonic in our framework is not related to the presence of 
the Frankel/Romer-type externality. This assumption is employed so as to allow the economy to achieve 
long-run growth. The difference with Bhattacharya and Qiao (2007) is our assumption that individuals try 
to improve their health status during old adulthood. This is what actually eliminates the possibility of 
endogenous cycles in our model. For a similar assumption on health costs being incurred during old 
adulthood, see Gutiérrez (2008).   7 
Consequently, by virtue of (19) and Proposition 1, the fertility rate declines towards its 
long-run  value  of  ( ) ( / )/[1 (1 ) ] n γ q β δε γ .  More  importantly,  the  decline  in 
fertility  is  an  outcome  solely  associated  with  the  supportive  effect  of  public  health 
expenditures on the effectiveness of private health expenditures. This argument can be 
formally established if we compare our existing results with the outcomes that transpire 
when  either  1 1   t εt   or  0 δ ,  in  which  case  the  term  1 ln( ) t h   from  the  utility 
function  disappears  and  1 0 t x   in  equilibrium.  In  both scenarios,  variations  in  the 
economy’s capital stock will not impinge on childrearing decisions and the fertility rate 
will be constant. We can summarise these results in   
 
Proposition 2. Given  0 0 k , the economy grows at a positive rate whereas the fertility rate declines. 
The  decline  in  fertility  is  associated with  the  complementarity  between  public  and  private  health 
expenditures.  
 
5   Discussion 
We have established a new mechanism in explaining a salient feature of demographic 
transition. In particular, the decline of fertility during the process of growth is attributed 
to the complementary effect of public health spending on private health expenditures – 
an effect that has been introduced in the manner of the seminal analysis of Bhattacharya 
and  Qiao  (2007)  (see  equations  (4)-(7)).  As  the  economy  grows,  the  public  capital 
available for health services increases and improves the effectiveness of private health 
expenditures. Old adults will find that it is optimal for them to increase the resources 
they  devote  towards  the  improvement  of  their  health  status.  In  order  to  ensure  the 
availability of these resources, they reduce their expenditures when young in order to 
save more. Given that childrearing costs are among these expenditures, reproductive 
young adults will reduce the number of children they give birth to. Consequently, the 
fertility rate declines.     
               
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3.  Dyson, T., and Murphy, M. 1985. “The onset of fertility transition”, Population and 
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synthesis”, American Economic Review, 52, 995-1022 
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Endogenous Fertility in a Growth Model with Public and Private Health Expenditures

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Endogenous Fertility in a Growth Model with Public and Private Health Expenditures

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