sumption. A contemporary philosopher, Derek Parfit (1984) has termed this the ... chapter, we try to avoid the repugnant conclusion of Parfit by putting an upper.
Optimal Control of AgeStructured Populations in Economy, Demography, and the Environment Edited by Raouf Boucekkine, Natali Hritonenko, and Yuri Yatsenko
Routledge Taylor & Francis Group LONDON AND NEW YORK
1 The genuine savings criterion and the value of population in an economy with endogenous fertility rate Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P Sethi Introduction We study an economy in which the rate of population change depends on population policy decisions. This requires population as well as capital as state variables. By showing the algebraic relationship between the shadow price of the population and the shadow price of the per capita capital stock, we are still able to depict the optimal path and its convergence to the long-run equilibrium on a twodimensional phase diagram. Moreover, we derive explicitly the expression of genuine savings in our model to evaluate the sustainability of the system. We study the issue of sustainability with endogenous population in a dynamic economic growth and consumption framework. The problem of population and economic growth has been extensively discussed and continues to be of interest. Ehrlich and Lui (1997) have surveyed the studies exploring relations between per capita income and population level and the studies on related policies of improving human conditions. Different opinions have formed in the community toward the long-term effects of population change on economic growth. The classical view of Malthus (1798) was that population growth hinders economic growth because resources are devoted to satisfy the need of the growing population. However, many studies (e.g., Kuznets, 1971; Simon and Gobin, 1980) using cross-country data in the 1960s and 1970s indicate a lack of any negative correlation between the growth rate of population and that of economy, leading to the "population debate." Several authors believe that population growth can induce technology advancement, which stimulates economic growth (Kuznets, 1971; Simon, 1981; Kremer, 1993). Conversely, there are also arguments that strong economic performance may induce an increased population growth rate through natural increase or migration (McNicoil, 1984). Blanchet (1988) examines such an inverse effect of economic improvement on demographic change to explain the insignificant correlations between economic and population growth rates. Interestingly, Kelley and Schmidt (1995) provide evidence that the impact of population growth on the economy has changed since 1980. They find that the impact of population growth on the per capita output varies with the level of economic development. It is negative in less developed countries and sometimes positive in developed countries.
The genuine savings criterion and the value of population in an economy 21
Recently, Strulik (2005) has developed a model involving human capital demand from production, quality research, and variety research. He shows that economic growth depends positively on the rate of human capital accumulation. Moreover, economic growth depends negatively (resp. positively) on population growth if households maximize utility of consumption per capita (resp. of their dynasty). Bucci (2008) considers positive, negative, or no effects of technological change on human capital accumulation. He examines how the per capita income growth depends on the population growth under certain types of technical progress and argues that population growth is "neither necessary nor conductive" to long-run growth in per capita income. In all of these models, the growth rate of the population is assumed to be exogenous. In studying endogenous fertility decisions, Grossman (1972) and Ehrlich and Chuma (1987) investigate the demand for health and longevity by recognizing the linkage between the two. Ehrlich and Chuma (1990) argue that no myopic rules can be used to derive behavioral propositions concerting the demand for health. In a recent work, Murphy and Topel (2006) study the value of health improvement by considering individual's maximization of lifetime expected utility, which increases in both longevity and quality of life. Their analysis of the medical research in the US during the period 1950-2000 suggests that the returns to basic health-related research may be quite large, so that substantially greater expenditures may be worthwhile. There are also attempts to integrate fertility into a model of the economy with (e.g., Razin and Ben-Zion, 1975; Eckstein and Wolpin, 1982) or without (e.g., Willis, 1986) altruism. Barro and Becker (2008) model endogenous choices of population growth and intergenerational transfers. They show that the rate of population growth in steady state is positively related to the degree of altruism toward children and the long-term interest rate, and it is negative related to the rate of growth between generations in per capita consumption. In this chapter, we investigate economic growth and consumption with an abstract endogenous population growth model by extending the analysis of Arrow et al. (2003, 2007). The first paper (2003) studies a one-sector model of an economy with exogenous non-exponential population growth and provides an analysis of the role of varying population in the measurement of savings. This is accomplished by recognizing population as another form of capital and formulated as a state variable of the system in its optimal control formulation. Arrow et al. (2007) have provided a more detailed analysis of the problem formulated in Arrow et al. (2003). By showing that the co-state of the population is only algebraically related to the co-state of the capital stock, they develop a two-dimensional phase diagram of the problem. Monotone properties of the optimal trajectories and a computation algorithm are also discussed in that paper. Our objective in this chapter is to extend the analysis in Arrow et al. (2007) to an economy where population change is endogenous. We do this by introducing population policy measures as decisions in addition to consumption/investment decisions over time. While our main purpose is to develop a methodology
22 Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P Sethi
of analysis, our discussion covers the case of a country with naturally declining population. We should mention here that population is already declining in Japan and Germany, and other countries (e.g., Italy) are expected to soon follow suit. Now imagine that the country under consideration is interested in encouraging indigenous population growth by such measures as education and baby bonuses. There is strong empirical evidence that financial incentives such as child subsidies can lead to a significant increase in fertility (see, e.g., Cohen et al., 2008). We will term such expenditures as population policy expenditures. Thus, in our model, the output at each instant needs to be optimally allocated between consumption, population policy measures, and investment. Even though the population change is endogenous in our model, we continue to follow the tradition of "total utilitarianism" articulated by Henry Sidgwick and Francis Edgeworth in the 1870s. Thus, we maximize the integral of the total societal consumption utility over time. We are well aware of the ethical issues it raises. In particular, it leads to the view that, if the cost of encouraging population growth is low, then the ideal can be a very large population with very low per capita consumption. A contemporary philosopher, Derek Parfit (1984) has termed this the repugnant conclusion. But Parfit also raises a similar argument against average utilitarian standards. According to him, it may also lead to absurd results. In this chapter, we try to avoid the repugnant conclusion of Parfit by putting an upper bound on the population growth rate. It is even possible to choose a zero growth rate as the upper bound. We use dynamic programming approach to solve the problem. A steady state analysis is conducted, which represents a nontrivial extension of the analysis in the classical case of the exponential growth of the population (Arrow and Kurz, 1970). The analysis involves a study of a system of differential equations in capital and its co-state as a function of the population. We show that it is not the population itself, but its rate of growth that reaches a-steady state. Of course, this rate may be negative, positive, or zero depending on the parameters of the problem. Using the algebraic relation between the co-states of the population and the capital stock, we are able to analyze the optimal trajectory in a two-dimensional phase diagram involving only the capital stock and its co-state. Our phase diagram reveals a structure similar to that in the classical model of Arrow and Kurz (1970). Furthermore, we show that both the optimal expenditure on population policy and the optimal consumption increase with the capital stock. The co-state of the population also increases with the capital stock. The plan of the chapter is as follows. We first develop the notation and the model. Here the state variables are aggregate capital and population. The control variables are consumption and population policy expenditures. The objective is to maximize the present value of the society's utility of consumption over time. The model is then transformed to per capita variables. We use dynamic programming to study the problem. The steady state analysis is carried out and a phase diagram analysis is presented. We further relate our analysis to the maximum principle formulation of the problem. We also obtain the expressions for genuine savings and conditions for sustainability. Finally, we conclude the chapter.
The genuine savings criterion and the value of population in an economy 23
Model description We introduce the following notation: K(t): N(t): k(t): c(t): m(t):
total stock of capital: a state variable; population: a state variable; per capita stock of capital: a state variable; consumption per capita: a control variable; population policy expenditure per capita: a control variable; F(K, N): production function, concave with constant returns to scale; u(c): utility of consumption, u(c) = 0, u'(c) = oo, u"(c) > 0 for c> c for some C? 0; 8: population decay rate; r: discount rate of utility; g(m): fertility rate function, g'(m) > 0, g"(m) < 0, g(oo) > b; f(k): per capita production function, f(0) = 0, J(k) > 0, f'(k) > 0 and f"(k)0;f'(0) >randf (oo) c units per unit time is u(c). We should note that c> 0 would correspond to the idea that sufficiently low level of consumption is worse than nonexistence and so justify the notion of finite optimal population. In the tradition of total utilitarianism, which
24 Kenneth J. Arrow, Alain Bensoussan, Qi Feng, andSuresh P Sethi
argues for treating people more or less equally, the objective becomes one of maximizing the total utility of the society given by J(c(•), m(•)) =
fo
e-" Nu(c)dt.
(1.3)
Note that in (1.3), we have weighted people by their futurity (discounting) but not according to number of their contemporaries. The problem is to select c(t) > c and m(t) > 0, t > 0, so as to maximize J(c(.), m(.)), subject to the condition that K(t) > 0, t > 0. Per capita model
Let k denote the per capita capital stock N . Since we have assumed that the production function F(K, N) is concave with constant returns to scale, we have F(K,N)=NF(N, 1) = NF(k, I)
Nf(k).
Notice that k= —KAT
=f(k)—c—m—k[g(m)— 8].
Then the state equations (1.1) can be rewritten as follows: k k=f(k) — [g( ni ) —6] — c — m, k(0) = k0
4°
(1.4)
We use dynamic programming for our analysis. As is standard, we shall let k(0) = k and N(0) = N. Then we can write the value function as v(k, N) =
max Je t N(t)u(e(t))dt.
c>> c.n(.)>o
(l.5)
0
subject to k=f(k)—k[g(m)-6]—c—m, N=
k(0) = k,
N[g(m)—S], N(0) = N.
(1.6) (1.7)
The initial conditions k and N are of course positive. We expect k(t) > 0, Vt. We need to impose the restriction that c(t) = 0 and m(t) = 0 when k(t) = 0. But it is known that since u'(c) = c, we must have c(t) > c This implies that k(t) > 0. Note that v(k, N) =N max
c(.)>c. m ( . ) 2 0j
u (c(t))dt.
(1.8)
The genuine savings criterion and the value of population in an economy 25 In the classical models with the exponential growth v = g(m) — d, a constant, the condition r > v is required for the value function to be finite. In the absence of this condition, the discount rate is less than or equal to the rate of the population growth, and the value function v(k, N) becomes infinite fork > 0, N> 0. The generalization of the condition r> v in our case is the condition that
^e'N(t)dt 0. Note, however, that if k(0) = 0, then the optimal consumption is c(t) = 0 for t >0. Dividing (1.15) by W'(k) and rearranging terms, we obtain
1 _ W(k) m) W'(k).
(1.16)
k+ g'(
Relations (1.14) and (1.16) suggest the definitions of the adjoint variables p(k): = W'(k) = u'(c),
(1.17)
W(k) W'(k)
(1.18)
k
I k+ g'(m)
In view of the Envelope Theorem (see, e.g., Derzko et al., 1984) we can differentiate the Bellman equation (1.13) with respect to k, and obtain the adjoint equation ' k
k
k
6
P ( ) k) + k6] + P( )[ "( ) + ]
b
( + r)p(k) — cP'(k)
which can be written as p , (k)
p(k)[r f(k) — k[g(m) — 6] — c — m
=
(1.20)
Furthermore, differentiating (1.18) with respect to k and using (1.17), (1.18), and (1.20) gives
rk +1 —r+(r+o _ ) ^ _
I
4m 2m
p(k)= — p
1
l+
k
+
V 2m
(6+r— m)lnp
(i(k)[f'(k) — r] +f(k)] — k[g(m) — 6] — c — in
(1.21)
f(k) — k[g(m) — 6] — c — m Next we show thatp(k) and ii(k) are linked by an algebraic relation. To see this, we substitute (1.17) and (1.18) into (1.13) to obtain the relation (/1(k)p(k)[r + 6 — g(m)] — u(c)
p(k) =
f( k ) — k [g( m ) — b] — c — in
(1
22 )
Dividing both sides of (1.22) by p(k), we obtain
f(k) — k(g(m) — 8) — c — m — (/i(k)(6 + r — g(m)) + — L(c) = 0,
(1.23)
The genuine savings criterion and the value of population in an economy 27 where _ u(c) — u(c) ' ( c ) p(k)
(1.24)
L(c) u
The quantity L(c) is interpreted as value of life, which we will explain later. From (1.21) and (1.23), we get _q (k)— J
(k)[f'(k) + 6 — g(m)] — L(c)
f( k) _ k [g( m ) _ 6 ] _
c
_m
(1.25) 1.25
In (1. 17), we express c in terms ofp. Since yr(k) is related to p(k) algebraically, we can also express in in terms of p. In fact, we can use (1.17) and (1.18) in (1.23) to obtain m+
r+S —g(m)
f(k)—rk—c+L(c)
g'(m)
f(k) — rk — in u' - '(p) -f-
u u (p)) P
(1.26)
Steady state analysis The initial condition is obtained at the steady state for which the numerator and
the denominator of the right-hand side of the differential equation (1.20) vanish. In doing so, we must also observe the maximization conditions (1.17) and (1.26). These provide us with four equations in k, p, c, and m. By bringing in the condition (1.18), we can rewrite the steady state relations as follows: f'(k) = r,
(1.27)
f(k) —(g(m)-6)k —c — m = 0,
(1.28)
u'(c) = p,
(1.29)
pgi(g(m) — 6 — r) + u(c) = 0,
(1.30)
t/i =k+ g,^m).
(1.31)
If there exists a solution of these equations, then this solution, denoted as k, c, n^ yr, and p represents the steady state values of the per capita capital, the consumption rate, the population policy expenditure rate, and the marginal valuationsp and V. Note that the population N(t) does not have a constant value in the steady state. Rather, it grows at the constant rate of g(m) — 6. We now attend to the question of the existence of a solution to the system of steady state relations (1.27)—(1.31). This analysis will also provide us with the
28 Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P Sethi conditions required for existence. We shall treat two cases depending on whether or not in = 0 is enforced. Tu e classical model of exponential population growth and no population policy With in = 0, equation (1.7) reduces to
N= [g(0) — b]N,
N(0) = No.
The usual assumption is that r > g(0) — b, so that the objective function J remains bounded. In this case, only (1.27), (1.28), and (1.29) are relevant with in = 0. These relations reduce to
.f '( kj =r,.f(kj—[g(0)-6]k , —c0,
=0 u' c ,
( j = p_
where k^, cx and p x are the equilibrium values of per capita capital stock, per capita consumption, and the co-state variable associated with the capital. The analysis of this classical economic growth model is well known (e.g., Arrow and Kurz, 1970), and will not be repeated here. Next we study a model with population policy measures. We introduce assumptions that simplify the exposition. More general cases can also be analyzed; their analysis is similar but tedious. The model with population policy This is a case with in > 0. From (1.27) and the conditions on f(k), we obtain k= f(1)(r)>0.
(1.32)
From (1.29) and (1.31), we can obtain p and yr in terms of k, c, and nx, if c and in exist. Thus we need to study only the existence of c and in. Let us define the function c(m) =f(k) + k8 — kg(m) — m.
(1.33)
corresponding to (1.28). Using (1.24) and (1.31) in (1.30), we obtain L( m)) = [ s + r — g( n► )] [ k
+g ].
(1.34)
which is an algebraic equation for nz. From (1.8), it is desirable to have d+r—g(m)> Oatthe equilibrium so that the value function is bounded. Thus, we look for solutions such that the right-hand side of (1.34) is positive, which implies L(c(m)) > 0. In view of u'(c) > 0, c> c? 0, this condition on L implies u(c(m)) >0 and c(m) > c.
The genuine savings criterion and the value ofpopulation in an economy 29 Remark 1.1. For u(c)=cy,00 for in>0. We can now prove the following result. Theorem 1.1. Assume r>
6
g( 0 ) - ,
c c
(
(1.38)
0.
(1.39)
Then there exists a unique m, such that c(m) = f(k) + k6 — kg(m) — m = c.
(1.40)
Assume in addition that r + (5— g(^) > 0. Then there exists a solution (k, c, yr, p) of (1.27) -(1.31). Moreover, c > c> 0,0< io < in, p > 0 and yr > 0. Furthermore, in the interval [0, m], m is uniquely defined and the other steady state values are also unique. Proof. From (1.38) and (1.39), we can easily conclude that u(c(0)) > 0. Hence, c(0) > c. But c(co) c> 0 and, therefore,
u(ff(k)+k6—kg(m)—m)>0,0'n) + m
u u
p
(1.48)
Note that
u u'^ - `^^^ p,
i4' (m)m k
=f
(k) — r.
tP r 1 (m) = — ([r + 8— g)m)]gT„(m))l ([g 1 ( m )] 2 ) .
(1.49) (1.50) (1.51)
We are interested in a solution m satisfying r + (5— g(m) > 0. In this case, we have the following properties.
32 Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P Sethi Lemma 1.1.
If
there is an m(k, p) E [0, g '(r + b)] that solves (1.48), then
4J (m) > 0 and ,n (k, p) < 0,
m k (k, p)>0fork k. Since yr(m) increases on 0 < in < m = g '(r + d), the corresponding (k, p) must satisfy I(0)^J(k)—rk—u'( t )(, )+
u(u' p
>^
^L,(;n).
(1.52)
Under the condition (1.52), there exists a unique solution m(k, p) to (1.22). Now de fi ne u(u'(- 1 xp)) H(p) = p Since H(p) is decreasing in p, the inverse of H exist. Thus, the requirement of (1.52) is equivalent to H'(i(h)—J(k)+rk) p < H'(^r(0)—J(k)+rk).
(1.53)
Optimal trajectory p(k) Substituting the solution m(k, p) of (1.22) into (1.20) together with the steady state values (k, p—) define the optimal trajectory p(k). Note that both the denominator and the numerator in the right-hand side of (1.20) vanish at the steady state. In order to determine the curve p(k), we also need to derive the value p'(k). Let k X ( k, p) = .1( ) + kb — uK-Oxr) — in — kg(nz).
Then, Xk
=
_
Xp
n) '(k)+a—g(r I
u ^ ( u^(-1)(n))
( 1 +kg'( m))mk' (I+kg(m))ml,.
For (k, p) in the neighborhood of (k, p), we have X ( k, p) z Xk ( k , P)(k — k ) + X,,C k , P)P'(k) ( k — k)
The genuine savings criterion and the value of population in an economy 33
[(r+6 -gym))-P'(k) - u (u ,I ^)(r)) +(l +kg'(m))mT, (k-k) In deriving the last equation, we have used the fact that By a perturbation argument, we have
m=
0.
Pr t ( )
(r +
-- -- . 6 -g(m))-p'(k)[- .....-^u (it 1c wf) +(1--+kg'(nm))mJ
(1.54)
That is,
-
1
u^ u' ( i)(i)
+(i
+
m)) nz_r ( p" ^ k))^+ r+ S -g^ ni ) ) p' ( k ) g'^
+ pf"(k) = 0 It is easy to see that the above equation has one positive root and one negative root. We take the negative solution of p'(k) because of the following consideration. With the negative solution, we can prove that the ordinary differential equation (1:20) has a smooth solution such that p'(k) < 0. This is discussed in Theorem 1.2. Theorem 1.2. The optimal trajectory p(k) defined by (1.20) is decreasing in k
within the range (1.53). Moreover,
u' ' --
m k J(k) - k [g( m ( k , p(k))) - 6] ( , p( k)) > 0 for k < k, f(k) - k[g(m(k, p(k))) - 5] - u' ( - ' Mk» - m(k, p(k)) > 0 for k < k.
Proof. We fi rst prove the result for k < k. Define 7r(k) =J(k) - kg(m(k, p(k)) + k8 - u'(-'' > - m(k, p(k)). Since p'(k) < 0, we have p(k - e) > p(k) for a small positive e. Also since p > 0. and f '(k) < 0, equation (1.54) implies
= f'(k)+b-g(rrt)-p'(k) -
,( U
(u
+(l+kg'(rn))m^ 0. This implies f(k) — k[g(m(k, p(k))) — 6] — u'^ ' ^ Suppose there is a point k 0, we have 7T'(k) > 0. On the other hand, r(k) = 0 in (1.20) implies p'(k) = —oo, which, in turn, implies i'(k) = — cxj in (1.55). This leads to a contradiction. Thus, we have proven the result for k < k. The result for k> k follows in a similar way. The details are omitted here. Corollary 1.1. Along the optimal trajectory p(k) defined by (1.20), the optimal population policy expenditure rate m(k, p, (k)) satisfies
dm(k,p(k))
>0.
dk Moreover, the co-state of the population yr defined in (1.18) is also increasing in k.
Proo f. From (1.49), (1.50) and (1.20), we have
dm
dk =mk +mpp'(k) p
u
► ( u ^(- )( ) )
- f'( k) -r + p Vi' ( m )
2
p(f'(k)-r)
WV (m) f(k)-k [_g(m)-3 ] -u'(
f'(k)- r f (k)—kg(m)+k6—m—u'(-')(P)+L(
N ' ( rn)
f(k) —k[g(rn) - 6]_u'L'
_-p'(k) [f (k)—kg(m)+kb—m—u i p 4 (m)
l)u))
_m
1)O))
—m
+L(u
)].
The genuine savings criterion and the value of population in an economy 35 From Theorem 1.2, we have p'(k) < 0. We also have yr'(m) > 0 and p = u'(c). We need to show that the term inside the parenthesis is positive. By (1.48), m J( k) - kg[( ) + (k(5 - m - u '(- ' > " + L(u'(->(P)) X
=J(k) — rk — u't- I P > + L(u' (-' XP) ) + k(r + 5— g(m))
=m
+1 1 + ,
1
g(m)
](r+d—g(m)).
Under the condition (1.53), we have r + (5? g (m). Hence, we conclude that dk > 0. Also from (1.18), we deduce that
_ g'(in ( k,p,(k))) din ^ dk2 >0. dy/=1+ dk
(g (m))
This concludes the proof. Curves k(t) = 0 and p(t) = 0 Along the curve x(k, p) = 0, we have k(t) = 0. Differentiating with respect to k, we obtain Xk
+ p'(k) = 0.
Thus, the curve k(t) = 0 is defined by f'(k)+5 —g(m)—(1+kg'(m))mk
p' ( k ) = u^
u^(-^)(n) (
+(l +kg'(m))mn )
with the bounda ry p(k) = p. Note that x(k, p) increases in p. Thus, k(t) > 0 for any _ below the curve. point above the curve, and k(t) The curve p(t) = 0 is defined by k = f ' = k. We have p(t) 0fork>k. and Examples. Consider our previous example with g(m) = I, f(k) = u(c) = Inc. The curves k(t) = 0 and p(t) = 0 are defined as 1 _ p'(k)
2 -.J
—rk+l
—r+(r+S—^)
p2
1+
4m
1 rn
+
k
2m
2m
1(b+r—^)lnp
36
Kenneth J. Arrow, Alain Bensoussan, Qi Feng, and Suresh P Sethi 1 k=--. 4r,
The optimal trajecto ry is given by 1
p
r
pk= k—kl sf
—S j
2^
---m(p,(k),k) P
where m(p, k) is the solution of
[
m+2 m r+b—g(m)] =^—rk---- lnp .
P P We consider an example with a positive population growth rate in Figure 1.1 and one with a negative rate in Figure 1.2.
0.75 p(k)
;
The classical curve with }
N(1)=0.116N
I
t ....-. 4
0.72
I
I
t
0.69
^`
I ( k , p)
= (4,0.67)
1—
0.66
(k.,P.) _ (4,0.65) ^I ^'
I
I
0.63 III
^
1
°.,,k(t)=0
^" t
\ ,
\ \
`
0.60 2
3
4
k Figure 1.1 The phase diagram when r = 0.25 and 8 = 0.1.
5
6
The genuine savings criterion and the value of population in an economy 37
0.60
I
^
p(k)
"
s
The classical curve with N(t) = 0.022N
0.56 II I ^•
p
0.52
I
k(t)=0
(k,p) = (4,0.51)
-^ ^
^
S S
`^ \1
(k^,p^)=(4,0.48)
0.48
In
I '^
p(f)=0
Iv
1
k
Figure 1.2 The phase diagram when r = 0.25 and 6 = 0.4.
When r = 0.25 and 6 = 0. 1, the steady state values are (k, c, in, ,fir, p) z (4, 1.49, 0.047, 2.97, 0.67). Since g(m) — (5 z 0.116 > 0, the population keeps increasing exponentially in the long run. In Figure 1.1, we also plot the classical curve for which the population grows at a constant rate of 0.116. Without population policy expenditure, we have p x (z 0.65)
