Withagen, C.A.A.M.

English

Research Papers in Economics

ELSEVIER  Economics Letters 53 (1996)  1-6 
economics 
letters 
Sustainability  and investment  rules 
C.A.A.M.  Withagen 
Department of Mathematics and Computing Science, Eindhoven University of Technology, 5600 MB Eindhoven, 
Netherlands 
Received  13 March 1996; accepted  12 June 1996 
Abstract 
We develop conditions under which an economy aiming at sustainability has to keep the value of the total stocks 
of capital constant over time. 
Keywords: Sustainability; Investment rules 
JEL classification: (26; QO 
1.  Introduction 
There  exists  an  abundant  literature  on  the  subject  of sustainability,  which  is  commonly 
defined as constant per capita well-being over time) We refer to Asheim (1986, 1994b, 1995), 
Dasgupta  and  M~iler (1993),  Dixit  et  al.  (1980),  Hartwick  (1977, 1978, 1994a, b, 1995a, b), 
Hartwick  and  Van  Long  (1995),  M/iler  (1991),  Pezzey  (1994),  Sefton  and  Weale  (1994), 
Solow  (1974,  1986, 1993), Vellinga and Withagen  (1996),  and  Weitzman  (1976, 1995).  This 
literature aims at finding rules for sustainability. A common feature of such rules seems to be 
that the value of capital is constant over time (beginning with the well-known Hartwick rule). 
These results were obtained for special cases and particular models. We will show in this paper 
that a  considerable generalization is possible in  a  control-theoretic framework.  The central 
idea can be sketched as follows. 
The  state  in  which  an  economy  finds  itself  at  a  particular  instant  of  time  can  be 
characterized  by  its  stocks.  Examples  are  the  stocks  of  natural  exhaustible  resources, 
renewable natural resources, man-made physical capital, human capital, international financial 
wealth,  the  stock  of  labor  and  the  stocks  of  pollutants.  Some  of  these  stocks  influence 
~We  are  aware  of the  fact  that  this  anthropocentric  interpretation  and  opcrationalization  of sustainability is 
subject to some criticism, but for our purpose this is irrelevant. See also Toman et al. (1995) and Asheim (1994a). 
Published by Elsevier Science S.A. 
Pll  S0165-1765(96)00887-7 2  C.A.A.M.  Withagen  /  Economics Letters 53 (1996) 1-6 
instantaneous well-being directly, such as those pollutants that have negative health effects. 
Others have an indirect impact, because well-being is derived from using their services, which 
is the case when raw materials are extracted from the exhaustible resources and when human 
capital is used as a  factor to produce consumer commodities. It is usually assumed that the 
motion of the  stocks in  the  economy can  be  described  by  a  set  of differential  equations. 
Moreover, technological and other constraints can be modelled in another set of equations. 
With  the  particular  concept  of  sustainability  given  above  in  mind,  maximal  constant 
instantaneous  per  capita  well-being  can  be  found  as  the  outcome  of  an  optimal  control 
problem with  a  (generally)  non-constant  rate  of discount.  This  is  the  c~se  in  most of the 
papers  mentioned  above.  The  next  step  is  then  to  show  that  the  current  value  of  the 
corresponding Hamiltonian just equals the constant value of well-being, corrected for non- 
autonomous elements in the constraints.  If such elements are absent, then the current value 
Hamiltonian is constant, which implies that the total value of net investments in the stocks 
involved should be zero. It is to be understood that the stocks are evaluated at their shadow 
prices originating from  the optimal control problem.  These prices  need not  prevail on  the 
actual markets, especially not when negative externalities are present. With non-autonomous 
differential  equations,  caused  for  example  by  exogenous technical  change,  varying  world 
market prices or non-constant exogenous interest rates, the situation becomes more compli- 
cated in the sense that no easy economic interpretation can be given. However, it is clear from 
a mathematical point of view what corrections should be made to achieve sustainability. This 
is particularly useful when we  are looking for  a  national income concept that  incorporates 
sustainability. See Sefton and Weale (1994), Vellinga and Withagen (1996) and Johanson and 
L6fgren (1996). 
2.  The main result 
We consider an optimal control problem with n  state variables, r  control variables and m 
constraints. There are given an open set X C R", a set U C R' and T "-R+. We shall only be 
interested in state variables x: T-, X  having piece-wise continuous derivatives and  control 
variables,  u:T-,U  being  piece-wise  continuous.  Also  given  are  the  functions 
p: T~ R+, f0: X × U-, R, f  :-- (f~, f2, • • •, f,):  X  x U × T~ R n and g := (g~, g,,...,  gin): 
X × U × T-, R'. We assume that f0, f  and g  are continuously differentiable. We shall also 
assume that 
(re  ) 
It(t)  "= exp  -  PO') dv  ---,  0 as t---,  oo 
Now suppose that (.~, t~ ) solves the following problem: 
max  ~r(t)fo(x(t  ), u(t)) dt  (1) 
subject to C.A.A.M.  Withagen  I  Economics Letters 53 (1996) 1-6  3 
~(t) = f(x(t), u(t), t)  (t E  r), x(O) = x o given,  (2) 
g(x(t), u(t), t) ~  0  (t E  T) .  (3) 
It is well known that in order to apply the usual necessary conditions, some kind of constraint 
qualification must be satisfied.  It is assumed here that this is indeed the case. We define the 
current value Hamiltonian and the current value Lagrangian as follows: 
~(x, u, t, A):= fo(X, U) + A" f(x, u, t) , 
~(x, u, t, A, I~) := ~(x, u, t, A) + Ix" g(x, u, t ) . 
Here dots between variables refer to the inner product.  Now the following holds: there exist 
A" T---, R" and ~"  T--, R~' with A continuous and piece-wise continuously differentiable and/~ 
piece-wise continuous, such that 
O~/Ouj =O  (j=  l,2,.  . . ,m;t~.  T) , 
Ix(t)" g(~(t),  a(t),  t) =  0  (t E  T)  , 
-)ti(t) = O.~/Ox~ -  p(t)Ai(t)  (i= 1,2,...  ,n; tE  T) . 
Here all the partial derivatives  are evaluated  in the optimum. 
We now state and prove the main result of the paper. 
Theorem.  Suppose  that  fo(~(t), ~(t)) = a,  a  constant  (t E  T), 
t~(t), t, A(t))--,0 as  t-, oo.  Then 
l  ff  [  Of($(s),(t(s),s)  ~(,~(t), ti(t), t, A(t)) = a  -  7r(t-"')  ,r(s)  A(s)"  Os 
(4) 
(5) 
(6) 
and  that  7r(t)~(~(t), 
Og(~(s), fi(s), s) ]  + Ix(s)"  Os  J ds. 
Proof.  Let ft and gt denote the partial derivatives of f  and g  with respect to time.  It follows 
from Seierstad and Sydsaeter (1987,  p.  277,  note 2) that 
d~*  0~* 
dt-  Ot 
at all continuity  points of u,  where ~* "= ~'~.  Hence, 
:~ = p[£#- a] + A" ft +/z • g, 
at all continuity  points  of u.  Therefore,  in shorthand: 
f:  ~(t)  ~(t)  7r(t)  Ir(s)[p(s)a  -  A(s)" f~ -  g(s)"  g,] ~  , 
1 
as is easily verified.  Moreover, 
1  ff 
ds  D 4  C.A.A.M.  Withagen  /  Economics Letters 53 (1996) I-6 
The following is now immediate: 
Corollary.  Suppose that f  and g are autonomous.  Then A(t). ~(t)= 0  (all t ~  T). 
Proof. 
~(t)  =  a  +  A(t)" f(.~, t~, t) =  a  +  A(t)" .~(t) =  a.  V1 
The condition that the present value Hamiltonian converges to zero as time goes to infinity is 
satisfied if f  and g are autonomous. However, the condition is indeed required when f  and g 
are non-autonomous. 
3.  Discussion 
The result presented above strongly resembles the findings of Dixit et al. (1980). The major 
difference is that our result is obtained in an optimal control framework, whereas they work in 
the context of a competitive equilibrium. 
It would go too far to verify for each mode' employed in this line of research whether or not 
it satisfies the assumptions we made. We shall therefore restrict ourselves to some important 
examples. 
In the Solow (1974) model there are two stocks: a stock of an exhaustible natural resource 
and a stock of man-made capital. Capital and the raw material are used in a Cobb-Douglas 
production process that yields a consumer commodity. In the case at hand, the maximum rate 
of consumption can be calculated explicitly and it is easily seen that this is also the outcome of 
an optimal control problem with a  decreasing rate of time preference. 
Also, the recent work of Hartwick and Van Long (1995) fits into our general framework. 
They consider constant consumption, C, in an economy which derives instantaneous profits, P, 
from  for  example  selling raw  material  from  exhaustible  resources  on  world  market.  The 
optimal profits are an exogenous function of time. Then the accumulation of financial capital, 
W,  is given  by 
W(t) =  P(t)  +  r(t)w(t)  -  c(t), 
where r  is the exogenous world market interest rate. The maximand is 
fo' ~r(t)C(t) dt. 
it is easy to see that along an optimum the rate of time preference must equal the interest rate. 
Moreover, it follows immediately from our theorem that with a constant rate of consumption 
ff we define V as the current value at time t of all future profits: C.A.A.M.  Withagen  /  Economics  Letters 53 (1996)  1-6  5 
1  ~ ® 
V(t) "-  ~r(t)  Ir(s)P(s)  ds  , 
then it readily follows that 
W(t) =  -  I/'(t)  7r(t)  Ir(s)[f(s)W(s)  + V(s)] ds 
which is exactly the investment rule derived by Hartwick and Long. 
4.  Conclusion 
In this paper we hav~ derived a general investment rule that must be satisfied in a situation 
of sustainable economic development. To the best of our knowledge it can be applied to all 
the  models  used  in  the  literature  on  this  subject.  As  was  already  mentioned  in  the 
Introduction, the investments should be weighted by the optimal shadow prices, which do not 
necessarily prevail in reality. This implies that even if the value of net investments measured in 
actual prices equals zero at all times, this does not guarantee that the economy finds itself in 
sustainable development. Another caveat applies to the interpretation of the Hamiltonian. It 
is tempting to identify the (linearized)  Hamiltonian with  national income and  hence to see 
national income (after correction, for example, for the extraction ot exhaustible resources) as 
an indicator for sustainability.  However, in view of the above this might lead to conclusions 
that are at variance with reality. 
Acknowledgements 
I have received very helpful comments from Geir Asheim, Peter Alders and Paul de Hek. 
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