The concept of genuine saving appeared for the first time in a proof of a now well known theorem in Weitzman (1976). It was reinvented and used as a local welfare indicator by Pearce and Atkinson (1993). The purpose of this paper is to generalize this welfare measure to a stochastic Brownian motion context. We will use a stochastic version of a growth model introduced by Ramsey (1928). The particular model was developed by Merton (1975). Although the model is simple, it is enough to understand what its welfare results will look like in a general case.Welfare measures under growth and uncertainty; diversified risk versus undiversified risk

Li, Chuan-Zhong

Löfgren, Karl-Gustaf

English

Research Papers in Economics

  1 
Umeå University and Uppsala University,  
June 2009           
Dept. of Economics 
           
Abstract: Genuine Saving under Stochastic Growth by Chuan Zhong Li and 
Karl-Gustaf Löfgren 
 
The concept of genuine saving appeared for the first time in a proof of a now well known 
theorem in Weitzman (1976). It was reinvented and used as a local welfare indicator by 
Pearce and Atkinson (1993). The purpose of this paper is to generalize this welfare measure to 
a stochastic Brownian motion context. We will use a stochastic version of a growth model 
introduced by Ramsey (1928). The particular model was developed by Merton (1975). 
Although the model is simple, it is enough to understand what its welfare results will look like 
in a general case. 
Keywords: Welfare measures under growth and uncertainty, diversified risk versus 
undiversified risk. 
JEL-Codes D 6, O40 
 
Genuine Saving under Stochastic Growth     
 
By Chuan-Zhong Li, Karl-Gustaf Löfgren 
1 
 
It has been known for quite a while that in a comprehensive deterministic dynamic growth 
model of the Ramsey type Genuine Saving (GS) is a welfare indicator. More precisely, the 
sum of net investment of all capital goods is a perfect local welfare indicator in the sense that 
positive GS means that welfare is increasing. Who first derived this result is not quite clear to 
us
2, but it shows up indirectly in Weitzman (1976). He derives the result as a step in the proof 
of the main theorem; the proportionality between the Hamiltonian and the present value of 
future utility. Important theoretical observations on its implications for sustainability are 
found in Pearce and Atkinson (1993), Asheim (1994) and Pezzey (1995).The measure has 
been popularized by Hamilton (1994), and used in practice by among others, Hamilton and 
Lutz (1996) and Hamilton and Clemens (1999). The purpose of this paper is to generalize this 
welfare measure to a stochastic context. We will use a simple version of a stochastic dynamic 
growth model introduced by Ramsey (1928). The model was developed by Merton (1975). 
Here we use a slightly modified version, where the model is optimized by choosing 
                                                 
1 The authors acknowledge research grants from Formas and the Swedish Environmental Agency. A first version 
of the paper has been commented upon by Senior Lecturers Bengt Lundquist and Anders Muszta Department of 
Statistics, Umeå University, and Professor Hans Wallin, Department of Mathematics, Umeå Universtity. They 
helped a lot, and the remaining errors are our own.   
2 It is almost like the Folk Theorem in cooperative game theory; known to many, first proved by Aumann (1959), 
and understood by only a few.     2 
consumption rather than the rate of saving. Although the model is simple, the derivations are 
enough to understand how the result generalizes to a multi-sector version of the model.  
 
The model    
Merton treats the asymptotic properties of both the neoclassical growth model developed by 
Solow (1956) and Swan (1956), as well as the Ramsey (1928) optimal growth model, when 
the growth of the labour force follows a geometric Brownian motion process. We will 
concentrate on the Ramsey model and mainly deal with the one dimensional case.  
 
Let 
22 ( ( ), ( )) ( ) F K s L s C R
   be a net production function (i.e., depreciation has been 
accounted for) of degree one, where  () Ksdenotes units of capital input, and  () Ls denotes 
units of labour input at time s. The capital stock evolves according to 
 
  K(s) F(K(s),L(s))-C(s) L(s)F(k(s),1)-C(s)     (1) 
 
where ()
dK
Ks
dt
 ,  and  () Csdenotes consumption. The last equality follows from 
homogeneity of the production function. Now let  ( ) ( )/ ( ) k s K s L s  ,  ( ) ( )/ ( ) c s C s L s  , and 
assume that  ( ) (0)
nt L t L e  ,  (0) 0 L  , 01 n  , and  differentiate totally with respect to time. 
Again, using the degree one homogeneity of the production function and 
putting ( ,1) ( ) F K f k  , it follows that 
 
  k(s) f(k(s))-c(s)-nk(s)           (2) 
 
Equation (2) is a variation of the Solow-Swan neoclassical differential equation of capital 
stock growth under certainty. Now suppose that growth of the labour force follows a 
geometric Brownian motion
3 of the following shape 
 
  ( ) ( ) ( ) ( ) dL s nL s ds L s dB s                                                                           (3) 
 
                                                 
3 Geometric Brownian motion is used to guarantee that the labour force remains positive. Note, however, that 
this does not result in an equation for the capital stock per capita that is Geometric Brownian motion.       3 
The increment  () dB s is the stochastic part.  () Bsis a Brownian motion process defined on 
some probability space with values in R . The drift of the process is governed by the expected 
rate of labor growth, n. In other words, over a short interval of time, ds  the proportionate 
change of the labor force ( / dL L) is normally distributed with mean nds and variance 
2 2 2 () dB s ds    
 . 
 We can now use Ito’s lemma to transform the uncertainty of growth in the labour force into 
uncertainty about the growth of the capital labour ratio. Non-standard calculations
4 yield  
 
 
2 [ ( ( )) ( ) ( dk(s) f k s c s n )k(s)]ds k(s) dB(s)           (4) 
 
In other words, we have translated uncertainty with respect to the growth rate of the labour 
force into uncertainty with respect to capital per unit of labour and, indirectly, to uncertainty 
with respect to output per unit of labour, ( ) ( ( )) ( ( ),1) y s f k s F k s  .The optimization problem 
is to find an optimal consumption policy, and the stochastic Ramsey problem is typically 
written 
 
   
() max
T
ss c
s
V(s,k )=  E u c( ) e d

 
  

                                                                   (5)                                                           
subject to 
 
2 ( ) [ ( ( )) ( ) ( ) ( )] ( ) ( ) dk f k c n k d k dB                      
                                                                                                                               (6)   
  () s k s k     
           ( ) 0 c            
where  s E  denotes the mathematical expectation taken at time s . The function  ( ( )) uc is the 
instantaneous utility function which is assumed to be twice continuously differentiable. e
   
is the discount factor, and   is the utility discount rate. T  is the first exit time from the 
solvency set
5  { ( ); 0} G k k    , i.e.  inf{ ; ( ) } T s k G      . In other words, the 
process is stopped when the capital stock per capita becomes non-positive (when bankruptcy 
                                                 
4 Ito calculus means that the first order differential contains second order terms. The details are available in the 
appendix. 
5 G is simply the real positive line (0, )  .   4 
occurs). The stochastic differential equation in (6) is not Geometric Brownian motion and we 
cannot guarantee that  () k  stays non-negative, i.e. that bankruptcy does not occur
6.  
 
Genuine saving under growth  
In most contexts it is realistic to assume that the control process  () c   is allowed to be 
conditioned solely on past observed values of the state process () k  . In such a case, 
mathematicians would say that the control process is adapted to the state process. Here, it is 
assumed that the optimal control function is a time autonomous Markov control of the 
following type 
 
           ( ) ( ( )) c c k                                                                            (7) 
 
Equation (7) means that the control at time   only depends on the state of the system at this 
time. In particular, it does not depend on the starting point or time as a separate argument.  
 
Starting from the value function in present value   
 
       ;
T
s
s
J s, k,c E u c( ) e d
 
    
                                                      (8) 
 
which is maximized with respect to  () c  and subject to equation (6), the stochastic differential 
equation for the capital stock, and  () k s k  , we obtain 
 
() ( , ) { ( ( ( ))) } ( , )
T
ss
s
s
e V s k E u c k e d W s k
   
                                                   (9) 
 Where  ( ( )) ck 
 is the optimal control,  ( , ) V s k is the optimal present value function, and 
( , ) W s k  is the optimal current value function. We now prove that 
  
Observation 1:  ( , ) (0, ) ( )
ss V s k V k e W k e
   , where the endogenous time spent in the 
solvency set  inf{ 0; ( ) } . G t k t G T s        
 
                                                 
6 A hard question is whether it occurs with probability one.    5 
Proof: The optimal control is a Markov control, i.e., it depends only on the capita stock  () kt 
at time t.  
Putting  st  means that the optimal control 
( ( )) ( ( )), 0, c k c k s t s t T t T s   
          and  ( ) (0) . k s k k   
The time spent in the solvency set,  G  =T-s, is, for a given experiment st     , endogenous, 
and the solvency set does not depend on the rescaling, i.e , 
{ ( ); 0} { ( ); 0} s t s t G k k k k        . Therefore  
( ) ( )
0
0
( , ) max [ ( ( ( )) ] max [ ( ( ( ))) ]
(0, ) ( )
G T
s s s s t s
s c c s
ss
V s k e E u c k e d e E u c k s t e dt
e V k e W k

    


      

   

  
  
The second equality follows since substituting  st kk     into the time autonomous stochastic 
differential equation (6), we get a process that starts at 0 (0, ) xk  , has the same probability 
law on an equivalent solvency set as the process that starts at ( , ) s x s k  , and the  optimal 
control is Markov. The remaining two equalities follow from definitions. 
   
Observation 1 means that the current value function, ( ) ( , ) (0, )
s W k e V s k V k
  , does not 
depend on the starting time. This implies that 
    ( , )
s
s
d
V s k e W
ds
     ()
s e W k
 
         
 
and the Hamilton-Jacobi-Bellman,  HJB, equation  
2 ( , ))
s 2 2
s k kk c
V1
-V (s,k)) Max u(c(s))e V (s,k))h(k,c; ,n) k V s k
s2
 
          
   (10) 
can be rewritten in the following manner   
 
2 ()
22
k kk c
1
W(k)) Max u(c(s)) W (k))h(k,c; ,n) k W k
2
        
             (10a)  
          .   
Here
22 ( , ; , ) ( ) ( ) h k c n f k c n k      ,  / k W dW dk  and 
22 / kk W d W dk  . We can now 
define a co-state variable  () ps as 
   6 
  ( ) ( )) k p s W k                                                                                                     (11) 
 
and its derivative 
 
 
() dp s
dk
=  () kk Wk                                                                                                   (12) 
 
Given the optimal consumption policy, equation (10a) can be written (neglecting the time 
index to save notational clutter) as   
           
2c * * 2 2 1 dp dp
W k u(c ) ph(k,c ; ,n) k H (k,p,  )
2 dk dk
  

                               (13) 
i.e., the total expected future welfare is proportional to a stochastic version of the 
deterministic Hamiltonian
7.  
To derive a local welfare measure like GS we start the optimal value function  
 
( ) ( ) ( , )   { [ ( )] } { ( ( )) }
TT
ss
ss c
ss
W s k Max E u c e d E u c e d
       
                     (14)                                                        
Differentiating with respect to time (the lower integration level) yields 
 
( )) ( ( )) ( )
W
W k u c k W k
s

 
   

                                                               (15) 
 
Now, using equation (13), i.e., the HJB-equation for the time autonomous problem,   
we get after substituting for  () Wk   
                
2 2 2
2 2 2
1 ( )
( ) ( )) ( ( ( ), ( ); , )
2
1
( )[ ( ( )) ( ( )) ( )] ( ))
2
k kk
dp s
W s p s h c k s k s n k
dk
W k f k s c k s n W k k




  
    
                  (16)  
  
The interpretation of the co-state variable  () psis the derivative of the optimal value function 
with respect to the initial capital stock, and  () h is the drift in net investment along the optimal 
path. The second term in the expression originates from Ito calculus and the sign of this 
                                                 
7 This may have been known by many but was pointed out by Aronsson and Löfgren (1995). Hence, Weitzman’s 
theorem from 1976 follows directly from the HJB equation.    7 
second order derivative of the value function with respect to the capital stock,  kk W , or, what 
amounts to the same, the derivative of the co-state variable (the shadow utility value of net 
investment) from an increase in the capital stock. For a “well behaved” maximization problem 
this entity should be negative. For  0    equation (16) collapses to the static GS 
measure
8.This means that we would under a stochastic Ramsey problem expect that a positive 
net investment value would not be enough to indicate a local welfare improvement. Net 
investment has to be large enough to compensate for the variance component. In the variance 
component we interpret  ( ( )) kk W k t  as the price of risk, and 
22 k   as the “quantity of risk”.  
 
The reason why this particular component appears is that an Ito integral is constructed from 
forward increments. An alternative, well known, way of constructing a stochastic integral is 
the Stratonovich integral
9, which picks the middle of the increment to weigh the components 
of the sums that approximates the integral. For a whole economy, where risk cannot be 
diversified away the Ito integral seems reasonable.  However, if risk can be diversified a 
stochastic integral which leaves out the risk component in expressions like (16) is more 
relevant. In Weitzman
10 (2003) it is shown that under a Stratonovich integral the variance 
component disappears. 
 
A general conclusion 
To find the solution in the general time autonomous case with n consumption goods and m 
capital goods the above procedure can be generalized. We will only have to change into a 
general HJB-equation. The derivative of the value function will look like the one in equation 
(15). In other words, we are left with the following result. 
 
Observation 2: In a stochastic time autonomous Ramsey problem with n consumption goods 
and m capital goods the derivative of the value function with respect to time is given by the 
following expression ( ( )) ( ) W k s HJB u
  c .  
         
If the problem is not time autonomous extra first order terms will be added in the HJB 
equation and change the time derivative accordingly. An example would be exogenous 
                                                 
8 Equation (13) collapses into Weitzman’s theorem on the proportionality between the Hamiltonian and future 
welfare 
9 The seminal reference is Stratonovich (1966). 
10 The Theorem is found in Chapter 7  page 321,    8 
technological progress, which would add net value to the GS component. Another example 
would be negative externalities that would deduct net value from the GS component. Under a 
Stratonovich integral the variance component is exactly netted out also in the general case. 
 
Finally, a Markov control may seem overly specific. A more general control would be to 
allow the control at time t to be conditioned on the whole process from start up to t, i.e., the 
control function is t F adapted  . Such controls are called closed loop or feed back controls. 
Under an integrability condition and a smoothness condition on the set G it is possible to 
show that the optimal value function for the Markov control coincides with the optimal 
control for the open loop control for any starting point in G . Hence, the Markov control is not 
particular restricted
11.     
 
References 
Aronsson, T. and Löfgren, K.G. (1995) National Product Related Welfare Measures in the 
Presence of Technological Change, Externalities and Uncertainty, Environmental and 
Resource Economics 5, 321-332. 
Asheim, G.B. (1994) Net National Product as an Indicator of Sustainability, Scandinavian 
Journal of Economics 96, 257-65. 
Aumann, R.J. (1959) Acceptable Points in General Cooperative n-Person Games, in 
R.D.Luce and A.W.Tucker (eds.), Contributions to the Theory of Games IV, Annals of 
Mathematics Study 40, 287-324. Princeton: Princeton University Press.  
Hamilton, K. (1994), Green Adjustment to GDP, Resource Policy 20, 155-68. 
Hamilton, K. and Clemens, M. (1999) Genuine Saving in Developing Countries, World Bank 
Economic Review 13 (2), 333-56.  
Hamilton, K. and Lutz, F. (1996) Green National Accounts: Policy Uses and Empirical 
Experience, paper No.039, Environmental Economic Series, Environmental Department, 
World Bank, Washington, DC.  
Hamilton, K. and Clemens, M. (1999) Genuine Saving in Developing Countries, World Bank 
Economic Review 13 (2), 333-56.  
Merton, R. (1975) An Asymptotic Theory for Growth under Uncertainty, Review of Economic 
Studies 42, 375-93.  
                                                 
11 See e.g. Öksendal (2003) Theorem 11.2.3..   9 
Pearce,  D.  and  Atkinson,G.(1993)  Capital  Theory  and  the  Measurement  of    Sustainable 
Development, An Indicator of Weak Sustainability, Ecological Economics 8, 103-108. 
Pezzey, J. (1995) Non-declining Wealth is not Equivalent to Sustainability, Dept. of 
Economics, University College, London (mimeographed).  
Solow, R. (1956) A Contribution to the Theory of Economic Growth, Quarterly Journal of 
Economics 70, 65-94 
Stratonovich, R.L.(1966) A New Representation of Stochastic Integrals and Equations, 
Journal of Siam Control and Optimization 4, 362-71 
Swan, T.W.(1956) Economic Growth and Capital Accumulation, Economic Record 32, 334-
61. 
Weitzman, M.L. (1976) On the Welfare Significance of National Product in a Dynamic 
Economy, Quarterly Journal of Economics 99, 1-13. 
Weitzman, M. L. (2003) Income Wealth and the Maximum Principle, Cambridge: Harvard 
University Press. 
Öksendal, B. (2003) Stochastic Differential Equations, Oslo: Springer (6
th edition). 
 
Appendix: Derivation of equation (4) in the main text   
We start from: 
[ ( , ) ] [ ( ) ] dK F K L C dt Lf k C dt
dL nLdt LdB 
   

 
Letting  / k K L  , Ito’s lemma yields  
2 2 2
22
22
2 2 2
23
1
[ ( ) 2 ( )
2
1 1 2
( ) ( ( ) ) ( ) [ ( ( )) ( ) (
2
k k k k k k
dk dt dL dK dK dLdK dL
t L K K K L L
K
nLdt LdB Lf k C K L dt f k s c s n )k(s)]ds k(s) dB(s)
L L L
   
     
      
      
          
 
 

A Contribution to the Theory of Economic Growth,

Acceptable Points in General Cooperative n-Person Games,

An Asymptotic Theory for Growth under Uncertainty,

Genuine Saving in Developing Countries,

Green National Accounts: Policy Uses and Empirical Experience, paper No.039, Environmental Economic Series, Environmental Department,

Income Wealth and the Maximum Principle, Cambridge:

Net National Product as an Indicator of Sustainability,

Non-declining Wealth is not Equivalent to Sustainability,

Öksendal

On the Welfare Significance of National Product in a Dynamic Economy,

Stochastic Differential Equations,

Genuine Saving under Stochastic Growth

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