Hartl, R.F.

Kort, P.M.

English

Research Papers in Economics

09:14  - 1 -  10.03.05 
Optimal Investments with Increasing Returns to Scale: 
a Further Analysis 
Richard F. Hartl1, Peter M. Kort2 
1  Institute of Management, University of Vienna, Brünnerstraße 8, A-1210 Vienna, Austria 
2  Department of Econometrics and CentER, Tilburg University, P.O.Box 90153, 5000 LE, 
Tilburg, The Netherlands  
Abstract. This paper considers a capital accumulation model that was previously 
analyzed by Barucci (1998). The specific feature of the model is that revenue is a 
convex function of the capital stock. We extend Barucci's work by giving a full 
analytical characterization of the case where a saddle point with a positive capital 
stock level exists. Furthermore we also analyze the other cases.  
1  Introduction 
In this paper subject of study is a standard capital accumulation model of the firm, 
where the objective is to maximize the discounted profit stream. The profit rate 
equals the difference between the revenue and the costs of investment. Revenue is 
obtained by selling goods on the market. The firm needs a capital stock to produce 
these  goods.  The  higher  the  capital  stock  it  owns  ,  the  more  goods  the  firm 
produces, which in turn leads to a higher revenue. The firm can increase capital 
stock by investing. Technically spoken, this model is an optimal control model 
with one state variable, the capital stock, and one control variable, the investment 
rate. 
The  study  of  this  framework  goes  back  to  the  sixties,  and  started  out  with 
Eisner and Strotz (1963). In this contribution the revenue function was assumed to 
be concave and investment costs were convex. Using standard methods of control 
theory it is easily shown that optimal firm behavior subscribes convergence to a 
unique long run equilibrium at which marginal revenue equals marginal costs. 
Later  it  was  recognized  (Rothschild  (1971))  that  arguments could be  found  in 
favor of a (partly) concave shape of the investment cost function. The problems 
(chattering controls!) that then occur in the maximization problem were subject of 
study in Davidson and Harris (1981) and Jorgensen and Kort (1993).  
On the other hand it can also be the case that the revenue function is convexly 
shaped for some intervals of capital stock values. Such a scenario was studied in 
Dechert (1983) and again Davidson and Harris (1981). From these contributions it 
can  be  concluded  that  partly  convex  revenue  functions  can  lead  to  multiple 
equilibria. It then depends on the initial level of the capital stock to which of the 
equilibria it is optimal for the firm to converge to. In this sense  we can speak of 
history dependent equilibria. Barucci (1998) studies the case where the revenue Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 2 -  10.03.05 
function is strictly convex troughout. He considered a framework where both the 
revenue function and the investment cost function are quadratic. As a result the 
isoclines, on which state, control, and co-state variables are constant, are linearly 
shaped, so that exactly one steady state exists. This means that multiple equilibria 
are ruled out. Barucci (1998) identifies the case where a saddle point equilibrium 
occurs for a positive level of the capital stock. He shows that convergence to this 
saddle point is the optimal policy. 
Fascinated by the fact that such a simple optimal solution exists for the model 
with a fully convex revenue function, in this paper Barucci‘s framework is studied 
once  again.  We  extend  Barucci  (1998)  by  (1)  determining  a  full  analytical 
characterization of the case with the saddle point with positive capital stock, and 
(2) by determining which other cases are also possible if the parameter values are 
different.  
The contents of this paper is as follows. The model is formulated in Section 2. 
After  establishing  the  necessary  optimality  conditions  in  Section  3,  the 
equilibrium and its stability properties are studied in Section 4. In Section 5 all 
possible cases are studied, while some ideas for future research are outlined in 
Section 6. 
2  Model formulation 
Following Barucci (1998), the model we consider is the following: 
  [ ] ￿
¥
- -
0
) ( ) ( max dt u c k r e
t
u
r ,  (1) 
  k u k m - = ￿ ,   k(0) = k0,  (2) 
where k denotes the capital stock and u is investment. The revenue function is 
given by r(k) while the investment costs are c(u). The discount rate is r while m 
denotes the depreciation rate. 
Although  Barucci  did  not  impose  this  constraint,  for  economic  reasons  we 
assume that investments are irreversible: 
  0 ³ u .  (3) 
In order to be able to obtain a full analytical solution, like Barucci (1998) we 
assume quadratic revenue and costs functions: 
  r(k) = ak + bk
2 ,  c(u) = cu + du
2.  (4) 
We require  all parameters a, b, c, d, m, and r to be positive. Hence, as already 
explained in the Introduction, the revenue function exhibits increasing returns to 
scale. Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 3 -  10.03.05 
3  Necessary conditions 
Hamiltonian: 
  H = ak + bk
2 - cu - du
2 +q[u - mk].  (5) 
maximum principle: 
  0 = ¶ ¶ u H     i.e.    
d
c q
u
2
-
= .  (6) 
If (3) is imposed, then (6) holds only for u > 0 i.e. for  q > c. Otherwise we have 
  u = 0  if  q £ c.  (6a) 
The adjoint equation is 
  ( ) bk a q k H q q 2 - - + = ¶ ¶ - = m r r ￿ .  (7) 
From (5), i.e.,  q = 2du + c  and (6) we get: 
  ( ) ( )
k
d
b
d
a c
u
d
q
u -
- +
+ + = =
2 2
m r
m r
￿
￿ .  (8) 
4  Equilibrium and its Stability 
The  (unbounded)  linear  DE-system  (2)  and  (8)  has  the  following  unique 
equilibrium: 
  ( )
( ) m r m
m r
+ -
- +
=
d b
a c
k
2
1
     ( )
( ) m r m
m r m
m
+ -
- +
= =
d b
a c
k u
2
.  (9) 
On the other hand, the original canonical system (2) and (7) has the unique 
equilibrium  
  ( )
( ) m r m
m r
+ -
- +
=
d b
a c
k
2
1
     ( ) m r m
m
+ -
-
=
d b
ad bc
q   (10) 
which is the same as Barucci's result on p. 794 except for a sign error in the 
second formula. 
For  economic  reasons  only  positive  equilibria  k   make  sense.  Then  also 
k u m =  is positive. From (5) it follows that then also  c u d q + = 2  is positive. 
Proposition 1: The unique equilibrium is in the relevant region (k > 0, u > 0, 
first quadrant), iff the sign of  ( ) a c - + m r  equals the sign of  ( ) m r m + - d b . 
The Jacobian of the linear DE-system (2) and (7) is 
  ￿
￿
￿
￿
￿
￿
+ -
-
=
m r
m
d
b J
1
  and  ( ) d
b J + + - = m r m det , Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 4 -  10.03.05 
so that the equilibrium is a saddle point iff  
  ( ) b d > + m r m   (11) 
as was also found by Barucci (1998); see (i) on p. 794.  
The  0 = k ￿ -isocline is 
  k u m =   (12) 
and the  0 = u ￿ -isocline is 
  ( )
( ) ( )
k
d
b
d
a c
u
m r m r
m r
+
+
+
- +
- =
2
.  (13) 
Comparing this with (11) it follows that the  0 = k ￿ -isocline is steeper than the 
0 = u ￿ -isocline iff the equilibrium is a saddle point. 
5  Solution in the 4 different cases 
From Proposition 1 we obtain, that the signs of the expressions  ( ) a c - + m r  
and  ( ) m r m + -d b  are crucial for the outcome of the model. Consequently we can 
distinguish four different cases. 
5.1  Case 1:  ( ) a c - + m r  < 0 and  ( ) m r m + - d b  < 0 
In this case, from (9) we get a positive equilibrium, which is, by (11), a saddle 
point. 
The phase diagram looks as follows: 
k
u
0 = k ￿
0 = u ￿
 
Fig. 1. The phase diagram in Case 1 where  ( ) a c - + m r  < 0 and  ( ) m r m + -d b  < 0. Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 5 -  10.03.05 
This clearly illustrates the reverse accelerator feature of the stable investment 
path as expressed in Dechert (1983). 
We now compute the trajectories u(t) and k(t) along the saddle point path and 
evaluate the objective function: 
First, we have to get the eigenvalues: 
 
2
) ( 4 4 2
1
d
b - + + -
=
m r m r r
l , 
2
) ( 4 4 2
2
d
b - + + +
=
m r m r r
l .  (14) 
Then we need the eigenvector [k*, 1]' associated with the negative eigenvalue 
1 l  and get the solution: 
 
t e
k
k
k k
u
k
t u
t k
1
1
*
* ) (
) ( 0 l
￿
￿
￿
￿
￿
￿ -
+ ￿
￿
￿
￿
￿
￿
= ￿
￿
￿
￿
￿
￿
.  (15) 
The eigenvector is easily computed as follows: 
  ￿
￿
￿
￿
￿
￿
= ￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
- + -
- -
0
0
1
* 1
1
1 k
d
b l m r
l m
, 
which yields: 
 
1
1
*
l m +
= k .  (16) 
From (15) and (16) we generate the following expressions: 
  ( )
t e k k k t k 1
0 ) (
l - + = ,  (17) 
  ( )( )
t e k k u t u 1
1 0 ) (
l l m + - + = .  (18) 
Then, with  k k - = 0 a  and   1 l m b + = the profit rate is 
 
2 2 ) , ( du cu bk ak u k - - + = p  
[ ] [ ] [ ]
2 2 2 2 2 2 1 1 2 2 b a a ab ab a a
l l d b e u d c k b a e u d u c k b k a
t t - + - - + + - - + =  
Evaluating the objective function (1) using this expression, we get: 
  [ ] = - = P ￿
¥
-
0
0 ) ( ) ( ) ( dt u c k r e k
t r  
  .
2
2 2 2
1
2
1
2 2
a
l r
b
a
l r
b b
r -
-
+
-
- - +
+
- - +
=
d b u d c k b a u d u c k b k a
 Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 6 -  10.03.05 
Using  k k - = 0 a  we get the profit P(k0) as a function of the initial state: 
.
2 2
2
2 2
2
2 2
) (
2
0
1
2
0
1
2
1
2
1
2
1
2 2
0
k
d b
k k
d b u d c k b a
k
d b
k
u d c k b a u d u c k b k a
k
l r
b
l r
b
l r
b b
l r
b
l r
b b
r
-
-
+
￿
￿
￿
￿
￿
￿
￿
￿
-
-
-
-
- - +
+
￿
￿
￿
￿
￿
￿
￿
￿
-
-
+
-
- - +
-
- - +
= P
 
Apparently, profit P(k0) is a quadratic function of k0.It is clear, that the NPV of 
the  profit  is  higher  the  larger  the  initial  capital  stock  is.  This  means,  that  the 
coefficient of 
2
0 k  is positive, and that the unbounded minimum of P(k0) occurs for 
k0. £ 0. 
Extensive numerical experiments have shown, that in Case 1 the NPV of the 
profit is always positive, although it seems difficult to derive this result from the 
above  formulas  analytically.  If  one  wants  to  minimize  P(0)  under  all  the 
constraints that hold in Case 1 and under the positivity of the parameters, then one 
can make P(0) a positive number arbitrarily close to zero, while d becomes very 
large compared to all the other parameters and k  approaches zero.  
This confirms Barucci's (1998) result (Proposition 4.1) that approaching the 
equilibrium is always optimal in Case 1. 
5.2  Case 2:  ( ) a c - + m r  < 0 and  ( ) m r m + -d b  > 0 
In this case the equilibrium is not in the first quadrant and it is not a saddle 
point: Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 7 -  10.03.05 
0 = k ￿
0 = u ￿
k
u
 
Fig. 2. The phase diagram in Case 2 where  ( ) a c - + m r  < 0 and  ( ) m r m + -d b  > 0. 
The equilibrium with negative k and u is an unstable focus. 
If (3) is imposed, u = 0 and k ® 0 could be expected to be optimal when 
looking at the figure. However, for economic reasons it is clear that this is not 
true. Case 2 is characterized by large values of the parameters a and b occurring in 
the revenue function. In this case, approaching k = 0 is certainly not optimal.  
In fact, no optimal solution exists, since the objective is unbounded. This will 
now be verified by showing analytically that constant or proportional investment 
rates can yield arbitrarily high values of (1) 
5.2.1  Constant Investment 
We first consider a constant investment policy 
  u(t) = u*  for all  t. 
Then the capital stock develops according to 
 
t e
u
k
u
t k
m
m m
-
￿ ￿
￿
￿
￿ ￿
￿
￿
- + =
* *
) ( 0 . 
Evaluating the profit using these expressions, we get: 
 
2 2 ) , ( du cu bk ak u k - - + = p  Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 8 -  10.03.05 
 
.
*
* *
2
*
* *
* *
2
0
2
0 0
2
2
2
￿ ￿
￿
￿
￿ ￿
￿
￿
- +
￿
￿
￿
￿
￿
￿
￿ ￿
￿
￿
￿ ￿
￿
￿
- + ￿ ￿
￿
￿
￿ ￿
￿
￿
- +
￿
￿
￿
￿
￿
￿
￿
￿
- - + =
-
-
m
m m m m m
m
m
u
k b e
u
k
u
b
u
k e du cu
u
b
u
a
t
t
 
so that the objective function (1) becomes: 
  [ ] = - = P ￿
¥
-
0
0 ) ( ) ( ) ( dt u c k r e k
t r  
m r
m
m r
m m m
r
m m
2
* * *
2
*
* *
* *
2
0 0 0
2
2
2
+
￿ ￿
￿
￿
￿ ￿
￿
￿
-
+
+
￿ ￿
￿
￿
￿ ￿
￿
￿
- + ￿ ￿
￿
￿
￿ ￿
￿
￿
-
+
- - +
=
u
k b
u
k
u
b
u
k du cu
u
b
u
a
. 
The terms with u*
2 are 
 
m r
m
m r
m
r
m
2
1 1
2 2 2 2
+
+
+
-
- b b d
b
( )( )
( )( ) m r m r r
m r m r
2
2 2
+ +
+ + -
=
d b
. 
Thus,  the  objective  (1)  can  be  made  arbitrarily  large,  if  the  constant  u*  is 
chosen large enough, provided that  
  ( )( ) ( )
2
2
1
2
3 2 2
2
r rm m m r m r + + = + + > d
d
b . 
Barucci shows that the objective is unbounded for 
  ( ) rm m + >
2 d b  
which is a weaker condition.  
5.2.2  Proportional Investment 
We now consider a constant investment policy 
  u(t) = [m + e]k(t) for all t. 
Then the capital stock develops according to 
 
t e k t k
e
0 ) ( =    and   [ ]
t e k t u
e e m + = 0 ) ( . 
Evaluating the profit function using these expressions, we get: 
 
2 2 ) , ( du cu bk ak u k - - + = p  Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 9 -  10.03.05 
  ( ) [ ] ( ) [ ]
t t e d b k e c a k
e e e m e m
2 2 2
0 0 + - + + - = , 
which is again used to evaluate the objective function (1): 
  [ ] = - = P ￿
¥
-
0
0 ) ( ) ( ) ( dt u c k r e k
t r ( ) ( )
e r
e m
e r
e m
2
2
2
0 0 -
+ -
+
-
+ - d b
k
c a
k  
provided that 2e < r. 
However (1) is infinite for 2e ³ r. In particular, it is +¥ if  ( )
2 e m + > d b  and it 
is -¥ if  ( )
2 e m + < d b . 
So (1) is unbounded, provided that  ( )
2 e m + > d b  and 2e ³ r, i.e., if 
  ￿
￿
￿
￿
￿
￿
￿
￿
+ + = ￿
￿
￿
￿
￿
￿ + >
4 2
2
2
2 r
rm m
r
m d d b . 
This lower bound is better than that obtained for constant investment, but still 
above the Barucci boundary.  
5.3  Case 3:  ( ) a c - + m r  > 0 and  ( ) m r m + -d b  < 0 
In this case the equilibrium is not in the first quadrant and it is a saddle point: 
u
0 = k ￿
0 = u ￿
k
 
Fig. 3. The phase diagram in Case 3 where  ( ) a c - + m r  > 0 and  ( ) m r m + -d b  < 0. Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 10 -  10.03.05 
This case is characterized by small values of the parameters a and b occurring 
in the revenue function. In this case, approaching k = 0 is certainly optimal.  
The question remains whether u = 0 throughout, or whether it looks like shown 
in the above figure. 
5.3.1  Allowing for reversibility of investment 
Here it makes sense to consider the scenario where constraint (3) is replaced by 
the state constraint k ³ 0. Then the optimal trajectory in the phase diagram would 
converge to the k = 0 within finite time. Note that disinvestment occurs here. 
u 0 = k ￿
0 = u ￿
k
 
Fig. 3a. The phase diagram in Case 3 where investment is reversible. 
The proof is the same as in Example 8.8 on p.219 in Feichtinger and Hartl 
(1986). 
5.4  Case 4:  ( ) a c - + m r  > 0 and  ( ) m r m + -d b  > 0 
In this case the equilibrium is in the first quadrant and it is not a saddle point: 
Fehler! Es ist nicht möglich, durch die Bearbeitung von Feldfunktionen 
Objekte zu erstellen. 
Fig. 4. The phase diagram in Case 4 where  ( ) a c - + m r  > 0 and  ( ) m r m + -d b  > 0. Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 11 -  10.03.05 
The equilibrium is an unstable focus. Except for the fact that the equilibrium 
now occurs for  positive values of k and u this case is identical to Case 2. 
No  optimal  solution  exists,  and  that  the  objective  is  unbounded.  The 
calculations  in  Section  5.2  concerning  constant  and  proportional  investment, 
respectively, also apply to this case. 
6  Ideas for future research 
In this paper the standard capital accumulation model, but then with a strictly 
convex revenue function, was studied. Due to the quadratic specifications of the 
revenue and investment cost function, interesting results could be generated. A 
straightforward extension is to make the revenue function a third order polynomial 
in  which 
3 k   is multiplied  with  a  negative  parameter.    In  this  way  a  convex-
concave revenue function arises which makes it possible to redo the calculations 
of Dechert (1983) and Davidson and Harris (1981). As already mentioned in the 
Introduction, they arrived at solutions with multiple equilibria. They could identify 
levels of the capital stock, which we now call DNS (Dechert Nishimura Skiba)-
points, where the firm is indifferent concerning to which equilibrium it should 
converge. Hopefully, it is possible to generate additional insights concerning these 
DNS-points in case we study the model with such a third order polynomial as 
revenue function. 
A second interesting extension is to combine the just described convex-concave 
revenue function with introducing adjustment costs on changes in the investment 
rate (see Jorgensen and Kort (1983), Section 3.4.2). In such a model investments 
will be introduced as a second state variable, and the rate of change of investment 
is  the  control  variable.  Hence,  the  resulting  model  now  contains  two  state 
variables  and  one  control  variable.  It  can  be  expected  that  also  here  multiple 
steady states exist. Depending on the location in  the (k,u)-plane, it is optimal to 
converge to one of these steady states. It would be interesting to study whether so-
called DNS-curves (which are DNS-points in a one-state-variable-model) exist, on 
which the firm is indifferent concerning to which steady state to converge to.  
7  References 
Feichtinger, G., Hartl, R.F.: Optimale Kontrolle ökonomischer Prozesse: Anwendungen des 
Maximumprinzips in den Wirtschaftswissenschaften, Berlin: de Gruyter 1986 
Barucci, E.: Optimal investments with increasing returns to scale. International Economic 
Review 39, 789 - 808 (1998) 
Davidson, R., Harris, R.: Non-convexities in continuous-time investment theory. Review of 
Economic Studies 48, 235-253 (1981) 
Dechert,  A.:  Increasing  returns  to  scale  and  the  reverse  flexible  accelerator.  Economic 
Letters 13, 69 - 75 (1983) 
Dixit,  A.K.,  Pindyck,  R.S.,  Investment  under  Uncertainty,  second  printing,  Princeton: 
Princeton University Press 1996 Optimal Investments With Increasing Returns To Scale: Further Analysis 
09:14  - 12 -  10.03.05 
Eisner, R., Strotz, R.H., Determinants of Business Investments in Impacts of Monetary 
Policy, Englewood Cliffs, N.J.: Prentice Hall 1963  
Hartl, R.F., Kort, P.M.: Capital accumulation of a firm facing environmental constraints. 
Optimal Control Applications & Methods 17, 253-266 (1996) 
Jorgensen,  S.,  Kort,  P.M.:  Optimal  dynamic  investment  policies  under  concave-convex 
adjustment costs. Journal of Economic Dynamics and Control 17, 153-180 (1993) 
Rothschild, M.: On the cost of adjustment, Quarterly Journal of Economics 85, 605-622 
(1971) 
 

Optimal investments with increasing returns to scale: A further analysis.

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