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- 1 -
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 we study 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 describes 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
- 2 -
function is strictly convex throughout. 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 dtuckret
u
ρ,(1)
kukµ−=
&
, 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 ρ while µ
denotes the depreciation rate.
Although Barucci did not impose this constraint, for economic reasons (see,
e.g., Dixit and Pindyck (1996)) we assume that investments are irreversible:
0
≥
u.(3)
In order to be able to obtain a full analytical solution, like Barucci (1998) we
assume quadratic revenue and cost functions:
r(k) = ak + bk2 , c(u) = cu + du2.(4)
We require all parameters a, b, c, d, µ, and ρ to be positive. Hence, as already
explained in the Introduction, the revenue function exhibits increasing returns to
scale. To illustrate the importance of analyzing this framework, we can refer to,
e.g., Hartl and Kort (1996). In this paper a variant of the current model was studied
where the revenue function is concave and pollution was included. Also a second
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 3 -
control in the form of abatement expenditures was added. For this model it turned
out that, after solving for abatement expenditures, an optimal control model results
for which in one particular case the objective is strictly convex in k. In Hartl and
Kort (1996) this scenario was not analyzed because it seemed to complicated at
that time.
Investment costs include costs of acquisition and adjustment costs. In the next
section it turns out that the strictly convex shape of the investment cost function
implies that u is continuous over time.
3. Necessary conditions
To obtain the necessary conditions for optimality we start out by presenting the
current value Hamiltonian:
H = ak + bk2 - cu - du2 +q[u - µk]. (5)
From the maximum principle it is derived that:
0
=
∂
∂
uH i.e. d
cq
u2
−
=.(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)
Since the Hamiltonian is strictly concave in u we know from, e.g., Feichtinger
and Hartl (1986) that u is continuous over time. The adjoint equation is
(
)
bkaqkHqq 2
−
−
+
=
∂
∂
−
=
µ
ρ
ρ
&
.(7)
From (6), i.e., q = 2du + c and (7) we get:
( )
(
)
k
d
b
d
ac
u
d
q
u−
−+
++== 22
µρ
µρ
&
&.(8)
4. Equilibrium and its stability properties
The (unbounded) linear DE-system (2) and (8) has the following unique
equilibrium:
(
)
( )
µρµ
µ
ρ
+−
−
+
=db
ac
k2
1
(
)
( )
µρµ
µρµ
µ+−
−+
== db
ac
ku2.(9)
On the other hand, the original canonical system (2) and (7) has the unique
equilibrium
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 4 -
(
)
( )
µρµ
µρ
+−
−+
=db
ac
k2
1
( )
µρµ
µ
+−
−
=db
adbc
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 kuµ=
is positive, which in turn implies that then also cudq
+
=
2 is positive too.
Proposition 1: The unique equilibrium is in the relevant region (k > 0, u > 0,
first quadrant), iff the sign of
(
)
ac
−
+
µ
ρ
equals the sign of
(
)
µ
ρ
µ
+
−
db .
The Jacobian of the linear DE-system (2) and (8) is
+−
−
=µρ
µ
d
b
J1 and
(
)
d
b
J++−= µρµ
det ,
so that the equilibrium is a saddle point iff
(
)
bd
>
+
µ
ρ
µ
(11)
as was also found by Barucci (1998); see (i) on p. 794.
The 0=k
&
-isocline is
ku
µ
=
(12)
and the 0
=
u
&
-isocline is
(
)
( ) ( )
k
d
b
d
ac
uµρµρ
µ
ρ
+
+
+
−
+
−= 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 four different cases
From Proposition 1 we obtain, that the signs of the expressions
(
)
ac
−
+
µ
ρ
and
(
)
µ
ρ
µ
+
−
db are crucial for the outcome of the model. Consequently we can
distinguish four different cases.
5.1 Case 1:
(
)
ac
−
+
µ
ρ
< 0 and
(
)
µ
ρ
µ
+
−
db < 0
In this case, from (9) we get a positive equilibrium, which is, by (11), a saddle
point. Figure 1 clearly illustrates the reverse accelerator feature of the stable
investment path as expressed in Dechert (1983). This means that investment is
lower the larger the difference between the steady state level and the current level
of capital goods. The economic intuition behind this is that in this model marginal
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 5 -
revenue increases with the capital stock so that investment is more profitable if the
capital stock is large.
k
u
0=k
&
0
=
u
&
Fig. 1. The phase diagram in Case 1 where
(
)
ac−+ µρ < 0 and
(
)
µρµ+− db < 0.
We now compute the trajectories u(t) and k(t) along the saddle point path and
evaluate the objective function. This is normally impossible, but here it can be
done due to the fact that the functions r(k) and c(u) are quadratic (cf. (4)).
First, we have to obtain the eigenvalues associated with the Jacobian of the
dynamic system (see section 4):
2
)2(4
2
1d
b
−+−
=µρρ
λ,
2
)2(4
2
2d
b
−++
=µρρ
λ.(14)
It is easily obtained that 1
λ
is negative while 2
λ
is positive. Since the solution
of Figure 1 is stable only 1
λ
must be considered.
Then we need to compute the eigenvector [k*, 1]' associated with this negative
eigenvalue 1
λ
and get the solution:
t
e
k
k
kk
u
k
tu
tk1
1
*
*
)(
)( 0λ
−
+
=
.(15)
Taking once again the Jacobian of the dynamic system into consideration, the
eigenvector is easily computed as follows:
=
−+−
−−
0
0
1
*
1
1
1k
d
bλµρ
λµ,
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 6 -
which yields:
1
1
*λµ+
=
k.(16)
From (15) and (16) we generate the following expressions:
(
)
t
ekkktk1
0
)( λ
−+= ,(17)
(
)
(
)
t
ekkutu1
10
)( λ
λµ+−+= .(18)
Then, with kk −= 0
α and 1
λ
µ
β
+
=
the profit rate is
22
),(ducubkakuk−−+=π
[
]
[
]
[
]
222
2
22 11 22 βαααβαβαα λλ dbeudckbaeuduckbkatt −+−−++−−+=
Evaluating the objective function (1) using this expression, we get:
[ ]
==Π∫
∞−
0
0),()( dtukek tπ
ρ
.
2
22 2
1
2
1
22 α
λρ
β
α
λρ
ββ
ρ−
−
+
−
−−+
+
−−+
=dbudckbauduckbka
Using kk −= 0
α we get the profit Π(k0) as a function of the initial state:
.
22
2
22
2
22
)(
2
0
1
2
0
1
2
1
2
1
2
1
22
0
k
db
kk
dbudckba
k
db
k
udckbauduckbka
k
λρ
β
λρ
β
λρ
ββ
λρ
β
λρ
ββ
ρ
−
−
+
−
−
−
−
−−+
+
−
−
+
−
−−+
−
−−+
=Π
Apparently, profit Π(k0) is a quadratic function of k0. It is clear, that the net
present value 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 that the coefficient of k0 is
positive, i.e. the minimum of Π(k0) occurs for an infeasible k0 < 0. This is shown in
Appendix 1.
Extensive numerical experiments have shown, that in Case 1 the net present
value of profit is always positive. Unfortunately it turned out to be too difficult to
derive this result analytically. If one wants to determine the parameter values a, b,
c and d such that Π(0) is minimized, taking into account all the constraints that
hold in Case 1 and the positivity of the parameters, then one can make Π(0) a
positive number arbitrarily close to zero. In this case d is very large compared to all
the other parameters and k approaches zero. If this Π(0) would not have been
positive the solution that converges to the saddle point would have been dominated
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 7 -
by a policy of zero investment throughout. The outcome of this exercise does not
contradict Barucci's (1998) result (Proposition 4.1) that approaching the
equilibrium is always optimal in Case 1.
5.2 Case 2:
(
)
ac
−
+
µ
ρ
< 0 and
(
)
µ
ρ
µ
+
−
db > 0
In this case the equilibrium is not in the first quadrant and it is not a saddle
point:
0=k
&
0
=
u
&
k
u
Fig. 2. The phase diagram in Case 2 where
(
)
ac−+ µρ < 0 and
(
)
µρµ+− db > 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.
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 8 -
Then the capital stock develops according to
t
e
u
k
u
tkµ
µµ
−
−+= **
)( 0.
Evaluating the profit using these expressions, we get:
22
),(ducubkakuk−−+=π
.
*
**
2
*
**
**
2
0
2
00
2
2
2
−+
−+
−+
−−+=
−
−
µ
µµµ
µ
µ
µ
µ
u
kbe
u
k
u
b
u
kaeducu
u
b
u
a
t
t
so that the objective function (1) becomes:
[ ]
==Π∫
∞−
0
0),()( dtukek tπ
ρ
µρ
µ
µρ
µµµ
ρ
µ
µ
2
***
2
*
**
** 2
000
2
2
2
+
−
+
+
−+
−
+
−−+
=
u
kb
u
k
u
b
u
ka
ducu
u
b
u
a
.
The terms with u*2 are
µρ
µ
µρ
µ
ρ
µ
2
11
2222
+
+
+
−
−bbd
b
(
)
(
)
( )( )
µρµρρ
µρµρ
2
22
++
++−
=db .
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
ρρµµµρµρ++=++> d
d
b .
Barucci shows that the objective is unbounded for
(
)
ρµµ+> 2
db
which is a weaker condition.
5.2.2 Proportional Investment
We now consider a constant investment policy
u(t) = [µ + ε]k(t) for all t.
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 9 -
Substitution of this expression into (2) yields that the capital stock develops
according to
t
ektkε
0
)( = and
[
]
t
ektuε
εµ+= 0
)( .
Evaluating the profit function using these expressions, we get:
22
),(ducubkakuk−−+=π
(
)
[
]
(
)
[
]
tt edbkecakεε εµεµ2
2
2
00 +−++−= ,
which is again used to evaluate the objective function (1):
[ ]
==Π∫
∞−
0
0),()( dtukek tπ
ρ
(
)
(
)
ερ
εµ
ερ
εµ
2
2
2
00 −
+−
+
−
+− db
k
ca
k
provided that 2ε < ρ.
However (1) is infinite for 2ε ≥ ρ. In particular, it is +∞ if
(
)
2
εµ+> db and it
is -∞ if
(
)
2
εµ+< db .
So (1) is unbounded, provided that
(
)
2
εµ+> db and 2ε ≥ ρ. Then it holds that
++=
+> 42
2
2
2ρ
ρµµ
ρ
µddb .
This lower bound is better than that obtained for constant investment, but still
above the Barucci boundary.
5.3 Case 3:
(
)
ac
−
+
µ
ρ
> 0 and
(
)
µ
ρ
µ
+
−
db < 0
In this case the equilibrium is not in the first quadrant and it is a saddle point:
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. Note that
k = 0 will not be reached in finite time.
Although in this particular case the parameters in the revenue function are small
compared to those of he investment cost function, small investment expenditures
will still be profitable if k is sufficiently large. This is true because marginal
revenue increases linearly with k. Therefore, u will be positive for large values of
k, but definitely zero for low values of the capital stock; see Figure 3.
5.3.1 Allowing for reversibility of investment
Since Figure 3 shows that u = 0 at a final time interval, it makes sense to
consider the scenario here, where constraint (3) is replaced by the state constraint k
≥ 0. Then the optimal trajectory in the phase diagram would converge to k = 0
within finite time. This is sketched in Figure 3a. Note that disinvestment does
occurs here.
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 10 -
The proof is the same as in Example 8.8 on p.219 in Feichtinger and Hartl
(1986).
u
0
=
k
&
0
=
u
&
k
Fig. 3. The phase diagram in Case 3 where
(
)
ac−+ µρ > 0 and
(
)
µρµ+− db < 0.
u
0=k
&
0
=
u
&
k
Fig. 3a. The phase diagram in Case 3 where investment is reversible.
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 11 -
5.4 Case 4:
(
)
ac
−
+
µ
ρ
> 0 and
(
)
µ
ρ
µ
+
−
db > 0
In this case the equilibrium is in the first quadrant and it is not a saddle point:
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 the objective is unbounded. The calculations in
Section 5.2 concerning constant and proportional investment, respectively, also
apply to this case.
u
0
=
k
&
0
=
u
&
k
Fig. 4. The phase diagram in Case 4 where
(
)
ac−+ µρ > 0 and
(
)
µρµ+− db > 0.
6. Directions 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 k3 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
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 12 -
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
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)
Appendix 1
We now show that in Case 1 the profit Π(k0) is increasing in k0.
We first show that that the coefficient of 2
0
k is positive:
1
2
2λρ
β
−
−db > 0 ⇔ (note that 02 1
>
−
λ
ρ
because of (14))
Optimal Investments With Increasing Returns To Scale: Further Analysis
- 13 -
2
)2(
4β>
d
b ⇔ (using 1
λ
µ
β
+
=
and (14))
d
b
d
b4
)2(2
42−+−+> µρµρ ⇔
µρµρ2
4
)2(
42+>−++ d
b
d
b ⇔ (since both sides are positive)
( ) ( )
2
2
22
4
)2(
4
2
4
2
4µρµρµρ+>−++−++ d
b
d
b
d
b
d
b ⇔
0
4
)2(2>−+ d
b
µρ which is clearly true.
We now show that that the coefficient of k0 is positive:
k
dbudckba
1
2
12
2
22
λρ
β
λρ
ββ
−
−
−
−
−−+ > 0 ⇔
(
)
(
)
(
)
(
)
( )( ) ( )( )
( )
( )
[]
0222
2222
1
2
11
1
2
1
>−−−−−+−−
=−−−−−−+
kdbdbca
kdbudckba
λρβλρβµλρβ
λρβλρββ
This is true because of:
(
)
(
)
(
)
(
)
02)(211
>
−
+
−
>
−
−
λ
ρ
µ
ρ
λ
ρ
β
caca in Case 1 (first term)
and using 1
λ
µ
β
+
=
the bracket in the second term can be written as:
(
)
(
)
(
)
(
)
=−−−−−−−− 11
2
1
2
11
222 λρµλλµλρµλµdddbddb
(
)
(
)
( ) ( )
=−−−++++−+
++−+−−
2
1
3
11
2
11
2
1
2
2
11
2
11
2
22
222
µλλλµλρµλρλρµρ
µλλµλρµλρµρ
dddbdddb
ddbddb
[
]
1
2
11
2λλρµρλµ−+++− d
b
d = 0
since from Section 4 we know that the eigenvalue satisfies
(
)
0
2=++−− d
b
µρµρλλ
Thus, profit Π(k0) is higher, the larger the initial capital stock is.
