=Paper=
{{Paper
|id=Vol-1623/paperme13
|storemode=property
|title=On One Multicriteria Optimal Control Problem of Economic Growth
|pdfUrl=https://ceur-ws.org/Vol-1623/paperme13.pdf
|volume=Vol-1623
|authors=Alexey Zakharov
|dblpUrl=https://dblp.org/rec/conf/door/Zakharov16
}}
==On One Multicriteria Optimal Control Problem of Economic Growth==
https://ceur-ws.org/Vol-1623/paperme13.pdf
On One Multicriteria Optimal Control Problem
of Economic Growth
Alexey O. Zakharov
Saint Petersburg State University,
7/9 Universitetskaya nab., St. Petersburg, 199034 Russia.
a.zakharov@spbu.ru
Abstract. An optimal control problem of an economic system with two criteria
is considered in the paper. The Solow model of economic growth is used. Rate of
consumption is a control variable, quality vector criterion is a discounted utility
in the long-term and near-term. Pontryagin’s maximum principle is applied to
the single criteria problem with weighted sum of two functionals. The existence
of the solution is investigated. The law of Pareto-optimal control, such that the
system converges to the balanced growth path, is derived.
Keywords: muticriteria problem of optimal control, the Solow model, the
Pareto set.
1 Introduction
The model of economic growth, proposed by R. M. Solow, plays an important role in
neoclassical theory [1, 4, 15] and originally is determined by labor and capital. Later the
contribution of endogenous parameters, technology and population, was investigated
by P. M. Romer [13], P. Aghion and P. Howitt [2], G. M. Grossman and E. Helpman [6],
C. I. Jones [7], D. Acemoglu [1]. According to these research technological advance is the
result of R&D activity, governmental actions (e.g., taxation, regulations of international
trade), financial markets, and imperfect competition.
Golden Rule saving rate was established by E. S. Phelps [10], and it could be
expressed in terms of the Solow growth model as follows: it is possible to maximize
the size of the consumption in the steady–state mode by choosing the rate of saving.
For this purpose we actually use the rate of consumption, which is naturally connected
with the rate of saving.
The model considered in the paper is based on the differential equation described
changes in capital, where development of technology and population are not taken into
account. Single and multiple criteria problems of optimal control built on the Solow
model were studied in [9, 3, 14]. Quality factors characterized a discounted utility of
consumption and an ecological impact were introduced, and a consumption or a rate
of consumption played the role of control variable in the aforementioned references. To
Copyright ⃝
c by the paper’s authors. Copying permitted for private and academic purposes.
In: A. Kononov et al. (eds.): DOOR 2016, Vladivostok, Russia, published at http://ceur-ws.org
On One Multicriteria Optimal Control Problem of Economic Growth 657
solve these problems Pontryagin’s maximum principle is used. As a result, the system
converges to a balanced growth path under constructed optimal control laws.
In the paper we investigate an optimal control problem with two criteria. First
component of vector criterion describes a discounted utility from an infinite planning
horizon to the present time. However, in such case a near-term outlook is not taken into
account. A discounted utility from a given finite time T in the future to the present time
is introduced as a second component of vector criterion. Then we transform derived
multiple criteria problem to a single criteria problem with weighted sum, which is a
piecewise functional. So, the problem is investigated first on interval [0, T ] and second
on interval [T, +∞).
Firstly, for each interval we study the existence of solution to the problem and
obtain constraints on parameters of the model. Secondly, we use Pontryagin’s maximum
principle to derive a form of Pareto-optimal control and then establish a stationary
point under this control. Summing up, we give a theorem that shows a Pareto-optimal
control law on the entire interval [0, +∞), and thus it propels the system to the balanced
growth path.
2 Model of Optimal Control with Two Quality Criteria
Let us consider a single-sector Solow model of optimal growth. We suppose that one
product, which could be consumed and invested, is produced. The following external
parameters describe the model: Y (t) is a gross domestic product (GDP), C(t) is a
nonproduction consumption, K(t) is basic production assets (capital), and ρ(t) is a
rate of consumption, i. e., the part of GDP going to the consumption: C(t) = ρ(t)Y (t),
where t > 0. In such notation the difference 1 − ρ(t) means a saving rate. It should
be mentioned that variables are not time invariant and technology is not changed.
Here, function ρ(·) plays a role of control variable, and the set of admissible control is
Uρ = {ρ(t) ∈ P C([0, 1]) | t > 0}.
The relation between GDP and capital is carried out via production function F (·).
Here we will use the Cobb-Douglas production function F (K) = AK α (t), where A > 0
is capital productivity and α ∈ (0, 1] is the elasticity of GDP with respect to capital.
According to [9] a capital is changed by the following equation:
K̇(t) = Y (t) − C(t) − δK(t), K(0) = K0 , (1)
where K ∈ G, G is a given bounded open set, t > 0, δ > 0 is a depreciation rate of
the stock of capital, K0 > 0 is the initial value of capital. From formula (1) one could
obtain the equation of Solow model
K̇(t) = (1 − ρ(t))AK α (t) − δK(t), K(0) = K0 , t > 0. (2)
Consider the logarithmic function of utility
u(ρ, K) = ln(ρAK α ) = u0 + ln(ρ) + α ln(K),
where u0 = ln(A) is a constant.
658 A. Zakharov
In book [9] Solow model was investigated with integral criterion of quality in the
long-term outlook. Let us introduce a multicriterion problem with vector criterion
I(ρ) = (I1 (ρ), I2 (ρ)):
∫ +∞ ∫ +∞
I1 (ρ) = e−rt u(ρ, K)dt = e−rt (u0 + ln(ρ) + α ln(K))dt,
0 0
∫ T ∫ T
I2 (ρ) = e−rt u(ρ, K)dt = e−rt (u0 + ln(ρ) + α ln(K))dt,
0 0
where r > 0 is a discount rate and T > 0 is a fixed given time. The first component of
vector criterion I1 (·, ·) indicates a discounted utility in the long-term outlook, and the
second component I2 (·, ·) indicates a discounted utility in the near-term outlook (until
time T ).
Thus, we obtain the following optimal control problem with two criteria:
I(ρ) = (I1 (ρ), I2 (ρ)) −→ max
ρ(·)∈Uρ
∫ +∞
I1 (ρ) = e−rt (ln(ρ) + α ln(K))dt,
0
∫ T (3)
I2 (ρ) = e−rt (ln(ρ) + α ln(K))dt,
0
K̇(t) = (1 − ρ(t))AK α (t) − δK(t), K(0) = K0 ,
ρ(t) ∈ Uρ , t > 0.
Without loss of generality we omit the component u0 , because it is constant and does
not effect on the result. Here, the notation max is formal.
Problem (3) with only criterion I1 was solved in book [9], i. e., a control ρ∗ (t),
which maximizes the functional I1 and corresponding balanced growth path K ∗ (t)
were established. To solve problem (3) one should find the Pareto set P I (Uρ ) (see,
e. g., [11]), since the problem is multicriterion. The Pareto set P I (Uρ ) is the set of
control ρ∗ (t) from admissible set Uρ , such that it does not exist any other control ρ′ (t)
from admissible set Uρ which satisfies inequality I(ρ′ (t)) ≥ I(ρ∗ (t)). Thus,
P I (Uρ ) = {ρ∗ (t) ∈ Uρ | @ ρ′ (t) ∈ Uρ : I(ρ′ (t)) ≥ I(ρ∗ (t))}.
The control ρ∗ (t) is called Pareto-optimal control. Here, inequality I(ρ′ (t)) ≥ I(ρ∗ (t))
means that inequalities I1 (ρ′ (t)) > I1 (ρ∗ (t)), I2 (ρ′ (t)) > I2 (ρ∗ (t)) hold, that is more
I(ρ′ (t)) ̸= I(ρ∗ (t)).
So, to solve problem (3) we should find the Pareto set P I (Uρ ). For that purpose we
use Pontryagin’s maximum principle, which will be considered in the next section.
3 Use of Pontryagin’s Maximum Principle to Weighted Sum
of Criteria
Consider the set I = I(Uρ ). It could be easily proved that the set I is convex. According
to Proposition III.1.5 from [8] if the set I is convex, then for any function ρ∗ (t) ∈ P I (Uρ )
On One Multicriteria Optimal Control Problem of Economic Growth 659
there exists such parameter µ = (µ1 , µ2 ) ≥ 02 that ρ∗ (t) = arg max φ(ρ(t)), where
ρ(t)∈Uρ
φ(ρ(t)) = µ1 I1 (ρ(t)) + µ2 I2 (ρ(t)).
Also one could establish that any admissible control ρ∗ (t) maximizing functional
φ(ρ) is a Pareto-optimal control with respect to criterion I. Thus, multicriteria prob-
lem (3) is equivalent to the problem with one criterion φ(ρ) with nonnegative param-
eters µ1 , µ2 :
φ(ρ) −→ max
ρ(·)∈Uρ
(4)
subject to (2), ρ(t) ∈ Uρ , µ ≥ 02 , t > 0.
3.1 Solution to the Problem on Interval [0, T ]
The weighted sum φ(ρ), as an optimization criterion of problem (4), is piece-wise
defined: on interval [0, T ] and on interval [T, +∞). Let us initially consider problem (4)
on finite interval [0, T ]. Put φT (ρ) = φ(ρ) .
t∈[0,T ]
∫ T
φT (ρ) = e−rt (µ1 + µ2 )(ln(ρ) + α ln(K))dt −→ max
0 ρ(·)∈Uρ
(5)
subject to: K̇(t) = (1 − ρ(t))AK (t) − δK(t), K(0) = K0 ,
α
ρ(t) ∈ Uρ , µ ≥ 02 , t ∈ [0, T ].
Conditions of existence of solution to optimal control problem on infinite interval
was established in [3], [14]. Let us investigate the corresponding conditions for prob-
lem (5). Firstly, one could obtain that there exists such constant L > 0 that inequality
K((1 − ρ)AK α − δK) 6 L(1 + K 2 )
holds for all K ∈ G, ρ ∈ Uρ (condition (A1) in [3]). Secondly, condition (A2) [3] of
convexity of set
Q(K) = {(z 0 , z) : z 0 6 (µ1 +µ2 )(ln(ρ) + α ln(K));
z = (1 − ρ(t))AK α (t)−δK(t); ρ ∈ Uρ }
is valid for all K ∈ G due to linearity of the equality with respect to ρ and convex-
ity of logarithmic function. Then from conditions (A1), (A2) and Filippov’s theorem
(Theorem 9.3.i from [5]) there exists an optimal pair (K ∗ , ρ∗ ) for problem (5).
As aforesaid, if some control ρ = ρ∗ (t) is an optimal control with respect to func-
tional φ(ρ), and hence it is a Pareto-optimal solution with respect to criterion I, then
according to Pontryagin’s maximum principle [12] such control maximizes Hamilton
function.
Let ψ0 and ψ be the adjoint variables. Construct a Hamilton function to problem (5):
H(t, K(t), ψ0 , ψ̂(t), ρ) = ((1 − ρ)AK α − δK)ψ̂ + ψ0 (µ1 + µ2 )(ln(ρ) + α ln(K)),
660 A. Zakharov
where ψ̂(t) = ert ψ(t). Also without loss of generality we can assume that ψ0 = 1.
Hamilton function could be written in this way according to [3].
Let ρopt ∈ P I (Uρ ). From equation ∂H/∂ρ = 0 we obtain constrained maximum
0, K α ψ̂ < 0,
µ1 +µ2
ρopt = AK , K α ψ̂ > µ1 +µ
A ,
2
(6)
α ψ̂
1, 06K α
ψ̂ 6 µ1 +µ
A ,
2
and also the adjoint equation is the following:
˙ 1
ψ̂ = ((r + δ) − α(1 − ρ)AK α−1 )ψ̂ − α(µ1 + µ2 ) , ψ̂(T ) = 0. (7)
K
Using the same arguments as in Theorem 10.1 [3] we can establish that the adjoint
variable ψ̂(t) > 0 for all t ∈ [0, T ). It leads to the impossibility of the equality ρopt = 0
in formula (6).
As mentioned above derived control (6), which depends on nonnegative parameters
µ1 , µ2 (µ1 + µ2 > 0), is a Pareto-optimal control of problem (5).
3.2 Constructing Balanced Growth Path on Interval [0, T ]
From the economic point of view consideration of steady-state mode K(t) = = K ∗ ,
C(t) = C ∗ has a practical interest since it shows an equilibrium state of the economic
system.
µ1 +µ2
Consider the case when ρ = AK α ψ̂
. Then using formula (6) we change variable ψ̂ by
variable C(t) = ρ(t)Y (t) in equation (7). So, we have the following Hamilton system:
K̇(t) = AK α (t) − C(t) − δK(t),
(8)
Ċ(t) = C(t)(αAK α−1 (t) − (r + δ)).
Conditions K̇(t) = 0, Ċ(t) = 0 give a stationary solution (K ∗ , C ∗ ) of equations (8)
( ) α−1
1 ( ) α−1
1 ( ( )α )
∗ r+δ ∗ r+δ r+δ
K = , C = · A −δ . (9)
αA αA αA
It is easy to see that K ∗ > 0 and C ∗ > 0, if the inequality K ∗ < (A/δ)
1/(1−α)
is
valid. One can show that if ρ ≡ 1 on interval [0, T ], then steady-state mode is K ∗ ≡ 0,
C ∗ ≡ 0. Obviously, this case does not have a practical sense.
Lemma 1. Stationary point (K ∗ , C ∗ ) (9) of Hamilton system (8) is a saddle point.
The proof is based on linearization of Hamilton system (8) in a neighborhood of sta-
tionary point (K ∗ , C ∗ ). Moreover, eigenvalues of Jacobian of system (8) are real, and
its product is negative.
On One Multicriteria Optimal Control Problem of Economic Growth 661
3.3 Solution to the Problem on Interval [T, +∞)
Now consider problem (4) on interval [T, +∞). Put φ+∞ (ρ) = φ(ρ) .
t∈[T,+∞)
∫ +∞
φ+∞ (ρ) = e−rt µ2 (ln(ρ) + α ln(K))dt −→ max
T ρ(·)∈Uρ
(10)
subject to: K̇(t) = (1 − ρ(t))AK (t) − δK(t), K(T ) = K1 (T ),
α
ρ(t) ∈ Uρ , µ2 > 0, t ∈ [T, +∞),
where K1 (t) is the solution of equation (2) on interval [0, T ]. According to [14] one
could derive that the following result is valid.
Lemma 2. A solution to optimal control problem (10) exists if A > δ and
( ) 1−α
1
A
0 6 K0 6 . (11)
δ
The proof is based on checking the implementation of conditions (A1), (A2), and (A3)
from [3] and conditions of Filippov’s theorem (Theorem 9.3.i from [5]).
Construct a Hamilton function to problem (10):
H(t, K(t), ψ0 , ψ̂(t), ρ) = ((1 − ρ)AK α − δK)ψ̂ + µ2 (ln(ρ) + α ln(K)),
Let ρopt ∈ P I (Uρ ). Simultaneously Subsection 3.1 we establish constrained maximum
0,
K α ψ̂ < 0,
ρopt = AK2α ψ̂ , K α ψ̂ > A2 ,
µ µ
(12)
1, 0 6 K ψ̂ 6 A ,
α µ2
and the adjoint equation
˙ 1
ψ̂ = ((r + δ) − α(1 − ρ)AK α−1 )ψ̂ − αµ2 . (13)
K
Also if conditions A > δ and (11) are hold, the adjoint variable ψ̂(t) > 0 for all
t ∈ [T, +∞) (see Theorem 10.1 [3] and Lemma 4.1 in [14]). So, the equality ρopt = 0 in
formula (12) is impossible. We establish that derived control (12), which depends only
on parameter µ2 > 0, is a Pareto-optimal control of problem (10).
3.4 Law of Optimal Control for System (3)
µ2
Similarly to Subsection 3.2 if we assume that ρ = AK α ψ̂
, then we will have the same
∗ ∗
Hamilton system (8) with stationary solution (K , C ) (9).
Since a stationary solution K ∗ ≡ 0, C ∗ ≡ 0 is yielded if ρ ≡ 1, this control could
not propel the system to the balanced growth path K(t) = K ∗ , C(t) = C ∗ , which is
assigned by formula (9).
We prove that the asymptotic behavior of Pareto-optimal control is determined by
the following theorem.
662 A. Zakharov
Theorem 1. Let inequalities A > δ and (11) hold, point (K ∗ (·), ρ∗ (·)) is a Pareto-
optimal point of problem (3), and variable ψ̂(·) is the adjoint variable. Then there
exists such τ > 0, that
µ1 + µ2
ρ∗ (t) = for all t ∈ [τ, T ], if τ 6 T ;
Aψ̂(t)(K ∗ )α (t)
µ2
ρ∗ (t) = for all t ∈ [T, +∞), if τ 6 T (14)
Aψ̂(t)(K ∗ )α (t)
( for all t ∈ [τ, +∞) , if τ > T ),
where parameters µ1 , µ2 > 0, µ1 + µ2 > 0.
The proof is based on Lemma 4.2 from [14].
Thus, Pareto-optimal control (14) asymptotically propels the system (3) to the
balanced growth path K(t) ≡ K ∗ ,C(t) ≡ C ∗ (9). Moreover, the situation, in which the
control ρ ≡ 1 on interval [0, τ ] could also be used, is to be investigated.
Acknowledgements
The author is grateful to Profs. V.D. Noghin and A.V. Prasolov for the statement of
the problem.
This research is supported by the Russian Foundation for Basic Research, project
no. 14-07-00899.
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