On One Multicriteria Optimal Control Problem of Economic Growth

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

Introduction

Copyright ⃝c by the paper's authors. Copying permitted for private and academic purposes. 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 in nite 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 nite 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 rst on interval [0; T ] and second on interval [T; +1).

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; +1), 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 (t) is a rate of consumption, i. e., the part of GDP going to the consumption: C(t) = (t)Y (t), where t > 0. In such notation the difference 1 (t) means a saving rate. It should be mentioned that variables are not time invariant and technology is not changed. Here, function ( ) plays a role of control variable, and the set of admissible control is U = f (t) 2 P C([0; 1]) j t > 0g.

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 (t), where A > 0 is capital productivity and 2 (0; 1] is the elasticity of GDP with respect to capital.

According to [9] a capital is changed by the following equation:

K_ (t) = Y (t)

C(t) where K 2 G, G is a given bounded open set, t > 0, > 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 where u0 = ln(A) is a constant.

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( ) = (I1( ); I2( )):

I1( ) =

I2( ) = ∫ +1 0 ∫ T 0 e rtu( ; K)dt = e rtu( ; K)dt = ∫ +1

0 ∫ T 0 e rt(u0 + ln( ) +

ln(K))dt; e rt(u0 + ln( ) + ln(K))dt; where r > 0 is a discount rate and T > 0 is a xed given time. The rst 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: 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 (t), which maximizes the functional I1 and corresponding balanced growth path K (t) were established. To solve problem ( 3 ) one should nd the Pareto set P I(U ) (see, e. g., [11]), since the problem is multicriterion. The Pareto set P I(U ) is the set of control (t) from admissible set U , such that it does not exist any other control ′(t) from admissible set U which satis es inequality I( ′(t)) I( (t)). Thus, P I(U ) = f

I( (t))g: The control (t) is called Pareto-optimal control. Here, inequality I( ′(t)) I( (t)) means that inequalities I1( ′(t)) > I1( (t)); I2( ′(t)) > I2( (t)) hold, that is more I( ′(t)) ̸= I( (t)).

So, to solve problem ( 3 ) we should nd the Pareto set P I(U ). 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(U ). 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 (t) 2 P I(U ) there exists such parameter = ( 1; 2) 02 that φ( (t)) = 1I1( (t)) + 2I2( (t)).

Also one could establish that any admissible control (t) maximizing functional φ( ) is a Pareto-optimal control with respect to criterion I. Thus, multicriteria problem ( 3 ) is equivalent to the problem with one criterion φ( ) with nonnegative parameters 1; 2: (t) = arg max φ( (t)), where

(t)2U φ( ) The weighted sum φ( ), as an optimization criterion of problem ( 4 ), is piece-wise de ned: on interval [0; T ] and on interval [T; +1). Let us initially consider problem ( 4 ) on nite interval [0; T ]. Put φT ( ) = φ( ) t2[0;T ]

. 2 U (condition (A1) in [3]). Secondly, condition (A2) [3] of holds for all K 2 G, convexity of set

Q(K) = f(z0; z) : z0 6 ( 1+ 2)(ln( ) +

ln(K)); is valid for all K 2 G due to linearity of the equality with respect to and convexity of logarithmic function. Then from conditions (A1), (A2) and Filippov's theorem (Theorem 9.3.i from [5]) there exists an optimal pair (K ; ) for problem ( 5 ).

As aforesaid, if some control = (t) is an optimal control with respect to functional φ( ), 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 0 and be the adjoint variables. Construct a Hamilton function to problem ( 5 ): )AK

K) ^ + 0( 1 + 2)(ln( ) + ln(K)); ( 4 ) ( 5 ) where ^(t) = ert (t). Also without loss of generality we can assume that Hamilton function could be written in this way according to [3].

Let opt 2 P I(U ). From equation @H=@ = 0 we obtain constrained maximum Using the same arguments as in Theorem 10.1 [3] we can establish that the adjoint variable ^(t) > 0 for all t 2 [0; T ). It leads to the impossibility of the equality opt = 0 in formula ( 6 ).

As mentioned above derived control ( 6 ), which depends on nonnegative parameters 1, 2 ( 1 + 2 > 0), is a Pareto-optimal control of problem ( 5 ). 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.

Consider the case when = A1K+ 2^ . Then using formula ( 6 ) we change variable ^ by variable C(t) = (t)Y (t) in equation ( 7 ). So, we have the following Hamilton system:

K_ (t) = AK (t) C_ (t) = C(t)( AK

C(t) 1(t)

K(t); (r + )): Conditions K_ (t) = 0, C_ (t) = 0 give a stationary solution (K , C ) of equations ( 8 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ) is 0, K = ( r +

A The proof is based on linearization of Hamilton system ( 8 ) in a neighborhood of stationary point (K ; C ). Moreover, eigenvalues of Jacobian of system ( 8 ) are real, and its product is negative.

Solution to the Problem on Interval [T ; +1) Now consider problem ( 4 ) on interval [T; +1). Put φ+1 ( ) = φ( ) t2[T;+1) . φ+1 ( ) = ∫ +1

T e rt 2(ln( ) + ln(K))dt 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 > and opt = 80; > <

2 ;

AK ^ :>1;

K ^ < 0, K

^ > A2 , 0 6 K

^ 6 A2 , 0 6 K0 6 ( A ) 1 1 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 ): Also if conditions A > and ( 11 ) are hold, the adjoint variable ^(t) > 0 for all t 2 [T; +1) (see Theorem 10.1 [3] and Lemma 4.1 in [14]). So, the equality opt = 0 in formula ( 12 ) is impossible. We establish that derived control ( 12 ), which depends only on parameter 2 > 0, is a Pareto-optimal control of problem ( 10 ). Theorem 1. Let inequalities A > and ( 11 ) hold, point (K ( ); ( )) is a Paretooptimal point of problem ( 3 ), and variable ^( ) is the adjoint variable. Then there exists such > 0, that (t) = (t) =

A ^(t)(K ) (t) ( for all t 2 [ ; +1) , if > T ); ( 14 ) where parameters 1; 2 > 0, 1 + 2 > 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 1 on interval [0; ] 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.

References

1. Acemoglu , D. : Introduction to Modern Economic Growth. Princeton University Press ( 2009 )

2. Aghion , P. , Howitt , P. : Endogenous Growth Theory . MIT Press, Cambridge MA ( 1998 )

3. Aseev , S.M. , Kryazhimskii , A.V. : The pontryagin maximum principle and optimal economic growth problems . In: Proceedings of the Steklov Institute of Mathematics . vol. 257 ( 1 ), pp. 1 { 255 ( 2007 )

4. Barro , R.J. , Sala-i Martin , X.: Economic Growth . The MIT Press, London, England ( 2004 )

5. Cesari , L. : Optimization { theory and applications. Problems with ordinary differential equations . Springer, New York ( 1983 )

6. Grossman , G.M. , Helpman , E.: Innovation and Growth in the Global Economy . MIT Press, Cambridge, MA ( 1991 )

7. Jones , C.I. : Growth:with or without scale effects . American Economic Review 89 , 139 { 144 ( 1999 )

8. Makarov , I.M. , Vinogradskaya , T.M. , Rubchinskiy , A.A. , Sokolov , V.B.: Teoriya vybora i prinyatiya recheniy (Theory of choice and decision making) (In Russian) . Nauka , Moscow ( 1982 )

9. Noghin , V.D.: Vvedenie v optimal'noe upravlenie (Introduction to optimal control) (In Russian) . Yutas, Saint-Petersburg ( 2008 )

10. Phelps , E.S. : Golden Rules of Economic Growth: Studies of Efficient and Optimal Investment . W. W. Norton, New York ( 1966 )

11. Podinovskiy , V.V. , Noghin , V.D.: Pareto-optimal'nye resheniya mnogocriterial'nyh zadach (Pareto-optimal solutions of multicriteria problems) (In Russian) . Fizmatlit , Moscow ( 2007 )

12. Pontryagin , L.S. , Boltyanskii , V.G. , Gamkrelidze , R.V. , Mishechenko , E.F. : The Mathematical Theory of Optimal Processes . John Wiley & Sons, New York / London ( 1962 )

13. Romer , P.M.: Endogenous technological change . Journal of Political Economy 98 , part II , 71 { 102 ( 1990 )

14. Rovenskaya , E.A. : Model' economicheskogo rosta i svyazannogo s nim kachestva okruzhayushchey sredy (economic growth model and related quality of environment) (in russian) . Matematicheskaya teoriya igr i ee prilozheniya 3 ( 3 ), 67 { 84 ( 2011 )

15. Solow , R.M. : Growth Theory: An Exposition . Oxford University Press, 2 edn . ( 2000 )