A differential game among sectors in a macroeconomy

Automatica, Vol. 11, pp. 473-485. Pergamon Press, 1975. Printed in Great Britain

A Differential Game Among Sectors in a Macroeconomy* Un Jeu Diff6rentiel Entre Secteurs dans une Macro6conomie Ein Differentialspiel zwischen Sektoren in einer Makro6konomie L. F. PAUl"

Macroeconomic planning requires that power relationships among economic agents be taken into account and an example of the corresponding open-loop Nash differential game, based on the Danish economy, is solved by a new algorithm. Snmmary--Tbe purpose of this paper is primarily to model the Danish economy as a differential game among the sectors, and to solve this game using an algorithm (A) published in full detail elsewhere. This numerical algorithm (A) is first described briefly. It is used for the approximation of open-loop Nash-Cournot equilibrium controls in a differential game of fixed duration and initial state; it is based upon a hierarchical decomposition of the differential game into optimal control problems, with a fictive referee for the Nash-Cournot playing rule. Each constrained optimal control problem is solved by means of the generalized reduced gradient using cbnstraint co-ordination. The algorithm has been applied to a nonlinear dynamic sectoral model of the Danish economy, which is described in detail. The control functions are: investments, labour, wri'te-offs, marginal tax rates in each sector. The state variables are" foreign debt, state budget excess. Some results for the 1947-52 period are given and they show that the open-loop Nash equilibrium controls obtained are somehow closer to the actual historical controls than those yielded by maximizing classical welfare criterions.

describe in detail the sectoral model used for the Danish economy, together with some numerical results. The background for this work may be found in the following three methodological remarks. Optimal control theory receives more and more applications in the field of dynamic national planning, partly because of the steering concept, but also because planners m a y now want to specify their models with nonlinear functions, criterions and constraints. This approach is discussed by Shell [25], Hadley and K e m p [7], and further references are listed by Peston [23] and Kendrick [15]. Our point of view is, however, that realistic planning on the basis of such models can only be accomplished if the conflicting interests of the most important decision-makers are explicitly contributing to the computation o f growth plans (Albouy [3] and Pau [19]). In other words, why not try to include the economical power relations into such plans ? In national planning it should be necessary to account for the economic-political rules applying within each country or community of nations. By economic-political rules are meant here the rules governing the relations a m o n g economic agents, as actually used or accepted by them. Depending upon local political and business traditions, the legislative bodies will have to tell the economic agents whether they may achieve their many different individual goals by daily duels, or simply by waiting for collective improvements. The duel would correspond to a Maximin kind of rule, while the simplest type of collective improvement schemes would be related to the Pareto equilibria as described by Telser [30]. The economic-political rule investigated in the present paper and our example is the N a s h - C o u r n o t equilibrium. Assuming that differential games are therefore required for macroeconomical planning purposes,

INTRODUCTION This paper discusses the application of a new algorithm to the numerical solution of N-person nonzero-sum differential games of a type which is to be used in the formulation of a national development plan. The chief interest is presently to expose very briefly this algorithm for the computation of oRen-loop Nash equilibrium controls, and to

* Received 28 December 1973; revised 2 December 1974; revised 15 March 1975. The original version of this paper was presented at the IFAC/IFORS International Conference on Dynamic Modelling and Control of National Economies which was held in the University of Warwick during July 1973 and has been published in [ll]. It was recommended for publication in revised form by associate editor D. Tabak. t L. F. Pau, Professor, presently with ENS des T616communications (B-230), 46 rue Barrault, F 75634 Paris Cedex 13, France; and Laboratoire d'automatique th6orique, Tour 14-24 V, Unlversit6 Paris VII, 2 Place Jussieu, F 75005 Paris 5, France. 473

474

L . F . PAU

one runs into numerical problems related to the computation of conflicting or co-operating equilibria in N-person nonzero-sum differential games as indicated by Ho and Starr [8]. Very little attention has been given to the development of computational techniques to solve such problems, and especially to find open-loop or closed-loop equilibrium controls, except may be in linearquadratic games; a survey paper on this topic has been given recently by Tabak [29]; the main known references are Starr [28], Holt and Mukundan [9], Simaan and Cruz [26] and Pan [20], [21], all based on necessary conditions for having a Nash equilibrium and all restricted in fact to open-loop controls in a neighborhood of an initial guess control. Much more effort should therefore be made finding better algorithms for this purpose, especially because there may be fairly many decision centers j = I, N and control functions, which adds up to the difficulties related to singularities, to non-uniqueness and to the existence of such equilibrium controls. Open-loop Nash-Cournot equilibria, notations and necessary conditions are introduced in Section 1. The main algorithm (A) is sketched out in Section 2. A macroeconomical sectoral model of the Danish economy is described in Section 3. Finally, numerical results of (A) are given in Section 4, together with the experimental findings relevant to economic planning theory and practice. The Appendix describes briefly an algorithm (B) for the computation of weighed optimal mean square controls; the solution of (B) is used in the application as initial guess control for (A). 1. NASH EQUILIBRIUM CONTROLS IN AN N-PERSON NONZERO-SUM DIFFERENTIAL GAME In this section, we describe the formulation which will be used throughout this paper. 1.1 Formulation of an N-person nonzero-sum differential game The system to be controlled in open-loop by the N players has an n-dimensional state vector X(t) zx [xx(t)... x,~(t)]T~Rn, t~[0, T], where T is the fixed time horizon. The global control vector for all N players is U(t)ZX[ul(t)... UN(t)]T~R "v at time t~[0, T]. The state equation is

)? = f ( X , U, t),

X(O) = X 0 given t~[0, T].

(1)

Each player j, j = l . . . . . N, has one single real control function and only one, namely us(. ) defined on [0, T]; all results in the following sections may be extended to the case of each player j having more than one single real control function, which is the case in Section 3. The global control function U(.) ~ [ul(. )... UN(.)]T is assumed to be piecewise continuously differentiable.

Define for any t~[0, T] a closed bounded and compact subset/z(t) in R ,v by N scalar continuously differentiable functions rj, j = 1..... N:

U(t)Etz(t)-.-~r~(uj(t ), t) <~0

Vj = 1..... N.

/~ (t) will be called the set of feasible controls at time t, and/z(.) the set of feasible control functionals on [0, T]. Each player j--- 1, ..., N has an individual final time performance criterion gj(U(.), X(O),T) that he wants to minimize within the set of all feasible control functionals U(.) at X(0). Elements about the theory for optimization of functionals, and Gateaux-Frechet differentials, may be found in Luenberger [18]. 1.2 Open-loop Nash equilibrium controls The control functional U*(.) is an open-loop Nash optimal control at X(0), iff U*(.)~/~(.) is feasible and if it satisfies the Nash-Cournot equilibrium conditions [2] for the terminal costs of the N players V I = 1, ...,N,

g,( u*(. ), x(o), r ) ~<~,(u,o(.), x(o), 73, where

oA

/ (2)

U~ = [ul* • ..U(I_I) * UIu(I+I) * . . . / , / N * ] T

and

r,(u~(. ), . ) <-.O. A Nash optimal control is secure against unilateral attempts to minimize the individual terminal costs. But, when the player 1 plays U~°(.) instead of U*(.), it may very well happen that some other players see their own costs decrease. Here we assume no coalitions involving two or more rivals. U*(.) is an open-loop control, which is only time dependent, as opposed to closed-loop strategies solution of (2), which are time and state dependent; we will return later to this comparison. 2. ALGORITHM (A) FOR SEARCHING NASH OPTIMAL CONTROLS BY HIERARCHICAL DECOMPOSITION AND CO-ORDINATION THROUGH A FICTIVE REFEREE Algorithm (A) is described in this section. A more complete and rigorous presentation of this algorithm can be found in references [11] and [20]. Initially, a necessary Nash optimality condition will be presented, followed by the description of the (A) algorithm. A solution of related optimal control problems will be treated in the last subsection.

2.1 Necessary Nash equilibrium conditions The idea used here is to try to find a single performance functional, aggregating the terminal

475

A differential game among sectors in a macroeconomy costs gj(U(.), X(O), T) of the N players j = 1.... , N, together with the features of a Nash equilibrium (2) among these. This is done by increasing the dimension of the control space, and by defining an apparent overall criterion function g for any

te[O, T) g(U(.), v(.), x(o), t) = ;=~, .

<

~g~(V?(.), x(0), t)

--/a,,(.),u,(.)-,,,(.)>, /)

V~ [vl..... vN]%

~o~ [v~

... ~'(t-1) uz v ( , + l , ...

l

/

v~] r, ) (3)

where U(.), V (.): feasible control functionals on [0, T],

~g~/auz(.): C_fftteauxpartial derivative mapping of g~ with respect to u~.); the argument indicates where this mapping is computed, ( . , •>z: pre-hilbertian scalar product for our control functionals uz(.) on [0, T ], i.e.

< un('), uz=(') > z~

F

ua(t ) u~=(t)dt. (4)

o

The apparent overall criterion g(., V(.), X(0), T) is designed for the purpose of letting it represent the value of the actual aggregated performance flmctional, defined as g(., U (.), xO), T), as it would appear to the N players if the equilibrium control was V(.) instead of the true Nash equilibrium control U*(.). This is due to the following result, proved in Pau [20] and in [11]. Let U*(.) be a feasible Nashoptimal control for (1) at X(0); assume N>_.2 and that a growth condition holds for the cost functions gt(U(.), X(0), T), l = 1, ..., N; then: there exists a neighborhood D(U* (.)) of U* (.), such that the aggregated performance functional g(., U*(.), X(0), T) is extremal at U*(.): (i) for any feasible control functional U(.)eD(U* (.)), the sign of g(U(.), U* (.), X(0), T) is constant, (ii) g(U* (.), U* (.), X(O), T) = O,

only approximate it by a non-equilibrium control V(.) appearing in the apparent overall criterion function. 2.2 General description of the (A) algorithm It is now essentially assumed that each of the N players allows nobody else other than himself to obtain knowledge about his own terminal cost function g;, l = 1.... ,N. Though a referee is introduced and he requires full knowledge of all functionals gj, j = 1,N, he may be fictive, be represented by a third body or be represented eventually by one of the original N players. Our basic idea is that such an N-person differential game with imperfect information may be viewed as a two-level system (S) (Fig. 1) based on constraint co-ordination, and in which (a) each infimal subsystem j = 1, N is a genuine player trying to minimize his terminal cost gj(U(.), X(0),T) for U(.)e/~(.); (b) the single supremal subsystem is the referee j=0 quoted above, having g ( . , U * ( . ) , X(0), T) as criterion function: this aggregated performance functional g expresses primarily the respect of the Nash rule (2) accepted by the N infimal players, and subsidiary an overall cost minimization among these. Referee

j,O

Optimum glU(°),U~*}IX(O),T) 1 U(*)I D( Ut( * ))

. . . . . . .

Suprem°l pl°yer

. . . . . . .

Hierorchico] ~ystem (S}

r

Min g (u[o), o-{o}, x(o),T)

su=,.~ I UI')'DIU*~'l)g .... {°T(*~'x3"}'J'L ..... N} level

Is,g.

g (U{o), 0=( .I, X(O ),T ) Sign g (O~ e ), a''l( ° ) . x(O) ,T) E Refereej. 0 ~*

V

~" = = X) *),U I [.) a I x;(o).p~ (o),~: (o)

gj (U(.), X (o), I") ,.,,mo,

(5)

(iii) VI~[1,N]: ag(U* (.), U* (.), X(0), z31auL ) = o. As a result, we may say that a necessary Nash equilibrium condition at X(0) for U* (.), is that the aggregated performance functional

g(., u* (.), x(o), T) is extremal (5); the (A) algorithm will use this property. However, U*(.) is not known at the initiation of the computations, and the N players

,eve,

[ u ( . ) .a;(.) I 0~*). O(U*(.))ond ,he

I |

jet, ...~N

FIG. 1. Information flows in the (,4) algorithm. T h e system (S) as a w h o l e is hierarchical

as

discussed by Jumarie [13], in the sense that any constraint co-ordination performed by the (N+ 1) players must first aim at satisfying player O's interest, before taking care of the N others. If there exists a Nash optimal control U* (.), the algorithm (A) performs a direct search, where the referee carries the constraint co-ordination role, and the N infimal players perform their individual constrained

476

L.F. PAU

optimizations. It might instead have been conceivable to let the referee carry the optimization responsibility. From above, we also deduce the fact that the initial guess control used to start (A) will first be treated at the supremal level. Constraint co-ordination is carried out iteratively in (A) by a co-ordination control

am(t) ~=[a~m(t)... aNr'(t)] "r~ R "~, t ~[0, T], representing the best current approximation of U*(.) at stage m, and limam(.)= U*(.) if convergence is achieved (see flow-chart; Fig. 2), m--*-+- ~ .

'a'( ,1

I

(G)$otves in ~Orollel the N ODtimOI Coil rol OroOlems(6) j.~, .., N I

XT(.),U?(.) j,l,. ,N

the opnn~ co~rol p r o ~ ) t e m ( 7 ) ~ I at(G)the~olv,$suprerna~ ~eve~~ or : o Newton

I_I

~

(U)~

~:'(U) [U~

p~

~+,

g(am+l(. ), am(. ), X(O), T) Min g(U(. ), am(.), X(0), T) u(.) ~ (D(U*)n ~,(.)) (7) This is achieved by decomposing the Lagrangian function Lm corresponding to (7), Lm including the inbalance between: (i) the current control candidate U(.) and the best approximation am(.) of U*(.); (ii) the actual optimal trajectories X~m(.) of the infimal players I = 1, N given am(.), and the approximated equilibrium trajectory associated to U(.), X(0) by (1).

E

Rol~lson Orgor~t~m 5olves t~e system of non-decentralized equations(g)

infimal players. The improvement of the apparent overall performance will express a better agreement with the Nash playing rule, and this optimization problem is the optimal control problem of the referee j = 0. Decomposition is used in order to ease the computation of this optimal control problem for the supremal player j = 0, given the infimal responses (uim (.), X i m ( . ); j = 1.... , N}

~.x~l-Kulm II Igocillqra *ntegrote$ ~n porotlel the decentcalized

di fferentiol equati(~'~governing the mulhphers r,20]

I

Lm(U (.), t) z x d g ( u ( . ), am(.), X(O), t)

|

'

--fk( Xl', Uz=, t)] j

0

I Sto~)pln~ ruJe

Oll(*} ~oes Oot

{*} veri float*on of trle I open-food Nash equilibrium proaer~V

=

+ ~ (A/n(uJ(t)-aJ(t))

exisl

j=l, N

Urn(.) * a~*q(,) End

+ p/nrj(uj(t), t)),

FIO. 2. Row-chart of the (A) algorithm; implementation on a digital computer.

Define x i m ( . ) zx [xi~ (.); k = 1..... n] T, the trajectory obtained from (1) given 2"(0) and the feasible ideal control functional

U]m(.)Zk[u~im(.), 1= l, ...,N] T, solution of the following optimal control problem at the infimal level, in which am(.) is given

g~(Utm(. ), X (O), T) ~ Min g~( U (. ), X (O), T), U(. )~(D(U*(. ))nt,(. ))

!

co-ordination constraint:

|

(6)

!

ui(. ) = aim(.) = ut~m(. ) ) Expression (6) means that each player j~[1,N] responds to the binding u~(.)--aim(. ) by giving some local information concerning the decisions of all remaining ( N - 1 ) players that player j would regard as being optimal for himself. The referee j = 0 will then try to revise the current approximation am(.) of U*(.) by minimizing the apparent performance functional g( U (. ), am(.), X (O), T) with respect to feasible controls U(.), chosen close to the controls previously requested from the

(8)

where ~bikm, A/~, pim, pjm/>0 are time-dependent scalar Lagrange multiplicators, and X(am(.), X(O), t) is the state vector computed by (1) on the basis of the control am(.) and X(0). Provided that U(.) is not on the boundary of (/z(.)n D(U*(.))), we may write the necessary Euler-Lagrange optimality conditions for Lm at U(.); these conditions are given in Pau [20], [11 ]; all of these relations, except a single one, can be decentralized among the N infimal players, i.e. each of them involve only controls, states or .multiplicators specific to one single player at the time. This leaves at the supremal level j = 0 mainly the last non-decentralized relation in which coupling appears: ~3Lm d ~L m Out(.) dt O~t(.) = Arm

-

0 Z q',k" ZZrT'~fk(~ m, u, "~,t)

4=1,

UUI~ , ]

n.

d

-~ Out(. ) dt g( U(" ), am(. ), X(O), T) +plm ~ 0

rt(ut(t),t) = O, U(.)~D(U*(.)) (9)

A differential game among sectors in a macroeconomy Instead of solving (7) directly, it is easier to solve the system consisting of (9) and of the decentralized relations, and let am+X(.) be the corresponding solution. If this new co-ordination control am+X(.) yields a value of lg(am+X(.), am(.), X (O), T) I smaller than [g(am(.), am-X(.), X(O), T) [, then the procedure is started again for the stage (m + 1), using am+X(.) as a new approximation of U*(.). If U*(.) exists and if the sequence {am(.)} converges towards a control O(.)e/~(.), then 0 ( . ) can be identical to U*(.) if a°(.)eD(U*(.)). However, one is urged always to verify directly whether U(.) actually is an open-loop Nash equilibrium control, by using either definition (2) or the so-called sufficiency conditions of Stalford and Leitmann [27]. It is reasonable to ask about the convergence rate of this algorithm (A), especially in the presence of control constraints tz(.). Singular controls may occur, but the general answer is that the convergence towards U*(.) (if it exists) in the neighborhood of. a°(.), will be fairly rapid; the reason is simply the definition of the apparent overall criterion function g(., a~(.), X(0), T): its magnitude must decrease according to (5), which requires both the partial derivatives of the criteria gx and the iteration steps to decrease (3), for all 1 = 1,..., N, because of the product form. 2.3 Numerical procedures (G) to solve optimal

control problems In all optimal control problems formulated in (A) as local optimization problems in the control space /~(.), direct search algorithms have been used and they require that a feasible start control is given. The basic (G) algorithm is a modified version of the generalized G R G algorithm as stated by Abadie [1] and Abadie and Bichara [2]. This algorithm handles optimal control problems such as (6) and (7) rewritten as a set of difference equations, with separable upper and lower bounds on all control variables and separable constraints on all state variables. For a given system (6) the programming has been done in such a way that (G) may very easily be adapted to different separable constraint sets/z(.), and to different criteria of the integral type (see Pau et al. [22]): in all numerical examples solved up to date, the criterion functions were of this type (see Section 3). It is possible to solve optimal control problems in which the control functionals of some players are tied down to equal given functionals: see the coordination constraints in (6). If all control constraints are separable, this is done by narrowing the slack between the upper and lower bounds, until this slack is only determined for rounding-off errors, and adding the norm of the slack as a

477

penalty to the cost criterion. However, if only a few control functions uj(.) are kept relaxed, it may be useful to use differential dynamic programming, as described by Jacobson and Mayne [12], instead of GRG. But the main troubles are due to non-separable substitution constraints on the state and control vectors simultaneously, many of them appearing in Section 3.5

s(X(t), U(t),t) = O, w(X(t), U(t),t)<<,O. (10) The corresponding singular arcs are still not handled quite satisfactorily, even after having implemenied in (G) the following procedures: (a) Equality constraints in (10) are treated by introducing supplementary state variables x,(.) such that

dx,/dt = s(X(t), U(t), t), xs(O) -- 0,

Ix,(t) l -<~,

where e is small: this is again the standard case for (G) and GRG. (b) Inequality constraints are treated by penalty techniques, but this is far from efficient and always very time consuming with respect to computer usage. 3. SECTOKAL MODEL OF THE DANISH ECONOMY The present study was initiated as a research project dealing with a sectorwise macroeconomic analysis of the Danish economy. Several large nonlinear planning control problems, with up to 25 control functions and 8 state variables, were solved in order to cover the periods 1947-52 [22] and 1967-72 [33]. The concept of optimal growth in models involving several sectors is discussed by Gale [6], Uzawa [31] and Shell [25], but besides the previous references, numerical results have only been reported by Kendrick [14] and Abadie [2]. Interpretation of the numerical results of [22] with the actual handling tactics showed that the maximization of welfare functionals was often made at the expense of labour in Danish agriculture. This sector happened to be the most important at that time; but the production capacities and the exports from most other sectors were at relatively low levels, while rising very rapidly and contributing largely to the new welfare ideal. Especially during the latter years of the 1947-52 period, people were increasingly indicating interest in having a way of living close to what it could be in industrial countries at that time. Maximization of the discounted private consumption in which the consumptions of goods from each sector were weighed according to the shares of each of these sectors in the 1952 gross national product, would therefore clearly request agricultural labour to be somehow the smallest acceptable. This effect

478

L . F . PAU

was, in the model, apparently much stronger than the financial contributions of agriculture to the expansion of the growth sectors. It also appeared unrealistic to use only bounds on the labour force or on the unemployment rate in order to regulate the welfare optimizing controls. It therefore seemed necessary to let each sector carry its own defense through the differential game's approach to planning. In the actual case investigated, co-operative equilibria such as Pareto equilibria might have been interesting; but due to research interest and to the behaviour of economic agents in Denmark, it appeared more useful to focus attention upon Nash equilibria. Simultaneously, this approach fits into the scope of the coming long-range plan for the Danish economy, which for the first time includes the private sectors ("Perspektivplan II"); follow-up studies to the present work are therefore conducted for 1973-78, in parallel with the application of optimal control to a new monetary model [32]. In this section, we describe the model tested on the 1947-52 period, and most data related to it can be found in Pau et al. [22]. The algorithm used for solving this differential game is the (A) algorithm described in the previous section. All quantities in the model are supposed to be in fixed prices (t = 0). 3.1 Control functions The model shown in Table I has split the Danish economy up into six interrelated economical sectors

with substitution effects, either through resources, production functions or constraints. The sectors j = 1,6 each have their own terminal criterion g~(U(.), X(O), T) to be maximized at the end T of the planning period [0, T]. The global control U ( . ) = [ul(.),...,uv(.)] T is made up of the following instrument variables, where the subscript is the sector index: (a) Gross investments in the sector: 81(.), 82(.),

~3(.), ~4(.), ~5(.), 86(.). (b) Workers employed in the sector: LI(.), Lz(.),

Ls(.), L4(.), Ls(.), (/;d.) = 0). (c) Write-offs (depreciation used for selffinancing): AFSI(. ), AFSz(. ), AFS3(. ),

AFS4(. ), AFSs(. ), (AFSn(. ) = 0). (d) Marginal tax-rates on profits and wages: hi(. ), h2(.), h3(. ), h,(. ), h5(. ), (h s parameter). (e) Total free import of goods and services into the country: m(.). U(.) can be partitioned into six subsets of control functions which are those of the individual sectors: (a) For each private sector j = 1, 2, 3, 5(3j(.),

Lj(.), AFSA.)). (b) For the public sector j = 4

(84(.),L4(.),

AFS4(. ),hl(. ),h2(. ),hz(. ),h4(. ),hs(. ),m(. )). (c) For the housing sector (3s(.)). In practice there will clearly be problems about how to decompose the economic activity into sectors, and to extract significant statistical time series for each of these. Anyhow, sector-partitioned models are of great interest, not least because of their role in some national planning procedures, especially in Eastern Europe.

TABLE 1

Sector j 1 Agriculture

Terminal criterion of sector j at time T

NCF1

Present value of net

cashflow

2 Industry+ handicraft + NCF~ building industry

Present value of net cash-

NCFs

Present value of net cashflow

3 Transportation

4 Public services and

We have introduced two state variables, which • are Xl(t): foreign debt year t of the Danish economy as a whole (XI(0) known), X2(t): excess of the state budget (X2(0) known). An extended version of the model computes the gross product generated by each sector. Xl(t), Xz(t) are expressed by the following difference equations.

flow

See Section 4.4

Present

5 Services, financial institutions, liberal employments

NCF5

6 Housing (rentals)

forqr(t) (1 + rr)-tdt

value of net cashflow Present value of net cashflow

works

3.2 State equations

Present value of output

Foreign debt accumulation. Xl(t+ 1) = (I + 0) Xl(t)+m(t ) 6

+ Z (d# qj(t) - ej(t) + zri3j (t)), 5=1

where 0: interest rate on foreign debt, djj: diagonal element in the diagonal matrix D of the marginal propensities to import for the production qj(t) : (0- I 15; 0.1620; 0.198; 0.065; 0.013; 0), at(t) production level of sector j given by the production function, in fixed prices, ~ $ x

/

A differential game among sectors in a macroeconomy e#(t):

export from sector j, proportional to the production qj(t): ej(t) = ~7#q#(t) for some sectors, zr~.: marginal differential propensity to import for capital formation: (0, 112; 0, 186; 0, 165; 0, 038; 0; 178; 0). The exports e~(t) include visible as well as invisible revenues. The coefficient ~-~,d~ are difficult to estimate, and in the best case one can only obtain the mean propensities and not the marginal ones.

State budget excess. X2(t + 1) = X2(t) + h a qs(t ) + AFS4(t) + F~(t) + ~ ha(t ) Woy(1+ Aj)tLj(t) Y-L 5

-w~l + Al)t(Lo(t )- Z Li(t)) J-L 5

+

Z

hs(t)[(l-a)Fj(t)

j-1,~,3,5

-- AFSi(t)]- ~,(t), where gross profit on sales of sector j year t (see below), Wo: yearly wage by employee in sector j year 0, A;: rate of increase of the wages in sector j in fix~l prices, a: fraction of the gross profits F~(t) carried forward into a free legal reserve account, or free capital surplus account, wz: yearly wage paid as social welfare by the state to unemployed workers in year O, ~: rate of increase of the wages of the unemployed workers, Lo(t): maximal total Danish labour capacity on the basis of national manpower, given exogeneously. Notice that the tax rates on wages and corporate profits are here identical within each sector, because entrepreneurs are supposed to be taxed in the same way as employees. But tax exemptions may be different from sector to sector, so that the mean tax rates are different. The wage increase rates A~ are here exogeneous, but might be made functions of the unemployment according to a Phillips curve as the one Livesey [17] used. F4(t ) will be negative because of high costs for social help, public expenses and civil servant's salaries FjCt):

(seebelow). 3.3 Technical relations The state and control variables used hereupon are interrelated through the following relations.

Production functions. Following some empirical work, we assume that the production functions are

479

of the Cobb-Douglass type, and that the write-offs are reinvested

qs(t) = Kj.~y(8~(t)+ AFS~(t)) (Lj(t))%

K~: constant depending upon the added value, the profit margin and the technological progress, ffj: second-degree polynomial estimated by regression or curve fitting from historical data, %.: elasticity parameter for labour, estimated by regression: (-0.08; 0.52; 0.18; 0.10; 0.30; 0). Notice that we have not introduced the capital stocks, because of considerable estimation difficulties for these. For medium-range planning (T = 5), ~bj. seems appropriate as construced in [22]. Introducing the investments into the production functions means that they will trim much more closely than usually the outputs and exports. Anyhow, it would be better to take into account the installation time lag, i.e. by assuming a fixed delay on [Sj(t) + AFSs(t) ]. One major estimation problem also lies in the fact that the constant Kj is aggregated for all goods or services produced by sector j. The estimation has been performed iteratively in two steps: (a) Find Tchebycheff regression polynomials

given (b) Minimize the estimation variance found in (a) with respect to ~.

Input-output relation. In column vector notation we write q(t) + Dq(t) + rh(t) + A(t) = Aq(t) + B 8( 0 + e(t) + c(t). Each vector has six co-ordinates j = 1,6, where the new symbols are defined as follows fns(t ) = m(t)qj(t)/(~ qy(t),j = 1,6): untied competitive import distributed here proportionally to production; we assume that

~6(t)

--

0,

A: Leontief input--output matrix, provided by the Danish National statistics department for 1947, B: capital coefficient matrix, cj(t): private consumption of goods and netto increase of all stocks or reserves for sector j ; j f f i 1,6, Aj(t) = aFj(t) + (1 - h~(t)) [(1 - a) F~(t) - AFSj(t)] after-tax earnings distributed for savings or consumption, and tax-free increase in the legal reserve carried forward. The matrices A and B are here unfortunately fixed, simply because there have only been made two input--output analyses in Denmark (1947 and 1966, published in 1973); they should be made time dependent if possible. Finally c(t) being computed as a residual will include all stock variations

480

L . F . PAu

including the speculative ones, which makes it sensitive to time-lag effects and stochastic variations.

time horizon T will be approximated by

gj(U(.),X(O),T) = f: (l +ry'

Gross profits on sales in the private sectors

[(1 - hi(t)) ((I - a) FI(t)

j = 1,2,3,5. The gross profit on sales in sector j, year t will be evaluated in fixed prices

- AFSi(t)) + Hj(t ) + AFSi(t)]dt

Fi(t) = qj(t)- Z a.iqt(t)-d~iqj(t) l--l, 8

(1)

(2)

(3)

- ff'o/(l + Al)tLt(t) - pl(t) -- St(ql, Lt), (4) (5) where

a. i = {ati} column j in the Leontief matrix A, Sj: linear function expressing other taxes (or grants) from production and employment, ff'0~: yearly wage by employee in sector j, year t = 0, including social benefits and administration expenses, pl(t): repairs, given exogeneously, (1): gross sales, (2): interflows between sectors of raw or finished goods, (3): import for production, (4): labour wages, (5): repairs and other taxes on production.

Gross profit in the public sector j = 4. The gross profit F4(t ) will clearly be negative, because the incomes q4(t) other than taxes will be small, and because the social, educational and wage expenses, etc., will be high. LL(t): total Danish population, given exogeneously; ~4: a function expressing all social, educational, defense, municipal and administration expenses as a function ofLL(t); this function can roughly be estimated from the national and local government budgets: F,(t) = q,(t) - • auq,(t ) - d ~ q , ( t ) - p4(t) l--1, 6

- l'Po4(1+ •,)tL,(t) - dP,(LL(t)). 3.4 Criteria gi(V(.), X(0), T), j = 1, 6 Criterion for the private sectors j = 1,2, 3, 5. The criterion function for these is the present value of the netto cash-flow generated through production of goods and services. For this purpose, a simple model of the main accounting rules is being used, based upon letting each sector control the write-offs and to a certain extent the taxes paid on tax liable profits. The investments year t are supposed to be written off partially within the same year. At most a fraction a of the gross profit Ft(t ) may be carried forward and used tax-free, and this capability is employed at the upper limit. The present value of the netto cash-flow generated by sector j over the

where ri: time preference discount rate for sector j, hi(t): marginal tax rate on tax liable profits in sector j, Hi(t): balance of the capital surplus account after addition of Fj(t- 1): Hi(t+ 1) = H~(t)

+a~(t). Criterion for the public sector j = 4. The criterion first evaluated was

g4'(U(.),X(O),T) = f:(l +q)m [d.q4(t) + dP4(LL(t)) + t-i. y" ea~q'(t)+wz(l+Al)'(L°(t)-i-~l,L'(t))]dt' which is the total surplus of public services and aid to other sectors [see equation F4(t)]. An alternate criterion used was the Houthaker addilog utility index for consumption:

x(o), r) =

jo

r(l

a~(c~(t))DJdt, j-l, S

where aj: weighing coefficient for the consumption c~(t) in sectorj, according to this sector's share in the 1952 gross national product, •flj:expenditure elasticity connected parameter 0~
+g4"(V(.), X(0), T). The elasticities fli, J - - l, 6, and the weight cr were found to be critical parameters influencing largely the results of optimal control problems using g4"' as single criterion for the whole economy. This high sensitivity is unsatisfactory, namely because the private consumption elasticities fit can only be estimated within broad tolerance intervals. 3.5 Common state and control constraints A number of exogeneous constraints on the state and control variables were introduced; among these, the lower bound on the private consumptions

A differential game among sectors in a macroeconomy cj(t), and the upper and lower bounds on the foreign debt Xl(t ) and on the state budget excess X,.(t). The investments and write-offs had to be non-negative, and there was a time-dependent upper bound on 8s(t) for each sector j = 1,6. All these constraints are separable because they apply to one single variable at the time, and at a given instant. Moreover, substitution constraints were introduced on

Investments. A certain degree of self-financing is assumed for the private sectors j = 1, 2, 3, 5, due to the relation between loans (B 8, DS) and cash-flow 8j(t) ~<(1 + 0a) {(1 - hi(t)) [(1 - a) F~(t) - AFSj(t)]

+ Hi(t) + AFSj(t)), where Pa: constant close to 0; if /.h>0, foreign and external loans are accepted.

Labour. Upper and lower bounds are put on the unemployment rate, which may be negative, in the case of immigration -- l~l-~(t) ~ (Lo(t)- ~ Lj(t)) <~I~1.~(t). J-l, 5

Regularity in labour force changes. The labour Lj(t) in sector j = 1,5 is allowed to differ at the most by 100Aj% from the labour Ls(t-1) the year before. Feasible labour controls will be those fulfilling the following constraints, besides all the previous ones (1 - a ~ ) L j ( t - 1) <<.Lj(t) <~(1 + Aj)Lj(t- I).

Write-offs (deprecation). Depreciation is limited according to the balance method with regard to the current legislation on the subject; a single aggregated maximal rate for write-offs is assumed in each sector 0 <~AFSj(t) <<.~j Sj(t), Sj(t + 1) = Sj(t) + 8j(t)-AFSj(t),

Sj(O) given.

Marginal income tax rate h4(t ) in the public sector. This is to be majored by the mean value in the private sectors j=1,2,3,5:

0~4h4(t ) ~<(hx(t) + h~(t) + ha(t ) + hs(t)).

this is to prevent the migration of employees away from the public sector. Contrary to Kendrick [14], it was found unrealistic for planning purposes to require that some state variables fulfill terminal pointwise boundary conditions. It was found to be more fruitful to study the influence of the current time-dependent restrictions on the criteria and on the open-loop

481

Nash equilibrium strategies. Furthermore, this approach proved to cause less computational difficulties than the terminal target approach as explained in [22]. 4. C O M M E N T S

ON THE RESULTS

4.1 Numerical problems It can easily be conceived that guessing a set of feasible control functions in order to initiate the (A) algorithm used above is a difficult problem. Because the present analysis is of the ex-post type, the actual historical controls for the 1947-52 period were used as initial controls. However, in order to give some significance to the nature of the neighborhood in which an openloop Nash equilibrium U*(.) was sought, the following two-stage initialization procedure was used: (a) Initiate the (B) algorithm sketched out in the Appendix with the actual historical controls for the 1947-52 period; let a°(.) be the weighed meansquare control functional, computed by (B). (b) Initiate the (A) algorithm using a°(.) as the initial guess, as provided by the previous stage (a); let U*(.) be the open-loop Nash equilibrium control functional, computed by (A). The purpose of this procedure was that we wanted to find, if it existed, an equilibrium control U*(.) belonging to a common neighborhood of the actual historical control and of the control a°(.) for which the average relative regret of all 6 sectors would be minimized. When some constraints were modified, it was necessary to perform a manual or least squares adjustment process. By modifying automatically the gradient search to a conjugate gradient search, as soon as the iteration steps became small, a slow convergence was avoided near the optimum of each control problem (6) or (7). The singular arcs still need a special treatment not yet implemented satisfactorily; numerical instability was usually caused by the substitution constraints of Section 3.5. . The reader should not forget that the computational difficulties related to applying (A) to the model of Section 3 were eased considerably for the simple reason that this model is in discrete time because of the sampled nature of the economic data used. U*(.) is therefore a sampled-data open-loop Nash equilibrium control. 4.2 Computational implementation on a hybrid system (A) and (G) were initially implemented on an IBM 370-65 computer, and run times were 1.5-2-5 hr-CPU in order to obtain a sampled-data Nash equilibrium control on the model of Section

482

L . F . PAu

3([22]). Since then it has been decided to make a specific analog implementation of all state equations of a continuous time version of this model, for planning purposes on the 1973-78 horizon; besides the state equations, some important decentralized Lagrange multiplier equations (Section 3.2) were also programmed on an analog computer. The (A), (G) and constraint co-ordination algorithms are still implemented on a digital computer, connected with the analog computer in order to build a hybrid system. Computation times have been reduced to 30-45 min-CPU. This approach also has the considerable advantage of making parametric sensitivity analysis possible, and moreover one may follow on a display the modifications imposed by (A) on the state trajectories, and this helps in understanding the reasons for nonconvergence.

criterion g4"(U(.)), X(0, T) or even by this sector's own profit criterion gx(U(.), X(0), T) as shown in Fig. 5. At the same time, the equilibrium controls U*(.) for growth sectors such as industry were reasonably indicating the same trends as the actual controls as indicated in Fig. 4. (c) Tight constraints on the State budget excess will usually be effective and cause fairly sharp oscillations in the investment and tax rate controls at equilibrium; this confirms similar results of Livesey [16]. (d) The income distribution policies, as expressed by the growth rates ;~j, proved to be very important; Investments In industry ,

1200

llO0 /

4.3 Economic problems

/ /

The following topics have been investigated (a) The main result is that the Nash optimal state X*(.) is somehow closer to the actual historical evolutions than those yielded by maximizing a single classical welfare criterion g4"(U(.), X(0), T), as indicated in Section 3.4 and Fig. 3.

/"

,"

.

, Welfare

,,optimal debt

1000

/

1947

2

Actual / \ \ debt

I--

I

I

1947 4 8

49

I

I

50

51

\ \ 52

Foreign debt, Xt(t ), I000 MKr (1947)

/ '!,",, "W,, |

< / \

oL 1947

48

',

~

49

50

49

r

700

I

50

5t

t947

Current value of the criterion 8~(U.t}/

48

49

50

51

Investments in agriculture 800--

,/

15

/

Itllllll//l

,i ii

,¢

700 -

03

¢.

'\

///"

•

.)/

/ / , /////// //

I0

,~mployment

/

I

48

FIG. 4. Investments 8 s ( t ) and labour L=(t) in industry: . . . . . : actual historical controls; : Nash optimal controls; . . . . . : controls U + maximizing the industry's own criterion.

J • % of capacity total labour l-or- Welfare optimal i ~unempioyment I ~, NASH l i optimal /, 0.5 I"- \ unemployment'/

800

I

I

]

20 _

3-/

i Labour force in industry - - in 1000 workers

90C 03

4-

I000

;

_--

51

52

5

Unemployment (as computed from Lo,LIj=I'5)

, //

z/i/ 1

1947 48 4 9

50

51

L

52

Fie. 3. Foreign debt (state variable) and unemployment (control variable) for three different equilibriums : actual historical values, Nash equilibrium and maximization of the consumption welfare functional of Houthakker (direct addilog utility index).

Even more important is that this equilibrium evolution was much more stable than the welfare optimal evolution, with respect to parameter changes in g4"(U(.), X(O), T). If relatively few constraints are saturated, it looks numerically as if the equilibrium damps much of the imprecision in the estimation of the criterion functions of Section 3.4.

(b) Simultaneously, the Nash equilibrium controls U*(.) proved, for example, to secure an efficient defense of the labour force in agriculture, as opposed to the reduction requested by a welfare

f

I

I

49

/947 48

i

51

50

/, /

500

Labour force In the agriculture 500

in I000 workers

400

i

/

:Write-offs

i

/ ,'

400 /

J7

:E

3OO

200

1

I

1947 48

1 r

49

50

f

51

200

--

)/ I

1947

:

~

49

i a9

i

'i I

50 5~

Fta. 5. The agricultural sector j = l ; investments, labour, depreciation: .... : actual historical controls: --: Nash optimal controls; . . . . . : controls U + (.) maximizing the criterion gz (U, T).

A differential game among sectors in a macroeconomy a weak aspect of the present model lies in the fact that hj,j = I, 5, is exogeneous in constant prices, so that the interplay between prices and wages is not reproduced here. (e) Increasing the time preference discount rates rj, j = 1,6, leads to labour force transfers and to a drop in imports. (f) In sectorj = 5, services, financial institutions, liberal employments, the investments 86(t) were often found to go to the upper bound, separable or substitution, when using the welfare criteria g4"(U(.), X(O), T) or g,'"(U(.), X(0), T); this is due both to a very high expenditure elasticity fls, and to the surprising very weak labour intensive characteristic of this s e c t o r j - - 6, in 1947-52. (g) In the industrial sectorj = 2, the growth rate for labour was found to be moderate at equilibrium, and close to the historical trend; this is mainly because of the high propensity to import (~rs,d~) turned down by the saturating foreign debt, and because indu.stry still required logistical and financial support from the other sectors. (h) Reducing the total labour force Lo(t) by 5% increases the agricultural output, industrial investments and labour, while the other sectors are not influenced. 4.4 Relevance "of the differential games approach to actual planning problems Because of modelling errors, it is not realistic to use the equilibrium controls as such as planning decisions for the future. The first important fact is, however, that results such as those of Figs. 3-5, and for future periods such as 1973-78, provide the economic planners with trend indications which are meaningful for the considered forecasts. The second practical advantage is to make possible a comparison of these trends if the State shifts from one policy to another, i.e. from a maximum welfare policy (cr = 0 in g4"') to a minimum foreign debt policy (cr = ~ in g~'"); this may be very important, because it is reasonable to assume that the individual economic sectors will not change their goals, which obliges the State to select not only a proper policy but also corresponding equilibrium controls having some desired impacts on the equilibrium controls of the private sectors. This represents an indirect action of the State on these sectors, without modifying the political relationships among all of them as synthesized by the Nash playing rule. One may conjecture that computing closed-loop Nash equilibrium strategies would have been more relevant in planning applications than the present open-loop equilibrium controls. This is by no way evident, since the results should help in selecting goals as much as control policies for the future. Another argument is that forecasting errors are of

483

such a magnitude that closed-loop strategies would be even more sensitive than open-loop controls. One major finding of this research is that economic agents in a country such as Denmark do not act according to fully co-operative Pareto rules such as welfare maximization. An interesting point would then be to model constraints measuring the defiance of these agents away from a co-operative equilibrium. CONCLUSION Regardless of the sector-partitioned macroeconomical model considered here, there seems to be a need to account for power relationships in policy sciences and economics, and to compute their consequences in planning models. Because very little research on such computational algorithms has been published, the method developed in the present paper should contribute to initiate similar efforts. The results stated herein have been the main evidence for carrying out further experimentation with the differential games approach to planning problems. At the present time the priority should be to improve the (A) and (G) algorithms, and to investigate the properties of the equilibria U*(.) which they compute. This contribution raises the question of finding out why Nash-Cournot equilibria may be close to actual evolutions in the economy, and whether other kinds of equilibria might be more appropriate. It is also necessary to investigate numerically and in economic terms the non-uniqueness of these openloop Nash equilibria, if they exist.

Acknowledgements--The author thanks Professor A. Blaqui~re, Laboratoire d'automatique th6orique, Paris University VII for helpful mathematical comments; thanks also to several of my former students for data collection and experimentation work; to members of the Board of Economic Consultants to the Danish Government for methodological discussions, and to the reviewers for lucid remarks and suggestions for improvements in this paper. REFERENCES [1] L ABADIE: Application of the GRG algorithm to

optimal control problems. In Integer and Non-linear Programming (J. ASXDIE, ed.), pp. 191-198. NorthHolland, Amsterdam (1970). [2] J. ASAD~ and M. BICHAI~,:Rrsolution num~ique de certains probl~mes de commande optimale. Revue Fran~aise d'Automatique, lnformatique et recherche opdrationnelle, 7, V-2, 77-105 (May !973). [3l M. ALBOUY, ]..,a Rdgulation I~conomique dans l'Entreprise. Dunod, Paris (1972). [4] A. BLAQr,nimE, F. GERAV.~ and G. LEITMANN: Quantitative and Qualitative Games. Academic Press, New York (1969). [5] A. BLAQt~.n~, L. JOlUC~xand K. E. WmsE: Geometry of Pareto equilibria in N-person differential games. In Topics in Differential Games (A. BLAQUX~E,ed.), pp. 271-310. North-Holland, Amsterdam (1973). [6] D. GALE: On optimal development in a multi-sector economy. Rev. Econ. Studies 34, 1-19 (1967).

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[7] W. HADLEY and A. KEMP: Variational Methods in Economics. North-Holland, Amsterdam (1972). [8] Y. C. Ho and A. W. STARR: Non-zero sum differential games. J. Opt. Theory Appl., 3, 184-206 (1969). [9] D. HOLT and R. MUKUNDAN: A Nash algorithm for a class of non-zero sum differential games. Int. J. Systems Sci. 2, 379-387 (1972). [lo] S. C. HUANG: Note on mean-square strategy for vector-valued objective functions. J. Opt. Theory Appl. 9, 364-366 (1972). [ 1 l] IFAC/IFORS : International conference on dynamic modelling and control of national economies, Univ. of Warwick, 9-12 July 1973, IEE Conf. publication No. 101, London (1973). [12] D. H. JACOBSON and D. Q. MAYNE: Differential Dynamic Programming. American Elsevier, New York (1970). [ 131 G. JUMARIE:Towards a theory of multilevel hierarchical games, pp. 270-281 in IFAC/IFORS: International conference on dynamic modelling and control of national economies. Univ. of Warwick. 9-12 July 1973, IEE Conf. publication No. 101, London (1973). 1141 - _ D. KENDRICK and L. TAYLOR: Numerical solution of non-linear planning models. Econometrica 38.453467 (1970). [IS] D. KENDRICK: Control theory in economic models, Paver W 7-l. IEEE Decisions and Control Conference. pp: 80-84, hiiami (1971). [16] D. A. LIVESEY: Optimising short-term policy. Economic J. 81, 525-546 (1971). [17] D. A. LIVESEY:Some further results for a model of the UK economy, pp. 95-108 in IFAC/IFORS: International conference on dynamic modelling and control of national economies. Univ. of Warwick. 9-12 July 1973. IEE Conf. publication No. 101, London (1973). [18] D. G. LUENBERGER: Optimization by Vector Space Methods. Wiley, (1969). [19] L. F. PAU: Control, conflict and cooperation in evolutive economic systems, 92 pp., IMSOR, Tech. Univ. of Denmark (1971). [20] L. F. PAU: Diffenxtial games and a Nash equilibrium searching algorithm, 8th Int. Symp. on Mathematical Programming, 27-31 Aug. 1973, Stanford Univ.; and SZiM J. Corttrol 13, August (1975). 1211 asoects in two-level _ _ L. F. PAU: Comoutational differential games. _ In Theory &zd Application of Differential Games (J. GROTE, ed.). D. Reidel, BostonDordrecht (1975), and NATO ASZ, 27 Aug.-6 Sept. 1974, Univ. of Warwick, Coventry. [22] L. F. PAU, H. JEPPESEN,N. JENSENand P. MAJLANDJENSEN: Control of Hierarchical Non-Linear Systems. IMSOR. Tech. Univ. of Denmark (1972) (in Danish). [23] M. F%&oN: Econometrics and codtrol,‘pb. 15-31 h IFAC/IFORS: International conference on dynamic modelling and control of national economies. Univ. of Warwick. 9-12 July 1973. IEE Conf. publication No. 101, London (1973). [24] M. E. SALUKVADZE: On optimization of control systems according to vector-valued performance criteria. 1972 ZFAC Congress, Paris, June (1972). j25] K. SHELL (ed.): Essays on the Theory of Optimal Economic Growth. MIT Press, Cambridge, Mass. (1967). [26] M. SIMAAN and J. B. CRUZ: Sampled data Nash controls in nonzero-sum differential games. Znr. J. Control 13, 1201-1209 (1973). [27] H. S~ALFORDand G. LEITMANN:Sufficiency conditions for Nash equilibria in N-person differential names. In Topics in Djfferential Ga&es (A. BLAQUI&RE~ ed.), pp. 345-376. North-Holland. Amsterdam. (19731. [28] A. W. STARR: Computation of Nash ‘equiibria for non-linear nonzero-sum differential games. Proc. 1st Int. Conf. on the Theory and Applications of Di’erential Games, 29 Sept.-l Oct. Univ. of Massachusetts, Amherst, Mass. (1969) pp. IV 13-18. [29] D. TABAK: Computed solutions of differential games. In Theory and Application of Differential Games

(J. GROTE, ed.). D. Reidel, Boston-Dordrecht (1975), and NATO ASI. 27 Au-6 Seot. 1974. Univ. of Warwick, Coven&y. a ’ [30] L. G. TELSER: Competition, Collusion, and Game Theory. Aldine-Atherton, Chicago (1972). [31] H. UZAWA: Optimal growth in a two-sector model of capital accumulation. Rev. Econ. Studies 31, l-24 (1964). [32] P. T. VAL~TORP FREDERIKSEN and R. AAGAARD SVENDSEN:Optimeringaf0konomisk-Poiitiske Modeller paa Grundlag of den Finanspolitiske Model SMEC ZZ. TR, IMSOR, Tech. Univ. Denmark, Aug. (1974). (In Danish.) [33] R. AAGAARDSVENDSEN:Macroeconomic Planning in a Sectorial Economy Through Control Theory. IMSOR, Tech. Univ. of Denmark (1973) (in Danish,). APPENDIX: ALGORITHM (B) LOCALIZING A SPECIAL WEIGHED MEAN-SQUARE CONTROL

In this Appendix, special weighed mean-square optimal controls are defined, while their relations to non-inferiority, Huang [IO], and to Nash equilibrium controls are proved in Pau [ll]. This reference also contains an algorithm (8) for computing these weighed mean-square controls. 1. DEFINITIONS

(a) Assume the existence of the ideal control functional Vi+(.) minimizing the terminal cost of playerj, j = 1, . . . . IV, for the system of Section 1 8j(uj’(.)T X(O), %2M~;

,gi(U(.), X(O), 0.

(1)

(b) o(.) is a non-inferior control, or a Pareto control (Blaquitre et al. [5]) iff there does not exist U(. ) feasible at X(0)

vj = 1, . ..) N

gj(v. )9m9, n

Ggj(“(*)9 X(“)9T)9

(2)

3 1a1, Nl, gr(W.), =w), T> *

q(W)9 no), n. I (c) o(. ) is a weighed mean-square

optimal control, iff rf( .) minimizes the weighed meansquare cost h aggregating the utilities of our N players h( U( .I, v9,

T)

sj(u(.),w%T) 19x(o), T)-

‘j.sTN [

gj(llj+(.

2

I’

’

h(O(.). X(O), T) “z,CMj;, ,h(U(.), X(O), T).

(3) (4)

When minimizing h( U(. ), X(O), T) with respect to U(.), the losses/regrets are equally distributed in relative value among all N players. In expression (3) it is assumed that the values gj(Uj+( .), X(O), 7’) are strictly non-negative, which may easily be achieved by adding up positive numbers to the actual terminal costs because p( .) is bounded. In the usual mean-square criterion (Huang [lo]), the weights l/gj(Uj+( .), X(O), T) would not appear.

A differential game among sectors in a macroeconomy 2. RELATIONS BETWEEN THE (B) AND (A) ALGORITHMS [I 1]

Our numerical experimentations [22] have shown that the usual mean-square aggregate criterion is inadequate for the cases where the values gs(Uj+(.), X(O), T) do differ significantly from one player to another. It is inadequate in the sense that the costs gj(U(.),X(O),T) are oscillating numerically in much differentiated patterns during the iterative approximation procedure of a meansquare optimal control. This remark is also relying upon the experience of Salukvadze [24] in a

485

trajectory optimization problem. We have therefore decided to replace the mean-square criterion by the weighed mean-square criterion in all numerical applications. The (B) algorithm solves the optimal control problem (4). Moreover, when applied to the economic model of Section 3, economists found it reasonable to use as an initial guess for (A) such a weighed meansquare optimal control, because it tends to allocate the relative regrets [ - 1 +gj(0(.), X(0), T)/gj(U~.+(.),X(O), T)] evenly among the economic sectors.