The Production Analysis by Dual Dynamic Programming

A C T A U N I V E R S I T A T I S L O D Z I E N S I S FOLIA OECONOMICA 152, 2000

I w o n a N o w a k o w s k a *

T H E P R O D U C T IO N ANA LYSIS BY DUA L D Y N A M IC P R O G R A M M IN G

Abstract. The aim of this paper is to present a concept of a duality theory for dynamic production processes with constraints i.e. production processes described by nonconvex dynamical mathematical models (models depending on time).

1. INTRODUCTION

I he p ro d u c tio n function plays the crucial role in econom ics. E nterprises aim to m inim ize their costs (for a given level o f p ro d u ctio n ), b u t the m ain m o tiv a tio n o f their action is to m axim ize the profits. A n analysis o f the co st p ro d u c tio n is very often led by the p ro d u c tio n fun ctio n. In static econom y, if we are going to analyse costs th ro u g h a p ro d u c tio n fu nction and prices then a du ality theorem is especially useful (see e.g. L e i d l e r , E s t r i n, 1989). T h e essence o f duality theory is th a t all elem ents o f the p ro d u c tio n technology (available for enterprises) which are im p o rta n t for econom ists can be sim ply placed in the cost function. T his statem en t has im p o rta n t consequences for the choice o f the q u a n tity o f ex p en d itu re and the level o f p ro d u ctio n . F o r exam ple, optim al q u a n tity o f p ro d u c tio n facto rs we can gain directly from the cost function. E m pirical inv estigations also acknow ledge a great interest in the study o f d u ality theories, fo r instance: very often it is difficult to collect credible in fo rm atio n co n cerning the p ro d u c tio n factors (capital, labour). All available d a ta con cern in g the lab o u r expenditure take into account the num ber o f the w orkers, b u t they d o no t tak e in to account their qualifications and their intensity o f w ork. Such problem s are m uch m o re serious in the case o f capital. A capital value is difficult to be m easured and in fo rm atio n co ncerning usefulness o f

the capital is alm ost unavailable. T h e enterprises have m uch m ore inform ation a b o u t th eir costs. T he duality theo ry allows us to infer p ro p erties o f the p ro d u c tio n fu n ctio n from th e co st fu n c tio n , w here the in fo rm a tio n is available, m o re readable and reliable. A b o u t the relation o f th e costs with p ro d u c tio n we can think in tw o ways: 1) we find m axim ized level o f p ro d u c tio n , when the cost p ro d u ctio n is con stan t, 2) we find m inim ized level o f the cost, when the p ro d u c tio n is co n stan t. In each case we have the sam e result. T h e analysed cost function has a du al form with relation to the p ro d u c tio n function. O f course, such a d uality theorem exists, up to now , but only for p ro d u c tio n processes described by static (linear or convex) m athem atical m odels i.e. m odels which do n o t d epend on tim e.

T h e aim o f this no te is to present a concept o f a du ality theo ry for dynam ic p ro d u c tio n processes i.e. p ro d u ctio n processes described by non- convex dynam ical m ath em atical m odels (m odels d epending on tim e). In th at dual m odel the co n stra in ts ap p e ar, which ca n n o t be tak en into ac co u n t in prim al m odel. Such co n stra in ts ap p ear every tim e in each p ro d u c tio n process. T h e com panies which are interested in their ow n p ro fits know , th a t they do n o t possess inexhaustible resources and th a t they do n o t have endless and unlim ited funds. T h eir scale o f p ro d u c tio n is co n stra in e d , like all th eir possibilities.

2. DUAL APPROACH

Let us consider the cost functional: T

J( x, u) = j L ( t 9 x)(t), u(t))dt + l ( x ( T ) ) (1) 0

d ependin g on the state x(t) an d con tro l u(t) which m easure th e level o f a cost o f a p ro d u c tio n or a q u a n tity o f the p ro d u c tio n (d epending on the w ay we th in k ab o u t the relations betw een the cost and th e p ro d u ctio n ). T h e state x (i) denotes the ex penditure o f an en terprise d ep en din g o n tim e i.e. they m ay change w ith tim e. W e adm it a possibility to c o n tro l by u(t) the types o f tran sitio n s o f x(£) in time. It is n a tu ra l to require, th a t state x (i) varies dynam ically, i.e. th a t x( t ) and u(t) are subject to som e differential equatio n:

* 0 ) = /(* » x(t), u(t)) a.e. in [0, T ] (2) w here / :[0, T ] x R" x R m—* R n is to m easure the speeds o f changes o f the expenditu re x(f) in time. We assum e th a t / , L :[0 , T ] x R n x R m—+ R and l : R n—* R are co n tin u o u s function and the con tro ls u :[0 , T ] —* U c R m are

m easu rab le fu nctions, r e [0 , 7]. W e shall also assum e th a t th e expen diture at tim e t = 0 has a given value с i.e.

x(0) = c, c e R n (3)

M oreo ver, we shall ad m it th a t the expenditure x(r) is also subject to som e constraints:

g ( x ( ) ) = 0 (4)

w here g : R n—* R k. A pair x(t), u(t) satisfying the co n stra in ts (2), (4) will be called admissible and co rre sp o n d in g x ( t) an admissible state o r an admissible trajectory, see ( F l e m i n g , R i s h e l , 1975).

O u r goal is to m inim ize the functional (1) in the space o f abso lutely c o n tin u o u s states x (i) and m easurable contro ls u(t) subject to th e co n stra in ts (2), (3), (4).

T h e classical m eth o d to study such problem s is to define in som e open set G<=R" + 1 o f the variables (t, x) , the value function o f o u r p roblem . T h e value function S(t, x ) in the classical ap p ro ach is defined as follows:

S(t, x ) = i n f |j L ( r ,

x(x),

/(x(T)) j,

w here the inferior is tak en over all pairs x ( t ) , u ( t ) , т е [ £ , Г ], w hose states sta rt at the p oin t (t, x ) e G and their graph s are co ntained in G. T h e next step is the following: if S(t, x ) is continuously differentiable then it m u st satisfy the partial differential equation o f Hamilton-Jacoby-Bellman type:

S,(t, x) + H(t, x, Sx(t, x)) = 0 (t, x ) e G,

w here H(t, x, y) = yf(t, x, u(t, x)) - y ° L ( t , x , u(t, x)), y, y° are m u ltipliers and u(t, x ) is an optim al control. T h e value functio n satisfies also the partial differential equation o f dynamic programming:

inf{S,(i, x) -1- S x(t, x) f( t, x, u) - y ° L ( t , x, u ) : u e U} = 0.

T his ap p ro ach has m any disadvantages. F irst o f all, it is a very ra re case th a t the value fu nction is contin uously differentiable in som e open set G when the co n stra in ts (especially state co n strain ts) are included in o p ­ tim ization problem s. T he second is th a t there is n o suitable d u ality th eo ry for p ro d u c tio n analysis with the above app ro ach . In fact th a t ap p ro ach c a n n o t be in general applied to the problem (1), (2), (4) ju st because o f (4).

A non-classical m ethod to study the problem (1), (2), (4) by d y n a m i­ cal ap p ro ach is to carry o u t all ex plorations co ncerning dynam ic p ro ­ gram m ing from the (t, x) - space to the space o f m ultipliers ( ( t , y ° , y ) - space). Let us explain it briefly. Let be given an open set P c j?"+2 o f the d u al space o f the variables (t, y°, y) = (í, p), y° 0 an d a function x ( t , p ) , defined in P, x ( t , p ): P cz R n + 2- * R n, such th a t x ( - , p ) satisfies (4) for each p, such th a t (t , p ) e P . T hen in the set P we define a dual value function'.

T> x ( t ) , u ( i) ) á t — y ° l ( x ( T ) ) j (5)

w here the inferior is tak en over all pairs x ( t ) , u ( r ) , t e [ i , 7], w hose states

sta rt at (t, x(t, p)) and th eir graphs are contained in the set o f values o f the m ap p in g (t, p ) —*(t, x(t, p)), (t , p ) e P . N ext we define a new function:

V(t> P) = - s b0> P) - x (i, p)y,

a b o u t w hich we assum e th a t it is subject to satisfy the co nd ition :

V(t, p) = Vy° (Г, p)y° + Vy(t, p)y = Vp(t, p)p (6)

where: - SD(t, p) = Vy°(t, p)y° , - x (i, p) = Vy(t, p), (t , p ) e P .

W e shall require th a t V[t,p) satisfies the dual partial differential equation o f Hamilton-Jacoby-Bellman type'.

V,(t, p) + H(t, - Vy(t, p ) , p ) = 0, (t, p ) e P (7)

an d the state co nstraint:

9( - vy ( ' , p ) = 0,

w here H( t , x, p) = y f ( t , x , u ( t , p)) + y ° L ( t , x , u ( t , p)), y, y ° are m u ltip liers and u(t, p) is a d u al optim al co n tro l. T h e function V(t, p) m u st satisfy also the dual partial differential equation o f dynamic programming-.

sup{K/£, p) + yf(t, - Vy(t, p), u) + y°L(t, - Vy(t, p),u) : u e U } = 0 (8) an d the state co n strain t:

S|>(i-P) = i nf ] - y ° $ L (

T h e non-classical ap p ro ach has several advan tag es. N o w we need n o t require th a t the set G has a n onem p ty interior. W e d o n o t req u ire the value function S(t, x ) to be differentiable in G. T h e sta te co n stra in ts are in a n atu ra l way included in the dynam ic p rogram m ing eq u a tio n . T h e m o st im p o rta n t ad v an tag e is th a t we have a duality th eo ry which associates the value functions: prim al an d dual.

3. A VERIFICATION THEOREM

In this section we will give the m ain theorem a b o u t th e d u al sufficient co n d itio n s o f optim ality.

Let G c ü" +1 den o te a set covered by the graph s o f all adm issible trajectories.

Let P t = i? " +2 be a set o f variables (i, p), te [ 0 , T], w ith y ° ^ 0 and have a n o n em p ty in terio r. T a k e a fu n c tio n x ( t , p ) d efined in P su ch th a t ( t , x ( t , p ) ) e G , (t , p ) e P and g ( x ( , p ) ) = 0.

Let the function x(t, p) satisfy the follow ing assum ptions:

1) for each adm issible trajectory x(t) lying in G th ere exists an abso lu tely co n tin u o u s functio n p(t) = (y°, y (t)), lymg in P such th at: x (t) = x(t, p(t)),

2) if all trajectories x(t) sta rt a t the sam e (t0, x 0), th en all the co rre s­ p o n d in g them trajectories p(t) have th e sam e first c o o rd in a te y°.

L et S D(t, p) be as in (5). W e see that:

S „ ( t , p ) = - y°S(t, x(r, p)), (t, p) e P.

N ow we will give the p roposition, w hich will be used in th e p ro o f o f the m ain theorem o f this section.

Theorem 1. Let W ( t , p ) = * — y ° Z ( t , x(t, p)) be a real-valued fu n c tio n in P such th a t W ( T , p ) — — y°l(x(T, p)). Let (£0, x 0) e G be given initial condition. S upp o se th a t fo r each abso lu tely c o n tin u o u s fu n ctio n p (i) = (y°, y( t) ), t e [ i0, T ] , w ith grap h lying in P, the fun ctio n x (i) = x(t, p(t)), i e [ i 0, T], x ( t 0) = x 0, is an adm issible trajecto ry lying in G and that:

W (t , p(t)) + y ° } L ( t, x ( t ) , u(x))dz

t

is non-decreasing on [t0, Т]. If p ( t ) = ( y ° , y ( t ) , t e [ t 0, T ] is abso lu tely co n tin u o u s function and if x ( t ) = x ( t , p ( t ) ) , t e [ t 0, T ] , x ( t 0) = x0 is an adm issible trajecto ry in G and is such that:

W (r, p (i)) + y ° JL ( t , x (t) , u(x))dx

(

is c o n s ta n t in [t0, T ], th en x ( t ) is an o p tim a l tra je c to ry an d W (t 0, p( t o) ) = S D( t 0, p ( t 0)), w here u ( t ) is an op tim al c o n tro l co rresp o n d in g to x ( t ) .

P ro o f. F o r an y fu n c tio n p(t), £б[£0, T ] d esc rib ed ab ov e: T

- y ° Z ( t 0, x 0) ^ - у ° \ Ц х , х ( х ) , u ( x ) ) d x - y ° l ( x ( T ) ) , w here u(t) is a c o n tro l

to _

feasible for x(t). F o r the fu nction p(t):

- y ° Z ( t 0, x 0) = - 7 0J L ( z , x(x), v ( x) ) d x - J ° № T ) ) to

so W{to> p0o)) = S D(t0, p ( t 0)) and x ( t) , u(t ) is an optim al pair for the problem г

i n f { - y0j L ( t , x ( T ) u ( r ) ) d r - y ° l ( x ( T ) ) : x(t), u(t), t e [ i0, T ] , are adm issible (

pairs with x(£0) = x 0 and x ( t ) lying in G}.

N ow we will form ulate the m ain theorem (sufficient optim ality conditions) w hich is a c o u n te rp a rt for the dual version o f the verification th eo rem from ( F l e m i n g , R i s h e l , 1975, T heo rem 4.4, p. 87).

Theorem 2. Let V ( t , p ) , ( t , p ) e P , te [0 , T ], be a continuously differentiable solution o f (8), (9) with the b o u ndary condition: y°Vy0( T , p ) = y ° l ( - Vy(T,p) ), (T, p)P, an d satisfying the relation:

V ( t , P) = Vp( t ,p)p, ( t , p ) e P (10)

L et x ( t ), u(t) be an adm issible pair w hose grap h o f the trajecto ry x(t) is contain ed in G = {(I, x ) : x = - V y(t , p) , (t , p ) e P } and such th a t th ere exists an absolutely co n tin u o u s function p(t) lying in P an d satis­ fying: x ( t ) = - V y( t , p( t )) (11) T hen: W ( t , p ( t ) ) = - y°Vy0( t , p( t )) + y ° \ L ( r , x ( t ) , u(x))dx (12) Г is a non-decreasing function o f t.

L et now T ( t ) , tf(t), te [ 0 , Г ], 7 (0 ) = с be an adm issible pair w ith Y ( t ) lying in G and let p(t), te [ 0 , T], be a non zero absolutely c o n tin u o u s fu nction lying in P such that: Y(C) = — Vy( t , p ( t ) ) , t e [ 0 , Т]. L et fo r all r e [ 0 ,7 ] :

v , ( t , m )

+

. -

) = о аз)

T h en x ( t) , ü ( t ) , te [ 0 , T ] is an optim al pair for the problem (1), (2), (4) relative to all adm issible pairs x (í), i/(t), ŕ e [0, T], x(0) = с w hose g rap hs o f trajectories x (t) are contained in G .

M o re o v er: S D( t , p ( t ) ) = - J ° S ( t , x ( t , p ( t ) ) ) = - y 0Vy0( t , j r ( t ) ) w ith x (l, p) — - V y(t, p) is the dual value function along p{t ).

P roof. L et us differentiate b o th sides o f (10) with respect to t alo n g p(t): V,(t, p(t)) = y°(d/dt)Vy„(t, К О ) + y(t)(d/dt)Vy(t, p(t)).

F ro m (2) and (11) we receive:

(d/dt)Vy( t , p ( t ) ) = - / ( г , - V y(t , p(t )), u(t)), and from (1 2) we have:

( d/ dt )y°V yo(t,p(t)) = — ( d / d t )W ( t , p(t)) — y ° L ( t , - V y( t , p( t )) , u(t)). H ence and from (8) we o b tain th a t ( d / d t ) W( t , p(t)) ^ 0 fo r alm o st all ie [ 0 , Т]. T h e above relations w ritten for p ( t ), to gether w ith eq u a tio n (13), im ply th a t fo r all r e [0 , T]:

- y ° V A t , m ) = - 7 °ÍL (t, ЗГ(т), ü ( r ) ) d r - у ° К - V , ( Т , п т ))). r

H ence we get th a t W { t , p ( t ) ) = — y ° l ( x ( T ) ) for all te [ 0 , Г ], i.e. W ( t , p ( t ) ) is a c o n sta n t fun ction. T h is togeth er w ith T h eorem 1 im plies the assertions o f the theorem .

R em ark 1. Solving (8), (9) we o b tain m u ch m o re in fo rm a tio n a b o u t o u r problem th a n in the classical dynam ic program m ing. T h e function: — Vy(t, p ) defines the whole space o f admissible states where our problem m athem atically m a k e s sense. T h e c o n d itio n (10) extrem ely im p o rta n t in physics an d m ath em atics, in econom y was n o t included into co n sid eratio n u p to now . It show s the real p ro d u c tio n costs, dynam ically ch an gin g in tim e, n o t only tho se w hich are placed into the cost functional. T h is co n d itio n tells us th a t the m ultipliers (y°, - y) arc o rth o g o n al to the epigraph o f th e m inim ized cost functional S(t, x) at. the p o in t (x(t, p), S(t, x(, p))). It m ay be in terpreted econom ically as follows: m ultiplier y, which is equal to: — S x(t, x (r, p))

(w hen S(t, x ) is differentiable w ith respect to x) equ als th e m arg in al cost in tim e t (or m arginal p ro d u c t) (com pare ( L c i d l e r , E s t r i n , 1989) in the static case).

In o rd e r to u n d ersta n d w hat the new fu nction V ( t , p ) m ean s let us com e back to the static problem o f p ro d u ctio n analysis. T h en the cost function al (1) reduces to the function l(x), we have not dy nam ical eq u a tio n s (2) bu t we have c o n stra in t (4). U sually to m ak e an analysis o f p ro d u c tio n through the costs and the level o f production the Lagrange function is formed:

and then suitable calculations on this function are m ade. O u r new fun ction fo r this sim ple case h as the form :

w here p = (y °, у ) and x(p) is a p aram etric description o f th e co n stra in t (4), b u t th e p a ram eter is ju st the m ultiplier p. In fact (15) is a d u al fun ctio nal exactly in the sam e sense as it is in linear p ro g ram m in g pro blem s (see A u b i n , (1979, 1997), S c h i l l e r (1989)).

U sually in d u a lity th eo ries m u ltip lier у m ean s th e prices o f som e q u a n tity x. B ecause у = ( y l , y"), so the d u al variable y*(i = 1, n) denotes (according to neoclassical theory o f econom y) the marginal productivity o f the i-th resource o f p ro d u c tio n . In (15) у can be in terp reted as a price o f the q u a n tity x(p), like for exam ple in L e i d l e r , E s t r i n (1989). T h a t is why: — V (p) is ju st a full cost o f the w hole p ro d u c tio n process. We observe th a t studying (14) we c a n n o t derive this type o f d u ality results (see L e i d l e r , E s t r i n (1989)).

3.1. Conclusions

b ( x , у) = К*) + yg(x) (14)

V (P) = y°Kx(p)) - yx(p) (15)

4. EXAMPLE

L et us consider the problem o f m inim izing th e cost functional: n

where:

l(x)n)) = 0, if x ( n ) = 0,

-boo on the co n tra ry , (17)

bu t we assum e, th a t expenditure o f an enterprise changes in tim e and we ad m it a co n tro l o f them . E xp en d itu re is described by the follow ing dynam ic:

T h e co n stra in ts (21) are defined as follows:

Let g be an in d icato r function o f the set D, i.e. it eq uals zero o n th e set D an d equals one o u t o f D (on the plane R 2), w here the set

It m eans th a t if the grap h o f x(t), ie[0 ,rc] lies in D, then g(x( )) = 0. I he co n d itio n (17) m ean s th a t all adm issible tra je c to rie s (fo r o u r p roblem ) m u st be in the p o in t я equal zero.

T o find an op tim al co n tro l we can use P o n try a g in ’s M ax im um Principle (necessary optim ality co nditions) for a problem (16)-(20) - c o m p are ( F l e ­ m i n g , R i s h e l , 1975) - we can also sim ply guess a certain fam ily o f the trajectories, which we “ su spect” o f the extrem e, which is d ep e n d en t on changing initial conditions.

So, we receive the follow ing functions: x(t), u(t), p(t) = (y°,y(t)) 1) x(t, Cj) = Cjsint, u(t, C |) = CjCosf, y° = — e, y(t, e c j) = eCjCOSi,

x (t) = B(t)u(t) a.e. in [0, л] (18) where: fo r ie [ 0 , 7t], for t = 0, u ( t ) e [ 0,1], t e [0, n] (19) x (0) = с (

20

(

)

B ecause o u r trajectories m ust satisfy co n stra in ts (21) so, th e ab ov e (unctions x (i), u(t), p(t) reduce to:

1) x(t, Cj) = c j siní, u(t, Cj) = cycost, y° = - e, y(t, e c ,) = e r b o s t , w here í e ^ C j ) , л], c xe ( —1,1), » - \

w here t( c ,) is a solution o f e q u a tio n c , siní = í2 with respect to í in [0,7t] d ep en d in g on c ,;

2) x (i, e) = 0, u(l, e) = 0, y° = - e, y(t, e) = 0, where t e [0, я], е е

W e can easily check th a t the trajectories:

1) x(i, c t ) = c t sini, where t e [t(ct ), к], c , e ( — 1, 1), 2) x (t, e ) = 0, w here í g [0, л ] , е е

satisfy co n stra in ts (2 1).

Let us define a co n tro l (tak in g into account above functions):

f - y / y ° ) , if t e [0,n ], | y | < ^ | c o s t | ,

u(t, У0, У) =

J

0, if t e [0,rt], y ° Gí - - . - - J , у = 0.

N ow we will define x ( t , y ° , y ) and V(t , y ° , y ) in the sam e set o f variables t and (y°, y) respectively as:

Х (£ ,У ° ,У ) = j ^ /y ° )tg i’ (23)

V ( t , y ° , y ) = i ^ 2l2y0)tgt’ (24)

S ub stitu tin g x ( t , y ° , y ) and K (£,y°, >;) to the assertions o f the T h eorem 2 we see th a t V( t , y°, y) defined by (24) and Vy (£, y°, y) = - x ( t , y ° , y ) defined by (23) satisfy these assertio n s, an d also th ese asse rtio n s arc satisfied by the p air x (t) = 0, u(t) = 0. So, from th e T h eo rem 2, this pair is optim al.

T h e abov e statem ent d enotes th a t, if expenditure sta rts from th e value zero and after tim e m u st be also equal zero in th e problem (16)—(21), so they m u st be all the tim e equal zero, w ith o u t actio n o f a c o n tro l i.e. a c o n tro l m u st be equal zero. Intuitively this fact is obv io us, b u t this exam ple proves th a t m ath em atically there is n o o th er possibility.

S. REFERENCES

A u b i n J. P. (1979), Mathematical Methods o f Game and Economic Theory, North-Holland, Amsterdam.

A u b i n J. P. (1997), Dynamic Economic Theory, Springer Verlag, Berlin.

F l e m i n g W. H., R i s h e l R. W. (1975), Deterministic and Stochastic Optimal Control, Springer Verlag, Berlin.

L e i d l e r D., E s t r i n S. (1989), An Introduction to Microeconomics, Simon & Schuster International Group, Hamel Hemptstcad, Hertfordshire.