A new look at static spatial price equilibrium models

Regional Science and Urban Economics 12 (1982) 519. 597. North-Holland

A NEW LOOK AT STATIC SPATIAL PRICE EQUILIBRIWM Michael FLORIAN l3niwr.M

de MonlrPal.

MODELS

and Marc LQS*

Quebec N3C 3JF. Canadu

Received February 1981, tinal version received Jun- 1982 I.3 this paper WC propose a new view and more general formulations of static single commodity spatial equilibrium models, that lead to simpler and more efkient algorithms than those previously employed for solving models of this type. Tb: proposed formulations incorporate general transportation networks and may be extended t.2 allow for multiple commodities. Solution algorithms ;~resuggested for the problem with multiple commodities, when there ebsts an equivalent optimization problem. We show that the multiple commodity problem majr be cast in the form of it variational inequality, wlhen there does no! exist an equivalent optimization problem and propose algorithms to solve this version of the problem as well.

1. Iatrduction

Samuelson (1952) formuiated the partial spatial eqrtilibriutr: model of interregional (or international) trade with one commodi:y, first stated by Cournot (1838) in the 19th century, as a mathematical program, by introducing an artificial objective function (‘net quasi-welfare function’ or ‘net social cost function’) and pointed out that the classical transportation (Hitchcock) problem of linear programming is a special case of this formulation. Takayama and Judge (I 964, 1971) later extended the model by considering multiple commodities, multiple time periods and showed also that there is a dual formulation of the problem where the prices, rather than the interregional flows, are the decision variables. They proposed solution algorithms based on quadratic programming formulations [set also Hactwick (1971)J that result from the assumption of linear demand and supply Functions in each region. MacKinnon (1975, 1976) reformulated rq more genen,l version of the problem as a fixed point problem and applied z particular Exed point algorithm for its solution. More recently, Rowse (19X1 ). considcrcd the solution of the problem as a non-linear program. The Samuelson, Takayama--Judge (S, T.-J) model with one commodity determines an interregional trade equilibrium in which demand pricss equal to the sum of supply costs and transport.ation costs, and in which thcrc is

0 1&X)462/82/ oooo-oooO/$Q2.75 !(.;;I1982 North-Holland

M. Floriun

t;81)

and M. Los. Stnric .c;patial price rqrrilihrium

models

a trade equilibrium c~[/ic~iertc~~:

is not estnblishcd as long as cc)nsumcrs in a given region pay different prices for the same product from different regions and:or as long as there are supply sources with lower total prices t h:\ln those actually paid. If these conditions were not satisfied, consumers would rearrange their demands in favour of the supply regions with the cheapest total price. Thus, the S, T-J model determines simultaneousi? the quantities produced and demanded in each region, the :rade flows and the regional prices at which goods are interregional produced by the suppliers or bought by the consumers in each region at

irtfer’rqi~trrd priw

equilibrium. Many empiG applications of the S. T-J model have been made in its quadratic programming formulation, in particular in the field of agricultural economics. Schmitt and Bawden (1973) study the international trade of wheat; the impact of changes in freight rates on Canadian wheat exports is

analyzed by Nagy et al. (1980). Kennedy (1974) applies the model to the determination of the state of the world oil market and numerous other applications can be found in Judge and Takayama (1973). In this paper we formulate a new and more general version of the static S, T--J model, in the case of one product. as the minimization of a convex function sub_ject only to non-negativity constraints on the interregional flows. This formulation, which is simpler than those previously suggested, leads the application of a wide class of non-linear programming algorithms for solving the models of this type. The imbedded transportation network may be extended to include real links (ports, terminals, truck routes, etc.) with nonlinear cost functions which permit the endogcnous determination of route choice and intcrrcgional transportation costs. Then we consider extensions that incorpor;lte multiple commodities. The resulting tnodels are non-linear programs or vari:&nnal inequalities which arc not as simple as the single product model: howcvcr they arc amenable to solution by computational techniques, mentioned, but not studied in detail in this paper; these permit the solution of versions of this problem which were not considered in the csistinp litcraturc.

2. Thv Samuelson, Takrryama-Judge model: A single commodity spatial price t~quilibriunl model WC consider

II rcpions which both supply ;mci consume ;t given product. Each xgion l~~\y be considcrcd its ~1market where the price pi for supplying a qu:antity f/i is pivcn by a supply function Sj(tli). I’ = I,. . ., II und the demand price TT~is given by Ii(l~i). j = 1,. , ., II, the invcrsc of the demand functi;,n ll is assumed that Si(Clj)is a continuous monotone increasing function of ‘Ii, tli Z 0, and that fj(hj) is a continuous bounded monotone decreasing function of /I+ hi > I). Interrrgional shipments fij, i,j = 1,. . ,, II, of the prcxiuct Gj(Ttjl*

M. Floriun and M. Los, Stalic

spatiulprice tyrilibrttrm

nmlcls

581

are possible and cost Cij per unit. In this section unit shipping costs abe constant; later we relex this condition. A spatial price equilibrium is established [see Samuelson (1952)] when the demand price equals the supply price and the transportation cost for all positive interregional flows; if the demand price is iess than the supply price and the trans~rtat~~~ cost, then

no interregional flow occurs. These e~~i~jbr~u~ conditions rna~ be simple rtated as Si{di)+ Cij

lj(bp)=O if fij > 8,

--

20

if

all (i,j).

(11

fij=O*

Note that i=j is permitted as well and cl, need not bc zero. In addition, the inter-regional flows and the equilibrium values of the amounts supplied and

demanded satisfy the following conditions: di=Ctij.

i-

I,...,n,

bj=Cli,, i

i

i-

I....,~I,

(2)

and fijz

0,

all (i,j).

(3)

The constraints (2) may be interpreted as conservation of flow constraints on a bipartite network, which has led to this p >olem being referred to as a generalized version of the classical transportati.>n problem [see for instance MacKinnon (1975)]. As shown by Samuelson (IW2) and then by T;tl.ayama and Judge (1964), this model has an equivalet;t optimization formulation which consists of stating the objective

(4)

subject to (2) and (3). It is straightforward to verify that the Kuhn-Tucker conditiL,ns [&CC Zangwill (I%9)] for this optimization problem arc ccluiv;ll(-ilt to (I , 12) and (3). sin:e the objective (4) is a sum of convex functions (fhc first it1111third term) and of linear functions and is thcrcforc convex, in which LOW..lhe Kuhn Tucker conditions ;uc ncccssarj ;tr~d sufficicnl for lhc cq~~~llhrlurll vdues (di), [I?,), ;ti;;. ‘The litcraturc of the

late ho’s ;lnd car/y 70’\ 1WC fijr in~\t;\nuo It tips: ;tnti ‘l‘akayama (1Y73)J contains speciahzed versions of this model v-hen tkc functions Si(di) and t’j(h,) are linear s id, by consequence, the objectitc (4) is il quadratic function, ‘Nhich makes it possible to apply standard rnctb~~ds for quadratic programming. MacKinnon ( 107.5) rcformuialCd t!liS prrlh ~111;I’*+l

M. Fbrian

!w

und M. Los. Stork

spatial price equilibriummodels

fixed point problem and adapted a particular technique [Kuhn and MacKinnon (19731 for computing fixed points for its solution. It is our aim here to demonstrate that the problem (4), (2) and (3) may be reformulated in a way that considers implicitly the conservation of flow constraints (2) and which leads to very simple solution methods based on feasible direction methods of non-linear programming [see for instance, Luenberger (1973)]. While this formulation was noted by Takayama and Judge (19&I), it was, surprisingly, not exploited for computational purposes. Unlike the classical transportation problem [Hitchcock (1941)], the quantities supplied, di, and consumed, bj, are determined endogenously; thus (2) may be interpreted as cletinitional constraints. Once the interregional flows tii are determined, (2) serves to find fdi} and (bi). Hence (2) may be substituted into (4) to obtain

subject to (3). This is a relatively easy non-linear programming problem, namely the minimization of a convex function subject only to non-negativity constraints on the variables tij. Hartwick (1970) observed this formulation however did not exploit it for computations. This formulation suggests the adaptation of classical non-linear programming methods such as the linear approximation method [Zangwill (1969, p. 152)], or the gradient projection method [Luenberger (1973, p. 247)]. That this reformulation is useful for computational purposes is demonstrated by the adaptation of the linear approximation method (LAP), which is presented below. The LAP method is one of the simplest methods for solving a convex programming problem subject to linear constraints. The genera1 algorithm is given in the appendix. A typical iteration of this algorithm requires the solution of a linear program, that finds the descent direction and the solution of a one dimensional minimization problem, which determines the step size for a given descent direction. Consider now the application of the algorithm at iteration 1. when a feasible solution Itji) is known. The descent direction for the objective (5) is obtained by solving for “ij which solves

(i )

minCS, i

i(‘i, ~

Let Sf = S&

CfZ

Cwij+CCCijWij-_fj 1

0.

i j

~lri)~~ij,

j

( i

i

all (ii).

tfj) and 1: = fjxi

(7)

tfj). Then (6) may be written as

M. Floriun

ad

M. Los, Static spatial prica equilibrium

models

583

The problem (g), (7) has a trivial solution: set w,~=O if its coefficient is po?:;itive; otherwise set Wi,- &, where cj is a suitable up!per bound on the interregional flow tl,. (This bound may be determined empiric;tlly for a giveG problem.) The descent direction thus obtained is dt,=(wij-tf,i) and the step length is obtained by solving for rZin

The implicit consideration of the constraints (2) results in a problem which is, as evidenced above, rather easy to solve even when the supply and i,nverse demand functions are not linear. In the following sections we consider extensions of the single product model which are nearly as e,asy to solve as this basic model. Although we will present only the adaptation of the LAP method, we do so in order to demonstrate the utility of our view of these problems. Faced with a particular problem, any of the fealsible dirlcction algorithms of non-linear programming mentioned a.bove is a, candidate for the development of solution algorithms, including variants of the LAP method which have better convergence rates than the basic method. 3. Extensioos of the single-commodity S, T-J

model to non-coustant unit

transportation costs

In many instances of real-world transportation problems the cost of shipping one unit of product for a given region pair (i,j) iis a non-linear function Cii(tij) of the quantity txansported, tij. We assumle that cij(tij) is a continuous, monotone non-de’creasing function. It may, in particular, be strictly convex or strictly concave. The f>rmer correspolnds to situations where congestion effects occur, while the latter corresponds to quantity discount:;. It is reasonable to model thr costs” in this way when the transportation market under consideration is segmented into as many submarkets as there are region pairs, and where each submcxket is captive to one transportation firm, which follows a pricing policy r Elected by the function cij(tij). It is implicitly assumed also that transport&x prices reflect the cost of supplying transportation services, or at least vary in proportion with these costs.

‘$4. fkian

584

and M. Los. Static spatial price quilibrium

Thus, the equilibrium conditions SAdi)+c’i~fij)-I,(~j)=O

r,o

(1) become

if

tij>O,

if

t,i=o,

all (ij),

Subject to eqs. (2) and (3), and it is straightforward equivalent minimization problem is

and

r,jgO+

modtds

(10) to verify that

the

all (i,j).

The solution method, based on the LAP method, outlined in the previous section, may be app%d to solve this version of the problem, since by the assumption of monotonicit y of the cost functions, the second term of (11) is convex as well. En the direction finding subproblem (g), (7), ‘ii is replaced by t-Jij = Cfj(f:j),

MacKinnon (1975) suggests a slightly simpler, but perhaps more interesting, extension. Suppose that a single firm controls the transportation market and that the price of a unit of transportation (say a ton-mile) is a function of the total volume of transportation services, T, T=CiCjsijt~~, where Iii is the amount of transportation service required to ship one unit of commodity between i and j. Let f(C,Criitij) be the price for one unit of transportation service where f(e) is a continuous monotone increasing function. The equilibrium conditions I 101 now become S,(dil+fijP

CCfijfij

( i j

-1jJbj) =O

if

f,j>O,

if

tij=O,

all (i.j)

(12)

j

20

with constraints (2) and (3). Again, there is an equivalent problem:

minimization

problem for this version of the

The solution may be obtained by a slight adap‘ation of the previous versions of the LAP method: C;j is replaced by rijP(T’), where F=xiCj~i,ffjThis approach may bc contrasted with the solution method proposed by MacKinnon t 19751 which requires the application of a fixed point method.

M. Floridn and M. Los, Static spatial prkv equilibrium models

585

The problems that we have considered above, (11) and (13), are extreme cases of the situation where there are several transportation, firms, each of which controls a part of the market that consists of a subset of the routes (i,j). For instance, suppose that there are two transportation firms, I and 11. and that each controls a disjoint subset of the routes. Thus tZ, (i,.i)~ I ate flows on the routes controlled by firm I and t$. (~,J)E IT are flours on th: routes controlled by grrn II. Each supplier of transportation services has ;:s own price function which depends only on the services it supplies. Thus, the equivalent minimization problem for such a situation is, by omitting the statement of the equilibrium conditions: %. j)f

min x’f”S,(x)dx i

0

+

FiJfiJ

J

p’(z)dz

0

(14) tij>=O, all (i,j).

4. Extension of the single-commodity S, T-J model to general transportation networks with endogenoustransportation costs

In many instances shippers have a choice between different mode-repute combinations and the shipper’s cost of transportation for a given pair (i,j) is dependent on the actual flows on the different mode-routes. For instance, in maritime transportation, many ports are congested and a mode-route combination which would be least cost for a given pair (i,j) without port congestion present, ceases to be an attractive route due to congestion delays and costs. Thus, it may be of interest to extend the model described in the previous sections to include a general transportation network that would model the transportation infrastructure in more detail. Such an extended model could be used as well to predict the volume of traffic of different ports or cargo terminals, while taking account o!’ the interregional trade of the commodity in question, when the flows of other commcdities that use the Tame infrastructure are considered fixed. The use of a multimodal network has been proposed in freight transportation studies by Rronzini and Millcl ( 1979). The ~c:;rral network considered is compwcd of’ arcs l’, 11 E .4. and of no&s n, HEN. Some of the links model the regions of interest, while other links and nodes model the transportation infrastructure. With each arc of the transportation network we associate a continuous, monotone non-dccrcasing cost function c,(t’,), where I., is the flow of the commodity on arc rl. This function may bc convex or concave. Let /I~,,, be the flow on the path k

M. Florian

5th

and M. Los, Static spatial price equilibrium models

between region pair (i,j) and let ;5ok.ijbe an indicator function which equals one if arc a belongs to path k,[j and zero otherwise. The equilibrium conditions are Si

(j ) Ctij

( 1

=O if

Ctij

+~Sa~,ij*C~(U,)-~j

h,ij>Og

1

,I

tij-Chk,ij=O*

(15)

(16)

all (ij),

k

and the definitional

all k,&

constraint

(18) This model has an equivalent minimization

problem, which is

Ejrij

minc 1 i

Si(.Y)d.X + 1 ‘i c,(Z)dz -C ‘1” Ifv)dy,

0

u

0

j

(19)

h

subject to (16), (17) and (18). It is again straightforward to verify that (19) is a strictly convex function and that the Kuhn-Tucker conditions of (16)--(19) correspond to the equilibrium problem (I S)-( 18). The adaptation of the LAP method to solve this problem is hardly more involved than the previous models. Consider an intermediate iteration I when a feasible solution ir:. ~fii is kno,wn. The direction finding sub-problem is

(21) (22) where ,-.’ k

ij=C,ijuk

In

ij’C’n(l’,)*

an ‘optimal solution of the ribove problem :III the flows \~i; would be

hi. Florian and M. Los, Static spatial price equilibrium models

allocated to the shortest path between i and j. Thus the problem reduces minCC(S’,+Zfj-~$‘~~, ij

Wij 20,

subject to

all (iJ),

587

to

(23) (24)

where ?;I is the length of the shortest path between i and j on the transportation network when the flows are (~fi). It is evident now that (23), (24) is equivalent to (8), (7), with the exception that the constant transportation costs have been replaced by the shortest path lengths i?ij. Thus, the only difference in the LAP method for the problem with a general transportation network is that it is necessary to keep track of the flows on the network and recompute the shortest paths at each iteration. It is a remarkable consequence of this formulation of the spatial price equilibrium problem, that the inclusion of a general transportation network hardly complicates the solution method, while making the model more realistic for certain applications. From a hlstorical perspective, the model (15)--(18) is a synthesis of the Samuelson (1952) model and of the network equilibrium model with variable demand formulated by Beckmann et al. (1956). It should be noted however that the latter formulation is based on Wardrop’s (1952) ‘user-optimizaiion principle of the equalization of transportation costs on all used paths, while in the more general model presented in this section it is the dem‘md prices, which include the unit supply cost and the transportation cost, which .-u-c equalized for all routes that supply a given market. 5. Multiple commodity spatial price equilibrium models In this part of the paper we consider the generalization of the models considered in the previous sections when the n regions considered supply and consume multiple products. Thus, each region may be considered as a market where the price #” for supplying quantity t/y of’ product M is given by a supply function S;“(dJ where di =(d;“> m = I,. . ., M) 20, is the vector of all the products offered. It is assumed that S;“(di) is a continuous function of d, and that its restriction’ with respect to all products except on’:, say r17, is a monotone increasing function of dl’. The demand hy. h;’ *_0, for product 1)1 in market j is given by a demand function Gy(nj) where Xj=(Ky, m ==1,. . ., AI) is the vector of demand prices. It is assumed that tij”(~$ is a continuous function of 7Cjand that its restriction with respect to all prices except IMUG .~iy upper bour;.c.led decreasing function of $“. In addition it is nrn. is a monotone

M. Florian and hf. Los. Static spatial price equilibrium models

588

assumed that the mapping Gjnj) admits a smooth inverse Xj=fjbj), with the property that the restriction of each I,“(bj) with respect to all demands except one, say f$“, is a monotone upner bounded decreasing function of by’. The interregional flows may use a general transportation network composed of of goods from the links ti, l:r=l...., A that permits the transportation producing regions to the consuming regions. Let c~(Y,J,m = 1,. . ., M, a = 1,. . ., A be the cost function for commodity m over link cs where u,=(u,“, m = 1,. . ., M). It is assumed that ct(u,J is a monotone increasing function of o, and that its restriction with respect to all commodity flows except one, say II,“, is a strictly increasing function. Let /I~, &EKt be the flow on path k between region pair (ij) of commodity m; K! is the set of paths available for commodity m and region pair (rj). The cost of path k is given by the expression C,( C’)=

c s, *c:(u,),

kE Kz

Vi,j,m,

(25)

a

where

6,, = t =0

if arc a belongs to path k, otherwise.

Since the demands in market j depend on all the demand prices, the equilibrium conditions for this multicommodity model are stated as follows: Sy(dF) + C,(c*) - I,“(/$) = 0

if hf > 0, ktz Kz

>O

Vi,j,m.

(26)

if h:=O,

The subdivision of the quantities supplied dy over the paths of the network must satisfy, as usual, flow conservation; thus fz.

c

h,=O

Vi,j,m,

(27~

keK;

where fz is the total quantity of product m shipped from region i to region j Thus

1‘2~ path flows are non-negative (30)

M. Florian and M. Los, Static spatial price equilibrium models

and the rink flows are given by the definitional

v~=~~1

In addition, the equilibrium demand prices; thus Z77=

Z,"(bj)

constraints

Va,m.

6,k-hk

demands

589

(31) are those tha:

correspond

Vj,m.

to the

(32)

We note that by our consumption mapping (32) is equivalent to

on the invertibility

lly = G~(Z7,) V j,m

of the demand

(3’3)

and we will use both (32) and (33) in our analysis. In order to determine the conditions for the existence of an equiv.&r,t optimization problem we convert (2;) into a variational inequality, akin fo the work of Smith (1979). For a given (i&m) the equilibrium condition of (25) may be rewritten as (s;“(@) + ck(v*) - Z,“(bf))(hk- h,*)2 0. where * indicates the equilibrium supplies, flows and demands and is ;tny other feasible flow. (34) and (25) are equivalent since if h: >O then Sy(@) + Ck(l:*)- Zjm(hf)=O as h, may assume values inferior to hf and if hz -0 then SF(@) + c,&*) - I,“@;) 2 0. By summing (34) over k, then over i,j, m and by using (27), (28), (29), (3 1) and changing the order of summation in the appropriate places, it rntily be rewritten as the variational inequality

(35)

The variational notation as

inequality

(35) may be

I .written

S(d*)(d- d*) + c(t*)(v - v*) - Z(b*)(b-

b* j 2

compactly. 0.

L)y using vuclor

(36)

It is relatively easy to show that (36) implies (34) by assuming (d*. v”, h*) known and by constructing another soIution (2, b, fi) which differs only cln the

590

M. Florian and M. Los. Sfafic spatial price equilibrium motiels

flow of the path k. keK;. Thus &=\I? and @=dy* (&-I$). For this solution (36) reduces to

+(&--h,*). !$‘=I-$*+

which is equivalent to (34). The existence of a solution to (36) is ensured of the demand functions are bounded and there is suficient supply to satisfy the maximum demands. The existence of a solution to a variational equality was studied by Hartman and Stampacchia (1966). The question of the existence of an equivalent minimization problem for (36) over the feasible set (27), (30) is determined by contrasting (36) with the optimality condition for minimizing a function f’(t) over a convex set 0, which is that Vf(~*#y-.u*)zO for all LEO. As is well known, [Hotelling (1932)] the function f exists when the Jacobian of S(d), C(D) and I(b) is symmetric. Special cases when this condition is satisfied occur when the supply functions are separable by product, that is S~(di)=fS~(~) and when the demand functions are separable by product, [see for instance, Rarros et al. (197’7)], it is unlikely that the demand functions would satisfy such a condition in practice. Thus, when the symmetry conditions are satisfied the equivalent minimization problem is

subject to (27), (30), where (q), (9) a:ld (5) are appropriately defined and $ denotes the line integral. A similar, but less general analysis for the symmetric case may be found in Takayama and Judge (197 1, p. 113), while Barros et al. (1977) present the equivalent minimization problem with general inverse demand functions, separable supply functions and linear trans,>ortation costs. In practice, it may often occur that the symmetry conditions on the Jacobians of S(cl), c(t) and n(h) arc not satisfied, in which case the equivalent minimization problem (37), (27), (30) does not exist. However uniqueness of the amounts supplied (rf) and amounts purchased (h) and the flows (c) is ensured if S(tf). C(P)and -G(n) arc strictly monotone mappings. [Recall that ,I is strictly monotone if (_f‘(s)- .f’(_#ou-- y)> O.] This may be demonstrated [SW for instance Smith (1979) and Aashtiani and Magnanti (1981)] by supposing that thcrc exist two distinct solutions (,d’, IT’,!I’) and (8, I’*, b*) and showing that the two must be equal. By this hypothesis, and by using our assumption about the invertibility of G(X)

M. Florian and Jr. L’s, Static spatial price equilibrium models

S(d2)(d1 - d2)+ c(v2)( u! - 02)- z(6z)(61- b2)2 0. By adding (38) and (39) and performing obtain

(S(d’)- S(dZl(dld) -

some algebraic

591

(39) manipulations

WC

+ (c(d)

C(V2))(U’ - u2)+ (I@‘) - z(62))(6’- h2)5 0.

(40)

By imposing the strict monotonicity assumptions it follows that d’ =d2. u’ = v2 and b1 =b2. We turn our attention in the next section to solution algorithms for the multi-commodity problem, first for the symmetric case, and then for the asymmetric case.

6. Computational approaches for the multiple commodity models

For the sake of generality we will discuss computational approaches for solving the minimization problem (37) subject to (27), (30), since the simpler versions of the problem, when the underlying network has special structure, are special cases, for which the method for solving the general problem would specialize in a straightforward manner. Since (37) is a convex objective function, and the ccnstraints (27), (3) are linear, it is possible again to adapt a general non-linear programming algorithm, such as the LAP method or others, for its solution. However, the added dimension of the multiple commodities provides two strategies for the development of solution algorithms: apply a general method directly to the objective (37) or decompose the problem by commodities. We discuss the latter approach first. The drcompositiun of the problem by commodities implies that the problem may be solved by a cyclical application of a single commodity algorithm to all commodities in turn, by keeping the supplies, flows and demands for all other commodities fixed at each step. Thus, one would suboptimize each commodity at a time and the computations would bc terminated, when improvement in the objective functions of all commoditic:( is no longer possible. The decomposition cll.qorithnl ma!’ be stated generally as follows: Stclp 0 Step I

(initialization). Determine an initial feasible solution (d, r+i~),set p=O, q = 0 and continue. (stopping test). If y = M, terminate; otherwise set I= p*mod( M)+ 1 and continue.

(st~lvo

subprohlcm

:I

commodity and

return

for

commo4ty

the

swbptobkm

set y=il+

for

1. p-t-p+

to step 1: othcrwisc solve the sinplc commodity

for (d’, t+‘. h’) by keeping (P, and return

If

I).

I is ~~ptimill (in spatial equilibrium)

1

problem

I, fixed set q= I, p-p+

P’,

1

to step 1.

method finds a local minimum: however (31) is strictly convex, and hence it hits il unique minimand. Thus the local and global minima coincide.

The intcrcst in this approach is motivated by the simple observation that the methods developed for the single commodity problem may be applied directly

to solve the multiple

suggested by Barros cwvcx

simplex

suhproblcms

commodity

problem.

ct al. (1977); in particular,

method

[Zangwil!

for each commodity.

(1969,

This

is the approach

they sl:ggest the use of the

p. 162)]

for

the solution

cfticicntly wlvcs the single commodity problem may be used. It is wtxthwhilc to point out that the dccompositiun succcssfull!? mud for solving network (1974) and Florian not

esplore

in

of the

although any other suitable method that

equilibrium

approach

was

problems, such as Nguyen

and Nguyen (1974). among others. In this paper we will

detail

alI

the

interesting

variants

of

the decomposition

approach; r;i t her we otjntcnt ourselves with prcscnting it as a possible solution strategy and in forthcoming work wc shall deal with particular ;ipplicuticms.

(41)

bf. Fbriun ad

bf. Im. Static spatial price quilikrium

models

593

those shortest paths be Z& the;i (41)-(43) yield

This prsbkm has naw the same trivial solution as the single ~o~~rn~~t~ subp~ob~~ (8). (7): set %% $ = 8 if its ~~~~ie~t is posi’6ive: otherwise set $ =r%, where 12 is a suitable: upper bound an the inter-regianal flow ‘5. 11 is clear that the other steps of the algorithm mrq be impteme~ted in a str3i~tfo~rd way. From the above disunion it should be clear that the ~~ultic~~rnrn~~dity spatial quibim problem is easy to solve, when the supply cost and inverse demand rnap~jn~ satisfy the symmetry conditions which foad to the equivalent optimization problem. However, these symmetry conditions arc not likely to be satisfkd in many situations in practice, and the solution approaches for the non-symmetric case may be of more practical importance. The developments of a solution algorithm for the asymmetric multicommodity version af the problem requires the solution of the variational inequality (36) over the feasible set defined by the constraints (23, (1-O).Thcrc arc many possible approaches which include projectic.m and relaxation algorithms. Pang and Chan (1981) provide a good survey of these possihlc approaches. At present, most of the computational cxpericncc ltvail;rbfc have been obtained with the Jacobi (or diagonalization) method [see for instance Hogan (197$), Florian (1977)J which is a rclax;ttic?n type algorithm. The foca! convergence of this method was studied by Ahn I 1078) anti F-fori;ln ;tr,tl

Spiess (19&l); global convergence of this method was studied by Dafermos (1980). In the following exposition of the Jacobi method WC shall itssumc thilt the supply, link cost iind demand mappings arc monatonc and hcncc the solution is unique. The Jacobi (diagonalization) algorithm is based on rathcr intuitive ideas and has been uwd in prrrcticc with rigcrrrlus convqp,wx fixing

proofs, Roughly

the asymmetries

fcasihlc sduiiotr

success cvcn before it was supported

(2, C. h)

in the supply, IS

h!,

spaking, this dpor~thm iq t~soc~ on cost anti cfcmanti mappings. gVcr1 d

fOllOWs:

140)

594

M. Florian

and M. Los, Stalic spatial

price eqbtii:brium

models

all i, m, c(u, 6) = C,“(r,“). all a, m and a(b,6) = iy(hjm), all j, m, have now diagonal Jacobian matrices and this diagonalized problem has an equivalent minimization problem which may be solved by using one of the methods that solve the symmetric multiple commodity version of this problein, The procedure is repeated iagain until convergence is obtained. The sufficient conditions for the convergence of this algorithm is more stringent than the monotonicity of the supply, cost and demand mappings; however convergence is ensured if the Jacobians of these mappings are diagonally dominant, that is, for each commodity, these functions depend primarily on the volume of that commodity and the cross-effects of the other commodities are of less importance [A matrix A =(Nij) is row and column diagonally dominant if ilii >Ci~~ij and Llii>Cjaij, allI i], The other computational approach which merits special attention is the dcconrposition algorithm, which was introduced earlier for the solution of the symmetric problem. This algorithm may be applied to solve the asymmetric problem as well, since in Step 2, the subproblem for commodity 1 has BT: equivalent optimization formulation, due to the assumptions that were madt on the restrictions of the demand, cost and supply functions. Thus, the algorithm could be implemented by solving a sequence of single commodity problems. Thr decomposition algorithm may be interpreted as the application of a block Gauss-Seidel method for solving the special variational inequality (37) subject to (27), (30). The convergence of this method for such problems was studied recently by Pang and Chan (1982). We anticipate that future work with these computational approaches will determine which approach is superior for multicommodity spa.tial price equilibrium problems. The

mappings

S(&>=

$“(dy),

7. Conclusiorr In this paper. WC propose a new view of and more general formulations of the static single commodity S, T-J model. This formulation leads to the application of a wide class of non-linear programming algorithms. Our praposcd computational procedures constitute an alternative to MacKinnon’s (1976) fixed point approach and extends the approach of Rowsc (IWI). The imbedded transportation network may be extended to inchrde transportation links that model facilities such as ports, terminals, truck routes and others, with general monotone increasing non-linear cost functions. This formulation may be extended to incorporate multiple commodities. The conditions that we give for the existence of an equivalent optimization problem 4th multiple commodiaies are more general than those given by ‘Paksyama and Judge (1971) since they allow for non-linear transportation

M. Florian

and M. Los, Static spatial price equilibrium models

595

costs. General transportation networks may be incorporated in these models as well. Solution algorithms for the equivalent optimization problem for multiple commodities are discussed as well. Finally, we examine the multiple commodity static spatial equiiibrium problem when no equivalent optimization problem exists. Since the existence of a solution is ensured if the demand functions are upper bounded, we point out sufficient conditions for the urlqueness of the solution. This version of the problem may be solved by se. era1 approaches, which we outlined in the previous section. In this respect, the variational inequality formulation of the asymmetric multicommodity version of this problem complements the work of Takayama and JL::lge (197 1). Appendix: The linear approximation algorithm Consider constraints:

the problem

minZ(X)

of minimizing

subject to

,4X26,

a convex function

subject to linear

X20.

The detailed steps of the Frank and Wolfe algorithm for solving this problem are: Step I

Given a feasible solution X’, set I= 1.

Step 2

Determine Y’ that minimizes VZ(X’)Y

Step 3

subject to

AY >=h, Y 20.

Set the descent direction d’ = Y’-- X’. If 1VZ(X’)d’J 5 E, terminate, where E is a suitable parameter. (X’ is the optimal solution.)

Strp 4

Find the optimal step length R’ that minimizes Z(X’+Ad’)

Step 5

convergcncc

for

O~;l~l.

Revise the current solution X’ + ’

= x! + ),‘J’;

set I= I+

1

and return to Step 2.

I is an iteration index. References Aashtiani,H.7,.and T.L. SIAM

Magnanti, 1921, E
trnnspoi’tation

network.

M. Florian and M. Los, Static spatid price equilibrium mdels

596

Ahn, B.H., 1978, Computation of market equilibria for policy analysis: ‘The project independence evaluation system approach, Ph.D. dissertation (Department of Engineering-Economic Systems. Stanford University, Stanford, CA). Barros, 0.. A. Gomez ard A. Weinrraub, 1977, Distribution equilibrium problems with elastic supply and demands, Publication no. 77/10/C (Departamento de Industrias, Universidad de

Chile, Santiago). Beckmann, M., C. McGuire and C.B. Winsted, 1956, Studies in economics of transportation (Y&z University Press, New Haven, CT). Bsonzmi, M.S. and R.C. Miller, 1979, Freight transportation energy use - Vol. II,. Methodology and program documentation, Report no. D0T-TSC-OST-79-1, II, prepared for the U.S. Department of Transportation (Washington, DC). Conmot, A& 1838, ‘Mathematical principles of the theory of wealth, Ch. X. Dafermos, S.. 1980, The general muiltimodai network equilibrium problem with elastic demand (Brown Universily. Providence, RI). Dafermos, S.. 1981. An iterative scheme for variational inequalities, to appear in Mathematical Progfammink I-Iorian. M.. 1977, A traffic equilibrium model of travel by car and pu’:llic transit modes, Transportation Science ii, 166- 179. t’hntan. XI. and S. Nguyen, 1974, A method for computing network equilibrium with elastic demands, Transportation Science 8, 321-332. Elonan. M. and H. Spiess, 1981. The convergence of diagonalization algorithm for fixed demand asymmetric network equilibrium problems. Publication no. 148, (Centre de recherche sur les transports. IJniversite de Mont&al, MontrCai) to appear in Transportation Research. Fufiran. W.H.. LG. Nagy and G.G. Storey. 1979, The impact on the Canadian rapeseed industry from change in transport and tati rates. American Journal of Agriculture Economics 61, 238.248. Guise. J.W.B.. 19?9. Ai expository critique of the Takayama- Judge models of interregional and intertemporal market equilibrium, Regional Science and Urban Economics 9, 83--95. Hanman. P. and G. Stampacchia., 1966. On some nonlinear elliptic differential functional equations. Acta Mathematics 115. 271- 310. Hartwick, J.M., 1970, A generalization of the transportation problem in linear programming and sp~liai price equilibrium. Discussic? paper no. 30 ilnstitule for Economic Research, Queen’s University. Kingston). tiartwick, J.M.. 1971. The generalized transportation problem as a quadratic profram. Discussion paper no. 35 (Institute for Economic Research. Queen’s University, Kingston). ffltchcnck F.L.. 1941, The distribution of a product from several sources to numerous localities, Journal of Mathematics and Physics 20, 224-230. Hogan. W.W.. 1975. Energy policy models for project independence, Computers and Operations

Research 2. Hotelling H., 1932. Fdgeworth’s taxation paradox and the nature of demand and supply functions. Journal of Political Economy 40, 577-616. Judge. G.G. and T. Takayama eds., 1973, Studies in economic planning over space and time (North-Holland~American Eisevier, Amsterdam). Kennedy. M.. 1974. An economic model of the world oil market, The Bell Journal of Economics and Management Science 5. Ktihn, H.W. and J.G. MacKinnon. 1975, The sandwich method for tinding fixed points. Journal of Optimization Theory and Applications. I uenberper. D.G.. 1973, Introduclion lo linear and nonline*+r programming (Addison-Wesley. Rrxdmg, MAI. !bl~Kinnon. J.
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