Cybernetics and Systems Analysis, Vol. 33. No. 6, 1997

SYSTEMS ANALYSIS

C O N V E R G E N C E OF A M E T H O D F O R C O M P U T I N G E C O N O M I C

EQUILIBRIA

V. I. Norkin , Yu. M. Ermol 'ev , and G. Fischer UDC 519.6

1. INTRODUCTION

The method Of sequential joint maximization (of consumer utility functions) [1] has been proposed as a heuristic

procedure for finding equilibria in applied economic models. The method exploits the equivalence of the model of equilibrium

with fixed consumer incomes to some optimization problem (see Eisenberg and Gale [2], Gale [3], Eisenberg [4], and

Polterovich [5]). It involves iterative nonlinear aggregation of consumer utility functions and solution of a sequence of

optimization problems with an aggregated utility function as the objective. The method is effective for solving fairly complex

temporal equilibrium models intended for integrated assessment of international environmental protection strategies (see Manne

[6], Manne and Rutherford [7]). In the present article, we investigate the convergence conditions of the method.

We show that this method fits the general scheme of Dafermos [8] for solving variational inequalities (the equilibrium

problem is reducible to variational inequalities). Convergence conditions are formulated in terms of a parametric aggregated

excess demand function.

In the consumer income space, the proposed method is a simple iterative method for Finding the fixed points of some

multivalued map. For Cobb-Douglas utility functions, the properties of the method can be studied analytically. We show, for

instance, that the convergence of the method is linked with the convergence of some homogeneous and nonhomogeneous

Markov chains. We also prove convergence of the method without assuming strict gross substitutability.

2. ECONOMIC EQUILIBRRYM MODEL

We introduce some notation (see [9-11]). Consider an economy consisting of m consumers and l producers. Each

consumer k is characterized by the utility function Uk(xk), the consumption vector x k E Qk C R n, the initial commodity stock

m

w k E R+n, and the shares Ski in the profit of producer i, ~ Ski " - 1. Producer i is characterized by the production activity k = l

vector Yi E Yi C R n and the vector production function gi(Yi) = = (gil(,Yi) . . . . . gin(Yi))" Let p E R+ n be the vector of

commodity prices in the economy, x = ( . I f 1 . . . . . Xm), Y = (Yl . . . . . Yt), Q = Q t x ... x Qm, Y = Y1 x ... x YI.

The demand for commodities in the economy is generated according to the maximum utility principles: consumer k

chooses the consumption vector x k by maximizing the utility function (1) subject to the budget constraint (2) and other constraints (3):

Uk,(xO --,.max, (1) xk

Xk ~ Qk C R n - ' (3)

Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 127-142, November-December, 1997. Original article submitted December 24, 1996.

854 1060-0396/97/3306-0854518.00 �9 Plenum Publishing Corporation

where the income function I k = It(y, p) is

l ??1

It(y , p)= pw/~ + ~ aki pgi(Yi), 2 ak.i = 1. i = 1 k =1

(4)

Here Pgi(Yi) is the scalar product of the vectors p and gi(Yi). This approach enables us to consider any demand functions

xk(l k, p) by choosing the corresponding utility functions Uk(xk). Producer i chooses the level of production activity Yi by maximizing the profit:

pgi(Yi) ~ max, (5 ) y,

Yi E Yi C R n. (6>

�9 , $

The vectors x*, y , and p constitute the competitive equilibrium of model (1)-(6) if the vectors x k are the solution o ~

of (1)-(3) given p = p*, y = y*, k = 1 . . . . . m; Yi is the solution of (5), (6) given p = p , i = 1 . . . . . l; and the following

material and f'mancial balances are satisfied:

m

2 " x~:~< W + G ( y ~ k = l

m

p x k = p w + a ( y * ) ) , k = l

(7)

(8)

m l

where W = ~ w k, G~) = ~ gi(Yi) (here summation is over the components). k=l i=l

We introduce the "market" as one of the players (see Zangwill and Garcia [12]), which given x and y solves the

following optimization problem"

p x k - W - G ( y ) ~ m a x , (9) k = l P

p 0, E ' j - '- j = l

(io)

The gradient of the objective function (9) with respect to the variables p

m

d -- x,-- w - a (y/ k = l

is the excess demand in this equilibrium model.

Consider the game of consumers (1)-(3), producers (5), (6), and the market (9), (10).

L E M M A 2.1. The competitive equilibrium x , y , p of the economic model (1)-(6) with prices p normalized so that

n

pj* = 1, is a Nash equilibrium in the corresponding game (1)-(3), (5), (6), (9), (10) with (m + l + 1) players. Conversely, j = l

the Nash equilibrium in the game (1)-(3), (5), (6), (9), (10) with consumer utility functions (1) that are strictly monotone

increasing in all variables is a competitive equilibrium of the model (1)-(6).

855

Proof. To prove the first assertion, it suffices to show that p* solves problem (9), (10) given x = x* and y = y . tn m

, * I * ( - �9 Indeed, by (7) p ( ~ x t -- W - - G ~ * ) ) _< 0 for a l lp that satisfy (10), and by (8) p ~ x k * - - W - - G ~ * ) ) = 0 k--I k=-i

Let us prove the converse assertion. By monotonicity of the utility function (1), the budget constraints (2) are satisfied

as equalities on the optimal solution, the aggregated financial balance (8) holds, and the optimal value of problem (9)-(10) is

zero. Thus, the material balances (7) also hold.

The competitive equilibrium x*, y , p is thus actually a Nash equilibrium in the corresponding (m + l + 1)-player

game.

In what follows, we use some assumptions:

(i) the utility functions Uk(xtc) are convex and continuous on Qk; n

(ii) the sets Qk are closed and convex, 0 E Qk c_ R+"

(iii) the production functions gO(yi) are concave, i = 1 . . . . . m, j = 1 . . . . . n;

(iv) the sets Y/, i = 1 . . . . . l, are convex compacta, Yi C R+n;

(v) there exist production activity vectors Yi ~- Yi, i = 1 . . . . . m, such that Wj + G/,y) > 0 for all commodities j =

1 . . . . . n .

The case of nonlinear functions g~{Yi) (instead of the traditional case g~(Yi) = Yi) is important for decomposition schemes

in land-use models (see, e.g., [13]) and also for the sequential joint optimization method considered below.

If the utility functions Uk(.), k = 1 . . . . . m, are positive homogeneous and the income functions lk(y, p) := tk, k =

1 . . . . . m, are constant, then the general equilibrium model reduces to an optimization model (see Eisenberg and Gale [2], Gale

[3], Eisenberg [4], Polterovich [5, 11]).

Def'mitioa 2.1. The nonnegative function U(x), x E Q, is called positive homogeneous of degree/3 on the cone Q E

ti n if for every x E Q and r > 0 we have U(rx) = raU(x).

The following positive homogeneous functions are often used in economic theory:

U(x) = x~l x ... x xff,,, ~ ~2 = 1, 0 ___ /7 2 _< 1 (Cobb-Douglas function); j= - I

U(x) = min {x 1 / a 1 . . . . . x n / an}, aj >_ 0 (Leontief function)" I " ~ i < n

U(x) " - ~ cixi, c i >__ 0 (linear function).

T H E O R E M 2.1. In addition to (i)-(v) we also make the following assumptions:

(vi) the function U k is positive homogeneous of degree ~7 k and is nonnegative on Qk, where Qk is a cone with its vertex

at the origin that contains a vector x k' such that Utc(xk') > 0, k = 1 . . . . . m;

(vii) the income function Ik(y, p) = t k, is constant, k = 1 . . . . . m.

Then the vectors x , y , and p constitute an equilibrium if and only if the vectors xg , k = i , . . . . m, Yk , i = 1 . . . . .

l, are solutions of the following aggregated optimization problem

m tk F t (x , y ) = ~ ~--s In Uk(xk)--, max,

k = l x , y

(11)

m

E Xk <~ W + G (y), (12) k = l

(13)

and p* is the vector of optimal Lagrange multipliers corresponding to inequalities (12).

This proposition generalizes the result of Polterovich [5] to the case of several nonlinear production functions gi and

is proved along the same lines. Particular cases of this theorem have been proved by Eisenberg and Gale [2], Gale [3], and

Eisenberg [4]. The proof of the theorem relies on the following lemma, which is also necessary in what follows.

856

LEMMA 2.2 (Polterovich [5] see also [10]). Assume that the function 9fix) is concave and positive homogeneous of

degree/3 > 0, the set Q is a closed convex cone with 0 E Q, the number t > 0, and p is a vector parameter. Then in the

optimal solution of the optimization problem

t in/" (x) --, max, (14) x

px <~ t, (15)

x E Q. (16)

(if it exists) the constraint (15) is satisfied as equality and the optimal Lagrange multiplier corresponding to the (budget)

constraint (15) is 1.

Consider the parametric optimization problem (11)-(13). Denote its solution set by Z(t) and the set of optimal Lagrange

multipliers (corresponding to (12)) by P(t). Construct the multivalued map

l ( t ) = r E R m l ~ l c = p Wk+ ct k , k = l . . . . . m; i = 1

p E P (t), z = (x, y) E Z (t)}.

(17)

The next theorem links the equilibrium of model (1)-(6) with the fixed points of the map l(t).

T H E O R E M 2.2. Assume that (i)-(vi) hold.

If x*, y , p constitute an equilibrium of the economic model (1)-(6), then the equilibrium income vector

iii , I11

t = t k = p + akigi(y , k = 1 . . . . . m. (18) i - 1

is a fixed point of l(t).

If t* is a fixed point of l(t) , i.e., t* E l(t*), then there exist z* = (x*, y*) E Z(t*) and p* ~. P(t*) that form an

equilibrium of the original model (1)-(6).

Proof. Let x ~ y , p be an equilibrium of (1)-(6). Construct t* from (18). Clearly, x*, , y p is also an equilibrium

of the model (1)-(6) given the income functions l(y, p) = tk*, k = 1 . . . . . m. Consider the optimization problem (11)-(13) with lit * * * * ~t *

t = t . By Theorem 2.1, x , y , p are contained in the solutions of (11)-(13), i.e., z = (x , y*) E Z(t*) and p E P(t*).

Hence

t" ~ . l ( t ' ) m = ~ 1 ~ I ~ k = p w k + aug~(y~) , k = 1 . . . . . m;

i = 1

p ~ P (t "), ~ = (x, y) ~ z (t ")] . , /

Let us now prove the converse. From t* ~. l(t*) and the definition of l(t) it follows that there exist p* E P(t*) and z*

= (x*, y*) E Z(t*) such that

�9 -( t i~ = p wk, + a kfli(y , k = 1 . . . . . m. (19) i = 1

857

By Theorem 2.1. . r . p . x constitute an equilibrium of the original model (1)-(6), where t k replaces Ik(Y, p), k =

/ 1 . . . . . m. But by (19) the budget constraint (2) can be rewritten as px k <_ t~. = p (w k + ~ Otkigi(y i )). This implies that x k

i--I solves problem (1)-(3) for consumer k given p = p and y = y . Q.E.D.

3. C O B B - D O U G L A S UTILITY FUNCTIONS

Note that the aggregated function (11) in Theorem 2.1 is actually the logarithm of the C o b b - D o u g l a s function

" t//~,(x~). U (x) = I--I u /,

k=l

It is therefore natural to analyze the capabilities of the computational procedures associated with Theorem 2.1 under

the following assumption.

(viii) Assume that the consumer utility functions are also of Cobb-Douglas form"

Uk(xt) = x~ xx~=x.., xx~ ~', (20)

x/, = (x a . . . . . % , ) ;, O,

where 0 < 13ki _< I, ~ r = 1, k = 1 . . . . . m. 1=1

These functions are positive homogeneous of degree 1.

Consider problem (11)-(13) for consumer utility functions (20):

V*(t) = max ~ t k In (xti . . . . . x~: n ), x, Y k = I

(21)

m

k = l

(22)

x~>0, y E Y. (23)

LEMMA 3.1. Assume that (iii)-(v) hold. Then in problem (21)-(23) with t k > 0 the optimal production activity vector

y* is the solution of the problem

max ~ t~/q In( + (y)). y E Y j= k = l (24)

The optimal vector of Lagrange multipliers p* has the form

f m pj =

0

m E > 0,

k = l otherwise

] = 1 . . . . . n. (25)

858

The optimal consumption levels xk" ' k = 1 . . . . . m, are computed as follows:

. �9 , i f 2 t :~ . j > O, j = 1 . . . . . n, k = 1 . . . . . m. ( 2 6 )

x k = Pk k=L

0 otherwise t i t t i t

Proof. Denote by J0 the set of j such that ~ t~/3# = 0. and by J+ the set of j such that ~ tk[3kj > 0. For j E J0 ~,-I k - I

we have t/,43k/ = 0. k = l . . . . . m. and by the necessary conditions of extremum pj = 0 and x~j = O, k = 1 . . . . . m. Take

t i t ,

j E J+ . i.e.. ~ tkt3ki > 0. Then for some L). we have tk/3k, j > 0. Note that-D,) are bounded. Denote by p = (Pl . . . . . P,,)

> 0 the vector of Lagrange multipliers corresponding to inequality (22). Then by the necessary condition of extremum

ni �9

Z (tl,/31,j/rl,) ,) - - PJ = 0 we have pj r O. Now for the case when t/,,, /~kj ;t~ O, k "-- l . . . . . m, given condition (v) the required a - I

assertion is obtained as follows:

m

+2 k=t j ~ J +

U "(t) = max rain tkln Uk(xk) - p x k - W - G (y) = y E Y , x>_O p ~ O k = l k = l

= max min ~ max < ,:On (xk/) - pixk/) + p (W + G (y)) = yEY p>_O k=l jE1+x,tj >0

= max rain kl . t :kJ - t : k j + p ( W + a (y)) = yE~Y p:>O k = l jEJ+ PJ

-max( : , , j . 0._o ~=, Tj + p j ( ~ +

2 tk.flk] ln[3kj + ~ (tk. In t k - t~:) = max ~~].t t~l:j In (Wj+ G.(y)) - k=l yEY j + k=l

- ~ 2 t In ~lq + ~ 2 tLflkj In ilk~ + 2 t/tin t/c" j ~ l + k = l k k = l j E J + k = l

The case when t3k7 = 0 or t k = 0 for some k, j is considered similarly" the result is obtained from the previous formula

by substituting 0 In 0 = 0. Q.E.D.

Consider the map l(t) for Cobb-Doug las utility functions.

L E M M A 3.2. With Cobb-Douglas utility functions (20) the multivalued map l(t) has the form

I (t) = {A (y)t I y E Y ( t ) } ,

m

where t = (t 1 . . . . . tin) r, Y(t) is the solution set of (24), and the elements of the matrix A(t) = {apq} p,q = t are

l

i =1 aoq(t) = Z Wj + G.:(y) fla.j"

/=1

(27)

(28)

859

Proof. By definition,

l l ( l ) = { t E R m Ir k = p ( w k + 2 akfli(yi)), k = 1 . . . . . m,

i = 1

p ~ t, (t) , z = (x, y) ~ z (t)},

where Z(t) and P(t) are solutions of (21)-(23). But by Lemma 3.1,

- tqflad, j = 1 . . . . . n, yff. Y(t) , e ( t )= p ~ R ~ S p j - W i+ G(y) q-Z

where Y(t) is the solution set of problem (24). Then for z = (z t . . . . . zp . . . . . zn) (7_ l(t) we have

7. 0 ---- 2 p] wp] + 2 Ctplgi1(Yi) --'-- IV. + G~y) tq wp] + E ctpggil(Yi) -- j--l i=1 ./=1 q=l ! i=t

1

w .+ 2 aptgii(Yi) m n Pl n i=l

= E 2 W.+ Gi(Y ) flq./ t = ~ a tq, q - I ]---1 ! q--I

where y E Y(t). Q.E.D. Remark. Note that the matrix A(y) in (27) has a special property: the elements in each column of A(y) sum to 1.

Indeed,

l

m m n P l

~ a 0 q = 2 2 l~.+Gj(y) -flq./= p=l p=l /=l m l m

n E woi + E gii(Yi)E aoi n - 2 : J - ' o=, w + %(y) = 2 = I.

./=t J ]=l

4. THE CASE W I T H O U T STRICT GROSS SUBSTrrUTABII~ITY

Y = y .

Let us compute the excess demand function assuming Cobb-Douglas utility functions and a given production program

Let p be the vector of prices. Each consumer k solves the problem

x ~ x . . . x x { ~ "-" max, x k

PXk <- P wk akfli(Y'i) = P-~k' xk >~ O,

l

where w k = w k + ~_, Otkigi(Yi). i - l

By Lemma 2.2 this problem is equivalent to the problem rl rt tl

(P ~k) ~ flkj In x/q - ~ , pjx/q + ~ pfff /q --, max. > 0 j = l j = l f = l xk t . . . . . x kn -

860

Its solution is

I xk4 = ~.. (p ~t)flt/, j = 1 . . . . . n.

Thus, the excess demand function t ip) = ~(p)} has the following components"

m

PJk=l 1

Let us check the gross substitutability condition. We have

04('19)0p i = ~jJk~l'Wk~k] >~

i.e., t ip) is a monotone map. If, for instance, %: > 0 and 3k = (3kt . . . . . 3in) > 0 for all k, then Ofi(p)/Opi > 0 for all i, j ,

i ~ j . Thus, the strict gross substitutability condition is satisfied and the map t ip) is strictly monotone. In this case the

/71

equilibrium in an (exchange) economy can be found by Walras's groping process. But if for some pair (i, j) we have ~ ~'ki3kj k=i

= 0, then Of/p)/,gpi = 0, and convergence of the Walrasian process is not guaranteed.

Consider the following simple numerical example.

Example . Consider an exchange economy with two consumers and two types of commodities.

The first consumer has the utility function Ul(x l) = x12 and an initial stock w I = (1, 1), i.e., the first consumer solves

the problem

xl 2 '" max, XII' XI2

PlXll + P2X12 ~< Pl -t- P2' Xll ' X12 >~ O.

/ The second consumer has the utility function U2(x 2) = ~/Xz~X =

consumer solves the problem

and an initial stock w 2 = (1, 0), i .e., the second

q X21 X22 -" max, x~. ~,,

Pl X21 + P2X22 < PI' X21, X22 ;a O,

llt * This economy has the following equilibria: p* = (0, 1), x 1 = (xll , 1), x 2 = (x21, 0) where x11, x21 a re arbitrary,

b u t 0 _<xll + x 2 1 _< 2.

The excess demand functions are

1

) I2Co/= 72 + p2 - 1.

Thus, Ofl (p)lOP2 = 0 and the condition of strict gross substitutability does not hold. In this case, the classical Walrasian

process of finding the equilibrium prices dp/dr = tip) does not converge in the sense that its first component a priori goes to

- c o . We will show that the sequential joint maximization method overcomes this difficulty.

861

5. SEQUENTIAL JOINT MAX~4IZATION METHOD

5.1. Sequential Joint MaximiTation

Method in the Price Space

The sequential joint maximization method (Rutherford [ 1, 14]) in application to model (1)-(6) consists of the following steps.

Step 0. Initialize price vector p0 > 0, ~ pjO = 1 and production vector y0 E Y. Set iteration index s = 1. ./zl

Step 1. Given are the price vector ps-1 and the production vector ys-1 E Y. Compute the weights t / = p S - l ( w t

+ ~ ~kigi(,yiS-1)). j--!

Step 2. Find the equilibrium x s, yS, pS of model (1)-(6) in which I t have been replaced with tk s. Under the conditions

of Theorem 2.1 this is equivalent to solving the optimization problem (11)-(13) in which t k have been replaced with tk s.

Step 3. Set s := s + 1 and go to step 1.

We will show that this method fits into the general scheme of [8, 15] for solving variational inequalities.

Consider a parametric equilibrium model of the form (1)-(6) in which inequalities (2) have been replaced with the inequalities

l

PXk ~ lk = Ik(Y' q) = qwk + q Z a k A ( Y i ) ' i=l

(29)

n n ~ and the objective function in (5) has been replaced with qgi(Yi), where the vectors p , q E S+ = {z E R + I zi = 1 } are

i ~ l

external fixed parameters. Denote by xk*(p, q, y) the solution of problem (1)-(3) and by Yi (q) the solution of problem (5), (6),

and form the parametric excess demand vector

m 4i

h (p, q) = ~2 x kCo, q, y *(q)) - W - G (y "(q)) k=l

(30)

Assume that the set Qk in (3) satisfies condition (vi). LEMMA 5.1. If the utility functions Uk(xk), k = 1 . . . . . m, are strictly monotone increasing in all arguments, then the

function h(p, p) satisfies the Walras law:

p h ( p , p ) = O V p E S n +" (31)

Proof. If p = q, the inequalities (29) reduce to (2) and are satisfied as equalities on the optimal solution of problem

(1)-(3). Summing the budget equalities over k from 1 to m, we obtain (31). Q.E.D.

LEMMA 5.2. The price vector p E S+ is an equilibrium price vector for the model (1)-(6) with strictly monotone

increasing utility functions Uk(xt), k = 1 . . . . . m if and only if

h(p p ~<0. (32)

Proof. If x*, y , p are an equilibrium of model (1)-(6), then (7) and thus (32) hold. Conversely, assume that p* E n �9 �9 �9

S+, satisfy (32). This means that there exist solutions Yi , i = 1 . . . . . l, of problems (5), (6) with p = p and solutions x k ,

k = 1 . . . . . m, of problems (1)-(3) where in the right-hand side of (2) p, y are replaced with p*, y* = (Yl . . . . . Yl ). Then (32)

implies (7). The Walras law (31) with p = p implies (8). Hence x = (x I . . . . x m y p are an equilibrium of the model

(1)-(6). Q.E.D.

LEMMA 5.3. The price vector p E S n + is an equilibrium price vector for the model (1)-(6) with strictly monotone

increasing utility functions Uk(Xk), k = 1 . . . . . m, if and only if the vector p satisfies the following variational inequality:

862

( p - p * ) . h ( p ~ p*)~<O V p E S n (33)

Proof. First note that by the Walras law (31) the variational inequality (33) is equivalent to the inequalities

* *) rl ph(p p <~ 0 V p E S +. (34)

p* Now assume that some vector (together with x* E Q, y E Y) is an equilibrium of the model (1)-(6), i.e., satisfies

condition (32). Then clearly (34) holds for all p E s+n. Conversely, assume that (34) holds for all p E s+n. Then, taking

p = (0 . . . . . 0, 1, 0 . . . . . . 0) with 1 in position i, we obtain that h:{p*, p*) <_ O, j = 1 . . . . . n. Q.E.D. ?I

Dafermos [8] has studied the following method of finding the solution p* E S+ of the variational inequality

Co - p ")t4 Co ") -< 0 v p ~ s ~ (35)

n n . _ , R n where S+ n is a convex compactum in ti n, H(p) = h(p, p), the map S+ x S+ is continuously differentiable in its n

arguments, and the matrix Vph(p, q) is positive definite for all p, q E S+.

In terms of the function h(p, q) this method has the following form:

initialize q0, set k = 1; n

given qk, find pk as the solution of the variational inequality (p -- qk).hfpk, qk) < 0 Yp E S+"

set qk = p~:, change k to k + 1, and return to the preceding step.

If the sequence pk converges to some p*, then this p* is a solution of the variational inequality (35). The next theorem

provides the conditions when pk is a fundamental sequence. Consider the symmetric matrix H(p, q) = 1/2(Vgh(p, q) + VphT(p,

q)), where Vphpr(p, q) is the transpose of the matrix Vqh(p, q). Let the matrix H 1/2 be such that H 1/2 x H 1/2 = H, and H-1/2

is the inverse of H 1/2.

T H E O R E M 5.1 (Dafermos [8]). Assume that

1 1

I l l V H - g C o ~ ~ )Vqh Co 2 Z )VpH - - 3 3) P ,q ,q 2(p ,q Ill < 1 (36)

n for all pX, ql, p2, q2, p3, q3 E S+, where il I" t II is the standard matrix norm. Then pk is a fundamental sequence in S+ n.

A necessary condition for (36) is strict monotonicity of the map H(p) (Dafermos [8]).

5.2. Sequential Joint MaximiTation Method in the Income Space

The sequential joint maximization method (Rutherford [ 1, 14]) may be regarded as an attempt to solve the inclusion

t E I(t) by the sequence of vectors t s = (tl s, ..., tins), s = O, 1, " m

t o is an arbitrary nonnegative vector, ~ tk ~ = 1 k=l

7 s +l ~ I (t s), (37)

t s + l - . (1 - -GS)I .S + ( 7 " : s + l , (38)

where the map I(t) is defined in (17) and the parameters a s > 0 play the role of stepping multipliers. To construct ~s +t, we

l

, = ~ ) } k = I" have to find the solutionxS, y s pS ofthe aggregated problem (11)-(13) with t = t s and set t s+l {pS(w~ + ~ s n i--1

If a s = 1, then the (full-step) process has the form

t s+l E i ( t s ) . (39)

863

This process is similar to the simple iteration method for solving the equation t = l(t). However in this case the norm

of the Jacobian of l(t) (or the norm of the matrix A in (27), (41)) may be I, which complicates the application of the standard

stability theory to investigate the convergence of the method. Numerical experiments show that the sequence t s (with some

value of the parameter a, 0 < a ___ a s < 1) converges to the fixed points of the map I(t) (equilibrium incomes) (see Rutherford

[1, 14], Manne [6], Manne and Rutherford [7]). The corresponding equilibrium of the model (1)-(6) may be obtained as the

solution of the aggregated optimization problem (11)-(13).

Let us investigate the convergence of this method in the particular case when the utility functions are Cobb-Douglas .

The method takes the form

m

t o ;, 0, ~ t o k = l ; k - I

(4o)

t T M =((1 - a s ) E + a s A ( Y S ) ) t s, yS E y ( t S ) , s = 0 , 1 . . . . . (41)

where E is the diagonal identity matrix and the matrix A(y) is defined in (28).

m

Note that starting in the unit simplex, i.e., ~ t ~ the method always remains inside this simplex, specifically k = i

m

t2--1, because the elements in the columns of A(y) sum to 1. k=!

If the set Y(t) consists of a single element y(t) , % = ~r, then (41) reduces to the following nonhomogeneous Markov

process"

t s +1 = ~ ( t s ) t s, "~(t s) = (1 - a ) E + aA ( y ( t S ) ) , s = O, 1 . . . . . (42)

Here the stochastic matrices A( t s) depend on vectors (probability distributions) t s. Thus, in some cases the convergence

of the sequential joint optimization method (for Cobb-Douglas utility functions) follows from known results on convergence

of Markov chains. In this article, we only note some direct results.

Proposition 5.1. If the functions Gj{y) are strictly concave and monotone increasing, a s >__ a > 0, then the

subsequences {tSl}, such that lim Itst +1 - - tS~l = 0, converge to an equilibrium.

Proof. Note that the solution Y(t) of problem (24) with strictly concave and increasing functions Gj(y) is unique and

depends continuously on t; the same is also true for l(t). Assume that tSl ~ t* and I tst +1 - - tstl --, O, s --, oo. Then ts t +1 --~ t*

and from ts~ +1 = (1 - - ast)Pt + as I(tS~) it follows that t* = l(t*). By Theorem 2.2, t* is the equilibrium income vector of the

original model (I)-(6). Q.E.D.

This proposition shows how to choose subsequences of points that converge to an equilibrium. But, in general, a

subsequence tst, that satisfies the conditions of this proposition does not necessarily exist.

In the following three cases a s = a > 0 and the matrices A(y) , y E Y(t), are independent of y. Then the process (40),

(41) is a standard homogeneous Markov chain with well-known conditions of convergence to a stable distribution (see, e.g.,

Gantmakher [ 16]).

Case 1. Consider a pure exchange economy, i.e., gi (Yi) = O, i = 1 . . . . . I. Then A(y), y E Y(tS), is a constant matrix

and has the form

�9 W . r ' 2 ] "'" j = l l j = l

1=1 1 ]=1 1

W.' -m] j = l

] = 1 1 . . . . . . o . . . . .

j=l J j=l J j=1 J

864

Case 2. If the production levels are given, i.e., Y consists of a unique vector, then A(yS) , yS E Y, is a constant matrix

and has the form (28).

Case 3. Assume that

ctoi = ct o, i = 1 , . . . , l;

W > 0 andGj(y) = 0 for j = 1 . n "

W. = 0 and G.(y ') > 0 for j = n ' + 1 .... n and some v' E r 1

(in particular, we may have Wj = 0 for all j = 1 . . . . . n), i.e., each commodity in the economy is either produced or not

produced (simply available in stock). Then the matrix A(y) is also constant and has the form . .

+ a, 2 ,j .-- Z j = l 1 j = n ' + l j = i 1 j = n ' + l

a ~ . . . . . . . . . . .

rl ' w . rl t l ' w . tl

+,,,. . . -

j = l 1 j = n ' + l j = l J j = n ' + l

L E M M A 5.4. Assume that one of the cases 1, 2, or 3 holds, a s = a > 0, and thus the matrix A ( y s) = A is constant.

If A has a positive row, then the matrix A = (1 -- a)E + aA is stable with maximum eigenvalue X a = 1. Therefore

limp "l= l imAS t ~ = t a, where t a is the unique eigenvector of the matrix A corresponding to X a = 1" A t a = t a. Then A t a =

t a, and by Theorem 2.2 t a is the equilibrium income vector.

Example (continuation of the example from Sec. 4). In this example, the matrix A is constant (as in case 1)"

a - -

m

It has a unique eigenvector t a = (1, 0) r, y~ (ta) k = 1, that corresponds to the maximum eigenvalue 1. The sequence k=l

m

t s+ l = A t s starting from any initial point t ~ y~ t~ converges to t a . k=!

6. C O N C L U D I N G R E M A R K S

We have investigated the convergence of the method of sequential joint maximization (of consumer utility functions)

that can be used to find solutions of the general equilibrium model. We have shown that even the particular case of

Cobb-Doug la s utility functions leads to new problems with convergence of nonhomogeneous Markov chains in which the

transition probability matrices depend on the current probability distribution of the chain. In particular, the method converges

for a pure exchange economy and for an economy in which every commodity is either produced or is not produced but is

available in a certain amount. Note that the convergence of the method is independent of the condition of strict gross

substitutability. Further convergence analysis requires a deeper analysis of the map l(t) and the matrices A(y) in (17), (27).

R E F E R E N C E S

T. Rutherford, "A sequential joint maximization algorithm for applied general equilibrium problems," Working paper,

University of Ontario (1992).

865

.

4.

5.

6.

'7

, ,

9. 10.

11. 12. 13.

14.

15. 16.

E. Eisenberg and D. Gale, "Consensus of subjective probabilities, the Pari-mutuel method, " Ann. Math. Stat., 30, No.

1, 150-170 (1959).

D. Gale, Theory of Linear Economic Models [Russian translation], IL, Moscow (1963). E. Eisenberg, "Aggregation of utility functions," Manag. Sci., 7, 337-350 (1961).

V. M. Polterovich, "Economic equilibrium and optimum," l~kon. Mat. Metody, 9, No. 5, 835-845 (1973).

A. S. Manne, "Greenhouse gas abatement -- toward Pareto-optimality in integrated assessments," Conf. on Market Instruments for International Environmental Policy, Stanford Univ., Nov. 29-Dec. 1 (1993).

A. S. Manne and T. F. Rutherford, "International trade, capital flows, and sectoral analysis: formulation and solution of intertemporal equilibrium models," in: W. W. Cooper and A. B. Winston (eds.), New Directions in Computational

Economics, Kluwer Academic Publishers (1994), pp. 191-205.

S. Dafermos, "An iterative scheme for variational inequalities," Math. Progr., 26, 40-47 (1983). S. A. Ashmanov, An Introduction to Mathematical Economics [in Russian], Nauka, Moscow (1984). M. M. Levin, V. L. Makarov, and A. M. Rubinov, Mathematical Models of Economic Interaction [in Russian],

Fizmatlit, Moscow (1993).

V. M. Polter0vich, Economic Equilibrium and Business Management [in Russian], Nauka, Moscow (1990). W. I. Zangwill and C. B. Garcia, Pathways to Solutions, Fixed Points and Equilibria, Prentice-Hall (1981). G. Fischer, Yu. Ermoliev, M. A. Keyzer, and C. Rosenzweig, "Simulating the socio-economic and biogeophysical

driving forces of land-use and land-cover change: The IIASA land-use change model," Working paper WP-96-O10, IIASA, Laxenburg, Austria (1996).

T. Rutherford, "Sequential joint maximization," Working paper, Univ. of Colorado (1995). S. Dafermos, "Exchange price equilibria and variational inequalities," Math. Progr., 40, 391-402 (1990). F. R. Gantmakher, Matrix Theory [in Russian], 4th edn., Nauka, Moscow (1988).

866