Computing equilibria with increasing returns

arc supplied

by

AQPCNI,rnd Hahn ( 1971 I]. To awid

unnecessarily complicating the proof, (4)

has kwt aw~rnc4. In addition to demanding factors and goods. the household sector supplies fk~ws. that

Tk

P$,(P.

suppI? r~f the ith factor will be denoted bv S,(P. S ). It is assumed the same conditions as E:(P.S) [that is. (11, (2) and

.ci, satisfks

I?!]. and that S,tP.Sb satisfies the same condition as @(P.S) ;I~wnptions arc quite standard. One fin;ll assumption abut household sector behavior ,warmcd that

[see @I)]. These is needed. It is

(5) hrvc A, II’_ $1 i3 the prcsGt of the ith industry. which cjf course depends on 15 I is simpl\- the budget constrain for the household SKI the

?&hand

side is total expenditure.

and the right-hand

side is

;e-wmcd th;rt thcrc arc !I - 111induatrics. each of I$ hich produces one of rncd only with behavior at the Iewl of the i\ twt specified. Wc do not spccif~ whether ;w m0nc~p&tic. monopolistically compctitke. re!@itcd monoWC, M wmpctitisc, 4nc’r’ an! of these structures rnq be compatible with is 3,wmed to possess input demand functions !,,i( P. S) for r~htcc,. 7 hat 15. ishen faced with prices P and supplies S. industr! j inc ho\{ much of each commodity. including I (f~+‘~~ and intcrmcdiatc gwds. it ~~141~s to purchase. This may. but k the rc4t 0f an optimizing process. One element of S is of ustr! j is -required’ to pro&w. it ic not

required that the @ minimize wider the vector

costs, the!

must clearI)

the production wctor for thr: jth indu$tr>. where inputs as USUA q&s yuantitics and outputs b! positive ones. For I” to h it mu%t belong to 1-i the production

set for the jth industry. We ZMUIIC that the input demand functions are such that this is the case. ~rrxJuc~ron u’t\ for the II - IYI industries. )-“‘- ’ to I’“. imply a ~WJ *

fix the whole ewnomy.

1: 1I is assumed that

positive mrmer. This assumption rules out the p;Abiht> of producing infinite outputs t’rom finite inputs. or positive out!wts lwm /cw where

B is a hitc

inputs. Following Koopnwns Cockaigne’ assumption. It will

(_I%! ) it wit!

bc rcfwrcd to as the‘ %o Land 01’

be convenient to denote the expenditure of the jth industr> on the

ith commodity b>

&,i(P*S)=PiDf(P, S). The industry expenditure and demand functions arc assumed to butt+ the same conditions as the housc3l~otdsector functions. ( I I through (4). in the Watrasian modct. each firm determines it3 outputh .;tnd input3 ;ts functions of the prices it Paces.These functions wit! obvioust~ tend to be \c’r! ill-behaved if there are increasing returns to scale. In the nwdct of thiy pqw. by contrast, each industry determines the inputs it wit! dem;rn~! ;u~d the price it will ask for its product as functions of input prices. the amount ir must produce and perhaps (if there are extcrnatitics) the amounts produced h> other industries. Firms pricing bchaGor is rqwcwntcd by tr,&irrg /“‘iC*C functions. Each industry j possesses an ;tsking price function ..I ,t I’. S 1. N hich is assumed to satisfy certain conditions. The ashing price ma! !w thought 01 as the minimum price which industry j wuid ch;qe and Ati hc Ctlin~ to supply Sj. or as the priw it would charge for S, 3 Gc’n the ni;irhr‘t 3triiww 01’ the industry.

Various c;~sesof ashing price functions will bc ~Iiwuwd

%i(P,s)=.-I,(f3)si-

i i

bctw.

E;(P.S)g(f.

(SI

1

(S) states that the asking price function for industry

j ih such t!l;tt

indwtr!

profits are new neg.;ltiw. This :tssumption wtih implicit in the discussion of the t~ouset~otd sector. hincc”ii profits could bc ncgtiw might be some wctors incomes. and hence

(P. S) at uilich t~ousdwld

wme hwwhold~

expenditures

would haw

c;u-hc~

tlwrc

nt’pt~w

could non be wntinuous

and

non-negative. \vit!j ttwc Many types of pricing beha\ior b> industric\ m-t: wnsistalt For example. if indzstr! i wrc :i assumptions about asking price functions. competitive industry with constant or dccreasins returns to MX!C. .-Ii mi~llt simply be marginal cost. An industry with incrcasin~ returns to scale CCW!~ profits I\ odd bc not set price qua! to n-mrginal cost. how\ a-. b~c;wc

rnrpw4bic. ttrrvb utwh thcac

N0wwr. such fun&w wwld \ iolate one or more a-king prke funclidw Ii;w to satisfy. For esampk. andO.in&la> A might set .& -= P,: but at WIIIC asking price functions ~‘ouid have tcl I idate (81. the be

non-nqiit

i w.

Hence

such

functions

c C~_BCJ rn which asking price functions

arc wc~uld

haw imposed upon them. in parCcular the cpf continuit>. This problem ma? welt arise when firms have s ac,.t as if the?- have monopoly

power only in one

aarkct for the good the; produce. there should be no 4% cn S&hi 41961 j. firms would simply set price to maximize &ing pnce \aouid be the profit-maximizing price or average [so that (8) i\ ~aMi~d]_ But if firms recognize that m wcral markets. including markets for goods

pawr

polistrc firma, the maximization

carmpiicad,

problems they face

and an easily have disconGnuous solutions [see n uWV~). The modci of this paper is not intended

Ei(P. S)=E4(P. S)+

Ej(fq S).

i j-m-

In equilibrium. this must bc equal to the wlite supplied. Thus for alI II commodities,

Ei(P.

I

IW

1

of the wnitiiodi~~

1h;t1 13

S)‘PiSi(f’* S).

I’$, = A;( Y. s )S,. Conditions (1 I ) and (12) are titxcswry for cquilihrium. but iIl3J ~ittkicnt Cjnlv if CI’CrF f’i and S, i3 posifiw. If a wmnlodil~ ha ;I /ctw ‘price. llia’r‘ must not bc excess demand for it. H~ncc:

If the amount supplied asking price. hiticc thcti positiw amount. Hence

Conditions

( 1 I ) through

3. The existence of equilibrium

0.1). A sintplicial shlirisiort of an (12’ - 1 )-simplex is a partitioning of the simplex so that each of the parts is also an (N - 1 )-simplex (or strh.sirrtpI~s), and so that each subsimplex has either no faces or one entire face of any dimension in common with any other subsimplex in the subdivision. We shall be concerned only with the regular subdivision of the standard simplex. The regular subdivision can be described by an integer scalar. D, the degree of the subdivision, which is the number of subsimplices along any one-dimensional face of the simplex. A very important property of the regular subdivision is that, if x1 and .Y’ belong to the same subsimplex, the ith co-ordinates of s’ and x2 cannot differ by more than l/D, for any i. For a proof of this proposition, and a much fuller discussion of simplicial subdivisions. see Kuhn ( 1968). A ~I=CIPLT krhellillg is the as:ipnment to each vertex in a simplicial subdivision of an integer label between 1 and IV, which satisfies the restriction that no vertex receives the label k if its kth co-ordinate is zero. Sperneis Lemma asserts that every properly labelled simplicial subdivision must contain at least one subsimplex whose N vertices carry all N labels, that is. a completely lahelled subsimplex. A proof of this lemma may be found in Tompkins ( 1964). We are now ready to prove the existence of equilibrium for the class of models described in section 2. The proof involves a number of steps, which it may be worthwhile to outline in advance. First. we will associate every vector (P,S) with a vector s belonging to a standard simplex of suitable dimension. and show that this accounts for every (P. S) vector which could possibly provide an equilibrium. Next, we will define a proper labelling rule which assigns a proper label to every point on the simplex. based on the excess demands and excess prices associated with that point. We will then call on Sperrker’s Lemma to show that a completely labelled subsimplex exists for any D, and that there exists a limit point of a sequence of completely labelled subsimplices as D tends to infinity. Finally, using the properties of the labelling rule and some continuity arguments, it will be shown that such a limit point must be an equilibrium, Let s denote any point in a standard (IV- 1 )-simplex. where N = 2n- III. We map from the simplex to (P. S ) as follows: up

l

to

(0. . . .,

Pi =

Si

for

Si__n+nr- -Q’*-Yi for

i=

l,...,n,

i=n$

l,..., IV,

(IS)

where Q” is a number larger than the largest sum of produced goods that the economy could conceivably produce at equilibrium (see below). Observe that ( 15, is a continuous mapping which associates every point in a standard (?#I- ill - 1)-simplex with a vector of 11prices and II - III supplies.

Since all demand and supply functions were assumed homogeneous of degree zero, prices may be normalized to sum to 1 - cTY,1+ , .~i. Since free disposal was assumed, negative prices are impossible at equilibrium. Hence all potential equiilibriwm price vectlors are associaid with p&in& on the simplex. Provided that Q” is picked appropriately, this will also be true of all potential equilibrium supply vectors. Let Q denote the largest sum of all produced goods that the economy could conceivably produce. By the ‘No Land of Cockaigne’ assumption, Q is finite. We may therefore choose Q” as any number greater than Q. Given such a choice, (15) does not rule out any supply vector which could possibly be an equilibrium vector. Thus every vector (P, S) which could possibly be an equilibrium is associated with some point on the standard (N - 1 )-simplex, by (15). In order to define the labelling rule, we must first construct a vector function C’(x). with N elements. according to the following rule: S)-P,Si(Py

l((X)=Ei(P.

S)

~(-~)=Si-,*+,(Pi-.+“~-Ai-“+*(P,S))

for

i=l

for

i=n+

, . . .. 11,

l,...,lV.

m-j)

A ycry important property of this function is that i; I$(s)=O. i ;‘ 1

(1-n

Thib may be proved as follows. Using ( 16).

The first term on the right-hand side of (18) can be broken into two parts. household expenditure and expenditure on intermediate goods. Using (5), it can be written as . m

C i -- I

I)iSi+

~ h y m + 1

%,+

(1% i .r 1 h .- m * 1

Simiiarly, using the definition of profits. the last term on the right-hand side of ( 18) can he rewritten as

When (19) and (20) arc substituted into

t I@, t 17)

is verified.

The continuity of &(I’, S), PiSi(PqS) and Ai(P, S)Si ensures that I/;:(X) is conGnuous for all i and for all X, provided that -ri is positive for at least one of i= I to II (since otherw, x all prices would be zero). It may be convenient lo write k&x) as S&X), where g(x) is either (Di-Si) or Q”(fi_n+m -A i_wTm)+and hence is also continuous. I3y (15), every point _in an (iv- 1 )-simplex represents a vector (P, S). In particular. the vertices of any regular subdivision of an (N - 1 )-simplex represent such vectors. In order to make use of Sperner’s Lemma, we must associate every vertex of the subdivision with an integer label between 1 and N. TWOdifferent rules will be used. The lirst of these, rule L,, is the labelling rule which is really of interest. It is defined by L,(S)= k

if s, >O

and

I/;(.s)~ b(x)

for all

i,

with

the folIowAng. For i with -Xi>O (there must be at least cf:lluate I;(s). Give the label k if Vi(s) is the smallest of these. or if there is ii tic for the sma.llcst P&Y) and k is the lowest indcs ammg those tied. Rule L, is clearly a proper labelling rule. It also ensures that if s gets the label k. t;(s) must be non-positive. To prove this. suppose the contrary, that I;(X) were greater than i!ero. Then ( 17) would imply that there must be some i with Ii(s) less than zero. That can only occur when .Yjis positive [since the continuity of gj(s ) implies that L#)=O whenever Si=O]. so L, would give \-ertcx s the label j instead of the label k. Hence if s gets the label k. &(sjgk Unfortunately, it is not possible to use L, to label every vertex of the simplical subdivision. since at some vertices all prices are zero, so that it will not be possible to compute demands and supplies, and hence V(X). In order to get around this problem, we introduce a second labelling rule, LL, which is to be used whenever ~~_ + , Si ZQ’. where Q’ is chosen SO that Q
This rule simpiy say

one since z:

, K, = 1)

L,(_X)=k

k4

if S,~Si

if Xk=Si.

for

i=n+

l,..., N.

(22)

Like LI _L2 is clearly a proper labelling rule. Thus if we label the vertices of a

subdit ision of the (.V- 1 )-simplex by L, when xyz ,,1e I Si -c Q’ and by IA2 otherwise, we will get a properly labelled simplicial subdivision. For a regular subdivision of any degree D, we may label the vertices bj- L1 and Lz as explained ;tibove. Sperner’s Lemma ensures that there wilt be at least one completely labelled subsimplex. Take any point interior to one of these, and call that point x,,. If D is allowed to increase without limit, an infinite sequence of points xD will be generated. &iri’tise the simplex is a compact set, there must exist a subsequence of this seiuence which converges to a single point, _Y *. By (15), x* defines a vector of prices and supplies (P*,S*). We now show that (P*, S*) is an equilibrium. Since s* is the limit of a sequence of points, all of which are interior to completely labelled subsimplices whose size goes to zero as D goes to infinity. s’ must be arbitrarily close to the vertices of a completely labelled subsimplex. We must first show that all of these vertices must have been labelled by I., . Suppose the contrary. that at some vertex x’. xy ,,,. I Sit.\-’)zQ’. Since Q’ is greater than the greatest feasible sum of supplies, at Y’ there must be excess demand for at least one factor. say the kth : t lws gk (A-’ I > (1 for at least c)nc CI between t and m C$ntinuity implies th;it ;rt all other ~rfice~ s” arbitrarily closes IO s*. g&’ )>O as iyell. Hence either l&C’) >I) 06 +O. ;md in both case\ .I-”is ineligible for the label k. Thus no wrtcv w;1r s* can h;l\e the label A. \4lich is a contradiction. since s* must t-w arhitruril~ close to vertices with all labels. It follows that A-*must be arbitrarily close to the icrtices of a subsimplex completely labelled by f,, , that is. to .I’ points vhre I ii.\- )sO for all i. By the continuity of V(s). this implies that I,;(.\-*)sO for all i, and by (171 that is only possible if P$~*)=O for 41. By (16) it follows immediately that E&P*, S*)=P”&(P*.S*)

for

i= 1,. ..,]I.

which is equilibrium condition (1 l), and that for

i=ru+

l,....rz.

which is equilibrium condition (12). If .uF is positive, l{(.u*)=O implies Ki(.~*)=0. Suppose instead that sT=O. Since .I-* must be arbitrarily close to vertices with all labels, it must be arbitrarily close to a point s’ with xi ~0 and 2,;(s’)~O. which implies that Ri(uY’)SO~1s well. By the continuity of gi(s). gi(_x*) must be non-positive 4~. Thus we have established that gi(S*) 5 0 for all i. By ( 16) it then follows that Di(~*,S*)iSi(P*,S*)

’ for

i= l....,~l,

which is equilibrium condition ( 13), and that Pr&4,-(P*,S*)

for

which is equilibrium condition This completes the proof.

i=nz+l,...,rz,

( 14).

4, The computation of equilibria There exist algorithms which prove Sperner’s Lemma constructively and can be used to find completely labelled subsimplices. The first such algorithms were Developed by Scarf (1973). and numerous others have been simple and effective technique is the developed since. A particularly Sandwich Method of Kuhn and MacKinnon (1975). For sufficiently large D, a completely labelled subsimplex provides a good approximation to a zero of whatever continuous function is being used to label the subdivision; the sense of the approximation is discussed in the two works just cited. This means that the labelling rules of the previous section, when combined with an appropriate algorithm such as the Sandwich Method. provide a straiphtforward technique for the computation of equilibria in models \Gth increasing returns to scale. This technique has been applied to several fairly small models to see IUN it performs. One of these models is the following. There are two factors and four goods. so that N = 10. The supplies of the two factors are 100.000 and 50.000 respectively. There is only one household. whose income, I. is the total value of factors supplied plus profits from the four industries. The household sector demand function for each of the six commodities has the form (23) which is simply a constant used were 111 =

1.0.

(ij = 2.0.

112

share demand function. The parameter

=o.o.

Li5= 2.0,

uj =

values

1.0.

II, = 1.5.

Each of the four industries has a C.E.S. cost function of the form 124)

The parameters of this function are 3~~.the returns to scale parameter, which is equal to one if there are constant returns and less than one if tlJerg aGe increasing returns, 7& a technology parameter, yik, the &are pafmeters (which sum to unity over k), and pi, the substitution parameter, which is equal to one minus the elasticity of substitution. The cost function (24) is dual to a C.E.S. production function, but the parameters have different interpretations. Input demand functions are obtained from (24) by the use of Shephard’s Lemma; the demand for the kth input is the derivative of (24) with respect to f,, the price of the kth input. The parameter values used were

5t.q= 0.95.

z?A=0.98,

35 =0.96.

%(,= 1.O,

7-q=O.&

-l; = 0.7.

7:; =0.6,

& =0.9.

p.3 =

-, 4 ih =

03.

=0.25,

1’4

=

-03.

/I(,

=O.l,

0.35

0.40

0.10

0.00

0.00

0.15

0.36

0.40

0.04

0.10

0.00

0.10

A,(P, S)=x,CJ+ A#,

f’s

S)=C,(

.~,,CP.S)=C,,

+o.iCJR

11.

--I

’

(25)

l.O+O.OOl, sy )‘S,.

s,,.

Thus industry 3 charges ten percent more than average cost, industry 4 charges marginal cost plus ten percent of the cost of producing one unit of output, industry 5 charges average cost times an increasing function of output, and industry 6 simply charges average cost (which means, since it has constant returns to scale, that it behaves like a competitive iqdustry). Using the Sandwich Method with D equal to 239, i48,,iF150the following solution was obtained:

P, = 1.2777948.

D, = s, = 1001NM).~!2),

\4lcrs .\I is the number of %,.‘sequal to - 1 (so that Z sums to xro). Using these labelling r&s. the Vector Sandwich Method sol\:~~l the abow probicm in approsimatel~ ’ 23 seconds of CPU time, requiring 294 labcllings. Thi3 solution was indistinglIist~~bie from the OIIC prcsentcd Awvc. to the number of digits gilen thele. Thus wctor labclling algorithms appartA> prwide a substantially nwrc efficient way to compute cquiiibria for modA of this w-t than do integer tabclting algorithms.’

5. Possible ek;tcnsions

l

16

non-increasing returns to scale can be dealt with in the usual way (without putting supplies on the simplex). 6. Conclusion This paper has proved the existence of equilibrium for a class of genera1 equilibrium models with increasing returns to scale which are likely to be of interest for applied analysis. It has also shown how recently developed techniques for computing equilibria may be applied to models of this class. These results should make it easier to apply genera1 equiIibtium analysis to a wide range of real-world problems for which increasinsg returns cannot realistically be assumed away.

Arrow. KJ. and F.H. Hahn, 1971, General competitiw

analysis (Oliver and Boyd, Edinburgh).

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J,.

1977. ikncral

nir~nor,c~l~sl~~quilihriuni

under

Economic Review 18,425 431. Td?mpkin$. C-B, 1964, Sperner‘s Icnm;t and wmc c\tcn&ws. combinatorial mathematics (Wilq. !+I$ York 1,

non-conve.uitics.

in: F.F. Heckenbach,

Intcrnatiwal cd., Applied