World Bank Reprint Series: Number 189

Roger D. Norton and Pasquale L. Scandizzo

Marxket EquilibriumComputationsin Activity Analysis Models

Reprinted with permission from Operations Research, vol. 29, no. 2 (March-April1981), pp. 243-62.

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Market Equilibrium Computations in ActivityAnalysis Models

ROGER D. NORTONUniversity of New Mexico, Albuquerque, New Mexico

PASQUALE L. SCANDIZZOThe World Bank, Washingto a, D.C.

(Received June 1979; accepted April 1980)

This paper shows how an approximate competitive market equilibrium maybe computed as the solution to a linear programming model, when productionpossibilities are described by activity analysis vectors and demand functionsare locally linear. Numerical examples show that the degree of approximationmay be expected to be very small in many cases. Prices and quantities ofgoods and resources and also incomes are explicit in the primal. (The definingcharacteristics of a market equilibrium are written as primal constraints.)Linearizations of inherently nonlinear expressions are attained by use of gridlinearizations and the associated interpolation variables.

THIS PAPER describes a simple procedure for computing competitivemarket equilibria by means of matheniatical programming models

with explicit representations of aggregate consumer behavior. Unlikeexisting procedures (see Ginsburgh and Waelbroeck [1975], and Manneet al. [1978]) which utilize sequences of linear programming solutions,this procedure requires only a single linear programming solution. It hasbeen designed for use with activity analysis models which contain mul-tiple production technologies for each good.

The major restrictions on the procedure are as follows. Consumerbehavior is specified via linear market demand functions. They may beregarded as local linearizations of nonlinear functions or, alternatively, asestimated market demand functions. In either case, the demand functionsare not necessarily integrable into meaningful utility functions for rep-resentative individuals. A set of numerical initial conditions is required,but they do not have to be market-clearing conditions. In building amodel of an actual economy, the data set normally offers starting valueswhich are appropriate in this respect. The initial demand functions mustbe such that their point elasticities, evaluated at the initial conditions,satisfy some requirements of consumer theory as discussed subsequently.Normally these requirements would be imposed in the estimation process.

243Operations Research 0030c364X/81/2902-0243 $01.25Vol. 29, No. 2, March-April 1981 i 1981 Operations Research Society of America

244 Norton and Scandizzo

Also, owing to the linearizations, the solution is approximate rather thanexact.

If departures from the initial conditions are not great, then the linearprogramming solution gives an extremely close approximation to theequilibrium, as the examples below illustrate. A measure of the degree ofapproxiniation error is given by the amount of slack in the householdbudget constraints, i.e., the amount by which expenditures understateincomes.

Since it makes use of mathematical programrming, with its inequalityconstraints, the procedure is capable of extension to cases where marketsare subject to multiple controls and interventions; therefore it appearsusable in the context of price-endogenous models for economic policyplanning. That was in fact the principal'motivation for development ofthe procedure. A companion study (Hong [1980]) reports the experiencein applying this procedure to large-scale models of a developing economy,in which institutional and other constraints to growth are represented. Inthe applied sense, perhaps the principal appeal of this approach is that itis capable of handling a large variety of economic specifications withoutrevising the solution algorithm. In each case, a linear programmingpackage is used.

In this paper we distinguish between goods and resources, and themodel's maximand is the difference between the valtie of expen(litures ongoods markets and the economic rents to fixed resources. If all goods andresources in the model, including the initial endowments, are regarded astradeable, the maximand may be interpreted as the market value of thesum of excess demands. Shubik (1977, p. 221) suggests a model in whichuse of this maximand guarantees the efficient outcome in prices andquantities, but he does not address computational issues. As in Shubik'scase, at the optimum the value of this model's maximand is zero.

In the present context, a competitive market equilibrium is defined(narrowly) as having the following properties: a) markets clear, b) priceequals marginal cost for all goods, and c) the value of expenditures doesnot exceed the value of resources. In addition to exhibiting these prop-erties, primal and dual consistency are required of the model.

The paper's exposition begins with a model stated in general nonlinearform, which is not necessarily computable. A proof is given in AppendixA that if a solution exists, it corresponds to the competitive equilibriuln.(In that general model there is no restriction on the shape of the demandfunctions.) In addition to a linear statement of the demand functions,creation of the linear programming version requires linearization of thehousehold budget constraints which are bilinear forms. To achieve this,change-of-variable transformations have been used. While the lineartransformations may appear awkvard initially, in practice they are easyto use,

Analysis Models 245

The structure of the paper is as follows: Section 1 presents the basicstructure of the model in nonlinear form, Section 2 describes the linearprogramming version, and Section 3 discusses some numerical examples.One appendix gives the aforementioned proof, and the other describes inmore detail the change-of-variable transformations required for the com-putable model.

1. THE BASIC NONLINEAR MODEL

The principal elements of any market equilibrium model are four: atechnology and producer behavior specification, including resource limi-tations; commodity balances for market clearing; a description of howincome is distributed; and representations of demand behavior. If thedemand functions are not homogeneous of degree zero, i.e., they are notnecessarily derived from individual or household utility maximization,then the household budget constraints will not be satisfied automatically.Therefore they must be added to the model explicitly. Included in theproducer behavior specification is an equation setting price equal tomarginal costs, to reflect the hypothesis of profit maximization. (Mark-up pricing could be specified instead by a simple modification of thatequation.)

The marginal-cost pricing defines output price from the supply side,and the demand functions do the same from the demand side. Equatingthese prices ensures that price equals marginal cost in the primal solution.(At the optimum, the corresponding dual values are proportional to theprimal values; the dual solution is discussed further in Section 3.)

The basic nonlinear model can be written as follows for n goods (indicesi, j), m resources (index k), v consumers (index h), and z productiontechnologies per good (index t):

Model A

Max Z- X,,h X,,h - Ek bkrk (1)

subject to

Eh Xi,h-Et q,,t s 0 commodity balances (2)

X,, d,,tq,, - bk resource constraints (3)

X.>- - P-c ;, = ala demand functions (4)

E, X,,hPi - yh C 0 household budget

constraints

Yh -Sk rkbh,k C 0 income distributiondefinition (6)

246 Norton and Scandizzo

pi-Ek di,trk C 0 marginal cost pricing (7)conditions

Xi,h, qit, pi, rk, yh :> 0. (8)

Definitions of Variables

pi = price of good i (i = 1, * , n)Xi,h = quantity consumed of good i by consumer group h(h = 1, .. , v)rk = marginal value product of the kth resource (k = 1, * * *, m)qi,t = quantity produced of good i under production technology

t (t = , *. * , z)Yh = income for consumer group h.

Parameters

bk = endowment of the kth resourcebh,k = endowment of the kth resource held by the hth household; Eh bh,h

= bk

di,k,t = amount of the kth resource utilized per unit of production of q,Sk,j,h = cross-price demand slopeFih = income demand slopeak,h = demand intercept.

A brief commentary on the equations follows:a) The objective function is defined as the excess of expenditures over

factor income. At the optimum, its value is zero. (See Appendix A for aproof.)

b) The commodity balances allow for surplus production. Theirshadow prices Ti turn out to be proportional to the corresponding p, atthe optimum.

c) For the resource constraints, the shadow price Ak equals the primalrate-of-return variable rk.

d) The income distribution constraint in effect defines ari asset distri-bution. It says that expenditure outlays yh for consumer group h cannotexceed the group's factor income flowing from its asset ownership, whichis defined as bh,k. In models with only one aggregate consumer group, thisconstraint is unnecessary.

e) Marginal-cost pricing is a fundamental attribute of a competitivemarket. It will hold as an equality only for those technologies in theoptimal basis. For the other technologies, factor costs will exceed theprice. In his "simple general equilibrium models," Jones (1965) alsorequires an explicit statement of the marginal-cost pricing relationship.

From these remarks, it may be seen that the primal equations describethe conditions which characterize a market equilibrium. One test ofwhether a solution corresponds to an equilibrium, in fact, is to check theprimal equations to see if there is any slack. If all are binding (except for

Analysis Models 247

(7) in the case of nonoptimal technologies), then the solution representsan equilibrium position.

As stated, Model A contains nv + nz + n + v + m variables and nv +nz + n + zv + m constraints. However, not all constraints will hold asequalities at optimality. In particular, it is likely that only n of the nzconstraints (7) will hold as equalities. If that is the case, then for there tobe more than one feasible solution, we must have nz > n + v. In practice,this requirement is almost always satisfied with room to spare.

Among other authors' formulations, the ones which are most closelyrelated to this one are those of Shubik, Yaron (1967) and Plessner (1957).In Yaron's case, however, income was predetermined, so his model waseffectively a partial equilibrium model for a given period, albeit updatedfor lagged income changes. Also, he did not include Equations (4), (5)and (6). Plessner's model also is close to ours in spirit. The essentialdifference is that income does not enter Plessner's consumer demandfunction, which it must if we are to depart from partial equilibriumformulations. Also, as a consequence, Equations (5) and (6) are notincluded in his system.

In one sense, the methods presented in this paper can be viewed as ageneralization of the Samuelson (1952) and Takayama-Judge (1971)partial-equilibrium optimization models. Under partial equilibrium as-sumptions (approximately fixed income levels), those authors define anobjective function which is a function of the quantity demanded andwhose maximum is attained at the market-equilibrium quantity. In thepresent paper the models are constructed so that the own-price responsivedemand functions are shifted as incomes and other prices change, andthe maximand now is a function -of incomes. Rental incomes to fixedresources are represented as primal (as well as dual) variables, and theirmagnitudes contribute to defining the location of the demand fuinetionsin the space of own prices and quantities. Another important featurewhich is not present in the partial equilibrium models is the explicitstatement of the consumer budget constraints and marginal cost pricingconditions (perfect competition) on the supply side of the market.

Linearizations already are used in Model A and its only nonlinearity,apart from the objective function, is in Equation (5), the householdbudget constraint. It would seem a simple matter to approximate thisequation linearly, thereby obtaining immediately a linear programmingmodel. A direct first-order approximation to (5) was tried with numbers,and the optimization procedure exploited the error in the approximationwhich increased as the solution diverged from the equilibrium so that,effectively, equation (5) did not even hold approximately. The resultwhich emerged was very far from the correct equilibrium. Therefore it isnecessary to utilize a better linearization, one which gives a close approx-imation to (5) over a large domain around the equilibrium values.

248 Norton and Scandizzo

An improved approximation may be derived by techniques of gridlinearization, which require prior specification of the relevant range ofvalues and use of variable interpolation weights on the grid points. Sucha procedure is utilized for the LP . --)del which follows. The interpolationweights are variables (wMA,) and their values are jointly constrained by aset of convex combination constraints. Further details of the grid linear-ization procedure are given in Appendix B.

2. THE LINEAR PROGRAMMING MODEL

Although in linear programming format the model looks very differeintthan Model A above, it should be borne in mind that it is the same modelsubject to the grid linearization needed to be able to handle Equation (5).As before the maximand has two components: the positive one consistsof the current value of expenditure on final goods, and the negative oneis a measure of rental income. These two magnitudes again turn out to beequal in absolute value in the optimal solution. When intermediate goodsare introduced, the maximand remains defined as the value of expenditureby those economic agents whose consumption behavior is described bydemand functions, less total rent to fixed factors of production.

In this simplest case there are n goods, m resoLurces, et consumers(households), and z production technologies per good. A new subscript sis introduced to denote segments of the demand functions.

The Basic Linear Model: Model B

Max Z A Pt,h,sWt,h,s - Ek bkrk (9)

S.t. Zh,,h,st,hs s - , q,,, < 0 commodity balances [7r,] (10)

,t dh,,tq1,t 5 bi resouLrce constraints [Ak] (11)

-Y0 Es OtAsWtAs demand functions [a,h]

+ Ej 7tq,j.hPlj + E,,hyhYh (12)= -1

'Ehl P A -h y< 0 household budget [cph] (13)constraints

Y > - k rkbh,k C 0 income distribution [8h] (14)definition

Pt - Sk d,,k,,rk C 0 marginal-cost pricing [a,.,] (15)>3s W - C,OYi,Yh convex combinationi

- q,q, .1phj)P,P constraints (16)= + in,q,1,

w,,h,,., q,.j, pt, yl,, rk 2! O. (17)

Analysis Models 249

Definitions of Additional Variables and Parameters

The symbols in brackets denote the dual variables;

Wi)i s = segment choice variable on demand function, good i,household h, segment s;

pi,h,s = parameter denoting expenditure by household h ongood i at segment s of the initial demand function(the price is invariant under demand curve rota-tion)-see Appendix B;

Oi,h,s = parameter denoting quantity of good i consumed atsegment s of the initial demand function;

(fi, i,,t,, Yh) = (l/pi,o, l/x,,,,o, l/yh.o), reciprocals of initial values;d,f,k = amount of the kth resource utilized per unit of pro-

duction of q,t;71i,j,h = price elasticity of good i with respect to the price of

good j;Ei,h = income elasticity of demand, good i. (The demand

functions may be given originally in constant-elastic-ity form. If they are given in linear form, then eitherthe elasticities can be computed from the slopes andinitial price, quantity, and income values, or Equa-tions (12) and (16) can be appropriately restated interms of demand slope parameters.)

Remarks on the Linear Programming Model

a) Both the objective function and the household budget constrainthave been transformed into linear expressions by means of a prior gridlinearization of the demand functions. The quantity demanded and thevalue of expenditure on consumption purchases are now expressed interms of the interpolation variables wi,h,,; X0 = Es Ot,h,sWO, The convexcombination constraints (16) govern the interpolation choices.

b) The commodity balances and other constraints need not be satisfiedin the initial conditions, but the demand parameters must be computedsuch that in the initial conditions the Engel aggregation, homogeneity,and Cournot aggregation restrictions are satisfied:

>i C,A, = 1 where a,,h = (Pi,0,w/Yho

} Th,j,h = -Ei,h

>i ai,h7t,j,h = -aiA,

where the point elasticities, as noted, may be computed from lineardemand functions if necessary. Also, it is assumed that all E,ha > 0. Noticethat, if the solution does not reproduce the initial conditions (in the

250 Norton and Scandizzo

budget shares), then these properties do not necessarily hold for thesolution values.

c) Cross-price effects are included explicitly in these demand functions,and in fact the demand functions are not required to be synmnetric inthose effects. This is an advantage of the linearizations.

d) Throughout, the relevant ranges of the demand functions are definedso that at least one x,o, > 0.

e) When the model is specified with only one aggregate consumer,Equation (14) is not needed and may be deleted.

f) It is assumed that Z = 0 is feasible.The modifications require(d for the introduction of intermediate goods

are to rewrite (10) and (15), respectively, as

Jh.s OtA,.sa,h,s - Z,t a ,,tqj,t c 0 (18)ip, + ' a* *Sd,,rk 0 (19)

where a*y. represents an element of a rectangular (I-A) Leontief matrixwhich has multiple technologies per good. (This specification includesgoods which are partly intermediate and partly final as well as purelyintermediate goods.)

When all the constraints in Model B are binding, except for (15) in thecase of technologies which are not utilized, then the properties of acompetitive market equilibrium hold by definition. The equilib)riurn prop-erties have been written into the primal statement of the model.

3. NUMERICAL EXAMPLES

In this section we report some solutionis of a numerical linear program-ming model of the type given by EqALuations (9)-(l9). Two kinds ofexamples are presented: one in which the model is expected to reproducethe initial conditions, and another kind in which the initial conditions areperturbed so that the model gives a new equilibrium. The perturbationscases are especially important from the viewpoint of applications, as theycan be used to represent situations in which the initial conditions are notfully consistent, i.e., situations of initial or continling disequilibrium. Ingeneral, they can be used for comparative statics analyses.

Except for the latter cases, the initial conditions were coonstructed sothat they represent a competitive market equilibriumn: markets clear,price equals marginal costs, and income equals the total value of re-sources. In a practical planning context, the raw or even p)ul)lishe(d datanever are fully consistent, either in this sense or in the sense of satisfyingall the required economic accounting identities. The construction of aconsistent data base always is an important starting point for an appliedmodel.

The numerical model used for the examples has two resources, twoconsumer groups, and three commodities. One of the commodities (no. 2)

Analysis Models 251

is purely an intermediate good, and the other two (nos. 1 and 3) haveboth intermediate and final uses. Each consumer group owns a share ofboth assets, although the proportions owned are different. Twenty-onesegment variables are used per demand function, but using twice as manyor half as many segments does not make a noticeable difference in theresults. The data were constructed for these illustrations on the basis ofrepresentative aggregate parameter values for developing countries.

The parameter values are as follows in the notation of Equations (9)-(19), with the first subscript indicating rows in the case of matrices:

b,, b2 = (3712.069, 5000.001)

- 3340.862 500.000bh,k = 371.207 4500.001

Pio= (0.59424, 1.61670, 1.31077)

351.99732 429.97232]xi,h,O 0 0

222.01861 291.90766

Yh,O (502.12763, 636.82786)

0.8 0.6E,,h 0 0

1.135495 1.257712

F 0.940 0 -0.244441a = j -0.064 0.58025 -0.17222

L-0.048 -0.24691 0.91667

where for a given good i, the column vectors for all t values are identical.The resource technology matrix d,,k,t is

i1 ==2 i-3

t=l t=2 t=3 t=1 t-2 t=3 t-= t-2 t=3

k = 1 1.0 1.2 0.8 2.0 1.8 2.4 3.0 2.7 3.3k = 2 2.0 1.8 2.28 3.0 3.5 2.3 3.0 3.25 2.72

The initial consumer demand fuinctions in price-quantity space are de-fined by the requirement that they pass through the points (pi,o, qi,o) andby a matrix of cross-price elasticities which satisfies the relevant condi-tionIs from demand theory. Given the Engel elasticities, any number ofsuch matrices can be generated by using Frisch's scheme (see Frisch[1959]) and varying the values of the money flexibility parameter. Forthe solutions reported here, the matrices of cross-price elasticities are

F -0.593856 -0.2061441[no 21 L-0.275153 -0.860(342 j j = 1, 3

F-0.464569 -0.135431(Th..2) = -0.344967 -0.912745 J y - 1, 3

252 Norton and Scandizzo

atd the consumer budget shares a,,h areL0.416569 0.4012181at,h = 0 0

0.583430 0.598782

Note that in aggregate terms, the good 1 parameters are representativeof basic commodities (e.g., food) and good 3 represents nonbasic com-modities. For computing the demand parameters, the relevant range of

TABLE IPRIMAI SOI.XTIONS OF THE SAMPLE LP MODEL

Initial Model VariantVariable Value Si 82 S3 S4 Is6

pi 0.59424 0.59424 0.57990 0.56986 0.59604 0.52842P2 1.61 670 1.61670 1.63584 1.64557 1.60787 1.71767p3 1,31077 1.31077 1.31077 1.31077 1.30905 1.36908xl, 351.997 351.998 397.026 420.413 405.339 396.303xl X 222.019 223.500 261 261 280.315 272.437 239.169xl 429.972 429.972 424.824 431.852 446.758 468.733.r32 291.908 290.913 279.103 286.035 315.718 316.647ri n.a. 0.13073 0.531 5Il= 0. 1 06355 0.12186 0.14014r2 n.a. 0.13073 0.11218 0.10377 0.13795 0.14014y, 502.128 502.126 573 335 608.673 598.219 538.241Y2 636.828 636.827 612.125 621.069 679.589 682.6292 -0.0018 -0.2959 -0.5030 -0.0767 -. l378

Relative 0.00000 -0.00025 -0.00041 -0.00006 -0IAX1437error

p 1,0000 0.9943 0.9902 1.0180 1.0259y/p 1138.953 1192.314 1241.879 1255.214 1190.048

Definition of solu tions:3S1 parameters as reported in the textS2 = b2 increased by 10 over its S1 valueS3 - b2 increased by 20% over its 81 valueS4 = b, increased by 30% over its S1 valueS5 = technical progress in producing good 1: a*,,, increased by 20%'r, all t.

(See text for definitions of expost variables p and y/p.)

n.a. = not applicable.

price variation was defined arbitrarily to be (0.5p,,O, 1.551,.). With thesespecifications, the illustrative model contains 27 e(quations (inclu(ling theobjective function) and 100 variables.

Solutions

The primal and dual solutions are r eportedi in Tables T and II SolutionS1 reproduces the competitive market equLilibrium in both the primal andthe dual, with extremely smnall approximation errors. Solutions S2

Analysis Models 253

through S5 represent a series of perturbations of the resource endow-ments, and accordingly the results show changes in prices, incomes, andquantities. The income concept in the model is nominal (in currentprices), and there is no mechanism for deter-mininig the overall price level.Its value is arbitrary. Hence the "real" income increase (y/p) is found &'xpost by dividing the nominal income increases by the change in an indexof the prices (p). For illustrative purposes, an aggregate price index wasconstructed ex post by using aggregate value shares in corisurnipt ion asweights. Good 2 has a weight of zero in this inidex.

For small changes in resource endowments the technology vectors inthe basis did not change, and therefore relative prices remained un-changed. The larger resource changes shown in Table I did alter relativepriceto. and as long as the available di-screte technologies permitted, rates

TABLE IID)UAL SOLUJTIONS OF THE SAMPLE LP MODE.L

VaibeInitial __Model VariantValue Si 82 S3 84 S5

771 0.59424 0.59424 .02180 .036396 .07586 .03215IT 1.61670 1.61669 .06166 .10867 .20464 .104587n.3 1.31077 1.31077 .04936 .08619 .16(661 .08354

Al n.a. 0.13073 A0581 .01125 .01551 .00852n.a. 0.13073 .00417 .00643 .01756 .00852

.03721 [ 06201 .11641 .06081p-ratio 1.0000 L.0381 J .0..688J L~.1168J L..0610J

Note: The p-ratio is the ratio to the corresponding primal va-riables in the same solution.n.a. = not applicable.

of return to the two resources were equjalizedl. When relative resourceendowments were ~increased sufficiently it was no longer possible tomaintain equal rates of return. Under a change in the productionl functionfor good 1 (solution S5), rates of return were equalized but relative outputprices varied.

The approximation error is manifested in more than one place. Onemneasure of it is found in the objective function va]Lue, which is zero at thecorrect. equilibrium. The amount by which the objective function departsfrom zero equals the sum of errors in the household inicome-expenditureconstraints. This may be compared to the total household income to gainan idea of the importance of the error. For example, in solution 812 theobjective function is -0.2959 and the total inicome is 1185.46. Hence theaggregate p)ercentage error in the incomle-expenditure identities is 0.025'r'(0.00025), as reported in Table I immnedliately under the Z values. Thissolution may be regarded as an appr-oximiate competitive market equillib-

254 Norton and Scandizzo

rium, subject to that degree of error in the income-expenditure identities.The dual solution values also indicate the closeness to the equilibrium.

Table II shows that the dual values are equal to their correspondingprimal values in SI and approximately proportional to the primal valuesin the other cases. The dual-to-primal price ratios ("p-ratios"), for ex-ample, ranged from 0.0372 to 0.0381 in case S2. The amount of dispersionin the p-ratios for a given solution appears to be related to the amount bywhich the objective function deviates from zero.

The objective function does not exactly attain a zero value becausethat value is infeasible. The household budget constraints cannot besatisfied everywhere along a set of linear demand functions. An iterativeprocedure can be envisaged which entails changing the demand func-tions, so that Z = 0 becomes feasible. If one knows the true nonlineardemand functions, then the new parameters of the linearized demandfunctions could be defined so that they are "closest" in some sense to thetrue functions, but that topic is beyond the scope of the present paper. Itshould be noted that if the model is applied in a recursive annual manner,or in a medium-term comparative statics sense, then relative factorendowments would not be changing very much and hence the approxi-mation errors could be expected to remain at the relatively insignificantlevels of Table I.

Although this paper does not report systematic explorations of thefeasible space, that space is quite large in all variables. To illustrate itsdimensions, the model's constraint structure was combined with theobjective function >i',h x,,,r p,. When this objective was then maximized, itattained the value 1698.604; when minimized, it became 132.217.

4. CONCLUDING OBSERVATIONS

The evidence presented in this paper strongly suggests that a kind ofcompetitive market equilibrium can be reproduced by a linear program-ming model, subject to some important requirements. The principalrequirements are linearity of the demand functions, availability of a setof initial conditions in numbers, and the use of activity analysis produc-tion functions. Also, the demand point elasticities, evaluated at the initialconditions, must satisfy three principal requirements of demand theory.

No claims are nmade for the algorithmic efficiency of the method; as ofthis writing at least two other linear programming methods of equilibriumcomputations exist (Ginsburgh and Waelbroeck, and Manne). If theprocedure of this paper has a comparative advantage, it is likely to lie inthe area of planning model applications, and more precisely in that subsetof applications which utilizes recursive annuLal analyses or short- andmedium-run comparative statics.

Analysis Models 255

APPENDIX A. EQUILIBRIUM PROPERTIES OF THE MODEL

In this appendix the following formal demonstration is made: a max-imization model can be constructed, using explicit consumer demandfunctions, for which the optimal solution exists and describes a competi-tive equilibrium. Nothing is claimed for numerical solution procedures;in the rather general form in which the model is written in this appendix,it is unlikely that existing algorithms could find the global optimum.

Although application of an algorithm typically requires sc mle convexityassumptions, as do some proof procedures (e.g., use of Kuhn-Tuckerconditions), the proof below does not need convexity, but compactness ofthe feasible set is assumed.

Consider v, consumers characterized by the following general demandcorrespondence:

- = fh(P, Wh + r'bh) = {Xh I Xh,PX,', P'Xh c wl, + r bl,) (Al)

for every Xh' in the feasible set, where xh is an n x 1 bundle of goods, bha k x 1 endowment vector of factors of production owned by the hthconsumer, and P and r, respectively, the n X 1 vector of prices of finalgoods and the k x 1 vector of factor returns. The symbol P (leniotes "ispreferred to." In addition to selling his services at market prices, the hthconsumer also has the possibility of receiving additional incolme t,, inform of a predetermined share of total excess profits from prodLCtion1:

Uih = Oh(P Q - r'»' b,) (A2)

D'Q - shi% bh c 0. (A3)

In (A2) and (A3), Q represents the total production vector, D is a k x nmatrix of input-output coefficients, and 0,, is the share of total profitsaccruing to the hth consumer. The variable ilvh is then a measure of excessprofits. Clearly Eh 0/, = 1 and 0 - Oh S 1. Note that, by construction, thedemand correspondence in (Al) is such that all income of the hthconsumer is in fact expended (Debreu [1959], p. 71):

(p)'Xh = Wh ± r'b,,. (A4)

Given (lefinitions (Al) to (A4), the following model provides a marketequilibrium at its optimum:

Max Z = whiv (A5)

subject to>)= wi(P, u',, + r'bh) - Q s 0 (A6)

Wl, - Oh (P'Q - r' bh,) = 0 (A7)

D'Q - >h-= be, < 0 (A8)

256 Norton and Scandizzo

P- D'r5 sO (A9)

wh, P, r, Q 2 O. (AIO)

The set S = {w, P, r, Q I (A6)-(AIO) are satisfied) is the feasible set,where w is the s x 1 vector {wh}. Notice that the household budgetconstraint in the text has been substituted into the income definition(A7).

LEMMA 1. For all (P, Q, r) C S, Z s 0.

Proof. Premultiplying (A9) by Q one gets, using (A8) as well:

Q'P c r'DQ => P'Q ' r' Sh-l bh. (All)

Therefore Wh c 0, h = 1, 2 ... v, implying Z c 0.

LEMMA 2. If max Z = Z 0 , then at the associated solution (P*, Q*,r *,w*) we have:

(i) Q* maximizes (P*)'Q over D'Q Y _ bh(ii) r* minimizes (r*)' , bh over D'r - P*.

Proof. Since Z* = 0 by assumption and Wh < 0 by Lemma 1, thenwh = 0. Therefore, by the definition of demand function:

(P*)'fh* = (r*)'b;, (A12)

where fh* fh(P*, r*'bh). From (A6) and (A10),

(P*)I 3h fh* .5 (p*)IQ* (A13)

Hence, (P*)'Q* Ž (r*)' Eh bA and, by comparison with (All):

(P*)'Q* = (r*)' >h b,,. (A14)

Now, from (A9) we have

P*'Q - r*'D'Q c 0, (A15)

and from (A8),

r* LD Q c r *' h bh, (A16)

so p Q c r* Eh bh. (A17)

Equations (A14) and (A17) together imply

p*Q Is p*1Q* (A18)

which establishes part (i) of Lemma 2.In similar fashion, part (ii) follows from (A8), (A9), (A17) and (A14).

THEOREM 1. The solution of problem (A5)-(A10) exists and is a com-petitive equilibrium (CE).

Analysis Models 257

Proof Following Debreu (p. 79) we define a CE as a set (x,,*, Q*, P*,r*) of points of R' such that:

(i) Xh* is such that {P*Xh* P*xhIP*xA h r*"bh + Oh(P*Q* rEh bh)} for every h

(ii) Q* maximizes (P*)'Q over D'Q c E bh(iii) r * minimizes (r*)' E bh over D'r Ž P(iV) >hXh = QCondition (i) says that, for the hth consumer, Xh* is both a feasible and

preferred consumption bundle relative to (P*, Q*) where wh = X*'bh +Oh(P*'Q - r*` >h bh); (ii) and (iii) imply that Q* is an equilibriumproduction relative to P*, r* and, (iv) asserts that the state (xi*, Q*)clears the markets. We first prove that the condition of Lemma 2, thatZ*-=0 must necessarily hold at the optimum. This is easily done bynoticing that, because of Lemma 1, this condition must be necessarilyoptimal if it is feasible. Since wh c 0 also by Lemma 1, furthermoreZ* = 0 implies wh = 0 for all h's. Substituting wh = 0 in expressions (A6)and (A7) (the only two constraints containing Wh) feasibility is thendirectly proved.

Proof of (i) now follows from the fact that the demand function for thehth consumer is directly used to specify his consumption in (A6) (so thatthe revealed preference axiom is necessarily respected for all points onthe demand curve), and (ii) and (iii) are given by Lemma 2. To prove (iv)note that by (A12) and (A14):

(P*)(h fh* - Q*) = 0 (A19)which compared with (A6) yields directly (iv).

Note also that the equilibrium described is a particular case of onetreated by Debreu and thus his existence proof (pp. 83-85) applies.Existence of a solution for the maximum model is assured by the conti-nuity of the objective function and the compactness of the feasible set.

APPENDIX B. CONSTRUCTION OF LINEAR PROGRAMMINGVARIABLES AND PARAMETERS

The procedures used for the linear model assume a knowledge of thedemand curves in a "base year." To facilitate dealing with a nonlinearproblem via LP, change-of-variable transformations are utilized, as wellas piece-wise linearizations. In particular, the LP model does not dealdirectly with the quantity variables Xi,h-

For a given good i, the initial demand function in the price-quantityspace must pass through the point (pi,o, x,o), as illustrated for onehousehold in Figure 1. (In the rest of this appendix, we omit the householdsubscript h.) First, the relevant range of the demand function is estab-

258 Norton and Scandizzo

lished and it is truncated at points a and b. Then the function ispartitioned into segments s = 1, -*, S. For each segment endpoint,

parameters Oi,S are defined to represent the cumulative quantity sold andthe cumulative value of consumer expenditure for that good. Figure 1shows how these parameters are defined from the demand curve D andthe revenue function R.

price

\a

pi,o - -- - f

I quantity

I I I \

I Iexpenditure |

i's --- 1

1,l i,s i,S quantity

Figure 1. Demand parameter construction.

The quantity consumed now can be described as a linear combinationof the 0,,,. Following Duloy and Norton (1975), interpolation weightvariables uw, are defined such that Es u', 1 in the base year. Thus ifincome is fixed at the base-year level, we have:

xi =s O'su",,, (a)

and Y. = s p,..w,.s (b)

Analysis Models 259

where yi denotes the value of expenditure on good i at current prices.These interpolation variables, or demand segment variables, are the keyto expression of the problem in LP format.

The model developed in this paper requires that the own price-respon-sive market demand curve be shifted (additively) or rotated (multiplica-tively shifted) endogenously as income and other prices change. Forpurposes of exposition, we focus only on the effects of income changes inthe immediately following paragraphs. Cross-price effects are handled

I price

G L E D

quantity

O0 K A i D

Figure 2. Demand curve rotation.

analogously. For the base-period income level, entering equations (a) and(b) in the LP model ensures that the quantities xi and values yi lie alongthe given demand function. For subsequent periods, or for new compar-ative statics equilibria, we want the xi and yi to be defined so that theyare consistent with the rotated demand function. (Hereafter we use amultiplicative shift.) This accomplished by specifying that

EsWi, 1 +iy' (c)

260 Norton and Scandizzo

where y is income, y' represents its percentage change, and Ei is theincome elasticity of demand.

If the second-order term is ignored, the percentage change in quantityconsumed can be decomposed into the components due to income changeand to price change:

Xi' = Ely' + Tllpt' (d)

where r1i < 0. The first right-hand term represents a movement of thedemand function and the second term represents a movement along thedemand function. Therefore, an alternative way of writing (c) would be

Es Wi0 s = 1 + Xi' - 7'ipr'. (e)

In Figure 2, the distance KD, as a proportion of OK, represents xi', andthe distance CG, as a proportion of OG, represents pi'.

If income changes but price does not change, then Equations (a) and(c) taken together ensure that the new value xi,, lies on the new demandfunction D2, as in the case of point H in Figure 2. When both price andincome change, the same effect occurs and also the computation ofexpenditure from equation (b) is exact-there is no approximation erroreven though second-order effects have been ignored in writing (e) andhence in (c).

In Figure 2 notice that the price is constant for a given demandsegment, regardless of the income level. Thus points F and H arerepresented by the same LP choice variable wi,,, S = j. Suppose that theinitial solution gave F as the equilibrium point, and in the subsequentsolution E was the equilibrium. In terms of segments, a movement fromsegment j to segment k has taken place. The level of expenditure on goodi in the new equilibrium should be given by the area ODEC. To see thatthis is indeed the case, Equation (b) can be given its geometrical. repre..sentation. For simplicity, assume that only one demand segment entersthe optimal basis. (By optimality, no more than two adjacent segmentswill enter.) Then, by Equations (b) and (c), the new expenditure levelcan be written

A l = Pik(l + EyY')

= OABC[KJHF/OKFG + 1]

= OABC[KJ/OK + 1] (f)

= OABC[AD/OA + 1]

-OABC[(ADEB + OABC)/OABC]

= ODEC,

as required.

Analysis Models 261

Equation (c), the convex combination constraint, provides for themovement of the demand function. Movement along the dermand functionis dictated by choice of segment. To define Equation (c) in the LP model,the percentage change in income, y', must be defined; the variable y isnot needed directly. Its percentage change from the initial value can bewritten

y'= (l/y-I)y - 1 (g)

where the parameter in parentheses is updated before each new solution.Employing comparable expressions for xi' and pi', we can write the LPequations

(l/yi-Oy1 - (l/pi,-J)p, - (l/x1 ,)xi = 0 (h)

and y = ly,. (i)

Ignoring savings, income y also is given from the demand segmentvariables as

y = S pi,sWi,s. (i)When income is defined independently from the factor side, Equation (j)becomes the income-expenditure identity. Likewise, the price variablesmay be stated explicitly in the primal program via the inverse demandfunction:

X,Xi - ThIhpi - Eiyy = 1- 7,i- E. (k)

When nonzero cross-price elasticities are introduced, similar construc-tion procedures are utilized to derive the equations of the text.

ACKNOWLEDGMENTS

Many individuals have contributed to the development of the ideas inthis paper; we are grateful to all of them and in particular we would liketo express our appreciation for the important contributions of WilfredCandler, Alan Manne, Yakir Plessner and T. N. Srinivasan. Efficientcomputational assistance was provided by John Ayler, Shu Kong Chanand Richard Inman. This research was supported in part by The WorldBank. The opinions expressed in this paper do not necessarily reflect theofficial views of The World Bank.

REFERENCES

DEBREU, G. 1959. Theoiy of Value, John Wiley & Sons, New York.DULOY, J. H., AND R. D. NORTON. 1975. Prices and Incomes in Linear Pro-

Models. Am. J. Agric. Econ, 57, 591-600.FRISCH, R. 1959. A Complete Scheme for Computing All Direct and Cross

Demand Elasticities in a Model with Many Sectors. Economnetrica 27, 177-196.

262 Norton and Scandizzo

GINSBURGH, V., AND J. WAELBROECK. 1975. A General Equilibrium Model ofWorld Trade, Part I, Cowles Foundation Discussion Paper No. 412, YaleUJniversity.

HONG, S. H. 1980. General equilibrium analysis of Korean taxation policy, Ph.D.dissertation, Stanford University.

JONES, R. W. 1965. The Strupture of Simple General Equilibrium Models. J.Polit. Econ. 73, 557-572.

MANNE, A. S., H. P. CHAO AND R. WILSON. 1978. Computation of CompetitiveEquilibriumr by a Sequence of Linear Programs, Department of OperationsResearch, Stanford University (January).

PLESSNER, Y. 1957. Activity Analysis, Quadratic Programming and GeneralEquilibrium. Int. Econ. Rev. 8, 168-179.

SAMUELSON, P. A. 1952. Spatial Price Equilibrium and Linear Programming. Am.Econ. Rev. 42, 283-303.

SHUBIK, M. 1977. Competitive and Controlled-Price Economies: The Arrow-Debreu Model Revisited. In Equilibrium and Disequilibrium in EconomicTheory, pp. 213-224, G. Schw6dianer (ed.). D. Reidel, Dordrecht, Holland.

TAKAYAMA, T., AND G. G. JUDGE. 1971. Spatial and Temporal Price andAllocation Models. North-Holland, New York.

YARON, D, 1967. Incorporation of Income Effects into Mathematical Program-ming Models. Metroeconomica 19, 141-160.

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