12. General Market Equilibrium

12. General MarketEquilibrium

A scenario of general economic equilibrium may be articulated accordingto various degrees of detail and complexity. The essential feature of ageneral equilibrium, however, is the existence of economic agents gifted withendowments and the will to appear on the market either as consumers or asproducers in order to take advantage of economic opportunities expressedin the form of offers to either buy or sell.

In this chapter, we will discuss a series of general equilibrium modelscharacterized by an increasing degree of articulation and generality. Theminimal requirements for a genuine general equilibrium model is the pres-ence of demand and supply functions for some of the commodities.

Model 1: Final Commodities

A general market equilibrium requires consumers and producers. We as-sume that consumers have already maximized their utility function subjectto their budget constraints and have expressed their decisions by meansof an aggregate set of demand functions for final commodities. On theproducers’ side, the industry is atomistic in the sense that there are manyproducers of final commodities each of whom cannot affect the overall mar-ket behavior with his decisions. It is the typical environment of a perfectlycompetitive industry.

Hence, consider the following scenario. There exists a set of inversedemand functions for final commodities expressed by p = c − Dx, whereD is a symmetric positive semidefinite matrix of dimension (n × n), p isan n-vector of prices, c is an n-vector of intercept coefficients and x is ann-vector of quantities. There exists also a set of supply functions for inputsdefined as s = b+Ey, where E is a symmetric positive semidefinite matrixof dimension (m ×m), s is an m-vector of quantities, b is an m-vector of

p = c - Dx

c

b

xx* y*

$

$

CS = x’Dx/2

A’yk k

PS = b’y + y’Ey/2

Σ sk k

b +- Ey

182 General Market Equilibrium

intercept coefficients and y is an m-vector of prices. This specification of theinput supply functions leads directly to a symmetric structure of the marketequilibrium problem, as seen in the monopsonist case. Finally, there exists alarge number of firms, say K, that transform inputs into final commoditiesby means of individualized linear technologies Ak, k = 1, . . . ,K. Each firmis too small to influence prices appreciably and, therefore, it is consideredas a price taker on the output as well as on the input markets.

(a) (b)

Figure 12.1 Consumer and producer surplus in a market equilibrium

This specification of the model provides one of the most stylized rep-resentations of a general equilibrium problem. Strictly speaking, it is morecorrect to view this model as a market equilibrium problem that allows thestudent to focus on a few essential features of a general equilibrium. In thismodel, consumers are represented by the set of inverse demand functionsfor final commodities. These relations are aggregate functions represent-ing the maximization of utility functions subject to a given level of incomemade by individual consumers and collectively rendered by the above setof inverse demand functions. The aggregation problem involved is assumedto be solved consistently. The assumption of symmetry associated with thematrix D is not necessary but it circumvents the integrability problem andwill be relaxed in subsequent models. Furthermore, the consumer demandfunctions specified above are conditional on a predetermined level of in-come. In order to reconcile theory with empirical modelling (for example,to satisfy the assumption of zero-degree homogeneity in prices and incomeof demand functions), it is convenient to think that the general specifica-

$

c - Dx

x

CS + PS

h + Hx

X*

General Market Equilibrium 183

tion of inverse demand functions is p = dI − Dx, where d is the vectorof income coefficients associated with an aggregate level of income, I. Byfixing income at a given level, say I, the vector c = dI becomes the vectorof intercepts in the original formulation of the inverse demand functions.

The final objective of a general equilibrium specification is that of find-ing commodity prices and quantities that clear input and output markets.The atomistic structure of the economy and the information given aboveguarantee that, under the working of the “invisible hand,” this objective isequivalent to maximizing consumer (CS) and producer (PS) surpluses.

Figure 12.1 illustrates the geometric structure of this problem. In theoutput diagram (figure 12.1.a), the consumer surplus ( 1

2x′Dx) is the tri-

angle above the economy’s marginal cost (A′kyk) and below the aggregatedemand function (c−Dx). The input diagram (figure 12.1.b) exhibits in-verted axes with respect to the traditional textbook representation and,therefore, the producer surplus, (b′y + 1

2y′Ey), is the trapezoid below the

inverse input supply function. This specification of the producers’ side isbased upon the technical discussion of the perfectly discriminating monop-sonist’s input supply carried out in chapter 9.

In order to justify the structure of the primal objective function asstated in figure (12.1), we review the consumer and producer surpluses inan auxiliary single diagram where the demand is (c−Dx) and the supplyis (h + Hx).

Figure 12.2 Auxiliary Diagram for consumer and producer surpluses

The matrices D and H are assumed to be symmetric positive definite.The consumer and the producer surpluses are represented in figure 12.2by the area (triangle) under the demand function (c − Dx) and abovethe supply function (h + Hx) and are measured as the difference between


the integral under the demand function and the integral under the supplyfunction.

Hence, in analytical form,

(CS + PS) =∫ x∗

0

(c−Dx)′dx−∫ x∗

0

(h + Hx)′dx

= [c′x− 12x′Dx]− [h′x + 1

2x′Hx].

We must now consider the fact that the supply functions in the general equi-librium problem of this section deal with inputs, and not final commodities.As computed for the perfectly discriminating monopsonist in chapter 9, thesecond integral has the familiar specification [g′s+ 1

2s′Gs], where (g+Gs) is

the vector of input supply functions. By letting y = g+Gs, as done in chap-ter 9, we can write s = b+Ey and, finally, [g′s+ 1

2s′Gs] = 1

2b′Eb+ 1

2y′Ey.

In Appendix 1, we discuss this topic in more detail. The term 12b′Eb is

a constant and does not enter into the optimization process. Therefore,it can be disregarded. We can, thus, state the primal specification of thegeneral market equilibrium as done in the following paragraph.

The dual pair of symmetric problems representing the market equilib-rium described in this section begins with the specification of the primalproblem

Primal

max (CS + PS) = c′x− 12x′Dx− 1

2y′Ey (12.1)

subject to Akxk − sk ≤ 0 (12.2)

−K∑k=1

xk + x ≤ 0 (12.3)

K∑k=1

sk − Ey ≤ b (12.4)

y ≥ 0, xk ≥ 0, sk ≥ 0, x ≥ 0

for k = 1, . . . ,K. The first constraint (12.2) represents the technologicalrelations of the kth firm. The individual entrepreneur must make decisionsregarding the purchase of inputs sk and the production of outputs xk ina way to respect the physical equilibrium conditions according to whichthe kth firm’s input demand (Akxk) must be less than or equal to itsinput supply sk. Notice, therefore, that the quantity sk is viewed bothas a demand (when the entrepreneur faces the input market) and as asupply (when the entrepreneur faces his output opportunities, or inputrequirements, (Akxk)). The second constraint (12.3) represents the marketclearing condition for the final commodity outputs. The total demand for


final commodities x must be less than or equal to their total supply∑Kk=1 xk

as generated by the K firms. The third constraint (12.4) represents themarket clearing condition for the commodity inputs. The total demandof inputs in the economy

∑Kk=1 sk must be less than or equal to the total

input supply (b + Ey).The dual of model [(12.1)-(12.4)] is obtained as an application of KKT

theory according to the familiar Lagrangean procedure. Let yk, f and y bethe vectors of dual variables associated with the three primal constraintsin [(12.2)-(12.4), respectively. Then, the Lagrangean function of problem[(12.1)-(12.4)] is

L = c′x− 12x′Dx− 1

2y′Ey +

K∑k=1

y′k(sk −Akxk) (12.5)

+ f ′(K∑k=1

xk − x) + y′(b + Ey −K∑k=1

sk).

Karush-Kuhn-Tucker conditions from (12.5) are the nonnegativity ofall the variables involved (y ≥ 0, xk ≥ 0, sk ≥ 0, x ≥ 0) and

∂L

∂xk= −A′kyk + f ≤ 0 (12.6)

K∑k=1

x′k∂L

∂xk= −

K∑k=1

x′kA′kyk +

K∑k=1

x′kf = 0 (12.7)

∂L

∂x= c−Dx− f ≤ 0 (12.8)

x′∂L

∂x= x′c− x′Dx− x′f = 0 (12.9)

∂L

∂sk= yk − y ≤ 0 (12.10)

K∑k=1

s′k∂L

∂sk=

K∑k=1

s′kyk −K∑k=1

s′ky = 0 (12.11)

∂L

∂y= −Ey + b + 2Ey −

K∑k=1

sk ≥ 0 (12.12)

= b + Ey −K∑k=1

sk ≥ 0


y′∂L

∂y= y′b + y′Ey − y′

K∑k=1

sk = 0. (12.13)

The derivatives with respect to dual variables yk and f were omittedbecause of their simplicity and the fact that they do not contribute tothe definition of the dual constraints. Relations (12.6), (12.8) and (12.10)are the constraints of the dual problem associated to the primal problem[(12.1)-(12.4)]. Using (12.7), (12.9) and (12.11) in the Lagrangean function(12.5), it is possible to simplify the dual objective function in such a waythat the dual problem in its final form reads as

Dual

min COA = b′y + 12y′Ey + 1

2x′Dx (12.14)

subject to A′kyk − f ≥ 0 (12.15)

−yk + y ≥ 0 (12.16)

f + Dx ≥ c (12.17)

y ≥ 0, yk ≥ 0, f ≥ 0, x ≥ 0.

The objective function of the dual problem states that the market asa whole minimizes the cost of achieving an optimal allocation, (COA).The dual objective function consists of two components identified with theconsumer and producer surpluses, as shown in figure 12.1. It would be inap-propriate, however, to interpret the dual objective function as minimizingthe sum of the consumer and producer surpluses. From the point of viewof producers and consumers, these measures should be clearly maximized.A more suitable interpretation of the dual objective function, therefore, isthat of an abstract general planner (or invisible hand) whose objective is tominimize the expenditure necessary to make the market economy operateas a competitive institution. In order to entice producers and consumerstoward a general equilibrium as articulated by the primal constraints, it isnecessary to quote prices of inputs (y, yk, f) and of outputs (p = c−Dx)such that, at the optimal solution of [(12.1)-(12.4)] and [(12.14)-(12.17)],the two objective functions are equal and, thus, the optimal level of expen-diture is equal to the optimal level of producer and consumer surpluses.

The first set of dual constraints in (12.15) states that the marginal costof each firm (A′kyk) must be greater than or equal to an overall marginalcost for the entire economy, f . Assuming that, at the optimum, every firmproduces some positive level of output, the first dual constraint (12.15) isequivalent to say that every firm must exhibit the same marginal activity


cost. The third set of dual constraints (12.17) links this marginal cost tothe output prices (p = c−Dx). Economic equilibrium requires that outputprices be less than or equal to output marginal cost. Finally, the secondset of dual constraints (12.16) asserts that the valuation of resources at thefirm level, yk, cannot exceed the marginal value of the same resources atthe economy level.

Model 1 can be reformulated as a linear complementarity problem byfollowing the usual procedure which requires the simultaneous considerationof all the KKT conditions. The structure of the LC problem can, therefore,be stated as

(x xk sk yk f y )

M =

D 0 0 0 I 00 0 0 A′k −I 00 0 0 −I 0 I0 −Ak I 0 0 0−I I 0 0 0 00 0 −I 0 0 E

, q =

−c0000b

, z =

xxkskykfy

where k = 1, . . . ,K. The vector variables carrying the k subscript mustbe thought of as a series of K vectors associated with a similar number ofmatrices which have been omitted from the above LC specification simplyfor reasons of space.

Model 2: Intermediate and Final Commodities

In Model 1, the supply functions of resources (b + Ey) are exogenous andbear no direct relation with either consumers or producers operating inthat economy. Toward a fuller understanding of the structure of model 1,it is possible to interpret the vector b as the aggregate vector of resourceendowments belonging to the economic agents, mainly consumers, operat-ing in that scenario. More difficult is to explain how the other componentof the input supply functions, (Ey), is generated. The discussion in thissecond model, therefore, is centered on a justification of the entire supplyfunction for inputs.

Let the aggregate supply of resources in a given economy be the sumof initial endowments of resources belonging to consumers and of the ad-ditional amount of the same resources developed by firms operating in theeconomy. Let bC represent the sum of all endowment vectors of resourcesbelonging to consumers. In this model, the kth firm is allowed to produceeither final commodities xk or intermediate commodities (inputs) vk, or


both. The firm’s incentive to expand the supply of resources is given bya profitable level of input prices which she takes as given. As for pricesof final commodities, input prices are generated at the endogenous marketlevel. We postulate, therefore, that the aggregate supply of inputs over andabove the initial consumers’ endowments is represented by (bV + Ey).

The mathematical programming specification of this scenario corre-sponds to a further elaboration of model 1 and it is stated as follows

Primal

max (CS + PS) = c′x− 12x′Dx − 1

2y′Ey (12.18)

subject to Akxk + Bkvk − sk ≤ 0 (12.19)

−K∑k=1

xk + x ≤ 0 (12.20)

K∑k=1

vk − Ey ≤ bV (12.21)

−K∑k=1

vk +K∑k=1

sk ≤ bC (12.22)

y ≥ 0, xk ≥ 0, vk ≥ 0, sk ≥ 0, x ≥ 0

for k = 1, . . . ,K. The interpretation of this primal version of the modelbuilds upon the explanation offered for the various components of model1. The objective function, thus, corresponds to the maximimazion of theconsumer and producer surpluses. The first set of constraints in (12.19)articulates the assumption that any firm may produce either final or in-termediate commodities, or both. The second set of constraints (12.20)describes the market clearing condition (demand, x, is less than or equalto supply,

∑Kk=1 xk) for final commodities. The third set of constraints

(12.21) defines the endogenous vector of input supply functions (bV +Ey)as the sum of input vectors vk produced by all firms. The presence ofan intercept such as bV is justified by assuming that these functions de-pend also on other factors not accounted for in the model. The fourth setof constraints (12.22) represents the market clearing condition for inputs(∑Kk=1 sk ≤ bC +

∑Kk=1 vk ≤ [bC + bV ] + Ey).

The dual of [(12.18)-(12.22)] is obtained by applying KKT procedures.Let (yk, f , y) and λ be the vectors of dual variables asociated with theprimal constraints [(12.19)-(12.22)], respectively. Then, the dual problemof [(12.18)-(12.22)] is

Dual

min COA = b′Cλ + b′V y + 12y′Ey + 1

2x′Dx (12.23)


subject to f + Dx ≥ c (12.24)A′kyk − f ≥ 0 (12.25)−yk + λ ≥ 0 (12.26)B′kyk + y − λ ≥ 0 (12.27)

y ≥ 0, yk ≥ 0, f ≥ 0, x ≥ 0, λ ≥ 0.

The economic interpretation of the dual specification begins with recog-nizing that the dual objective function minimizes the cost of achieving anoptimal allocation. The terms in vectors y and λ constitute the producersurplus while the quadratic form in x is the consumer surplus. The first setof constraints (12.24) establishes the familiar economic equilibrium condi-tions according to which marginal cost of final commodities must be greaterthan or equal to their market price, (p = c−Dx). The second set of dualconstraints (12.25) asserts that the marginal cost of producing final com-modities in any firm must be at least as great as the corresponding marginalcost for the entire economy. The third set of dual constraints (12.26) es-tablishes that the value marginal product of the kth firm cannot be greaterthan the rent (λ) derived from the ownership of the initial endowmentof resources. Finally, the fourth set of constraints (12.27) states that themarginal cost of producing intermediate commodities must be greater thanor equal to (λ−y), the difference between the marginal valuation of the ini-tial endowment of resources and the shadow price of the resources developedby the firms. An alternative interpretation would be that (λ ≤ B′kyk + y),that is, the rent (λ) derived from the ownership of the initial endowment ofresources must not be greater than the combined marginal cost of produc-ing additional intermediate commodities and the shadow price commandedby these resources in the economy. The rent on the initial endowment of re-sources, therefore, is bracketed by the value marginal product of any firm yk(which forms a lower bound) and the marginal cost of producing additionalresources (which forms an upper bound) as follows (yk ≤ λ ≤ B′kyk + y).

The structure of the linear complementarity problem corresponding toproblems (12.18) and (12.23) can be stated as

( x xk vk sk yk f y λ )

M =

D 0 0 0 0 I 0 00 0 0 0 A′k −I 0 00 0 0 0 B′k 0 I −I0 0 0 0 −I 0 0 I0 −Ak −Bk I 0 0 0 0−I I 0 0 0 0 0 00 0 −I 0 0 0 E 00 0 I −I 0 0 0 0

,q =

−c00000

bVbC

, z =

xxkvkskykfyλ

where k = 1, . . . ,K. As for the previous model, the subscript k indicates aseries of components of the same nature for k = 1, . . . ,K.


Model 3: Endogenous Income

A complete general equilibrium model requires the specification of a processfor the endogenous determination of consumers’ income. Models 1 and 2are deficient with respect to this important aspect. Suppose, therefore, thatconsumers’ demand functions are respecified to include aggregate incomeI in an explicit way, that is, p = c − Dx + gI, where g is an n-vectorof income coefficients. Aggregate consumer income is assumed to be thesum of individual incomes which, in turn, is generated as the rental value ofresource endowments, that is, I = b′Cλ, where λ is the vector of rental pricesfor the consumers’ endowments. In the process so specified, we assume thatthe associated aggregation problems are solved consistently.

It turns out that the consideration of endogenous income, as definedabove, precludes the formulation of a dual pair of symmetric quadratic pro-gramming models. In other words, it is no longer possible to define primaland dual optimization problems which constitute a symmetric structure.The reason for this unexpected result can be articulated in several alterna-tive explanations. The principal cause of the problem uncovered is the factthat income enters a general equilibrium model in an asymmetric way, asa determinant of consumers’ demands but not of input supplies. A subor-dinate reason is that knowledge of the rental value of resources is requiredfor defining income. This fact necessitates the simultaneous considerationof primal and dual constraints.

The impossibility of formulating this general equilibrium scenario aspair of optimization models does not preclude its formulation as an equi-librium problem. The linear complementarity model is, thus, the naturalframework for dealing with the problem of general equilibrium with endoge-nous income. The explicit statement of the LC model corresponding to thiseconomic scenario will contribute a further clarification of the asymmetryuncovered in this section.

The formulation of endogenous income developed above suggests thatthe difference between model 2 and model 3 consists of two elements: theexpansion of the demand function for final commodities to include aggregateincome, I, explicitly, p = c−Dx+gI, and the definition of aggregate incomeI = b′Cλ.


The relations that define the relevant equilibrium problem for thisscenario are those of model 2 incremented by the income distribution:

Dx− gI + f ≥ c final commodity pricing (12.28)A′kyk − f ≥ 0 final commodity marginal costB′kyk + y − λ ≥ 0 intermediate commodity marginal cost−yk + λ ≥ 0 resource valuation

I − b′Cλ = 0 income distributionAkxk + Bkvk − sk ≤ 0 technology constraints

−K∑k=1

xk + x ≤ 0 final commodity market clearing

K∑k=1

vk − Ey ≤ bV input supply function from firms

−K∑k=1

vk +K∑k=1

sk ≤ bC total input supply function

x ≥ 0, xk ≥ 0, vk ≥ 0, sk ≥ 0, I ≥ 0, yk ≥ 0, f ≥ 0, y free, λ ≥ 0.

The equilibrium problem is defined by the relations stated in (12.28) andthe associated complementary slackness relations. The variables which mul-tiply the relations in (12.28) to form the required complementary slacknessconditions are stated in the appropriate order in the bottom line of (12.28).For the benefit of the reader we recall that a linear complementarity prob-lem is defined as a vector z ≥ 0 such that Mz+q ≥ 0 and z′Mz+z′q = 0.

The structure of the linear complementarity problem associated withthe equilibrium problem stated in (12.28) (and the associated complemen-tary slackness conditions) corresponds to the following representation:


( x xk vk sk I yk f y λ )

M =

D 0 0 0 −g 0 I 0 00 0 0 0 0 A′k −I 0 00 0 0 0 0 B′k 0 I −I0 0 0 0 0 −I 0 0 I0∗ 0 0 0 1 0 0 0 −bC0 −Ak −Bk I 0 0 0 0 0−I I 0 0 0 0 0 0 00 0 −I 0 0 0 0 E 00 0 I −I 0∗ 0 0 0 0

,

q =

−c000000

bVbC

, z =

xxkvkskIykfyλ

where k = 1, . . . ,K. The matrix M is not anti-symmetric as are the cor-responding LC matrices of models 1 and 2. The zero elements with anasterisk as superscript in the matrix M indicate the location of the miss-ing terms which would make the matrix M anti-symmetric and, therefore,suitable for expressing the problem as a dual pair of optimization models.

In the above specification, we glossed over a rather important prop-erty of consumers’ demand functions, namely the fact that inverse demandfunctions satisfy the budget constraint: x′p(x, I) = I. When applied to thelinear demand functions specified above, this property implies the followingconditions: x′p(x, I) = x′c−x′Dx+x′gI = I and, therefore, x′g = 1 whilex′c−x′Dx = 0. The imposition of these two additional constraints can beperformed in alternative ways.

Model 4: Spatial Equilibrium - One Commodity

The solution of a competitive equilibrium among spatially separated mar-kets, in the case of linear demand and supply functions, was originallysuggested by Enke (1951) using an electric analogue. Samuelson (1952) ri-formulated the problem in a linear programming model that minimizes thetotal transportation cost of the commodity flow among regions. Takayama

p

s1d1

xs1xd1 x1 -->

s2

d2

<-- x2 xd2 xs2

p1

p2t12


and Judge (1964) riformulated the problem in a quadratic programmingspecification. Over the years, this problem has received a lot of scrutinywith many empirical applications dealing with international trade issues.

In a stylized formulation, the problem of finding the competitive equi-librium involving one homogeneous commodity that is traded among manyseparated markets can be stated as follows. Assume R regions which rep-resent separated markets. For each region, the demand and the supplyfunctions of a given commodity are given. Furthermore, the unit cost oftransporting the commodity over all the routes connecting the R marketsis also given. A competitive solution of this problem is constituted by thelist of prices and quantities of the given commodity that are supplied andconsumed in each market and the associated commodity flow among all themarkets so that consumer and producer surpluses are maximized.

Figure 12.3. Competitive solution between two separated markets

The competitive solution dealing with two markets is illustrated in theback-to-back diagram of figure 12.3. In region 1, the given commodity isproduced (supplied) in quantity xs1 but is consumed (demanded) in quan-tity xd1. In region 2, the commodity is produced in quantity xs2 and con-sumed in quantity xd2. To clear the markets, therefore, the excess supply(xs1− xd1) from region 1 will be shipped to region 2 in order to satisfy theexcess demand (xd2 − xs2). This event will happen at prices p2 = p1 + t12,where t12 is the unit cost of transportation from region 1 to region 2. The


consumer surplus of region 1 is the area under the demand function andabove the price line. The producer surplus of market 1 is the area abovethe supply function and under the price line. Analogous inference can bemade for market 2.

Before discussing a more general specification of spatial market equi-librium, we present the associated transportation problem, as analyzedby Samuelson. We assume that there are R regions endowed of a givenquantity (availability, supply) of a commodity xsr, r = 1, . . . , R, and re-questing a given quantity (demand) xdr for the same commodity. Fur-thermore, the available information regards the unit trasportation cost,tr,r′ , r, r

′ = 1, . . . , R, over all the possible routes connecting any pair ofregions. For r = r′, the unit transportation cost is, obviously, tr,r = 0.The problem is to satisfy the demand of each region while minimizing thetotal cost of transportation over all routes. This is a linear programmingproblem with two sets of constraints. The first set defines the requirementof supplying a given region with the quantity demanded. The second setguarantees the feasibility of that operation by preventing that the supplyschedule of each region exceeds the available endowment. Indicating byxr,r′ the nonnegative commodity flow between regions r and r′, the result-ing primal specification of the transportation problem is the minimizationof the total transportation cost (TTC) subject to the given constraints,that is,

Primal

min TTC =∑r

∑r′

tr,r′xr,r′ (12.29)

subject to xdr′ ≤∑r

xr,r′ demand requirement (12.30)

∑r′

xr,r′ ≤ xsr supply availability. (12.31)

The dual specification of problem (12.29)-(12.31) can be stated as themaximization of the value added for the entire set of regions, that is,

Dual

max VA =∑r′

xdr′pdr′ −

∑r

xsrpsr (12.32)

subject to pdr′ ≤ psr + tr,r′ . (12.33)

The dual problem, then, consists in finding nonnegative prices pdr′ andpsr, for each region, that maximize the value added (VA), that is the differ-ence between the total value of the commodity at all destination markets


minus the total cost of the same commodity at all the supplying markets.The decision variables are the prices at the demand sites, pdr′ , and at thesupply origins, psr, r, r

′ = 1, . . . , R. The economic interpretation of the dual,then, can be restated as follows: Suppose that a condition for winning thetransport job requires to purchase the commodity at the supply markets(origins) and selling it at the various destinations (demand markets). Atrucking firm, wishing to bid for the contract to trasport the commodityflow over the entire network of regional routes, will wish to quote pricesat destination markets and at supply origins that will maximize its valueadded. The dual constraint (12.33) states that the price of the commodityon the r′-th demand market cannot exceed the price in the r-th supply re-gion augmented by the unit transportation cost between the two locations,tr,r′ (marginal revenue ≤ marginal cost).

Notice that, in general, a transportation problem has multiple optimalsolutions. That is, there are multiple networks of routes to satisfy thedemand requirements of the various regions while achieving the same valueof the total transportation cost. The reason is that the (R × R) matrix offlows, X = [xr,r′ ], ha rank less than R.

Let us now formulate the spatial equilibrium problem in a more generalspecification. Instead of a fixed demand requirement and supply availabil-ity, as in the LP problem (12.29)-(12.31), each region is endowed with aninverse demand function, pdr = ar −Drx

dr , and an inverse supply function

for the same homogeneous commodity, psr = br + Srxsr, r = 1, . . . , R. The

parameters ar, Dr, Sr are positive and known scalars. Parameter br can beeither positive or negative. Therefore, the quantities xdr and xsr must nowbe determined as part of the equilibrium solution. This objective can beachieved by maximizing the sum of consumers’ and producers’ surpluses,CS and PS, in all markets, netted out of the total transportation cost, asillustrated in figure 12.3. The analytical task is represented by the follow-ing quadratic programming problem which is a direct extension of the LPproblem discussed above.

Primal

max N(CS + PS) =∑r′

(ar′ − 12Drx

dr′)x

dr′ −

∑r

(br + 12Srx

sr)x

sr

−∑r

∑r′

tr,r′xr,r′ (12.34)



∑r′


A solution of the spatial equilibrium problem (12.34)-(12.36), which is


regarded as a quantity formulation, is given by optimal values of all theprimal decision variables (xdr′ , x

sr, xr,r′) and by the optimal values of the

dual variables (pdr′ , psr), for r, r′ = 1, . . . , R.

The dual of problem (12.34)-(12.36) would involve both primal anddual variables, as seen in all the quadratic programming problems dis-cussed in previous chapters. However, in analogy to the demand and sup-ply quantity-extension of the primal LP problem (12.29)-(12.31), we wishto define the demand and supply price-extension of the dual LP problem(12.32)-(12.33). To obtain this specification, it is sufficient to express thedemand quantity xdr′ and the supply quantity xsr in terms of the respectiveprices. In other words, given the assumptions of the model, it is possibleto invert the (inverse) demand and supply functions to obtain the desiredinformation. Hence

xdr = D−1ar −D−1r pdr = cr −D−1

r pdr demand function (12.37)xsr = −S−1br + S−1

r psr = hr + S−1r psr supply function (12.38)

Replacing xdr′pdr′ and xsrp

sr (by appropriate integration) into the objective

function of the dual LP of the transportation problem (12.32)-(12.33),we obtain the “purified” dual of the spatial equilibrium problem (12.34)-(12.36), that is

“Purified” Dual

max VA =∑r

(cr − 12D−1r pdr)p

dr −

∑r

(hr + 12S−1r psr)p

sr (12.39)


The terminology “purified duality” is due to Takayama and Woodland(1970). The solution of the “purified” dual problem provides the same so-lution of the primal problem in terms of prices pdr , p

sr and quantities xdr , x

sr.

The quantity flow of commodities xr,r′ obtained from the primal problem,however, may represent an alternative optimal solution to the flow obtained(as dual variables) from the “purified” dual problem. This is due to thetransportation component of the spatial equilibrium model, as discussedfor the LP specification. A detailed discussion of this spatial equilibriummodel is given in Appendix A12.2.

Model 5: Spatial Equilibrium - Many Commodities

Suppose that each region deals with M commodities. The extension of thespatial equilibrium model to this case is straightforward: it simply requires


the inclusion of the commodity index m = 1, . . . ,M into the primal andthe “purified” dual models discussed in the previous section. Hence,

Primal

max (CS + PS) =∑m,r′

(am,r′ − 12Dm,rx

dm,r′)x

dm,r′

−∑m,r

(bm,r + 12Sm,rx

sm,r)x

sm,r

−∑m,r

∑m,r′

tm,r,r′xm,r,r′ (12.41)

subject to xdm,r′ ≤∑m,r

xm,r,r′ demand requirement (12.42)

∑m,r′

xm,r,r′ ≤ xsm,r supply availability. (12.43)

Similarly,

“Purified” Dual

max V A =∑m,r′

(cm,r′ − 12D−1m,r′p

dm,r′)p

dm,r′

−∑m,r

(hm,r + 12S−1m,rp

sm,r)p

sm,r (12.44)

subject to pdm,r′ ≤ psm,r + tm,r,r′ . (12.45)

The possibility of multiple optimal solutions for the flow of commoditiesamong regions, xm,r,r′ , applies also for this spatial equilibrium specification.It is possible also to conceive that the demand and supply functions of eachcommodity and each region may involve all the commodities in that region.In this case, the D and S parameters of the inverse demand and supplyfunction stated above should be considered as symmetric positive definitematrices. Then, the analytical development presented above applies also tothis more general scenario.

Numerical Example 1: General Market EquilibriumFinal Commodities

Example 1 deals with a scenario of general market equilibrium involvingonly final commodities and the allocation of resources among firms. This


is the problem discussed in model [(12.1)-(12.4)], which is reproduced herefor convenience:

Primal

max (CS + PS) = c′x− 12x′Dx− 1

2y′Ey (12.1)

subject to Akxk − sk ≤ 0 (12.2)

−K∑k=1

xk + x ≤ 0 (12.3)

K∑k=1

sk − Ey ≤ b (12.4)

y ≥ 0, xk ≥ 0, sk ≥ 0, x ≥ 0

The relevant data are given as:

Technology of Firm 1

A =

2.5 −2 0.5 1

1 3 −2.1 3−2 1 0.5 0.9


A =

1 −1 0.5 .3.5 3 2 .32 1 .2 9


A =

0.7 −1 5 2

1 3 4 1−0.5 1 6 .6

Market Demands

c =

23403430

, D =

8 −5 4 −3−5 7 3 3

4 3 5 −2−3 3 −2 4

Total Input Supply

b =

10

248

, E =

1.3 −.4 .2−.4 .8 −.3.2 −.3 .4

The solution of this problem is given for each firm, for the final commodity


market as a whole, and for the total input supply.

Consumer Surplus = 201.6512Producer Surplus = 148.3685

Total Surplus = 350.0197Total Cost of Resources = 179.9459

Demand and Prices for FinalCommodity Market

x =

x1 = 6.4941x2 = 4.9629x3 = 1.4109x4 = 6.9597

, p =

p1 = 11.0976p2 = 12.6181p3 = 0.0000p4 = 9.5763

Outputs and Inputs from Firm 1

x1 =

x11 = 0.4010x21 = 0.8539x31 = 1.4109x41 = 0.0000

, s1 =

s11 = 0.0000s21 = 0.0000s31 = 0.7574


x2 =

x12 = 0.0000x22 = 0.0735x32 = 0.0000x42 = 0.2449

, s2 =

s12 = 0.0000s22 = 0.2939s32 = 2.2774


x3 =

x13 = 6.0931x23 = 4.0355x33 = 0.0000x43 = 6.7149

, s3 =

s13 = 13.6594s23 = 24.9145s33 = 5.0179

Quantity and Shadow Prices of Total Input Supply

s =

s1 = 13.6594s2 = 25.2034s3 = 8.0527

, y =

y1 = 3.8262y2 = 3.8337y3 = 1.0938


Command File for GAMS: Numerical Example 1

We list a command file for the nonlinear package GAMS which solves thenumerical problem presented in example 1. Asterisks in column 1 relate tocomments.

**************************************************************$TITLE A General Market Equilibrium: Three Firms and One Market* with symmetric D and E Matrices - SQP*

$OFFSYMLIST OFFSYMXREFOPTION LIMROW = 0OPTION LIMCOL = 0option iterlim =100000option reslim = 200000

* OPTion nlp = minos5option nlp = conopt3option decimals = 7 ;

*SETS j Output variables / x1, x2, x3, x4 /

i Inputs / y1, y2, y3 /*

alias(i,k,kk);Alias(j,jj) ;

*parameter c(j) Intercept of demand functions/x1 23x2 40x3 34x4 30/

*parameter b(i) Intercept of total input supply/y1 10y2 24y3 8/

;*

Table A1(i,j) Technical coefficient Matrix in Firm 1x1 x2 x3 x4

y1 2.5 −2 .5 1y2 1 3 −2.1 3y3 −2 1 .5 .9;

*


Table A2(i,j) Technical coefficient Matrix in Firm 2x1 x2 x3 x4

y1 1 −1 .5 .3y2 .5 3 2 .3y3 2 1 .2 9;

*Table A3(i,j) Technical coefficient Matrix in Firm 3

x1 x2 x3 x4y1 .7 −1 5 2y2 1 3 4 1y3 −.5 1 6 .6;

*Table D(j,jj) Slopes of demand functions

x1 x2 x3 x4x1 8 −5 4 −3x2 −5 7 3 3x3 4 3 5 −2x4 −3 3 −2 4

*Table E(i,k) Slopes of the input supply function

y1 y2 y3y1 1.3 −.4 .2y2 −.4 .8 −.3y3 .2 −.3 .4

*Scalar Scale Parameter to define Economic Agents / .5 /;

** ************* * General equilibrium problem* *

variables CSPSs1(i)s2(i)s3(i)x1(j)x2(j)x2(j)x(j)y(i);

*positive variables s1, s2, s3, x1, x2, x3, x, y ;

*


equations objeqtecheq1(i)techeq2(i)techeq3(i)mktequil(j)inputsup(i)peq(j);

*objeq.. CSPS =e= sum(j, c(j)*x(j) )

- scale*sum((j,jj), x(j)*D(j,jj)*x(jj) )- scale*sum((i,k), y(i)*E(i,k)*y(k) )

;*

techeq1(i).. sum(j, A1(i,j)*x1(j)) =L= s1(i) ;techeq2(i).. sum(j, A2(i,j)*x2(j)) =L= s2(i) ;techeq3(i).. sum(j, A3(i,j)*x3(j)) =L= s3(i) ;mktequil(j).. x1(j) + x2(j) + x3(j) =E= x(j) ;inputsup(i).. s1(i) + s2(i) +s3(i) =L= b(i) + sum(k, E(i,k)*y(k) ) ;peq(j).. c(j) - sum(jj, D(j,jj)*x(jj) ) =G= 0 ;

*model genmktequil / objeq, techeq1, techeq2, techeq3, mktequil,

inputsup, peq / ;*

solve genmktequil using nlp maximizing CSPS ;*

parameter CS, PS, TotSurplus, totSupply(i), totcost, p(j);*

CS = scale*sum((j,jj), x.l(j)*D(j,jj)*x.l(jj) ) ;PS = sum(i, b(i)*y.l(i)) + scale*sum((i,k), y.l(i)*E(i,k)*y.l(k) ) ;TotSurplus = CS + PS ;totcost = (1/2)*sum((i,k), b(i)*E(i,k)*b(k)) +

scale*sum((i,k), y.l(i)*E(i,k)*y.l(k) ) ;totSupply(i) = b(i) + sum(k, E(i,k)*y.l(k)) ;p(j) = c(j) - sum(jj, D(j,jj)*x.l(jj) ) ;

*display CS, PS, TotSurplus, totcost, p,

x1.l, x2.l, x3.l, x.l, s1.l, s2.l, s3.l,techeq1.m, techeq2.m , techeq3.m, inputsup.m,y.l, totSupply;

************************************************************


Numerical Example 2: General Market EquilibriumIntermediate and Final Commodities

Example 2 deals with a scenario of general market equilibrium involvingboth the production of final as well as intermediate commodities and theallocation of resources among firms. Furthermore, this scenario includesa beginning stock of resource endowments held by consumers. This is theproblem discussed in model [(12.18)-(12.22)], which is reproduced here forconvenience:

Primal

max (CS + PS) = c′x− 12x′Dx − 1

2y′Ey (12.18)

subject to Akxk + Bkvk − sk ≤ 0 (12.19)

−K∑k=1

xk + x ≤ 0 (12.20)

K∑k=1

vk − Ey ≤ bV (12.21)

−K∑k=1

vk +K∑k=1

sk ≤ bC (12.22)

y ≥ 0, xk ≥ 0, vk ≥ 0, sk ≥ 0, x ≥ 0

The relevant data are given as:

Technology for Final and Intermediate Goods of Firm 1

A1 =

2.5 −2 0.5 1

1 3 −2.1 3−2 1 0.5 0.9

, B1

.25 .2 .5

.1 .3 .21

.2 .1 .5


A2 =

1 −1 0.5 .3.5 .3 2 .32 1 .2 9

, B2

.25 .2 .5

1 .3 .21.2 .1 .5


A3 =

0.7 −1 5 2

1 3 .4 1−0.5 1 .6 .6

, B3

.25 .2 .5

.1 .3 .21

.2 .1 .5


Market Demands for Final Commodities

c =

9403430

, D =

8 −5 4 −3−5 7 3 3

4 3 5 −2−3 3 −2 4

Total Input Supply

bv =

−.10

.02

.50

, E =

1.3 −.4 .2−.4 .8 −.3.2 −.3 .4

Consumers′ Initial Endowment of Resources

bc =

7

53

The solution of this problem is given for each firm, for the final commoditymarket as a whole, and for the total input supply.

Consumer Surplus = 145.8201Producer Surplus = 5.3914

Total Surplus = 151.2114Total Cost of Resources = 4.7704

Demand and Prices for FinalCommodity Market

x =

x1 = 2.1636x2 = 3.1372x3 = 4.2646x4 = 3.4029

, p =

p1 = 0.5277p2 = 5.8549p3 = 1.4169p4 = 21.9966

Outputs, Inputs and Shadow Prices from Firm 1

x1 =

x11 = 2.1636x21 = 2.5583x31 = 4.2646x41 = 0.0000

, v1 =

v11 = 2.9742v21 = 0.0000v31 = 0.0000

s1 =

s11 = 3.1683s21 = 1.1803s31 = 0.9582

, y1 =

y11 = 6.5649y21 = 3.1550y31 = 9.5198



x2 =

x12 = 0.0000x22 = 0.5789x32 = 0.0000x42 = 0.0000

, v2 =

v12 = 0.0000v22 = 0.0000v32 = 1.1578

s2 =

s12 = 0.0000s22 = 0.4168s32 = 1.1578

, y2 =

y12 = 4.6113y22 = 3.1550y32 = 9.5198


x3 =

x13 = 0.0000x23 = 0.0000x33 = 0.0000x43 = 3.4030

, v3 =

v13 = 0.0000v23 = 0.0000v33 = 0.0000

s3 =

s13 = 6.8059s23 = 3.4030s33 = 2.0418

, y3 =

y13 = 6.5649y23 = 3.1550y33 = 9.5198

Quantity and Shadow Prices of Total Input Supply

s =

s1 = 2.9742s2 = 0.0000s3 = 1.1578

, y =

y1 = 2.7042y2 = 1.9990y3 = 1.7917



**************************************************************$TITLE A General Market Equilibrium: Three Firms and one Market* with symmetric D and E Matrices - SQP and Intermediate Commodities*

$OFFSYMLIST OFFSYMXREFOPTION LIMROW = 0OPTION LIMCOL = 0option iterlim =100000option reslim = 200000

* OPTion nlp = minos5option nlp = conopt3option decimals = 7 ;


*SETS j Output variables / x1, x2, x3, x4 /

i Inputs / y1, y2, y3 /*

alias(i,k,kk);Alias(j,jj) ;

*parameter c(j) Intercept of demand functions/x1 9x2 40x3 34x4 30/

*parameter bv(i) Intercept of total input supply functions/y1 −.1y2 .02y3 .5/

;*

parameter bc(i) Initial consumers’ endowment of resources/y1 7y2 5y3 3/

;*

Table A1(i,j) Technology of Final Commodities of Firm 1x1 x2 x3 x4

y1 2.5 −2 .5 1y2 1 3 −2.1 3y3 −2 1 .5 .9;

*Table B1(i,k) Technology of Intermediate Commodities of Firm 1

y1 y2 y3y1 .25 .2 .5y2 .1 .3 .21y3 .2 .1 .5;

*Table A2(i,j) Technology of Final Commodities of Firm 2

x1 x2 x3 x4y1 1 −1 .5 .3y2 .5 .3 2 .3y3 2 1 .2 9;

*


Table B2(i,k) Technology of Intermediate Commodities of Firm 2y1 y2 y3

y1 .25 .2 .5y2 1 .3 .21y3 .2 .1 .5;

*Table A3(i,j) Technology of Final Commodities of Firm 3

x1 x2 x3 x4y1 .7 −1 5 2y2 1 3 .4 1y3 −.5 1 .6 .6;

*Table B3(i,k) Technology of Intermediate Commodities in Firm 2

y1 y2 y3y1 .25 .2 .5y2 1 .3 .21y3 .2 .1 .5;

*Table D(j,jj) Slopes of Final Commodities’ Demand Functions

x1 x2 x3 x4x1 8 −5 4 −3x2 −5 7 3 3x3 4 3 5 −2x4 −3 3 −2 4

*Table E(i,k) Slopes of the Input Supply Functions

y1 y2 y3y1 1.3 −.4 .2y2 −.4 .8 −.3y3 .2 −.3 .4

*Scalar Scale Parameter to define Economic Agents / .5 /;

** * ************** general equilibrium problem**

variables CSPSs1(i) Input supply of Firm 1s2(i) Input supply of Firm 2s3(i) Input supply of Firm 3v1(i) Intermediate commodities’ output of Firm 1v2(i) Intermediate commodities’ output of Firm 2


v3(i) Intermediate commodities’ output of Firm 3x1(j) Final commodities’ output of Firm 1x2(j) Final commodities’ output of Firm 2x2(j) Final commodities’ output of Firm 2x(j) Total quantity of final commodities’ demandy(i) Shadow prices of total resources;

*positive variables s1, s2, s3, v1,v2,v3, x1, x2, x3, x, y ;

*equations objeq Name of objective functiontecheq1(i) Name of technical constraints of Firm 1techeq2(i) Name of technical constraints of Firm 2techeq3(i) Name of technical constraints of Firm 3mktequil(j) Name of Market clearing of final commoditiesinputsup(i)] Name ofMarket clearfing of resourcesConsEndow(i) Consumers’ endowmentand intermediate outputs;

*objeq.. CSPS =e= sum(j, c(j)*x(j) )

- scale*sum((j,jj), x(j)*D(j,jj)*x(jj) )- scale*sum((i,k), y(i)*E(i,k)*y(k) )

;*

techeq1(i).. sum(j, A1(i,j)*x1(j)) + sum(k,B1(i,k)*v1(k)) =e= s1(i) ;techeq2(i).. sum(j, A2(i,j)*x2(j))+ sum(k,B2(i,k)*v2(k)) =e= s2(i) ;techeq3(i).. sum(j, A3(i,j)*x3(j)) + sum(k,B3(i,k)*v3(k)) =e= s3(i) ;mktequil(j).. x1(j) + x2(j) + x3(j) =E= x(j) ;inputsup(i).. v1(i) + v2(i) +v3(i) =e= bv(i) + sum(k, E(i,k)*y(k) ) ;ConsEndow(i).. -v1(i)-v2(i)-v3(i) +s1(i)+s2(i)+s3(i) =e= bc(i) ;

*model genmktequil / objeq, techeq1, techeq2, techeq3, mktequil,inputsup, consEndow / ;

*solve genmktequil using nlp maximizing CSPS ;

*parameter CS, PS, TotSurplus, totcost, totSupply(i), p(j);CS = scale*sum((j,jj), x.l(j)*D(j,jj)*x.l(jj) ) ;PS = sum(i, bv(i)*y.l(i)) + scale*sum((i,k), y.l(i)*E(i,k)*y.l(k) ) ;TotSurplus = CS + PS ;totcost = (1/2)*sum((i,k), bv(i)*E(i,k)*bv(k)) +

scale*sum((i,k), y.l(i)*E(i,k)*y.l(k) ) ;totSupply(i) = bv(i) + sum(k, E(i,k)*y.l(k)) ;


p(j) = c(j) - sum(jj, D(j,jj)*x.l(jj) ) ;*

display CS, PS, TotSurplus, totcost, p,x1.l, x2.l, x3.l, x.l, s1.l, s2.l, s3.l, v1.l, v2.l, v3.l,techeq1.m, techeq2.m , techeq3.m, inputsup.m,y.l, totSupply ;

******************************************************************


Numerical Example 3: Spatial Equilibrium - One Commodity

Example 3 deals with the computation of the spatial equilibrium among fourregions, (A, B, U and E), and involves only one homogeneous commodity.The equilibrium will be computed twice using the quantity and the priceformulations. In this way, the possibility of multiple optimal solutions of thecommodity flow among regions will be illustrated. The relevant models arethe primal [(12.34)-(12.36)] for the quantity specification and the “purified”dual [(12.39)-(12.40)] for the price representation. They will be reproducedhere for convenience.

Primal

max (CS + PS) =∑r′

(ar′ − 12Drx

dr′)x

dr′ −

∑r

(br + 12Srx

sr)x

sr

−∑r

∑r′

tr,r′xr,r′ (12.34)



∑r′


“Purified” Dual

max V A =∑r′

(cr′ − 12D−1r′ pdr′)p

dr′ −

∑r

(hr + 12S−1r psr)p

sr (12.39)


The relevant data are as follows:

Regions : A, B, U, E

Demand Intercepts Demand SlopesA 15.0B 22.0U 25.0E 28.0

A 1.2B 1.8U 2.1E 1.1

Supply Intercepts Supply SlopesA 0.4B 0.2U 0.6E 0.5

A 1.4B 2.4U 1.9E 0.6


Unit Cost of Transportation between RegionsA B U E

A 0.0 1.5 1.0 0.5B 1.5 0.0 0.2 0.8U 0.75 2.25 0.0 0.6E 3.0 5.0 4.0 0.0

The solution of this numerical example will be given for the price modeland the quantity model, in that order.

Price Model

Demand andSupply Prices

A 7.129B 6.829U 7.029E 7.629

RegionalSupply

A 10.380B 16.589U 13.955E 5.077

RegionalDemand

A 6.445B 9.708U 10.240E 19.608

Interregional TradeA B U E

A 6.445 3.935B 9.708 6.881U 3.359 10.596E 5.077

Implied Demand Implied Supply

Elasticities ElasticitiesA −1.327B −1.266U −1.442E −0.428

A 0.961B 0.988U 0.957E 0.902

Total Transportation Cost 9.701Objective Function 321.972

***************************************************************

Quantity Model


A 7.129B 6.829U 7.029E 7.629

RegionalSupply

A 10.380B 16.589U 13.955E 5.077

RegionalDemand

A 6.445B 9.708U 10.240E 19.608


Interregional TradeA B U E

A 6.445 3.935B 9.708 5.216 1.665U 5.123 8.931E 5.077


Consumer Surplus Producer SurplusA 17.310B 26.180U 24.964E 174.766

A 38.483B 57.333U 51.246E 21.482

Total Consumer and Producer Surplus 411.764

Notice that the interregional trade flow is different in the price and quantitymodels but that the total transportation cost is the same in the two mod-els. This occurrence is an example of multiple optimal solutions in spatialequilibrium analysis.



**************************************************************$TITLE WORLD commodity TRADE - ONE COMMODITY$OFFSYMLIST OFFSYMXREF

OPTION LIMROW = 0OPTION LIMCOL = 0option iterlim =2500

** Regions: Africa, Brazil, USA, Europe* QUANTITIES MEASURED IN MILLION TONS/YEAR* PRICES AND COSTS MEASURED IN $10/bale*


SETSR regions /A, B, U, E/

*Alias (r,rr) ;

*PARAMETERS AA(R) DEMAND INTERCEPTS/A 15.0B 22.0U 25.0E 28.0 /

*PARAMETERS DD(R) DEMAND SLOPES/A 1.2B 1.8U 2.1E 1.1 /

*PARAMETERS BB(R) SUPPLY INTERCEPTS/A 0.4B 0.2U 0.6E 0.5 /

*PARAMETERS SS(R) SUPPLY SLOPES/A 1.4B 2.4U 1.9E 0.6 /

*TABLE T(R,RR) UNIT COSTS OF TRANSPORT

A B U EA 0.0 1.5 1.0 0.5B 1.5 0.0 0.2 0.8U 0.75 2.25 0.0 0.6E 3. 5. 4. 0.0

***—————————————————–*** price dependent form*

VARIABLESPD(R) DEMAND PRICE (price Model)PS(R) SUPPLY PRICE (price Model)OBJ OBJECTIVE FUNCTION


*POSITIVE VARIABLE PD, PS;

*EQUATIONSVA OBJECTIVE FUNCTIONTRADE(R,RR) TRADE BETWEEN REGIONS (Price Model);

*VA.. SUM(R, (AA(R)-.5*DD(R)*PD(R))* PD(R) )-SUM(R, (BB(R)+.5*SS(R)* PS(R))* PS(R) ) =E= OBJ ;

*TRADE(R,RR).. PD(RR) =L= PS(R) + T(R,RR)) ;

*MODEL INTERREG /ALL/ ;

*SOLVE INTERREG USING NLP MAXIMIZING OBJ;

*PARAMETERREGDEM(RR) REGIONAL DEMAND QUANTITYREGSUP(R) REGIONAL SUPPLY QUANTITYDEMELAS(RR) IMPLIED DEMAND ELASTICITYSUPELAS(R) IMPLIED SUPPLY ELASTICITY;

*REGDEM(RR) = SUM(R, TRADE.M(R,RR) ) ;REGSUP(R) = SUM(RR, TRADE.M(R,RR) ) ;

*DEMELAS(RR) = -DD(RR) * (PD.L(RR)/REGDEM(RR)) ;SUPELAS(R) = SS(R) * (PS.L(R)/REGSUP(R)) ;

*DISPLAY PD.L, PS.L, TRADE.L,OBJ.L ,REGDEM, REGSUP, DEMELAS, SUPELAS ;

***—————————————————–*** quantity dependent form*

PARAMETERSA(RR) PD DEMAND INTERCEPTSD(RR) PD DEMAND SLOPESH(R) PD SUPPLY INTERCEPTSS(R) PD SUPPLY SLOPES ;

*A(RR) = AA(RR)/DD(RR);D(RR) = 1/ DD(RR) ;H(R) = -BB(R)/ SS(R);


S(R) = 1/ SS(R) ;*

display A, D, H, S ;*

VARIABLESQD(RR) DEMAND QUANTITY (quantity Model)QS(R) SUPPLY QUANTITY (quantity Model)X(R,RR) QUANTITY TRADE (quantity Model)OBJ2 OBJECTIVE FUNCTION

*POSITIVE VARIABLE QD, QS, X;

*EQUATIONSNSB OBJECTIVE FUNCTIONAGGD(RR) AGGREGATE REGIONAL DEMANDAGGS(R) AGGGREGATE REGIONAL SUPPLY ;

*NSB.. SUM(RR, (A(RR)-.5*D(RR)*QD(RR))* QD(RR) )-SUM(R, (H(R)+.5*S(R)* QS(R))* QS(R) )-SUM((R,RR), T(R,RR)*X(R,RR)) =E= OBJ2 ;

*AGGD(RR).. QD(RR)=L= SUM(R, X(R,RR) );

*AGGS(R).. SUM(RR, X(R,RR) ) =L= QS(R);

*MODEL INTERREG2 /NSB, AGGD, AGGS / ;

*SOLVE INTERREG2 USING NLP MAXIMIZING OBJ2;

*PARAMETER PD2(RR) Demand Prices (quantity model)PS2(R) Supply Prices (quantity model)transcostX Total Transportation cost (quantity Model)transcostT Total Transportation cost (price Model)CONSUR(RR) Consumer surplusPROSUR(R) Producer surplusTOTSUR Total Surplus;

*PD2(RR) = A(RR)-D(RR)*QD.L(RR) ;PS2(R) = H(R)+S(R)* QS.L(R) ;transcostX = sum((R,RR), T(R,RR)*X.l(R,RR));transcostT = sum((R,RR), T(R,RR)*Trade.M(R,RR));CONSUR(RR) = 0.5* (A(RR)-PD2(RR))* QD.L(RR) ;PROSUR(R) = 0.5*(PS2(R)-H(R))*QS.L(R) ;


TOTSUR = SUM(RR, CONSUR(RR))+ SUM(R, PROSUR(R)) ;*

DISPLAY CONSUR, PROSUR, TOTSUR ;DISPLAY QD.L, QS.L, X.L , OBJ2.L, AGGD.M, AGGS.M ;DISPLAY PD.L, PD2, PS.L, PS2, TRADE.M , X.L, OBJ.L, OBJ2.L,transcostX, transcostT ;

****************************************************************

Numerical Example 4: Spatial Equilibrium - Many Commodities

Example 4 deals with the computation of the spatial equilibrium amongfour regions, (A, B, U and E), and involves three homogeneous commodities.The equilibrium will be computed twice using the quantity and the priceformulations. In this way, the possibility of multiple optimal solutions of thecommodity flow among regions will be illustrated. The relevant models arethe primal [(12.41)-(12.43)] for the quantity specification and the “purified”dual [(12.44)-(12.45)] for the price representation. They will be reproducedhere for convenience.

Primal

max (CS + PS) =∑m,r′

(am,r′ − 12Dm,rx

dm,r′)x

dm,r′

−∑m,r

(bm,r + 12Sm,rx

sm,r)x

sm,r

−∑m,r

∑m,r′

tm,r,r′xm,r,r′ (12.41)

subject to xdm,r′ ≤∑m,r

xm,r,r′ demand requirement (12.42)

∑m,r′

xm,r,r′ ≤ xsm,r supply availability. (12.43)

“Purified” Dual

max V A =∑m,r′

(cm,r′ − 12D−1m,r′p

dm,r′)p

dm,r′

−∑m,r

(hm,r + 12S−1m,rp

sm,r)p

sm,r (12.44)

subject to pdm,r′ ≤ psm,r + tm,r,r′ . (12.45)


The relevant data are as follows:

Regions : A, B, U, E

Commodities : 1, 2, 3

Demand Intercepts Supply Intercepts1 2 3

A 15.0 25.0 10.0B 22.0 18.0 15.0U 25.0 10.0 18.0E 28.0 20.0 19.0

1 2 3A 0.4 0.1 0.7B 0.2 0.4 0.3U 0.6 0.2 0.4E 0.5 0.6 0.2

Demand Slopes Supply Slopes1 2 3

A.1 1.2A.2 2.1A.3 1.1

B.1 1.8B.2 1.6B.3 2.6

U.1 2.1U.2 0.9U.3 1.7

E.1 1.1E.2 0.8E.3 1.9

1 2 3A.1 1.4A.2 2.1A.3 1.7

B.1 2.4B.2 1.6B.3 1.8

U.1 1.9U.2 2.8U.3 2.1

E.1 0.6E.2 2.1E.3 1.2

Unit Cost of Transportation between RegionsA B U E

A 0.0 1.5 1.0 0.5B 1.5 0.0 0.2 0.8U 0.75 2.25 0.0 0.6E 3.0 5.0 4.0 0.0

The cost of transportation is the same for all the three commodities.The solution of this numerical example will be given for the price modeland the quantity model, in that order.


Price Model


RegionalSupply

1 2 3A 7.129 5.306 4.294B 6.829 5.500 3.994U 7.029 4.556 4.194E 7.629 5.156 4.794

1 2 3A 10.380 11.244 8.000B 16.589 9.200 7.490U 13.955 12.958 9.208E 5.077 11.429 5.953

RegionalDemand

1 2 3A 6.445 13.856 5.276B 9.708 9.200 4.615U 10.240 5.899 10.870E 19.608 15.875 9.891

Interregional Trade

commodity 1A B U E

A 6.445 3.935B 9.708 6.881U 3.359 10.596E 5.077

commodity 2A B U E

A 11.244B 9.200 6.881U 2.613 5.899 4.446E 11.429

commodity 3A B U E

A 5.276 2.724B 4.615 1.662 1.213U 9.208E 5.953


***************************************************************


Quantity ModelDemand andSupply Prices

RegionalSupply

1 2 3A 7.129 5.306 4.294B 6.829 5.500 3.994U 7.029 4.556 4.194E 7.629 5.156 4.794

1 2 3A 10.380 11.244 8.000B 16.589 9.200 7.490U 13.955 12.958 9.208E 5.077 11.429 5.953

Regional Demand1 2 3

A 6.445 13.856 5.276B 9.708 9.200 4.615U 10.240 5.899 10.870E 19.608 15.875 9.891

Interregional Trade

commodity 1

A B U EA 6.445 3.935B 9.708 5.761 1.120U 4.478 9.476E 5.077

commodity 2

A B U EA 11.244B 9.200 6.881U 2.613 5.899 4.446E 11.429

commodity 3

A B U EA 5.276 2.724B 4.615 2.875U 7.995 1.213E 5.953


Notice that the interregional trade flow is different in the price and quantitymodels for commodity 1 and 3 but that the total transportation cost is thesame in the two models. This occurrence is an example of multiple optimalsolutions in spatial equilibrium analysis.***************************************************************




***************************************************************$TITLE WORLD commodity TRADE* Multi-region Multi-commodity

$OFFSYMLIST OFFSYMXREFOPTION LIMROW = 0OPTION LIMCOL = 0option iterlim =2500

*** Commodities: 1,2,3** Regions: Africa, Brazil, USA, Europe*

SETSR regions /A, B, U, E/M commodities /1,2,3/

*Alias(r,rr) ;Alias(m,mm) ;

*Table AA(R,M) DEMAND INTERCEPTS

1 2 3A 15.0 25.0 10.0B 22.0 18.0 15.0U 25.0 10.0 18.0E 28.0 20.0 19.0

*Table DD(R,M,MM) DEMAND SLOPES

1 2 3A.1 1.2A.2 2.1A.3 1.1

B.1 1.8B.2 1.6B.3 2.6


1 2 3U.1 2.1U.2 0.9U.3 1.7

E.1 1.1E.2 0.8E.3 1.9

*Table BB(R,M) SUPPLY INTERCEPTS

1 2 3A 0.4 0.1 0.7B 0.2 0.4 0.3U 0.6 0.2 0.4E 0.5 0.6 0.2

*Table SS(R,M,MM) SUPPLY SLOPES

1 2 3A.1 1.4A.2 2.1A.3 1.7

B.1 2.4B.2 1.6B.3 1.8

U.1 1.9U.2 2.8U.3 2.1

E.1 0.6E.2 2.1E.3 1.2

*TABLE T(R,RR) UNIT COSTS OF TRANSPORTATION

A B U EA 0.0 1.5 1.0 0.5B 1.5 0.0 0.2 0.8U 0.75 2.25 0.0 0.6E 3.0 5.0 4.0 0.0

***—————————————————–*** price dependent form*


VARIABLESPD(R,M) DEMAND PRICE (price Model)PS(R,M) SUPPLY PRICE (price Model)OBJ OBJECTIVE FUNCTION

*POSITIVE VARIABLE PD, PS;

*EQUATIONSVA OBJECTIVE FUNCTIONTRADE(R,RR,M) TRADE BETWEEN REGIONS (Price Model)

;*

VA.. sum((R,M), AA(R,M)*PD(R,M))- sum((R,M,MM), .5*PD(R,M)*DD(R,M,MM)*PD(R,MM))- sum((R,M), BB(R,M)*PS(R,M))- sum((R,M,MM), .5*PS(R,M)*SS(R,M,MM)*PS(R,MM))=E= OBJ ;

*TRADE(R,RR,M).. PD(RR,M) =L= PS(R,M) + T(R,RR) ;

*MODEL INTERREG /ALL/ ;

*SOLVE INTERREG USING NLP MAXIMIZING OBJ;

*PARAMETERREGDEM(RR,M) REGIONAL DEMAND QUANTITYREGSUP(R,M) REGIONAL SUPPLY QUANTITYDEMELAS(RR,M,MM) IMPLIED DEMAND ELASTICITYSUPELAS(R,M,MM) IMPLIED SUPPLY ELASTICITY;

*REGDEM(RR,M) = SUM(R, TRADE.M(R,RR,M) ) ;REGSUP(R,M) = SUM(RR, TRADE.M(R,RR,M) ) ;

*DEMELAS(RR,M,M) = -DD(RR,M,M)* (PD.L(RR,M)/REGDEM(RR,M)) ;SUPELAS(R,M,M) = SS(R,M,M) * (PS.L(R,M)/REGSUP(R,M)) ;DISPLAY PD.L, PS.L, TRADE.M,OBJ.L , REGDEM,REGSUP, DEMELAS, SUPELAS ;

***—————————————————–*** quantity dependent form*

PARAMETERS


A(RR,M) PD DEMAND INTERCEPTSD(R,M,MM) PD DEMAND SLOPESH(R,M) PS SUPPLY INTERCEPTSS(R,M,MM) PS SUPPLY SLOPES ;

*A(RR,M) = AA(RR,M)/DD(RR,M,M);D(RR,M,M) = 1/ DD(RR,M,M) ;H(R,M) = -BB(R,M)/ SS(R,M,M);S(R,M,M) = 1/ SS(R,M,M) ;

*display A, D, H, S ;

*VARIABLESQD(RR,M) DEMAND QUANTITY (quantity Model)QS(R,M) SUPPLY QUANTITY (quantity Model)X(R,RR,M) QUANTITY TRADE (quantity Model)OBJ2 OBJECTIVE FUNCTION

*POSITIVE VARIABLE QD, QS, X;

*EQUATIONSNSB OBJECTIVE FUNCTIONAGGD(RR,M) AGGREGATE REGIONAL DEMANDAGGS(R,M) AGGREGATE REGIONAL SUPPLY ;

*NSB.. sum((R,M), A(R,M)*QD(R,M))- Sum((R,M,MM),.5*QD(R,M)*D(R,M,MM)*QD(R,MM) )- SUM((R,M),H(R,M)*QS(R,M))- sum((R,M,MM),.5*QS(R,M)*S(R,M,MM)* QS(R,MM) )- SUM((R,RR,M), T(R,RR)*X(R,RR,M)) =E= OBJ2 ;

*AGGD(RR,M).. QD(RR,M) =L= SUM(R, X(R,RR,M) );

*AGGS(R,M).. SUM(RR, X(R,RR,M) ) =L= QS(R,M);

*MODEL INTERREG2 /NSB, AGGD, AGGS / ;

*SOLVE INTERREG2 USING NLP MAXIMIZING OBJ2;

*PARAMETERPD2(RR,M) Demand Prices (quantity model)PS2(R,M) Supply Prices (quantity model)transcostX Total Transportation cost (quantity Model)


transcostT Total Transportation cost (price Model)CONSUR(RR,M) Consumer surplusPROSUR(R,M) Producer surplusTOTSUR Total Surplus;

*PD2(RR,M) = A(RR,M)- sum(MM, D(RR,M,MM)*QD.L(RR,MM) )

;PS2(R,M) = H(R,M) + sum(MM, D(R,M,MM)* QS.L(R,MM) ) ;transcostX = sum((R,RR,M), T(R,RR)*X.l(R,RR,M));transcostT = sum((R,RR,M), T(R,RR)*Trade.M(R,RR,M));CONSUR(RR,M) = 0.5* (A(RR,M)-PD2(RR,M))* QD.L(RR,M) ;PROSUR(R,M) = 0.5*(PS2(R,M)-H(R,M))*QS.L(R,M) ;TOTSUR = SUM((RR,M), CONSUR(RR,M))+ SUM((R,M), PRO-

SUR(R,M)) ;*

DISPLAY CONSUR, PROSUR, TOTSUR ;DISPLAY AGGD.M, AGGS.M ;DISPLAY PD.L, PD2, PS.L, PS2, QD.L, QS.L, TRADE.M ,X.L, OBJ.L, OBJ2.L, transcostX, transcostT ;

*****************************************************


Appendix 12.1: Alternative Specification of GME

In this appendix, we discuss an alternative specification of the GeneralMarket Equilibrium (GME) presented in Model 1: Final Commodities, atthe beginning of this chapter.

The known elements of this problems are as follows. There exists a setof inverse demand functions for final commodities expressed by p = c−Dx,where D is a symmetric positive semidefinite matrix of dimensions (n×n),p is an n-vector of prices, c is an n-vector of intercept coefficients and xis an n-vector of quantities demanded. There exists also a set of inversesupply functions for inputs defined as ps = g+Gs, where G is a symmetricpositive definite matrix of dimensions (m ×m) and where ps,g, s are m-vectors of prices, intercepts, and quantities supplied, respectively. Finally,the production system is represented by a large number K of competitivefirms each of which uses an individualized linear technology representedby Ak, k = 1, . . . ,K. The unknown (quantity, or primal) elements of theproblems are x,xk, sk, s, that is, the total quantity of final commoditiesdemanded by the market, the quantity of final commodities supplied bythe k-th firm, the quantity of inputs demanded by the k-th firm, and thetotal quantity of inputs supplied by the market, respectively.

A competitive market equilibrium, therefore, is a production and con-sumption allocation of commodities (inputs and outputs), and their corre-sponding prices, that satisfies consumers and producers.

A competitive equilibrium can be achieved (when the technical condi-tions allow it) by maximizing the sum of consumers’ and producers’ sur-pluses subject to the technology constraints that establish the transforma-tion of inputs into outputs and the associated market clearing conditions.Hence, given the demand and supply functions stated above, the primalspecification which is suitable to represent the described scenario can bestated as

Primal

max (CS + PS) = [c′x− 12x′Dx]−[g′s + 1

2s′Gs] (A12.1.1)

subject to −Akxk + sk ≥ 0 (A12.1.2)K∑k=1

xk − x ≥ 0 (A12.1.3)

−K∑k=1

sk + s ≥ 0 (A12.1.4)

x ≥ 0, xk ≥ 0, sk ≥ 0, s ≥ 0

for k = 1, . . . ,K. The first constraint (A12.1.2) represents the technologicalrelations of the kth firm. The individual entrepreneur must make decisions


regarding the purchase of inputs sk and the production of outputs xk ina way to respect the physical equilibrium conditions according to whichthe kth firm’s input demand (Akxk) must be less than or equal to its in-put supply sk. Notice, therefore, that the quantity sk is viewed both as ademand (when the entrepreneur faces the input market) and as a supply(when the entrepreneur faces his output opportunities, or input require-ments, (Akxk)). The second constraint (A12.1.3) represents the marketclearing condition for the final commodity outputs. The total demand forfinal commodities x must be less than or equal to their total supply

∑Kk=1 xk

as generated by the K firms. The third constraint (A12.1.4) represents themarket clearing condition for the commodity inputs. The total demandof inputs in the economy

∑Kk=1 sk must be less than or equal to the total

input supply (s).With a choice of Lagrange multipliers (dual variables) for the primal

constraints in the form of yk, f ,y, respectively, the Lagrangean function isstated as

L = c′x− 12x′Dx− g′s− 1

2s′Gs +

K∑k=1

y′k(sk −Akxk) (A12.1.5)

+ f ′(K∑k=1

xk − x) + y′(s−K∑k=1

sk).

The relevant KKT conditions (for deriving the dual problem) are thoseassociated with the primal variables. Hence,

∂L

∂x= c−Dx− f ≤ 0 (A12.1.6)

x′∂L

∂x= x′c− x′Dx− x′f = 0 (A12.1.7)

∂L

∂xk= −A′kyk + f ≤ 0 (A12.1.8)

x′k∂L

∂xk= −x′kA

′kyk + x′kf = 0 (A12.1.9)

∂L

∂s= −g −Gs + y ≤ 0 (A12.1.10)

s′∂L

∂s= −s′g − s′Gs + s′y = 0 (A12.1.11)

∂L

∂sk= yk − y ≤ 0 (A12.1.12)

s′k∂L

∂sk= s′kyk − s′ky = 0. (A12.1.13)

The simplification of the Lagrangean function, as the objective func-tion of the dual problem, takes place, as discussed already in many other


occasions, by using the information contained in the complementary slack-ness conditions (A12.1.7), (A12.1.9), (A12.1.11) and (A12.1.13). The finalresult is that all the linear terms disappear from the Lagrangean functionand, therefore, the dual problem can be stated as

Dual

min TCMO = 12s′Gs + 1

2x′Dx (A12.1.14)

subject to A′kyk ≥ f (A12.1.15)

+ y ≥ yk (A12.1.16)

f ≥ c−Dx (A12.1.17)

g + Gs ≥ y (A12.1.18)

y ≥ 0, yk ≥ 0, f ≥ 0, x ≥ 0, s ≥ 0.

The dual objective function is interpreted as the minimization of thetotal cost of market options (TCMO) which, on the primal side, correspondto the sum of the consumers’ and producers’ surplus. Constraint (A12.1.15)states that the marginal cost of the k-th firm, A′kyk, must be greater thanor equal to the marginal cost found in the entire economy, f . Constraint(A12.1.16) states that marginal valuation of the resources for the entireeconomy, y, must be greater than or equal to the marginal valuation ofthe same resources for the k-th firm, yk. Constraint (A12.1.17) meansthat the marginal cost of producing the final commodities in the entireeconomy must be greater than or equal to the market price, p = c−Dx, ofthose commodities. Finally, constraint (A12.1.18) states that the marginalvaluation of the resources in the entire economy, y, must be less than orequal to the market price of resources, ps = g + Gs.

We reiterate that the dual pair of problems discussed in this appendixrefer to a general market equilibrium scenario that was already discussedin Model 1 of this chapter. We, therefore, wish to establish a direct connec-tion between the two equivalent specifications, one of which (Model 1) cor-responds to a symmetric quadratic programming problem while the secondone (Appendix 1) corresponds to an asymmetric quadratic programmingproblem.

KKT condition (A12.1.10) and the mild assumption that s > 0, thatis the economy needs at least a miniscule positive amount of each resource,allow us to state that y = g + Gs = ps. Therefore, we can express thevector of resource supply functions in the form of quantity as a function of


price (recalling the assumption of positive definiteness of the G matrix)

s = −G−1g + G−1y (A12.1.19)= b + Ey

where b = −G−1g and E = G−1. Then, the producers’ surplus in thedual objective function (A12.1.14) in terms of input quantities, s, can bereformulated in terms of the resource price, y

12s′Gs = 1

2 (b + Ey)′E−1(b + Ey) (A12.1.20)= 1

2b′E−1b + b′y + 1

2y′Ey.

Analogously, the total cost function (in terms of resource quantities, s) inthe primal objective function (A12.1.1) can be restated in terms of inputprices, y:

g′s + 12s′Gs = g′(b + Ey) + 1

2b′E−1b + b′y + 1

2y′Ey (A12.1.21)

= g′b + g′Ey + 12b′E−1b + b′y + 1

2y′Ey

= −b′E−1b− b′E−1Ey + 12b′E−1b + b′y + 1

2y′Ey

= − 12b′E−1b + 1

2y′Ey

since g = −E−1b and G = E−1, by definition.By replacing s and g′s + 1

2s′Gs in the primal problem and 1

2s′Gs in

the dual problem with their equivalent expressions in terms of input prices,y, we obtain the dual pair of symmetric quadratic programming modelspresented in Model 1.

Appendix 12.2: A Detailed Discussion of Spatial Equilibrium

To facilitate the understanding of the spatial equilibrium model presentedin this chapter we discuss a two-region example in detail.

Each region is endowed with a demand and a supply function for afinal and homogeneous commodity

pdr = ar −Drxdr demand function (A12.2.1)

psr = br + Srxsr supply function (A12.2.2)

where pr is the regional price, xr is the regional quantity and r = 1, 2. Eachregion may either import or export the final commodity and requires thesatisfaction of the demand expressed by its regional consumers. The unittransportation cost matrix is known and stated as

T =[

0 t12t21 0

](A12.2.3)

xD1 xS

1

xS2

x11

x21

x22

t11=0

t12

t21

t22=0 xD2


where it is assumed that the unit transportation cost within a region isequal to zero.

Figure A12.2.1. Network of trade flow between two regions

Figure A12.2.1 presents the diagram of the possible trade flows betweenthe two regions. With the auxilium of this diagram, the necessary primalconstraints of the spatial equilibrium problem will be easily established.The objective function of such a problem is to maximize the sum of theconsumers’ and producers’ surpluses minus the total cost of transportingthe commodity from one region to the other. The formal specification is

max N(CS+PS) = [a1xd1 − 1

2D1(xd1)2] + [a2x

d2 − 1

2D2(xd2)2] (A12.2.3)

− [b1xs1 + 12S1(xs1)

2]− [b2xs2 + 12S2(xs2)

2]− t12x12 − t21x21

subject to Dual Variablesxd1 ≤ x11 + x21 pd1

xd2 ≤ x12 + x22 pd2

x11 + x12 ≤ xs1 ps1

x21 + x22 ≤ xs2 ps2

The symbol N(CS +PS) stands for the net sum of the consumers’ andproducers’ surpluses which have been netted out of the total cost of trans-portation. The first set of two primal constraints states that the regionaldemand must be satisfied with an adequate supply of the final commoditysupplied by the two regions. The second set of the other two primal con-straints states that the total shipment to the two regions from the samelocation cannot exceed the available supply at that location.

On the way to specifying the dual problem, the Lagrangean functionis stated as

L = [a1xd1 − 1

2D1(xd1)2] + [a2x

d2 − 1

2D2(xd2)2] (A12.2.4)


− [b1xs1 + 12S1(xs1)

2]− [b2xs2 + 12S2(xs2)

2]− t12x12 − t21x21

+ pd1(x11 + x21 − xd1) + pd2(x12 + x22 − xd2)+ ps1(x

s1 − x11 − x12) + ps2(x

s2 − x21 − x22).

The relevant KKT conditions deal with all the primal variables:

∂L

∂xd1= a1 −D1x

d1 − pd1 ≤ 0 (A12.2.5)

∂L

∂xd2= a2 −D2x

d2 − pd2 ≤ 0 (A12.2.6)

∂L

∂xs1= b1 + S1x

s1 − ps1 ≤ 0 (A12.2.7)

∂L

∂xs2= b2 + S2x

s2 − ps2 ≤ 0 (A12.2.8)

∂L

∂x11= pd1 − ps1 ≤ 0 (A12.2.9)

∂L

∂x12= −t12 + pd2 − ps1 ≤ 0 (A12.2.10)

∂L

∂x21= −t21 + pd1 − ps2 ≤ 0 (A12.2.11)

∂L

∂x22= pd2 − ps2 ≤ 0. (A12.2.12)

The corresponding complementary slackness conditions can be statedready to use for the simplification of the Lagrangean function

a1xd1 = D1(xd1)

2 + pd1xd1 (A12.2.13)

a2xd2 = D2(xd2)

2 + pd2xd2 (A12.2.14)

b1xs1 = −S1(xs1)

2 + ps1xs1 (A12.2.15)

b2xs2 = −S2(xs2)

2 + ps2xs2 (A12.2.16)

x11pd1 = x11p

s1 (A12.2.17)

x12pd2 = x12p

s1 + x12t12 (A12.2.18)

x21pd1 = x21p

s2 + x21t21 (A12.2.19)

x22pd2 = x22p

s2 (A12.2.20)

By replacing the terms on the left-hand-side of the above complemen-tary slackness conditions in the Lagrangean function, all the linear termsdisappear from it and the final result is the following simplified dual objec-tive function

min L = 12D1(xd1)

2 + 12D2(xd2)

2 + 12S1(xs1)

2 + 12S2(xs2)

2. (A12.2.21)


The dual problem of the spatial equilibrium example discussed in this ap-pendix is given by the minimization of the objective function (A12.2.21)subject to constraints (A12.2.5)-(A12.2.12).

There exists, however, a further and convenient simplification of thedual problem which was suggested by Takayama and Woodland who calledit “purified duality.” The “purification” stems from the elimination of allthe primal quantity variables from the dual specification in such a way thatthe resulting equivalent dual model is written exclusively in terms of pricevariables.

Consider the demand functions (which is equivalent to using the firsttwo KKT conditions (A12.2.5)-(A12.2.6))

pdr = ar −Drxdr , r = 1, 2

and the corresponding inverse functions

xdr = D−1r ar −D−1

r pdr , r = 1, 2. (A12.2.22)

Then, the first two terms of the dual objective function (A12.2.21) can bewritten as

12Dr(xdr)

2 = 12Dr[D−1

r ar −D−1r pdr ]

2 (A12.2.23)= 1

2Dr[D−2r a2

r − 2D−2r arp

dr + D−2

r (pdr)2]

= 12D−1r a2

r −D−1r arp

dr + 1

2D−1r (pdr)

2

= Kdr − [crpdr − 1

2D−1r (pdr)

2], r = 1, 2

where cr ≡ D−1r ar and the Kd

r parameters are constant coefficients.Similarly, consider the supply functions (which is equivalent to using

the KKT conditions (A12.2.7)-(A12.2.8))

psr = br + Srxsr, r = 1, 2

and the corresponding inverse functions

xsr = −S−1r br + S−1

r psr, r = 1, 2. (A12.2.24)

Then, the last two terms of the dual objective function (A12.2.21) can bewritten as

12Sr(x

sr)

2 = 12Sr[−S−1

r br + S−1r psr]

2 (A12.2.25)= 1

2Sr[S−2r b2r − 2S−2

r brpsr + S−2

r (psr)2]

= 12S−1r b2r − S−1

r brpsr + 1

2S−1r (psr)

2

= Ksr + [hrpsr + 1

2S−1r (psr)

2], r = 1, 2

where hr ≡ −S−1r br and the Ks

r parameters are constant coefficients.


By replacing the quadratic terms in the dual objective function (A12.2.4)with their equivalent expressions in equations (A12.2.23) and (A12.2.25)we obtain

min L = −2∑r=1

[crpdr − 12D−1r (pdr)

2] (A12.2.26)

+2∑r=1

[hrpsr + 12S−1r (psr)

2] +2∑r=1

Kdr +

2∑r=1

Ksr

and finally, because the constant parameters Kdr and Ks

r do not enter theoptimization process, the purified duality specification of spatial equilib-rium is stated as

Purified Dual

max L∗ = min (−L) =2∑r=1

[crpdr − 12D−1r (pdr)

2] (A12.2.27)

−2∑r=1

[hrpsr + 12S−1r (psr)

2]

subject topdr ≤ psr′ + tr,r′ .

This purified dual model is stated in more general terms in equations(12.39) and (12.40).


Bibliographical References

Enke, S. (1951). “Equilibrium Among Spatially Separated Markets: Solu-tion by Electrical Analogue.” Econometrica, 19, pp. 40-47.

Samuelson, P.A. (1952). “Spatial Price Equilibrium and Linear Program-ming.” American Economic Review, 42, pp. 283-303.

Takayama, T. and G. G. Judge (1964). “Equilibrium among Spatially Sep-arated Markets: A Reformulation.” Econometrica, 32, pp. 510-524.

Takayama, T. and A. D. Woodland (1970). “Equivalence of Price andQuantity Formulations of Spatial Equilibrium: Purified Duality inQuadratic and Concave Programming.” Econometrica 38, pp. 889-906.

Documents

12. General Market Equilibrium