[Lecture Notes in Economics and Mathematical Systems] Equilibrium Models in an Applied Framework...

Chapter 3

The Planner and the Market:

The Takayama Judge Activity Model

The linear programming formulation of the Leontief input–output model, established

as the linear activity analysis model, represents an advancement in the construction of

applied general equilibrium models, because it introduces a great deal of flexibility

into the basic linear input–output structure. The lack of price-induced substitution

was overcome by the development of the linear activity model. By allowing inequal-

ity constraints and the introduction of an endogenous mechanism of choice among

alternative feasible solutions, the effects of sector capacity constraints and primary

input availabilities may be investigated in the model.

However, the linear programming formulation retains the assumptions of hori-

zontal supply functions (up to the point where capacity is reached) and vertical final

demand functions for each sector as well as fixed proportion production functions.

Hence, the demand for commodities and supply of factors are assumed to remain

constant no matter what happens to prices. In the linear programming framework it is

natural to interpret the shadow prices that result as a by-product of the solution as

equilibrium prices. However, these prices cannot be interpreted as market-clearing

prices of general equilibrium theory because endogenous prices and general equilib-

rium interaction to simulate competitive market behaviour cannot be achieved using

the linear programming specification. Thus, by using a linear programming formula-

tion, without representing a realistic price system in which endogenous price and

quantity variables are allowed to interact, the interplay of market forces cannot be

described properly. These are simplifying assumptions which severely restrict the

usefulness of the linear programming formulation of the input–output model.

In linear programming problems, the solution is guaranteed to occur at one (or

more) of the vertices, of the feasible set. This implies that the optimal solutions are

always to be found at one of the extreme points of the feasible set, and the solution

will constitute a basic feasible solution of the linear programming problem. Conse-

quently, all we need is a method of determining the set of all extreme points, from

R. Noren, Equilibrium Models in an Applied Framework,Lecture Notes in Economics and Mathematical Systems 667,

DOI 10.1007/978-3-642-34994-2_3, # Springer-Verlag Berlin Heidelberg 2013

21

which an optimum solution can be selected.1 However, this constitutes a significant

drawback of the applicability of the model because the linear programming specifi-

cation restricts the field of choice to the set of extreme points. Unlike the points of

tangency in differential calculus, the extreme points are insensitive to small

changes in the parameters of the model. That reduces the attractiveness of the

model for comparative static experiments. In order to include some elements of

flexibility within the system and make the linear programming model more realis-

tic, it is desirable to allow for the inclusion of several resource constraints and to

work on a highly disaggregate level. On the other hand, this will substantially

increase the amount of data required to implement the model. A technique which

removes any of the short-comings mentioned above will greatly improve the

applicability of the model.

For this purpose a straightforward extension of the linear programming model,

incorporating demand by sector and factor supply functions, will be developed. From

a complete set of demand and factor supply functions with only the demand and factor

prices as endogenous variables, it is then possible to compute the set of prices and

quantities that determines an economic equilibrium. The incorporation of demand and

factor supply functions provides a more realistic description of the aggregate market

conditions faced by individual decision makers. The Harrington (1973) formulation

of the Takayama and Judge (1964a, 1964b, and 1971) quadratic programmingmodels

of spatial price equilibrium operate in this way and will be followed to provide a

linear activity model for modelling economic equilibrium. This approach represents a

structure, where the technological data and estimates required to implement the

problem are to a great extent compatible with traditional linear programming models.

3.1 The Quadratic Programming Problem

In the quadratic programming formulation of the linear activity model both the

prices and quantities are determined endogenously within the model. In an

optimisation approach, the model is formulated in terms of the maximisation of

the sum of consumers’ and producers’ surplus.2 Based on empirically generated

demand and supply relations, this formulation of the objective function is used to

replace the utility and welfare functions of conventional economic theory.

Given downward sloping final demand and upward sloping factor supply curves,

relative price changes occur between sectors. Constraints on the model’s solution in

the form of fixed proportion production functions, current capacities and primary

resource availability are retained. Given this specification, the existence of a two

way feed-back in which quantity can influence price and price can influence

quantity for each sector, is developed.

1 The simplex method of linear programming represents such a method.2 See Noren (1987). The numerical tables are also presented in Noren (1991).

22 3 The Planner and the Market: The Takayama Judge Activity Model

The feasible set for quadratic programming problems is completely similar to the

feasible set for linear programming problems. On the other hand, the optimum value

of the objective function might occur anywhere in the feasible set. An optimum

solution may be on the boundary on the constraint region, but not necessarily at a

vertex or an extreme point, as we would expect in linear programming. Hence, the

quadratic programming model must permit consideration of non-basic solutions.3

Consequently, the field of choice extends over the entire feasible set and not merely

the set of its extreme points. In contrast to the linear programming model, we do not

have to work with a highly disaggregated model to increase the number of the

extreme points, and hence, extend the field of choice in the economic model. In the

quadratic programming formulation of the linear activity model, a framework has

been developed, that firstly, attempts to capture the role of prices and the workings of

a competitive market system, and secondly, the solution is not necessarily an extreme

point. The latter property implies that the solution is not so insensitive to small

changes in the parameters of the model. In fact, two of the major shortcomings of the

linear programming model have been overcome.

The theoretical basis of the model that will be presented in this chapter was

outlined in 1952 when Samuelson pointed out that an objective function whose

maximisation guarantees fulfilment of the conditions of a competitive market exists.

Samuelson defined this function as the “net social payoff” to avoid any association

with conventional economic concepts. Samuelson was the first to mention the possi-

bility of maximising the sum of consumers’ and producers’ surpluses to compute a

competitive equilibrium through an optimising model by showing how the problem of

partial equilibrium within spatially separated markets, as formulated by Enke (1951),

could be solved through mathematical programming. In the 1964 papers, Takayama

and Judge using linear price dependent demand and supply functions to define an

empirically oriented “quasi-welfare function”, and hence, extended the Samuelson

formulation so that the spatial structure of prices, production, allocation and con-

sumption for all commodities could be determined endogenously within the model

with quadratic programming. This work was followed by articles by Plessner and

Heady (1965), Yaron et al. (1965), and Plessner (1967), which contributed to the

formulation of the quadratic programmingmodel. In the development of the quadratic

input–output model, Plessner’s (1965) formulation of the Walras-Cassel model as

a quadratic programming problem has been of particular methodological interest.

Harrington (1973) followed the contribution of Plessner by showing how an

input–output model can be solved as a quadratic programming model, hence the

quadratic input–output model. The resulting quadratic input–output model is a theore-

tical improvement over the Leontief input–output model by the direct inclusion of

the pricing mechanism endogenously in the model. Thus, the methodological contri-

bution is the incorporation of the pricing mechanism in the programming model.

3 The main disadvantage of most quadratic programming algorithms is the large number of

calculations required for convergence to a solution. This implies that the quadratic programming

formulation is considerably more difficult to solve numerically than the linear programming

model.

3.1 The Quadratic Programming Problem 23

The model is a linearised version of the Walras-Cassel general equilibrium model

(linearised factor supply and commodity demand functions) which utilises the basic

Leontief input–output structure as a production relationship. Given the linearised

factor supply and commodity demand functions, both the prices and quantities are

determined endogenously. In technical terms, the shadowprices are incorporated in the

objective function. The solution of the quadratic programming problem can be

characterised as a simulation of market behaviour under the assumption of

competition.

The quadratic programming model presented in this chapter is applied for the

evaluation of the pattern of domestic production and trade of the Swedish economy.

The evaluation of the pattern of comparative advantages of the Swedish economy is

carried out as an analysis of the choice between import and domestic production in

a temporary equilibrium framework with exogenously given world market prices,

exports and domestic production capacities.

3.2 Specification of the Model

In developing the model, Hotelling’s (1932) total benefit function, based on empir-

ically generated demand and supply relations, is used to replace the utility and

welfare functions of conventional economic theory. We assume aWalrasian system

of private expenditures and factor supply functions, where the demand and supply

quantities are given as linear functions of the commodity price pj and factor price wh

respectively. Given this specification, we treat the aggregate demand and factor

supply functions as if they could be generated by a single representative individual.

To incorporate price-dependent demand and supply functions and derive an

economic equilibrium, mathematical models can be formulated with an objective

of maximising the sum of consumers’ plus producers’ surplus. Consumers’ plus

producers’ surplus or net social benefit is measured as the area between the

compensated demand and factor supply curves (after adjustment to remove income

effects) to the left of their intersection. The most obvious reason for the use of this

objective function is that its behavioural implications are consistent with theoretical

economic behaviour of the participants by sector. An important, although obvious

point, is that sector commodity supply curves and factor demand curves are not

required as they are already accounted for in the system by the fixed factor

proportion production functions calculated from the input–output table.

When this objective function is maximised, subject to the fixed proportion pro-

duction functions, a perfectly competitive equilibrium solution results.4 Constraints

reflecting the production capacities of the production sectors may alter the result, but

in a manner which continues to maximise producers’ and consumers’ surplus. Thus,

4 Takayama and Judge (1964a) present an existence proof based specifically on a mathematical

programming model of a space-less economy. This proof establishes the existence of a perfectly

competitive equilibrium in a mathematical programming framework of the general equilibrium of

an economy.


the market is viewed as a mechanism for maximising the sum of producers’ and

consumers’ surplus. In technical terms, the shadow prices are incorporated in the

objective function. Hence, the solution of the quadratic programming problem can be

characterised as a simulation of market behaviour under the assumption of competi-

tion. Within the competitive framework, it is assumed that each domestic production

sector and the individual groups of consumers are composed of many competitive

micro units, none of which can individually influence quantity or commodity price.5

The concept of consumer’s surplus is defined as the difference between the

maximum amount the consumer would be willing to pay for the commodity and

what he actually does pay for it.6 In equilibrium, the consumption of the i:thconsumer is at the level at which the willingness to pay for the last consumed

unit is equal to its price.

The factor supply curve is upward sloping and measures the marginal cost of the

factor specific to the sector. Diagrammatically, the producer’s surplus is measured

as the area below the price and above the factor supply curve.7 This area has to be

identified with what Marshall (1925) called quasi-rent. Marshallian quasi-rent is

defined as the excess of the price over the marginal cost of the factor (labour) which

accrues to the producer or the factor owner as a profit in the short-run. Within the

short period, during which capital retains its sector specific form and the other

factor is fixed in price, the area above the supply curve as a measure of quasi-rent is

clearly relevant. Quasi-rents generally arise either because it takes time for new

firms to enter or because certain factor prices may be fixed over the short-run.

Generally, the term producer’s surplus is somewhat misleading, because it does not

identify which particular factor, and hence, factor owner to whom the rents are to be

imputed.8 Anyhow, economic rent can be defined to provide a measure of the

welfare change arising from a movement of factor prices, commodity prices

being constant; in exactly the same way that consumer’s surplus provides a measure

of the welfare change arising from a movement in commodity prices, factor prices

being constant.

In order to manage this problem computationally, we assume that linear

functions are acceptable approximations for the private consumption and factor

5 In this context the artificial nature of the objective function must be emphasised. As Samuelson

(1952) noted “This magnitude (the objective function) is artificial in the sense that no competitor in

the market will be aware of or concerned with it. It is artificial in the sense, that after an invisible

hand has led us to its maximisation, we need not necessarily attach any social welfare significance

to the result” (p. 288).6More rigorously, the difference between the money value of the total utility of the consumer’s

purchase and the money he actually pays for it.7 Strictly speaking, the producer’s surplus is the difference between total revenue from his sales,

minus the area under his marginal cost curve.8 Under perfect competition, the producers’ surplus is captured by the factor owner (owners of

specific capital equipment) in form of rent. In this model all the rents must be paid to the

households. Thus, it is possible to have a producers’ surplus and yet zero profit in competitive

equilibrium.

3.2 Specification of the Model 25

supply functions. This specification results in a quadratic net-benefit or, in the

terminology of Takayama and Judge, quasi-welfare function, and market equilib-

rium may therefore be computed by the techniques of quadratic programming to

obtain the optimum prices and quantities.

The final demand and factor supply functions are specified by the Cassel-Wald

(1951) specification, i.e. demand and factor supply functions are functions of

demand respective factor supply prices alone. As demonstrated by Harrington

(1973) the demand and factor supply functions specify, together with the

specifications of the industry supply system, a consistent system without loss of

generality of the Dorfman et al. (1958) specification of the Walras-Cassel model of

a perfectly competitive economy.

To understand the nature of the programming formulation,9 let the consumption

(private consumption) of the final commodity xj be a linear function of price such that:

xj ¼ γj � pj Σi; νij (3.1)

where we assume γj > 0 and νij > 0 for all j > 0. xj is the quantity of demand of the

desired commodity j, pj is the price of the sector’s product, γj is the intercept term,

the νij represents the slope coefficient. Note that the demand function is independent

of the sector activity, i.e. the income variable is dropped from the demand func-

tion.10 Alter-natively, the inverse of the demand-quantity function11 above is the

demand-price function:

pj ¼ αj � Σi; ωij xij (3.2)

Where we, as for Eq. 3.1, assume αj > 0 and ωij > 0 for all j > 0. αj is the

intercept term, ωji represents the slope coefficient and xij the i:th consumer’s

demand of the desired commodity. The matrix of slope coefficients is assumed to

be symmetric and positive definite for all j. The demand functions are continuous,

differentiable and monotonically decreasing functions of the consumed quantity xj,i.e. ∂(Dj((xj))/∂xj < 0 for all j > 0. The adjustment of prices according to the

9A general survey of techniques for formulation and solving multimarket general equilibrium

models in the mathematical programming framework have been spelled out in detail by Takayama

and Judge (1971).10 This formulation does not incorporate the income generated by the sector as a simultaneous

shifter of the model’s commodity demand function. If the sector under consideration is small

relative to the entire economy, this should not be a serious problem. However, if a major sector or

set of sectors is of interest the income generated within that sector (or sectors) may have a major

impact on aggregated consumer demand.11 In making the model operational, inverted demand and supply functions are applied. The

inversion simplifies the mathematical exposition of the model and the interpretation of the

solutions rather than the direct demand and supply functions. Dorfman, Samuelson and Solow

claim that this inversion is not admissible (Dorfman et al. 1958, p. 352). However, their argument

does not apply to the linearised Walras-Cassel model.


market means that the pj’s may be regarded as functions of the xj’s, in spite of

individual consumers considering the pj’s fixed.The area under these demand curves and above the price represent consumers

surplus for each desired commodity. Integrating the set of the demand curves to

determine the area under the curves, a market-oriented net benefit function, denoted

by W, for the economy (comprising all desired commodities) may be specified as a

strictly concave quadratic function:

Wðx�Þ �ðx�0

Σj

αj �Xi

ωijxij

!dxj (3.3)

Where x* is a vector. Given the specification above, ωij � ωj. Hence:

Σi; ωjixij ¼ ωjΣi; xij ¼ ωjxj (3.4)

This results in:

Wðx�Þ �ðx�0

Σjαj � ωjxj� �

dxj (3.5)

Dropping the superscript, we obtain:

WðxÞ � Σj; αjxj�1=2Σj; ωjx2j (3.6)

More compactly, the function (3.6) may be written as:

WðxÞ � α0x�1=2x0Ωx (3.7)

where the matrix of slope coefficients is a diagonal, with zeros as off-diagonal

elements.

Similarly, we assume that the supply of factor quantities rih (primary commodities)

depends on the market prices of its productive services. Hence, let the inverse factor

supply function of commodity h (rih the supplied quantity of the primary commodity

h owned by the i:th consumer) be given by:

wh ¼ βh þ Σi; ηihrih (3.8)

Where we usually assume βh > 0 and ηih > 0 for all h > 0. wh is the price of the

primary commodity h. rh is the supplied amount of the primary commodity h. βh isthe intercept term and ηih represents the slope coefficient. The matrix of slope

coefficients is assumed to be symmetric and positive definite for all h. The supplyfunctions are continuous, differentiable and monotonically increasing functions of

the supplied quantity rh, that is ∂(Sh((rh))/∂rh > 0 for all h > 0.


The area under the factor supply curves (comprising all factor supply curves) is

total cost and may mathematically be written as:

Wðr�Þ �ðr�0

Σh

βh þXi

ηihrih

!drh (3.9)

According to the specifications above, we have here a model which will simul-

taneously determine the market demand price on final commodities (consumed

quantities of xj) together with the input market equilibrium prices on its primary

commodities (factor supplies of rh).The sum of producers’ and consumers’ surplus is then found by computing the

difference between the area under the final demand curves and the area under the

factor supply curves.

Wðx; rÞ �ðx�0

Σjðαj � ωjxjÞdxj �

ðr�0

Σhðβh þ ηhrhÞdrh (3.10)

Thus, total net benefit (comprising all desired commodities and all factor supply

curves) for the stipulated economy is the line integral of individual demand and factor

supply relations of which consumer’s and producer’s surplus is a part. The model can

actually be looked on as combining Koopmans (1957) linear production model with

Walras’s conception of the market, in a quadratic programming formulation.

The matrix of substitution terms in the demand and factor supply functions must

be symmetric. These conditions are the so called integrability conditions. They playan important role in the formulation of the model. The integration process is known

to be feasible when certain symmetry conditions are satisfied by the functions being

integrated, provided that these functions are sufficiently smooth. Hence, the sym-

metry conditions are often simply called the integrability conditions. Given the

symmetry conditions, a utility and cost function exists from which a consistent

demand respective supply function can be derived.12

If the substitution termmatrices do not conform to the assumption of symmetry the

integrability conditions are not satisfied, then we are unable to construct the net

benefit function given above. From an application standpoint, this presents

difficulties. However, the implications of this requirement vary depending upon

whether we are concerned with supply or demand. The classical assumptions of the

theory of production yield the symmetry conditions of the supply functions (Zusman

1969). Takayama and Judge (1971) have pointed out that if the integrability

conditions do not hold, then the system is still solvable and interpretable in terms of

net social monetary gain which is defined as total social revenue minus total social

12 For details, see Varian (1984), pp. 135–139.


production cost. Only the connection to utility maximisation and cost minimisation is

lost by violation of the integrability conditions, not the solvability of the system.13

The symmetric condition is a necessary and sufficient condition for what is known

as path-independence. This implies that the cross-price effects (compensated) are

equal over all commodity pairs. In the present context, this means simply that in

whatever way the order of price changes is calculated the adopted measure of

consumer’s and producer’s surplus for the combination of these price changes is

uniquely determined. The symmetry of the substitution termmatrices (Slutsky terms)

is exactly the condition under which the integral W(x,r) is solely dependent on the

terminal price vectors, and thus, regardless of the order in which the price changes are

taken, i.e. independent of the path. However, given a demand function including the

income variable, the path-independence condition requires that the income

elasticity’s are identical across all commodities of interest. Given the property that

the weighted sum of the income elasticity’s, where the weights are the shares of

income spent on each commodity, sums to one, all income elasticity’s are equal, and

thus, equal to one.14 Unitary income elasticity’s are the demand functions derived

from homothetic indifference maps. This implies that all Engel curves are straight

lines through the origin, i.e. at all income levels, a constant proportion of total

expenditures is allocated to each commodity.

3.2.1 The Introduction of Foreign Trade

Most commodities can be supplied not only by domestic production, but also by

importation. A standard approach is to specify imports as an alternative source of

supply of commodities classified by the input–output sectors (Technically as an

alternative column in the input–output table). A different approach is to specify

imports as a primary input that is not produced in the economy (Technically as a

row in the input–output table).

In the first approach, imports are specified as competitive, here denoted Mj,commodities which can be produced within the country but which are, as an

alternative to domestic production, also imported. The imported commodity is

here viewed as a perfect substitute for the domestically produced commodity.

Consequently, those imported commodities which the agents are free to select for

domestic production are classified as competitive imports. In this context, any

particular commodity classified as competitive imports is assumed to be tradable

in the international market, and has identical characteristics, whether it is produced

at home or abroad. Formally, competitive imports are treated as if they were

13 Takayama and Judge (1971), pp. 121–126 and pp. 233–257.14 The path-independence condition is also fully satisfied if the income elasticity’s of demand of all

commodities are zero (McCarl and Spreen 1980). In this model the income variable is dropped

from the demand function. Thus, the path-independence condition is satisfied.


delivered to the corresponding domestic industries and then distributed by these

industries together with the domestically produced amounts. Thus, the inputs aijZjstate the sums of produced and imported amounts, and not merely the produced

amounts.15

In the second approach imports are specified as non-competitive, here

denoted mqjZj, and instead of perfect substitutes for domestic production, imports

are treated as a complementary input, completely different from domestically pro-

duced commodities. This type of imports consists of commodities which cannot be

produced within the country. Non-competitive imports including predominantly

those commodities which are technically infeasible, and commodities whose produc-

tion is economically unviable because of the present market situation compared with

their minimum scale of production. In our notation, mij denotes the input coefficient

of non-competitive imports and Zj the extent of which the process j is utilised.When a commodity is imported there is an outlay of foreign currency per unit of

imported amountMj respectivemijZj. If PW denotes the world market price in foreign

currency, �PWjMj and �PWjmijZj ex-press the outlay of foreign currency. On the

other hand, when a commodity is exported, denoted Ej, there is a receipt, expressedby PWjEj, of foreign currency earned per unit of exported amount Ej. Consequently,foreign currency is here an intermediate commodity, where the import process

requires foreign currency as input, and foreign currency is the output of the export

process. Thus, in this context there are also given resources, but of foreign currency

only. These resources are made up of net export earnings plus net foreign capital

inflow, denoted F. In this model the amount of net foreign capital inflow is assumed

exogenous. Given the exchange rate, denoted ER, it follows that foreign trade can bedescribed as to be carried out by means of processes with fixed relations. Compatible

with the assumption made for domestic production, it will be assumed that an import

process involves importation of one single commodity. This assumption re-places, as

for domestic production, an optimisation requirement.16 Consequently, we also

assume that an export process leads to the export of one commodity only.

The effects of transportation costs and tariffs are taken into consideration by

including transport costs and tariffs into import prices (tariff augmented world

market prices). Hence, the currency spent on importing a unit of a commodity is

generally somewhat larger than the amount earned by exporting it.17 If it were

smaller, this would mean that the price in the exporting country would exceed the

price in the importing country, which is not compatible with interregional general

equilibrium. In this model world market prices of traded commodities are assumed

to be given. The assumption of given world market prices (the small country

15 The exposition in this section is based on and similar to that of Werin (1965).16 Optimisation implies that the import process, given the smallest currency outlay, as well as the

production process, given the best technique available, is chosen.17 Statistically, imports are calculated in c.i.f. prices and exports in f.o.b. prices. Given this

specification, the currency outlay for imports will not be proportional to the existing world market

prices. This implies that the foreign exchange constraint will not correctly reflect the conditions

prevailing on the world market.


assumption) implies that the country is confronted with infinitely elastic demand for

its exports and supply of its imports, so what the level as well as the pattern of

imports and exports may be endogenously determined only subject to the foreign

exchange restriction.

Considering the assumptions made, the production system is re-presented by an

input–output model extended to include foreign trade as an alternative to domestic

production. Each commodity can now in principle be supplied by two different

activities. One of them is the production activity, the other the import activity,

which is the result of the outlay of foreign currency. This means substitution

possibilities between inputs for the supply of various commodities. A linear activity

model which takes foreign trade into account is, in certain respects, quite similar to

a neoclassical model.18

The foreign exchange constraint (Eq. 3.11) restricts the amount of foreign

currency that can be spent on imports. The supply of foreign currency is generated

through exports and net capital inflows. PWj denote the world market price of each

commodity classified by the input–output sectors. In this model, imports will betreated both as an alternative (and identical) source of supply of commodities

classified by the input–output sectors and as another input (composite) that is not

produced in the economy, analogous to capital and labour. Technically, competitive

imports are placed outside the inter-industry part of the input–output table, specified

by sector of origin, and non-competitive imports are kept within the inter-industry

part of the input–output table, specified by sector of destination.

Σj; Σi; PWjmijZj þ Σj; PWjMj � Σj; PWjEj þ F (3.11)

3.3 The Programming Formulation

Given the net benefit function, and the constraint set as specified above the problem

takes the following form, i.e. maximise:

Wðx; rÞ � Σj; αjxj � 1=2Σj; ωjx2j � Σh; βhrh � 1=2 Σh; ηhrh

2 (3.12)

Subject to

Zj þ Σj; mijZj þMj � Ej � Σj; aijZj � ΣiDij (3.13)

Σj; bhj Zj � Σi; rih (3.14)

18However, if the model does not include any further restrictions on exports and imports, the

assumption of constant returns of scale in production together with endogenous choice in trade

may lead to an unrealistic specialisation in either trade or domestic production.

3.3 The Programming Formulation 31

cij Zj � Kij (3.15)

Σj; Σi; PWjmijZj þ Σj; PWjMj � Σj; PWjEj þ F (3.16)

Zj � 0; Mj � 0; Ej � 0; Dij � 0; rih � 0; Kij � 0

Making use of the Kuhn-Tucker conditions, the necessary conditions which must

hold for the optimum xoij; roih; Zoj ; Mo

j ; poj ; woh; voij; ERo to be a non-negative saddle

point of the Lagrangean, are:

@Lo

@xij¼ αj � ωjx

oij � poj � 0

�00 � < 0 ) xoij ¼ 0 ð3:17Þ

@Lo

@rih¼ �βh � ηhr

oh þ wo

h � 0

�00 �< 0 ) roih ¼ 0 ð3:18Þ

The constraints of the domestic activities will be the same as in the linear

version. See the discussion in Chap. 2, Sect. 2.5. However, the inclusion of foreign

trade implies two other constraints in the quadratic model. The new constraints are

discussed below as constraint (3.23) and (3.24).

For a given vector of pre-equilibrium prices pj and wh, these prices are revised

until the shadow prices poj and woh associated with the commodity balance Eqs. 3.13

and 3.14. If so, the solution is an equilibrium solution. Thus, the dual variables from

Eqs. 3.13 and 3.14 equals the maximum price the consumers are willing to pay for

the consumption of the commodities available to them, and the minimum price at

which they are willing to supply labour service from their initial endowment of

leisure. If not, the demand and supply prices are revised and start a new function

evaluation. In this way shadow prices have a feedback effect on the demand and

supply prices specified in the objective function. As stipulated above, this is what

leads to the similarity between the market mechanism and the optimisation formu-

lation of the model. A planning authority can use the shadow prices generated by

the plan to decentralise decisions because they are signals of relative scarcity of the

constraint to which they are attached. However, when imposing a number of

additional ad-hoc constraints to make the solution more realistic, the constraints

result in distortions in the shadow price system. If such constraints can be justified

as additional system constraints that define a reasonable notion of economic

equilibrium, there is no theoretical problem to interpret the solution as reflecting

the operation of a market system (Taylor 1975).

Starting with the shadow demand price, denotedpoj ,when the consumption of the j:

th commodity is positive, must exactly be equal to the demand price pj, the maximum

price the consumers are willing to pay for the consumption of the quantity of the


http://dx.doi.org/10.1007/978-3-642-34994-2_2

commodity xj, which in turn are generated by the optimum demand quantity xoj .

However, if xoj ¼ 0, the shadow demand price is greater than or equal to the demand

price. Thus:

if xoj > 0; then αj � ωjxoj ¼ poj ð� 0Þ; (3.19)

if xoj ¼ 0; then αj � ωjxoj � poj ð� 0Þ; (3.20)

for all j.The factor supply equilibrium stipulates, that when the optimum supply quantity

of the h primary commodity is positive, the shadow supply pricewoh must exactly be

equal to the supply price (factor cost) wh, the minimum price at which the resource

owners (consumers) are willing to supply rh, where roh are generated by the optimal

supply quantities roh. However, if roh ¼ 0, the shadow supply price is less or equal to

the supply price. Thus:

if roh > 0; then βh þ ηhroh ¼ wo

hð� 0Þ; (3.21)

if roh ¼ 0; then βh þ ηhroh � wo

hð� 0Þ; (3.22)

for all h.The individual country becomes a price taker in the small open economy model,

because the world market prices of traded commodities are assumed to be determined

in the international market. The domestic economy will at the optimum adjust to the

relative world market price ratio. In a free trade economy,19 the direction of trade will

be determined by the requirement of equality between the domestic and the world

market price ratio. It is the difference between these ratios that leads to trade. Thus,

efficiency requires equality among world market prices, domestic prices, and produc-

tion costs. Since the world market prices are assumed to be given, these prices

determine the domestic shadow prices of tradables.

@Lo

@Mj¼ poj � ERoPWj � 0

�00 �< 0 ) Moj ¼ 0 ð3:23Þ

Next condition (3.23), relates to the alternative way of supplying a commodity,

namely by importation. Condition (3.23) state, that when the optimum import

19 Using the small-country assumption and also assuming that domestically produced and imported

commodities are perfect substitutes this specification leads to extreme specialisation in either trade or

domestic production whenever there are no established domestic capacity constraints. The sector-

specific capacity constraints in this model are used to limit this problem. This implies that the

domestic shadow price system is no longer a simple reflection of world market prices.

3.3 The Programming Formulation 33

activity Moj is positive, the shadow price poj of the imported commodity must be

exactly equal to the value (cost) of the outlay of foreign currency. If the shadow

price poj is lower than the imputed cost of importing the commodity no importation

of the commodity will take place. Production will expand until domestic production

costs rise to the world market price level, converted into a domestic price by the

shadow exchange rate ERo. Consequently, as long as domestic production costs are

lower than established world market prices, it will be profitable to expand domestic

production for exports. On the other hand, if the domestic price is greater than the

world market price, the commodity will not be produced. If the country can always

import at a cost of poj it is never optimal to produce at a marginal domestic cost

higher than poj . This leads to excess domestic capacity which is reflected by a

shadow price of zero for installed capacity. Since, our model only contains

tradables; the shadow exchange rate is simply defined as a conversion factor from

foreign exchange units to domestic commodity units, and has no significance in

terms of relative domestic prices.20

Finally, condition (3.24) below state, that if the optimum price of foreign exchange

is positive, the foreign exchange equilibrium requirement for the economy is exactly

met. Note, that for any positive activity the shadow exchange rate ERo can never be

zero because it is always possible to use foreign exchange to purchase commodities

from abroad.21 If the shadow price of foreign currency is zero at the optimum no

activity (production and importation) take place in the domestic economy. Given this

specification, there is the assumption of a flexible exchange rate system, in which

exchange rate adjusts continuously so as to maintain the foreign exchange constraint

in equilibrium.22 However, specifying tariffs on currency outlay for imports implies

that the domestic shadow prices would reflect the existing tariff structure, and the

tariff-ridden domestic market prices will not be proportional to the existing world

market prices. Hence, the foreign exchange constraint will not correctly reflect the

conditions prevailing on the world market.

@Lo

@ER¼ Σj;PWjE

oj þ F� Σi; Σj;PWjmijZ

oj � Σj;PWjM

oj � 0

�00 �> 0 ) ERo ¼ 0 ð3:24Þ

In the closed economy the basic technological and demand variables determine the

domestic shadow price system.23 However, the situation is quite different in a free

20With non-tradables, the shadow price of foreign exchange will reflect the relative scarcity of

tradables with respect to non-tradables.21 For a discussion of this mechanism, see Dervis et al. (1982), pp. 75–77.22 Assuming given world market prices, an increase in domestic prices implies a depreciation of

home currency. Conversely, a decrease in domestic prices implies an appreciation of home

currency. See further, Sodersten (1980), pp. 315–328.23 The discussion that follows is based on Dervis et al. (1982).


trade economy where the domestic market is small in relation to the world market.

Given the assumption of perfect substitutability between imported and domestically

produced commodities, the small-country assumption implies that the individual

country becomes a price taker facing exogenous world market prices. The theory of

international trade suggests that, as far as some commodities are actually imported or

exported, the domestic shadow prices among them tend to converge to their relative

world market prices.24 Consequently, world market prices determine the domestic

shadow prices of tradables, and a given commodity has (at equilibrium) the same

price whether it is imported or produced domestically. Hence, whereas supply and

demand determine domestic shadow prices in a closed economy, they will adjust to

world market prices in the small open economy.

3.4 A Temporary Equilibrium Specification

The static model as presented above has no formal link between capital formation

and production capacity. Capital commodities are assumed exogenous without any

correspondence to the effect that is created by the supply of investment from sectors

producing capital commodities (investment in final demand). However, a tempo-

rary equilibrium specification endogenises investment and considerably extends the

requirement of consistency in the model. The period output of the capital stock

requirement is inserted as a predetermined variable for the next period optimiza-

tion.25 Once capital stock requirement by sector of destination is established, its

sectoral allocation into a demand for investment commodities by sector of origin

must be specified.

Operationally, the solution for each period is used to create the next period’s

model parameters. Thus, the model is of the temporary equilibrium type. It will

solve the market for equilibrium prices and quantities for one period and then add

the solution obtained to the predetermined variables that are needed to obtain the

market equilibrium solution for the next period. The model does not take into

account future markets despite the fact it explicitly considers time. There is no

inter-temporal optimization26 and the agents have no expectations about future

prices. This concept of equilibrium as static and temporary implies that we are more

interested in the outcomes of the adjustment that yields a new temporary static

equilibrium position than in the dynamics of the adjustment process itself.27

24 Differences may exist due to transportation costs and tariff rates.25 Given the specification of the model, also private consumption is inserted as a pre-determined

variable for the next period optimization.26 In intertemporal models, agents have rational expectations and future markets are considered

when optimizing. Endogenous variables follow an optimal path over time and there are no

incentives to deviate from this path at any point of time.27 Hence, we can overlook the issue of adjustment.

3.4 A Temporary Equilibrium Specification 35

Investment is made up of two parts, replacement investment and net investment.

Replacement investment is that portion of the total which exactly maintains the

capital stocks while net investment is that portion which depends on the level of

demand. In this specification, only net (private) investment in buildings and

machinery is considered. Logically, we disregard depreciation. Another component

of capital formation is inventories. However, the model treats inventories as an

exogenously given component of final demand, and thus, does not incorporate

inventories in the investment concept.

The change in capital stock is by definition the amount of investment. As long as

domestic demand is unchanged, the capital stock is adequate and no investment is

needed. Increases in domestic demand, however, call for additional capital and net

investment is positive. Formally, we assume investment (given the assumption of

full capacity) to be linearly dependent on the current period’s request for newcapacity. This implies that investment adjusts immediately to changes in capacity

requirement within a single period.28 Nevertheless, there is certainly reason to

suspect that in the real world firms do not respond immediately. Hence, it is

assumed that each period is long enough for relative prices to adjust to clear

markets. In quantitative terms, the request for capital commodities by sector of

destination ΔKj is translated into a demand for investment commodities by sector of

origin Ik (producing sectors of capital commodities). Thus we have

IiðtÞ ¼ ΣjτkjΔKjðtÞ (3.25)

Where τkj denotes the matrix of sectoral investment allocation shares, i.e. the

proportion of capital stock in sector j originating in sector k. Note that Στij ¼ 1 for

all j (summation is taken over i). The matrix of sectoral investment shares is

compiled by the Ministry of Finance for the 1984 Medium Term Survey Model

of the Swedish economy.29

It is important to note that the model, in this version, only considers positive net

investments. In other words, given a decrease in the capital stock requirement by

sectors of destination (ΔK < 0) the net investments by sectors of origin are zero.

For this alternative, only sectoral capital stocks are adjusted (scrapping) for the next

period optimization. Moreover, fixed coefficients are used to allocate investment

among sectors. Thus, profitability across sectors is assumed fixed over time. This

implies that we have no allocation process explicitly modelled, in which investment

gradually adjust to equalize profitability across sectors. Hence, the workings of

financial markets in the investment allocation process are ignored. Technically, the

capital stock in each sector is a well defined aggregate of various commodities with

28 This is the famous accelerator principle. In its simplest form, the accelerator rest upon the

assumption that the firm or industry at each level of distribution seeks to maintain its optimal

capital stock at some constant ratio to sales.29 SOU 1984:7, LU 84 (The 1984 Medium Term Survey of the Swedish Economy), Appendix 17,

Table 2:18. Only 9 sectors produce investment commodities for domestic capacity expansion.


a fixed compositional structure (by sector of origin). Finally, there are assumed to

be fixed incremental capital-output ratio by sectors.30

3.5 Empirical Findings: Applications

As stipulated above, the model works stepwise from period to period, and solves the

market for prices and quantities. The solution for each period (four periods in total)

is used to create the next period’s model parameters. Hence, a sequence of

equilibria can be achieved. The period output of capital stock requirement, invest-

ment demand and private consumption are inserted as predetermined variables for

the next period optimization.31

The point of departure for the experiments below (here named applications)

is the version of the model which describes the techno-logical conditions,

labour costs, capacities and estimated demand relations representing the Swedish

economic situation in the year 1980 (benchmark equilibrium data set).32 This year

is selected since it con-forms with data availability, and capacity utilization during

the whole of 1980 on the average can be characterized as normal full capacity.

Thus, the 1980 data provide a comparative benchmark for the experiments in this

chapter. In all solutions, the same maximand is used, i.e. maximize the consumers´

surplus (Eq. 3.3), subject to the constraints (Eqs. 3.13, 3.14, 3.15, and 3.16). Given

the assumptions above, a foreign payments imbalance cannot arise. Moreover, we

assume that the labour constraint (Eq. 3.14) is binding, i.e. labour resources are used

to the maximum availability. In all solutions the total supply of labour resources is

given exogenously and assumed perfectly mobile and free to flow among all sectors

of the economy. Hence, labour moves across sectors until the value of its marginal

product is the same everywhere. This assumption, the value of that marginal

product of labour are equalized in all uses in equilibrium, permits labour payments

data by industry to be used as observations on physical quantities of labour in the

determination of parameters for the model.

In general terms, adjustment to structural equilibrium is a process where profit-

ability in the different sectors will adjust to a “normal” level of profitability for the

economy as a whole. For sectors where profitability is high relative to this normal

level, the adjustment to equilibrium implies an increase in domestic production

relative to other sectors. On the other hand, a sector where profitability is low

relative to the normal level, an adjustment to equilibrium implies a decrease in

30 The temporary equilibrium approach used in this study does not imply that the underlying

economic system is viewed as discrete. Instead, the discrete moments are simply approximations

(artificial to some extent) of the essentially continuous system being modelled.31 Adjustment costs for the installation of capital are not considered.32 The model of the Swedish economy comprises 24 sectors. These are defined in the Appendix, in

accordance with both the Standard Swedish Classification of Economic Activity (SNI) and the

code for the ADP system for the Swedish National Accounts (SNR).

3.5 Empirical Findings: Applications 37

domestic production relative to other sectors. Thus, a development which implies

that a country adjusts to its comparative advantages33 is characterized as an

adjustment towards equalizing the relative profitability between sectors. The results

of this adjustment are reflected in the direction of domestic production.

In technical terms, the domestic shadow prices adjust to the exogenous world

market prices in this model. Thus, the concept of a normal level of profitability for

the different sectors is determined by the relative world market prices. If the

domestic shadow price is greater than the world market price, the domestic produc-

tion of the commodity relative to other sectors will fall. If it is not possible to reduce

domestic production costs to the level of world prices, the commodity will be

imported altogether. On the other hand, if the domestic shadow price is lower

than the world market price, domestic production relative to other sectors will

expand at the expense of imports until domestic costs rise to the level of world

market prices.34 If this equality is not satisfied in the case when the adjustment to

equilibrium implies a zero import level, it would be profitable to expand domestic

production for exports.

Generally, due to the assumed linearity of the underlying technology, the

solution in the model imposes that fewer commodities will be produced domesti-

cally, but in increased quantities in the least-cost sectors. On the other hand, the

specialisation will lead to an increasing amount of import in the high-cost sectors.

In all experiments, it is the difference between the world market prices (here

assumed to be given)35 and the pre-trade domestic commodity transformation

rates that leads the model to take part in trade.

To obtain a reasonable pattern of specialisation, exports are assumed exogenous.

As exogenous values of exports we have maintained the 1980 figures. By this

assumption extreme specialisation is prevented. Unfortunately, these constraints

reduce the experimental attractiveness of the model.

Given the model specification, the equilibrium data of the former period provide

a comparative benchmark for each experiment (four experiments in total).36 Appli-

cation 1 is considered as the first period. It is important to emphasize that the results

have been obtained under strong simplifying assumptions. The results of the

experiments are presented in the Appendix 2 (Tables 3.3, 3.4, 3.5, and 3.6).

As a starting point for the experiments we assume an increase in the sectorally

fixed capital stocks by 10 %. This implies that domestic resources may be shifted to

33Given two sectors 1 and 2, the economy has a comparative advantage in sector 2 if the pre-trade

ratio of sector 2 costs to sector 1 costs is lower than the world price ratio.34 Following Norman (1983) a domestic sector is competitive if (and only if) its marginal cost is

lower or equal to its foreign competitor, measured in the same currency. To be compatible with the

concept of comparative advantage, and hence meaningful, marginal cost is here defined as long run

marginal cost. This implies that the concept of marginal cost includes payment to factors that are

fixed in the short run, e.g. capital.35 The world market prices are specified as unity prices.36 The first experiment (application 1) provides the benchmark data for the second experiment

(application 2) and application 2 provides the benchmark data for the third experiment (application 3).


the lowest-cost sectors (given the capacity restriction) and thus increase the effi-

ciency in resource allocation. Logically, the model chooses to import in some

sectors (Sector definitions in Appendix 2, Table 3.2) rather than utilize the existing

capital stock. As expected, we obtain an increase in engineering (15) and a total

contraction of the shipyards (16). Moreover, the result obtained shows a decline of

domestic production in the basic metal industries (14). As specified above,

the sectoral demand for capacity expansion, evaluated in the former period (appli-

cation 1), is translated into investment by producing sectors in the current period

(application 2). In this connection, the increase in some sectors of the index

representing sectoral demand prices should be noted. The demand prices

(Tables 3.3, 3.4, 3.5, and 3.6, column 13) of the private consumption variables

are expressed in terms of an initially established index, assigned as 1,000. The

explanation for this increase in demand prices is that investment required for

capacity expansion (given as input from application 1) have increased for most

sectors producing capital commodities. Consequently, in some sectors a decrease

(crowding out) of other demand components (here, private consumption only) is

necessary to make capacity expansion possible. At the beginning (application 1 and

application 2) the request for capacity expansion is considerable. However, a

continuing fall in mobility, due to the limited supply of labour resources (measured

in terms of wages), increasing capital stocks in the investment sectors, and the

linear specification of the model, will in the long run reduce the demand for

capacity. The diminishing welfare effect, due to reduced potential in resource

allocation, is the main factor behind this development. Thus, in the next two

experiments (application 3 and 4) it is quite obvious that the demand for net

investment by sectors will fall. These calculations are presented in Table 3.1.

Capacity expansion and the process of structural transformation is restricted to

the existing structure of production. The technological structure is kept the same.

Not unexpected, the results presented in Table 3.1 indicate that the resource

transformation process alone is not sufficient to sustain a high rate of growth in

industrial real capital formation. Successively increasing investments in new tech-

nology, introduction of new commodities, and in its extension, the formation of

new activities (operations), are strongly needed to maintain the capacity for indus-

trial renewal.

From an evolutionary theoretical point of view (Schumpeter is among the classics

in this field) the model, and theory,37 outlined here is in this respect inadequate to

capture the process of structural renewal, and hence, the specification of the

mechanisms that creates incentives for the entrepreneur to enforce new investments

to maintain the capacity for growth. In assessing these results it must be emphasized

that investment is restricted to capacity expansion, i.e. net investment. Moreover, all

investments are in established industries and hence, according to the specification of

the model, directed to the production of a given set of commodities. In the real world,

37 The perfect competition theory defines the equilibrium state and not the process of adjustment.

(Kirzner 1973, p. 130).

3.5 Empirical Findings: Applications 39

however, investments made to increase the total capacity as well as the replacement

and scrapping of old production units change the production characteristics.

Investments in new capacity embodying best-practice techniques will decrease the

sector’s input coefficient at full capacity. Thus, new capacity has in general

input–output proportions different from those of existing production units due to

changed relative prices and technical progress, which may be embodied or

disembodied (learning by doing). Furthermore, investments introduce input–output

combinations, and in the long run, production of commodities which cannot be found

within the initial production possibility set.

Returning to application 4, the equilibrium model does no longer choose to

establish agriculture and fishing (1) and the mining and quarrying industry (3) in the

Swedish economy. On the other hand, engineering (15), wood, pulp and paper industry

(8) and chemical industry (11) belongs to sectors38 highly exposed to foreign compe-

tition, where expansion of domestic production is requested. Besides manufacturing,

private services (23) indicate an increasing share of domestic production.

In all experiments labour is assumed to be an aggregation of different skill

categories. In other words, labour is specified as homogenous in the model. Hence,

we can not value labour services (labour productivity) by skill group. Nevertheless,

the chemical industry and engineering are particularly intensive in terms of

technicians and skilled labour. In this respect, it seems that the joint utilisation of

human and physical capital provides an important input in the Swedish industry.39

In a model that does not include any restrictions on trade, a commodity is either

imported or exported, but never both.40 The explanation of this is that the commodity

imported and the commodity exported is assumed identical in the model. The

Table 3.1 Net private

investment by producing

sectors million kr – 1,975

prices

Sector Application Request in

1 2 3 4 5

1 281 326 358 0 0

2 346 684 0 0 0

7 161 296 203 100 87

8 807 500 334 155 175

15 21,840 34,862 26,841 11,405 6,253

16 1,299 2,251 2,476 0 0

17 28 20 22 0 0

19 26,118 15,978 12,562 3,697 1,904

23 650 3,214 2,490 967 324

38 The engineering industry is usually analyzed in terms of five sub-branches, i.e. metal goods

industry, machine industry, electrical industry, transport equipment (excl. ship-yards), and mea-

suring and controlling equipment industry. The machine industry is the largest sub-branch

(measured in number of employees and value added respectively). The sub-branches for metal

goods, electrical equipment and transport equipment are all roughly of the same size.39 See also Flam (1981), pp. 97–101.40 It is important to note that the level of aggregation will affect the value of the measures of intra-

industry trade. The higher the level of aggregation, the greater will be the share of intra-industry


tendency for specialisation would be even more explicit, if we were to leave sectoral

capital stocks as endogenous variables.41 Needless to say, extreme specialisation in

production and trade conflicts with empirical evidence, which on the contrary, shows

relatively little specialisation on the sectoral level. However, as pointed out byWerin

(1965), the observed combination of domestic production and trade may be in

complete accordance with the theoretical model. First, the country under study

consists of many regions, which implies that a commodity may be imported to one

region and exported from another, but never be both imported to and exported from

one single region. Second, the same argument is applicable to the fact that the model

is specified to cover a period of some length. Hence, a commodity may be both

produced and traded at different points of time during the period of specification.

Finally, the commodities of the model are aggregates of different commodity

categories. For each of these commodities the theoretical requirement may be

fulfilled.

3.6 Comparative Advantages?

Whereas the Swedish economy, as expounded by the equilibrium experiments

above, tend to illustrate a comparative advantage in industries with large

requirements of human capital42 several empirical studies examine the net trade

patterns and the specialisation of production of Sweden with the EU (in the

beginning EEC) and other OECD countries, indicate a weaker market position in

human capital intensive industries (Ems 1988). Moreover, the R&D intensity did

not seem to influence the international competitiveness of the Swedish industry at

all. The pattern of change in the competitiveness of the Swedish industry versus the

EEC in 1970–1984 (Lundberg 1988) seem to reveal a comparative advantage in

industries requiring large inputs of physical capital and domestic natural resources.

Human capital intensity does not seem to have influenced net export ratios during

the period.

The discussion above has already stressed that a model that does not include any

restrictions on trade, a commodity is either imported or exported but never both.

However, during the post-war period there has been a marked increase in interna-

tional specialisation within the differentiated product groups and a substantial

trade (Grubel and Lloyd 1975). Although the share of intra-trade is reduced by disaggregation,

substantial two-way trade remains (Blattner 1977) on the most detailed aggregation level.41 A common approach to avoid unrealistic specialisation in multi-country trade models is to use

the Armington (1969) formulation, which treats similar commodities produced in different

countries as different commodities (commodity differentiation by country of origin). Bergman

(1986) makes use of the Armington formulation and applies a numerical solution technique in

order to solve the model.42 Nearly all available evidence indicates that Sweden has a comparative advantage in human

capital intensive production. A survey of these studies is given in Flam (1981), pp. 97–101.

3.6 Comparative Advantages? 41

growth in the share of intra-industry trade, i.e., imports and exports in the same

statistical commodity group.43 Thus, the increase in trade and specialisation is

dominated by reallocation on resources within rather than between industries.

The increase in intra-industry trade between Sweden and the EEC has been

particularly strong. Theoretical elements explaining the determinants of intra-

industry trade are based on the roles of product differentiation and economies of

scale. One point of departure in seeking to explain the growth of intra-industry trade

(Petersson 1984) has been the Lancaster (1980) theory which places central impor-

tance on product differentiation and scale economies specific to the product

(production runs). The adoption of a global production strategy and specialisation

within a limited range of commodities and product variants enables a country’s

producers to achieve long production runs. Similar opportunities for the producers

of other countries gave rise to a flow of import and an improvement in consumers’

choice. Hence, the existence of product differentiation (which is especially found in

consumer products) implies monopolistic competition which, from the consumer’s

viewpoint may correspond to a demand for variety in commodities.

Economies of scale with product differentiation normally prevails where

corporations make horizontal investments, i.e. to produce abroad the same lines

of commodities as they produce in the home market. Swedish firms which have

manufacturing affiliates abroad (multinational corporations) account for some

50 % of manufacturing employment in Sweden and almost 60 % of Swedish

exports (Swedenborg 1988). Moreover, they are dominating in engineering and

are highly internationalized. In 1986 less than 25 % of their total sales were sold in

the home market. Of the 75 % sold in foreign markets over half was produced

abroad. Empirical observation (Erixon 1988) suggest that the reduced market

share for Swedish exports may to a great extent be explained by the tendency for

Swedish multinational corporations to supply through local production in the

largest markets rather than through exports from Sweden. Thus, the size of the

market affects not only the volume of sales in a country but also leads to a higher

propensity to supply the market through local production (Krugman 1980).

Comparing the discussion above with the pattern of changes that emerges from

the experiments with the equilibrium model is interesting. In the equilibrium

model the necessary reallocation of sectoral resources is reached solely by an

adjustment in the structure of inter-industry trade. However, within industries

where the equilibrium experiments call for a substantial growth in domestic

production the economic gains is mainly intra-industry in nature. These

gains are in the form of economies of scale utilized to a great extent by

foreign production, rather than arising from reallocation of resources according

to comparative advantages. Thus, we have to be careful in interpreting the

obtained results in a too mechanical fashion.

43 The expansion of intra-industry trade in Europe which was particularly marked in the 1960s

appears to have largely halted in recent years. A somewhat similar situation is apparent for the US

(Hine 1988).


3.7 Concluding Remarks

To conclude this chapter, it seems reasonable to compare the mathematical pro-

gramming (linear and quadratic) models above with models developed within the

tradition of computable general equilibrium (CGE) modelling. In such a compari-

son the programming models seem to be based on overly restrictive assumptions.

For example, while most standard CGE-models incorporate technology

descriptions that allow for factor substitution, there are fixed coefficients in the

linear programming model. Generally, due to the assumed linearity of the underly-

ing technology, the solution in the model imposes that fewer commodities will be

produced domestically, but in increased quantities in the least-cost sectors. On the

other hand, the specialisation will lead to an increasing amount of import in the

high-cost sectors. To obtain a reasonable pattern of specialisation, exports must be

specified to vary within certain limits or be assumed exogenous. By this assumption

extreme specialisation is prevented, but it is still a serious deviation from reality,

especially when foreign trade is a large part.

Another serious restrictive assumption is the treatment of maximising behaviour

by agents in mathematical programming models. In this chapter as well as in the

previous the central planner is assumed to be the only maximising actor. Theoreti-

cally, that conflicts with the market equilibrium price system, where the demand

and supply decisions are made separately and independently by various economic

actors. While most CGE-models incorporate complete systems of final demand

functions, usually derived from explicit utility functions, the demand representation

in the mathematical programming models are based on linear demand functions

with no explicit relation to utility maximisation under a budget constraint. Hence,

no ad hoc assumptions in order to avoid unrealistic solutions will be needed.

Not unexpected, these constraints reduce the experimental attractiveness of the

programming models in our study of a market economy.

Appendix 1: The Reformulation of the Walras-Cassel Model

To provide the methodology for the reformulation of the Walras-Cassel general

equilibrium model as a quadratic programming problem, and hence, the basic

structure of the quadratic input–output model, Harrington (1973) linearises the

Walras-Cassel model and specifies the Walrasian factor supply and commodity

demand functions into inverse form.44 The inversion simplifies the mathematical

exposition of the model while retaining the generality of the Walrasian factor supply

and commodity demand functions. Dorfman, Samuelson and Solow (1958) claim that

44 The Walras-Cassel model is specified in Dorfman, R., Samuelson, P. A. and Solow, R. M.,

(1958), pp. 346–389. The Walrasian model of the market system was first sketched by the

nineteenth-century French economist Leon Walras (1874–7).

Appendix 1: The Reformulation of the Walras-Cassel Model 43

this inversion is not admissible because there is no mathematical reason for assuming

the existence of inverse demand or supply relationships in a model were prices

depend on quantities only.45 However, their argument, as demonstrated by

Harrington, is well-founded in the general case but does not apply to the linearised

Walras-Cassel model. The quadratic input–output model is a linearised version of the

Walras-Cassel general equilibrium model which utilizes the inter-relatedness of

production established in the input–output structure. In this context, it is shown by

Harrington that the conventional input–output model is a limiting case of the

linearised Walras-Cassel model. In the linear form of the Walras-Cassel model the

assumptions of homogeneity of degree zero of factor supply and commodity demand

functions can be relaxed because the homogeneity constraint is satisfied elsewhere in

the model formulation. Furthermore, the Cassel-Wald specification of commodity

demand quantities as a function of product prices alone, and factor supply quantities

as a function of factor prices alone (Wald 1951), specify a consistent linear system

without loss of generality of the Walras-Cassel model.

In order to understand the underlying structure of the model that constitutes the

framework of this study a mathematical exposition of Harrington’s (1973) contri-

bution is given in this section.46 Let A denote a matrix of fixed coefficient produc-

tion processes, homogenous of degree one, partioned into a primary factor

transformation m � n matrix, Ar, and an intermediate commodity transformation

n � n matrix Aq. Let G(w, p) denote a linear factor market supply function defined

over all factor prices w (m1 1) and commodity prices p (n � 1), and let F(w, p)denote a linear commodity market demand function defined over all factor prices

w and commodity prices p.47 Thus, the assumptions above linearise the Walras-

Cassel model. Note, that the factor supply and commodity demand functions are not

assumed to be homogenous of degree zero in w and p.48 Under the assumption of

linearity of the factor supply and commodity demand functions the G and Fmatrices (Gr (m � m), Gq (m � n), Fr (n � m), Fq (n � n)) may be partitioned as:

Grwþ Gqp ¼ r and Frwþ Fqp ¼ q (3.26)

where q specifies a vector of final demand quantities, and r a vector of factor supplyquantities. Transforming factors into commodities require the following condition

on primary factor transformations:

45 Dorfman et al. (1958), p. 352 (footnote).46 The exposition in this section is based on Harrington’s own presentation of the subject.47 The factor supply functions are specified in the factor markets, the commodity demand functions

are specified in the commodity markets, and the transformation matrices are specified in the

production sectors.48 It is impossible to meet both the specification of linearity and homogeneity of degree zero in the

same function. Since F and G are matrices of constants they are by definition homogeneous of

degree one.


Arz ¼ r (3.27)

Intermediate commodity transformations require:

½I� Aq�z ¼ q (3.28)

where z represents a vector of gross output per sector. [I � Aq] referred to as the

Leontief matrix, is based on the conditions of conventional input–output analysis,

hence, its inverse exists. Consequently:

½I� Aq��1q ¼ z (3.29)

Given the specification above, the condition of efficient pricing implies that the

final commodity price must equal the sum of factor costs and the cost of intermedi-

ate commodities required in the production of a unit of the final commodity. Thus:

A0rwþ A0

qp ¼ p (3.30)

The first term is the price component of rewards to primary factors and the

second term is the price component of rewards to intermediate commodities at their

market prices.49

Solving Eq. 3.30 for p gives:

A0rw ¼ p� A0

qp (3.31)

A0rw ¼ I� A0

q

� �p (3.32)

I� A0q

� ��1A0

rw ¼ p (3.33)

Substituting from Eqs. 3.29 and 3.33 into Eq. 3.26 gives:

Grwþ Gq I� A0q

� ��1A0

rw ¼ Ar½I� Aq��1q (3.34)

Frwþ Fq I� A0q

� ��1A0

r w ¼ q (3.35)

Pre-multiplying Eq. 3.35 by Ar ½I� Aq��1, direct and indirect factor

requirements, gives:

49 This equation is equivalent to the price formulation of input–output analysis. The price system

appears as the dual of the quantity system, and vice versa, and the two can be studied indepen-

dently. Following these principles, we obtain the transpose of Aq and Ar,, which is denoted by A0q

and A0r.

Appendix 1: The Reformulation of the Walras-Cassel Model 45

Ar ½I� Aq��1Frwþ Ar½I� Aq��1Fq½I� A0q��1

A0r w ¼

Ar½I� Aq��1q ð3:36Þ

It follows that:

Gq ¼ Ar½I� Aq��1 Fq (3.37)

Gr ¼ Ar½I� Aq��1 Fr (3.38)

Equations 3.37 and 3.38 specify the effects of commodity demand functions on

factor supplies (direct and indirect factor requirements) necessary for the efficient

production, (3.27) and (3.28), and the efficient pricing condition (3.30) to hold.

Equation 3.37 specifies these conditions on the commodity price matrix assuming

that Fq is specified, and Eq. 3.38 specifies these conditions on the factor price

matrix assuming that Fr is specified. Given the assumptionm ¼ n and the rank of Ar

is equal to n the generalized inverse50 of Ar exists. Thus, applying the generalized

inverse of {Ar [I � Aq]�1} to Eq. 3.38 gives:

Fr ¼ ½I� Aq�½A0r Ar��1A0

rGr (3.39)

Equation 3.39 specifies the generation of the income constraint on demand.

Similarly, Eq. 3.38 specifies the generation of the income constraint on the factor

supply functions. Hence, the commodity demand functions and the factor supply

functions may be specified by the Cassel-Wald specification:

FðpÞ ¼ q and GðwÞ ¼ r (3.40)

which together with Ar and Aq specify a consistent linear system without loss of the

generality of Dorfman, Samuelson and Solow specification of the Walrasian equi-

librium system. As a consequence, commodity prices can be expressed as function

of factor prices alone, using the non-substitution theorem of Samuelson (1951).

The Fr and Gq matrices of the linearised Walras-Cassel model are completely

specified by the Fq, Gr, Ar and Aq matrices together with the conditions of efficient

production, Eqs. 3.27 and 3.28, and the efficient pricing condition (3.30). Thus, the

information contained in Gq and Fr in the Walrasian specification is redundant.

Both functions (F and G) together with the specifications given above specify a

system homogeneous of degree zero in w and p. This implies, that the F and Gfunctions need no longer be specified with homogeneity of degree zero. The

equations in (3.40) can be converted to inverse form:

50 For details, see Penrose, R., (1955). A summary is given in Maddala, G. S., (1977).


w ¼ G�1ðrÞ and p ¼ F�1ðqÞ (3.41)

where G�1 and F�1 are the inverses of G and F, respectively. Hence, the objectionby Dorfman, Samuelson and Solow that this inversion is not admissible in general

does not hold for the linearised Walras-Cassel model.

Appendix 2: Tables 3.2, 3.3, 3.4, 3.5, and 3.6

Table 3.2 Sectors and their definitions in the model

Sector Definition Column Definition

1 Agriculture, fishing 1 Domestic production (Z)

2 Forestry 2 Non-competitive imports (m)

3 Mining and quarrying 3 Competitive imports (M)

4 Sheltered food industry 4 Exports (E), 1980 values

5 Exposed food industry 5 Change in domestic production

6 Beverage and tobacco

industry

6 Change in competitive imports

7 Textile and clothing

industry

7 Change in exports

8 Wood, pulp and paper

industry

8 Capacity utilization in percent of the sectorally

established capital stocks

9 Printing industry 9 Percentage share of domestic production

10 Rubber products

industry

10 Percentage share of competitive imports

11 Chemical industry 11 Net trade ratio (E � M)/(E þ M), 1 only

exports, �1 only imports, 0 balance

12 Petroleum and coal

industry

12 Private consumption (x)

13 Non-metallic mineral

products

13 Equilibrium prices (p) of the quadratic

variables (x) – indexed at 1,000

14 Basic metal industries

15 Engineering, excl.

shipyards

16 Shipyards

17 Other manufacturing

18 Electricity, gas, heating

and water

19 Construction

20 Merchandise trade

21 Transport and

communications

22 Housing

23 Private services

24 Foreign tourist services

Appendix 2: Tables 3.2, 3.3, 3.4, 3.5, and 3.6 47

Table of 1980 statistics – million Skr – 1975 prices

Column

1 2 3 4 5678 9 10 11 12 13Sector

1 14,202 1,863 1,007 1,174 000100 2.69 1.13 0.08 6,617 1,000

2 8,388 284 272 129 000100 1.55 0.30 �0.36 230 1,000

3 4,381 6,371 1,712 2,457 000100 0.81 1.92 0.18 43 1,000

4 23,915 38 1,484 773 000100 4.41 1.66 �0.32 16,549 1,000

5 12,769 645 2,664 758 000100 2.36 2.99 �0.56 9,333 1,000

6 12,149 383 256 93 000100 2.24 0.29 �0.47 11,285 1,000

7 14,439 154 7,636 2,599 000100 2.66 8.54 �0.49 17,549 1,000

8 44,252 51 2,625 19,680 000100 8.17 2.94 0.76 4,374 1,000

9 11,544 0 610 413 000100 2.13 0.68 �0.19 2,772 1,000

10 1,941 19 1,097 599 000100 0.36 1.23 �0.29 991 1,000

11 16,796 995 8,681 6,096 000100 3.10 9.73 �0.18 4,479 1,000

12 19,188 26 6,159 2,300 000100 3.54 6.90 �0.46 6,125 1,000

13 5,878 0 1,447 1,022 000100 1.08 1.62 �0.17 396 1,000

14 18,875 96 5,342 8,123 000100 3.48 5.99 0.21 – –

15 84,100 0 32,90238,045 000100 15.52 36.88 0.07 13,122 1,000

16 5,138 0 722 1,660 000100 0.95 0.81 0.39 1,363 1,000

17 2,908 0 1,045 506 000100 0.54 1.17 �0.35 2,125 1,000

18 11,571 0 110 108 000100 2.14 0.12 �0.01 4,386 1,000

19 49,971 0 0 0 000100 9.22 0 0.00 – –

20 50,818 0 1,230 1,561 000100 9.38 1.38 0.12 – –

21 35,208 0 3,487 7,685 000100 6.50 3.91 0.38 7,047 1,000

22 33,683 0 0 0 000100 6.22 0 0.00 31,459 1,000

23 59,752 0 2,860 3,258 000100 11.03 3.21 0.07 19,719 1,000

24 0 0 5,861 2,960 000– 0 6.57 �0.37 3,171 1,000

Total 541,86610,92589,209101,728000 163,134


Table

3.3

Application1:Tem

porary

equilibrium

–period1

Column

12

34

56

78

910

11

12

13

Sector

115,590

2,082

816

1,174

1,388

�191

0100

2.69

0.91

0.18

7,168

750

29,227

312

266

129

839

�60

100

1.59

0.20

0.35

238

750

34,819

7,008

1,271

2,457

438

�441

0100

0.83

1.42

0.32

68

750

426,306

42

2,642

773

2,391

1,158

0100

4.55

2.94

�0.55

19,453

750

514,044

709

2,549

758

1,275

�115

0100

2.43

2.84

�0.54

10,005

750

613,364

421

093

1,215

�256

0100

2.31

01.00

12,150

735

715,884

169

11,219

2,599

1,445

3,583

0100

2.75

12.50

�0.62

22,504

750

848,675

56

256

19,680

4,423

�2,369

0100

8.41

0.29

0.97

5,027

750

912,699

0465

413

1,155

�145

0100

2.19

0.52

�0.06

3,219

750

10

2,135

21

1,117

599

194

20

0100

0.37

1.24

�0.30

1,101

750

11

18,476

1,094

8,659

6,095

1,680

�22

0100

3.19

9.65

�0.17

5,222

750

12

21,113

29

5,789

2,300

1,925

�370

0100

3.65

6.45

�0.43

6,677

750

13

6,466

01,082

1,022

588

�365

0100

1.12

1.21

�0.03

456

750

14

16,422

84

7,426

8,123

�2,453

2,084

079

2.84

8.28

0.04

––

15

92,508

027,832

38,045

8,408

�5,070

0100

15.99

31.02

0.16

14,683

750

16

00

5,917

1,660

�5,138

5,195

00

06.59

�0.56

1,637

750

17

3,200

0967

506

292

�78

0100

0.55

1.08

�0.31

2,287

750

18

12,399

00

108

828

�110

097

2.14

01.00

4,744

226

19

50,637

00

0666

00

92

8.75

00.00

––

20

55,899

061

1,561

5,081

�1,169

0100

9.66

0.07

0.92

––

21

38,728

02,716

7,685

3,520

�771

0100

6.69

3.03

0.48

7,944

750

22

34,298

00

0615

00

93

5.92

00.00

32,074

181

23

65,729

01,687

3,258

5,977

�1,173

0100

11.36

1.88

0.32

21,460

750

24

00

6,991

2,690

01,130

0–

07.79

�0.44

4,301

750

Total

578,617

12,027

89,728

101,728

36,751

519

0182,418


Table

3.4

Application2:Tem

porary

equilibrium

–period2

Column

12

34

56

78

910

11

12

13

Sector

117,149

2,290

01,174

1,559

�816

0100

2.90

01.00

7,135

1,015

29,975

338

0129

748

�266

098

1.69

01.00

260

436

35,162

7,507

02,457

343

�1,271

097

0.87

01.00

64

1,042

428,937

46

199

773

2,631

�2,443

0100

4.89

0.22

0.59

18,933

1,045

515,448

780

1,557

758

1,404

�992

0100

2.61

1.75

�0.35

9,884

1,045

614,700

463

093

1,336

00

100

2.48

01.00

13,383

624

717,474

186

8,947

2,599

1,590

�2,272

0100

2.95

10.07

�0.55

21,616

1,045

849,430

57

019,680

755

�256

092

8.35

01.00

5,925

655

913,969

00

413

1,270

�465

0100

2.36

01.00

3,591

792

10

2,349

23

983

599

214

�134

0100

0.40

1.11

�0.24

1,081

1,045

11

20,323

1,204

7,383

6,095

1,847

�1,276

0100

3.43

8.31

�0.10

5,089

1,045

12

23,227

31

3,175

2,300

2,114

�2,614

0100

3.93

3.57

�0.16

6,578

1,045

13

6,933

00

1,022

467

�1,082

097

1.17

01.00

527

703

14

202

120,093

8,123

�16,220

12,667

01

0.03

22.62

�0.42

––

15

101,760

032,424

38,045

9,252

4,592

0100

17.20

36.50

0.08

14,403

1,045

16

00

6,855

1,660

0938

0-

07.72

�0.61

1,588

1,045

17

3,519

0427

506

319

�540

0100

0.59

0.48

0.08

2,258

1,045

18

12,429

00

108

30

00

91

2.10

01.00

5,057

324

19

41,088

00

0�9

,549

00

74

6.94

00.00

––

20

57,836

00

1,561

1,937

�61

094

9.77

01.00

––

21

42,602

00

7,685

3,874

�2,716

0100

7.20

01.00

8,023

978

22

34,870

00

0572

00

92

5.89

00.00

32,646

238

23

72,302

00

3,258

6,573

�1,687

0100

12.22

01.00

22,248

887

24

00

6,789

2,690

0�2

02

0–

07.64

�0.43

4,099

1,045

Total

591,683

12,926

88,832

101,728

13,066

�896

0184,388


Table

3.5

Application3:Tem

porary

equilibrium

–period3

Column

12

34

56

78

910

11

12

13

Sector

17,912

1,057

11,689

1,174

�9,237

11,689

042

1.30

12.28

�0.82

7,569

803

29,638

326

0129

�337

00

88

1.58

01.00

283

328

31,464

2,129

9,622

2,457

�3,698

9,622

026

0.24

10.11

�0.59

84

803

431,831

51

259

773

2,894

60

0100

5.21

0.27

0.50

21,220

803

516,992

858

0758

1,544

�1,557

0100

2.78

01.00

10,774

668

616,168

510

093

1,468

00

100

2.65

01.00

14,789

570

719,218

205

11,101

2,599

1,744

2,154

0100

3.15

11.67

�0.62

25,541

803

851,131

59

019,680

1,701

00

94

8.38

01.00

7,165

524

915,236

00

413

1,267

00

99

2.50

01.00

4,407

544

10

2,584

25

916

599

235

�67

0100

0.42

0.96

�0.21

1,168

803

11

22,359

1,324

6,211

6,095

2,033

�1,172

0100

3.66

6.53

�0.01

5,674

803

12

25,549

35

1,372

2,300

2,322

�1,803

0100

4.19

1.44

0.25

7,012

803

13

6,808

00

1,022

�125

00

87

1.12

01.00

636

547

14

00

21,104

8,123

�202

1,011

00

022.18

�0.44

––

15

111,940

017,665

38,045

10,180

�14,759

0100

18.34

18.56

0.37

15,631

803

16

00

7,296

1,660

0441

0-

07.67

�0.63

1,804

803

17

3,872

0250

506

353

�177

0100

0.63

0.26

0.34

2,385

803

18

12,842

00

108

413

00

94

2.10

01.00

5,404

250

19

37,409

00

0�3

,679

00

61

6.13

00.00

––

20

60,381

00

1,561

2,545

00

89

9.89

01.00

––

21

44,496

00

7,685

1,894

00

95

7.29

01.00

9,930

468

22

35,493

00

0623

00

86

5.81

00.00

33,269

170

23

77,067

00

3,258

4,765

00

97

12.63

01.00

26,187

434

24

00

7,679

2,690

0890

0–

08.07

�0.48

4,989

803

Total

610,387

6,578

95,164

101,728

18,704

6,332

0205,921


Table

3.6

Application4:Tem

porary

equilibrium

–period4

Column

12

34

56

78

910

11

12

13

Sector

121

321,893

1,174

�7,891

10,204

00.24

022.32

�0.90

8,446

602

29,943

337

0129

305

00

94

1.59

01.00

308

240

30

013,973

2,457

�1,464

4,351

00

014.25

�0.70

124

602

435,014

56

2,372

773

3,183

2,113

0100

5.60

2.42

�0.51

25,839

602

517,587

888

0758

595

00

94

2.81

01.00

12,200

469

617,787

561

093

1,619

00

100

2.84

01.00

16,348

524

721,140

225

17,057

2,599

1,922

5,956

0100

3.38

17.39

�0.74

33,451

602

852,694

61

019,680

1,563

00

94

8.43

01.00

8,748

393

916,752

00

413

1,516

00

100

2.68

01.00

5,457

413

10

2,842

28

908

599

258

�80

100

0.45

0.93

�0.21

1,343

602

11

24,591

1,457

5,474

6,095

2,235

�737

0100

3.93

5.58

0.05

6,856

602

12

28,102

38

02,300

2,553

�1,372

0100

4.49

01.00

8,114

500

13

6,318

00

1,022

�490

00

84

1.01

01.00

778

408

14

00

21,506

8,123

0402

0–

021.93

�0.45

––

15

118,694

00

38,045

6,754

�17,665

096

18.98

01.00

19,311

409

16

00

5,266

1,660

0�2

,030

0–

05.37

�0.52

2,239

602

17

4,258

0145

506

386

�105

0100

0.68

0.15

0.55

2,642

602

18

13,368

00

108

526

00

95

2.14

01.00

5,789

168

19

28,456

00

0�8

,953

00

69

4.55

00.00

––

20

62,939

00

1,561

2,558

00

95

10.07

01.00

––

21

47,320

00

7,685

2,824

00

97

7.57

01.00

12,285

343

22

36,152

00

0659

00

93

5.78

00.00

33,928

122

23

81,343

00

3,258

4,276

00

96

13.01

01.00

30,911

321

24

00

9,476

2,690

01,797

0–

09.66

�0.56

6,786

602

Total

625,322

3,653

98,070

101,728

14,935

2,906

0241,903


References

Armington P (1969) A theory of demand for products distinguished by place of production. IMF

Staff Pap 16:159–178

Bergman L (1986) ELIAS: a model of multisectoral economic growth in a small open economy.

IIASA, Laxenburg

Blattner N (1977) Intraindustrieller Aussenhandel; Empirische Beobachtungen im Falle der

Schweiz und Teoretische Interpretationen. Weltwirdschaftliches Archiv 113(1):88–103

Dervis K, de Melo J, Robinson S (1982) General equilibrium models for development policy.

Cambridge University Press, Cambridge

Dorfman R, Samuelson PA, Solow RM (1958) Linear programming and economic analysis.

McGraw-Hill, New York, pp 346–389

Ems E (1988) Exportindustrins framtid, SIND 1988:2. Stockholm

Enke S (1951) Equilibrium among spatially separated markets: solutions by electric analogue.

Econometrica 19:40–47

Erixon L (1988) Loner och konkurrenskraft – lonekostnadernas betydelse for Sveriges

marknadsandelar. I Ems (ed) Exportindustrins Framtid, SIND 1988:2

Flam H (1981) Growth, allocation and trade in Sweden, vol 12, Institute for International

Economic Studies, Monograph series. University of Stockholm, Stockholm

Grubel HG, Lloyd PJ (1975) Intra-Industry trade: the theory and measurement of international

trade in differentiated products. Macmillan/Halsted, London

Harrington DH (1973) Quadratic input–output analysis: methodology for empirical general equi-

librium models. Ph.D. dissertation, Purdue University, West Lafayette

Hine RC (1988) 1992 and the pattern of specialisation in European industry. Paper presented to a

workshop on the Nordic countries an the EEC, Lund

Hotelling H (1932) Edgeworth’s taxation paradox and the nature of demand and supply functions.

J Polit Econ 40:557–616

Kirzner IM (1973) Competition and entrepreneurship. The University of Chicago Press, Chicago

Koopmans TC (1957) Three essays on the state of economic science. McGraw-Hill, New York

Krugman P (1980) Scale economies, product differentiation and the pattern of trade. Am Econ Rev

70(5):950–959

Lancaster KJ (1980) Intra-industry trade under perfect monopolistic competition. J Int Econ

10:151–175

Lundberg L (1988) The Nordic countries and economic integration in Europe: trade barriers and

patterns of trade and specialization. Trade Union Institute of Economic Research, Stockholm

Maddala GS (1977) Econometrics. McGraw-Hill, New York

Marshall A (1925) Principles of economics, 8th edn. Macmillan, London

McCarl BA, Spreen TH (1980) Price endogenous mathematical programming as a tool for sector

analysis. Am J Agric Econ 62:87–102

Noren R (1987) Comparative advantages revealed. Experiments with a quadratic programming

model of Sweden. Ph.D. dissertation. Department of Economics, University of Stockholm.

Akademitryck, Edsbruk

Noren R (1991) A Takayama-judge activity model applied to the analysis of allocation and trade in

Sweden. Appl Econ 23:1201–1212

Norman VD (1983) En Liten, Apen Okonomi. Universitetsforlaget, Oslo

Penrose R (1955) Generalized inverse for matrices. In: Proceedings of the Cambridge philosophi-

cal society, Cambridge, pp 406–413

Petersson L (1984) Svensk Utrikeshandel 1871–1980. En studie I den den intraindustriella

handelns framvaxt. Lund Economic Studies, Lund

Plessner Y (1965) Quadratic programming competitive equilibrium models for the U.S agricul-

tural sector. Ph.D. dissertation. Iowa State University, Ames

Plessner Y (1967) Activity analysis, quadratic programming and general equilibrium. Int Econ

Rev 8:168–179

References 53

Plessner Y, Heady E (1965) Competitive equilibrium solutions with quadratic programming.

Metroeconomica 17:117–130

Samuelson P (1951) Abstract of a theorem concerning substitutability in open Leontief models. In:

Koopmans TC (ed) Activity analysis of production and allocation. Wiley, New York

Samuelson PA (1952) Spatial price equilibrium and linear programming. Am Econ Rev

42:283–303

Swedenborg B (1988) The EC and the locational choice of Swedish multinational companies.

Paper presented to a workshop on the Nordic countries an the EEC, Lund

Sodersten B (1980) International economics. Macmillan, London

Takayama T, Judge G (1964a) Equilibrium among spatially separated markets: a reformulation.

Econometrica 32:510–524

Takayama T, Judge G (1964b) An interregional activity analysis model of the agricultural sector.

J Farm Econ 46:349–365

Takayama T, Judge G (1971) Spatial and temporal price and allocation models. North-Holland,

Amsterdam

Taylor L (1975) Theoretical foundations and technical implications. In: Blitzer CR et al (eds)

Economy-wide models and development planning. Oxford University Press, Oxford

Varian RH (1984) Microeconomic analysis, 2nd edn. W.W. Norton, New York

Wald A (1951) On some systems of equations of mathematical economics. Econometrica

19:368–403

Werin L (1965) A study of production, trade and allocation of resources, vol 6, Stockholm

economic studies, new series. Almqvist & Wiksell, Stockholm

Yaron D, Plessner Y, Heady E (1965) Competitive equilibrium application of mathematical

programming. Can J Agr Econ 13:65–79

Zusman P (1969) The stability of interregional competition and the programming approach to the

analysis of spatial trade equilibria. Metroeconomica 21:55

Government Publications

SOU 1984:7. LU 84 (The medium term survey of the Swedish economy), Appendix 17.

SM 1974:52. Stocks of fixed capital 1950–73 and capital consumption 1963–73, Swedish Central

Bureau of Statistics

Statistical Reports SM N 1970:13. Input–output tables for Sweden 1964, Swedish Central Bureau

of Statistics

Statistical Reports. SM N 1981:2.5. Appendix 1, Final consumption expenditure, Swedish CentralBureau of Statistics

Statistical Reports. SM N 1981:2.5. Appendix 2, Capital formation and stocks of fixed capital,Swedish Central Bureau of Statistics

Statistical Reports. SM N 1981:2.5. Appendix 5, Employment and compensation of employments,

Swedish Central Bureau of Statistics


Date post:	07-Dec-2016
Category:	Documents
Upload:	ronny
View:	213 times
Download:	0 times

[Lecture Notes in Economics and Mathematical Systems] Equilibrium Models in an Applied Framework...

Documents