Abstract
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 inequality 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.
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- 1.
The simplex method of linear programming represents such a method.
- 2.
- 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.
- 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.
- 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).
- 6.
More 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.
- 9.
A 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.
- 12.
For details, see Varian (1984), pp. 135–139.
- 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.
- 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.
- 18.
However, 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.
- 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.
- 20.
With 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, Södersten (1980), pp. 315–328.
- 23.
The discussion that follows is based on Dervis et al. (1982).
- 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.
- 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.
- 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).
- 33.
Given 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).
- 37.
The perfect competition theory defines the equilibrium state and not the process of adjustment. (Kirzner 1973, p. 130).
- 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 measuring 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 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.
- 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).
- 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 Léon Walras (1874–7).
- 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.
- 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 independently. Following these principles, we obtain the transpose of A q and A r ,, which is denoted by A′ q and A′ r .
- 50.
For details, see Penrose, R., (1955). A summary is given in Maddala, G. S., (1977).
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Appendices
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.Footnote 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 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.Footnote 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) contribution is given in this section.Footnote 46 Let A denote a matrix of fixed coefficient production processes, homogenous of degree one, partioned into a primary factor transformation m × n matrix, A r , and an intermediate commodity transformation n × n matrix A q . Let G(w, p) denote a linear factor market supply function defined over all factor prices w (\( m\infty1 \)) 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. Footnote 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.Footnote 48 Under the assumption of linearity of the factor supply and commodity demand functions the G and F matrices (G r (m × m), G q (m × n), F r (n × m), F q (n × n)) may be partitioned as:
where q specifies a vector of final demand quantities, and r a vector of factor supply quantities. Transforming factors into commodities require the following condition on primary factor transformations:
Intermediate commodity transformations require:
where z represents a vector of gross output per sector. [I − A q ] referred to as the Leontief matrix, is based on the conditions of conventional input–output analysis, hence, its inverse exists. Consequently:
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 intermediate commodities required in the production of a unit of the final commodity. Thus:
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.Footnote 49
Solving Eq. 3.30 for p gives:
Substituting from Eqs. 3.29 and 3.33 into Eq. 3.26 gives:
Pre-multiplying Eq. 3.35 by \( {A_r}\ {{[\mathrm{ I} - {A_q}]}^{-1 }} \), direct and indirect factor requirements, gives:
It follows that:
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 F q is specified, and Eq. 3.38 specifies these conditions on the factor price matrix assuming that Fr is specified. Given the assumption m = n and the rank of A r is equal to n the generalized inverseFootnote 50 of A r exists. Thus, applying the generalized inverse of {A r [I − A q ]−1} to Eq. 3.38 gives:
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:
which together with A r and A q specify a consistent linear system without loss of the generality of Dorfman, Samuelson and Solow specification of the Walrasian equilibrium system. As a consequence, commodity prices can be expressed as function of factor prices alone, using the non-substitution theorem of Samuelson (1951). The F r and G q matrices of the linearised Walras-Cassel model are completely specified by the F q , G r , A r and A q 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 G q and F r 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 G functions need no longer be specified with homogeneity of degree zero. The equations in (3.40) can be converted to inverse form:
where G −1 and F −1 are the inverses of G and F, respectively. Hence, the objection by 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
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Norén, R. (2013). The Planner and the Market: The Takayama Judge Activity Model. In: Equilibrium Models in an Applied Framework. Lecture Notes in Economics and Mathematical Systems, vol 667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34994-2_3
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