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An Armington–Leontief model
Journal of Economic Structures volume 8, Article number: 25 (2019)
Abstract
We develop a novel linear equilibrium model with an Armington flavor. We provide (1) proof of the solvability of the model and of the existence and nonnegativity of the equilibrium solution and of the newly derived multiplier matrix; we also show (2) that the standard Leontief multiplier matrix arises as a special case of this new model and (3) that this model allows the computation of multiplier effects with no external output bias, which is particularly relevant for applied economic analysis.
Highlights

We incorporate the empirically relevant distinction between domestic and imported production.

We propose a more precise calculation of multiplier effects for empirical applications.

We provide a mathematical proof of the model’s economic consistency.
Introduction
Linear general equilibrium models provide a simple and transparent platform for economic analysis and policy evaluation. The bestknown linear model is the classic Leontief model (Leontief 1966; Miller and Blair 2009). This model has the nice property that yields a reduced form that allows for the calculation of output multipliers in response to demanddriven changes, such as those initiated by discretionary government expenditure policies. We sometimes fail to distinguish that the total supply of output is the aggregation of domestic and imported outputs. From an evaluation perspective, however, the relevant triggered effect that one wishes to measure is on the domestic component of output, not on total output. Indeed, domestic output summarizes the internal economic response to any changes originating in final demand, once the economy has absorbed all the general equilibrium interactions. Previous applied research, however, has focused in measuring the total output response using competitive or noncompetitive imports (Su and Ang 2013; Duman and Özgüzer 2014) or in extending the standard multipliers incorporating an ad hoc intermediate imports factor (Trinh 2008). Neither of these cases contemplates the interlocking technological relationship between domestic output and imports nor the fact that the relevant endogenous variable of interest should be domestic output rather than total output.
Armington’s (1969) principle—widely used in trade analysis and applied general equilibrium modeling—allows us to capture these missing characteristics. The Armington principle is based on the fact that the total supply of an economy is an aggregate of domestic and imported foreign outputs. In its most general formulation, domestic and foreign goods are substitutes and the final composition of total output responds to both market signals (relative prices) and technology (substitution elasticities). We therefore propose to incorporate the Armington principle into the linear Leontief model in a manner that makes it conformal with the classical structure of the linear model: we assume perfect complementarity between domestic and foreign outputs (no substitution is permitted). The contribution of the paper is twofold. From an empirical perspective, we use the Armington principle to reformulate the linear interindustry model in a way that allows for a more sensible measurement of the multiplier effects; this new model version, additionally, includes the standard linear model as a limit case. From a theoretical perspective, we provide the necessary mathematical arguments that guarantee the model’s inner consistency (i.e., existence and nonnegativity); additionally, we demonstrate the extension of Nikaido’s (1968, 1972) solvability propositions to the new setup.
In the next section, we formulate some preliminaries that we will need later on. In Sect. 4, we undertake the economics of the model, whereas in Sect. 5, we show the mathematics of the main theorem justifying the existence and nonnegativity of the equilibrium in this extension of the linear model.
Preliminaries
Definition 1
A linear economy is a pair (A, f) with A being a (n × n) nonnegative square matrix and f a (n × 1) nonnegative column vector.^{Footnote 1} Matrix A = (a_{ij}) represents the available technology with a_{ij} indicating the minimal amount of good i (as input) needed to generate a unit of good j (as output). Vector f, in turn, represents final demand for goods.
Definition 2
The economy (A, f) is in balance if for the given final demand vector f there is a nonnegative vector x such that \({\mathbf{x}} = {\mathbf{A}} \cdot {\mathbf{x}} + {\mathbf{f}}\). Vector x represents total output in the economy and \({\mathbf{A}} \cdot {\mathbf{x}}\) indicates the part of total output that is needed to produce it (intermediate demand). In a balanced state, total supply x is equal to total demand, i.e., the sum of intermediate \({\mathbf{A}} \cdot {\mathbf{x}}\) and final f demands.
Definition 3
The technology A is productive if for any nonnegative vector of final demand f there is a nonnegative output vector x such that the economy is in balance.
Property 1
These three statements are equivalent (Nikaido 1972, thm. 17.1):

i.
The technology A is productive;

ii.
The maximal eigenvalue of A satisfies \(\lambda ({\mathbf{A}}) < 1\);

iii.
The inverse matrix \(({\mathbf{I}}  {\mathbf{A}})^{  1}\) exists and is nonnegative.
Definition 4
An input–output table is a collection of economic data that satisfies \({\mathbf{x}} = {\mathbf{Z}} \cdot \vec{1} + {\mathbf{f}} = \vec{1} \cdot {\mathbf{Z}}^{\text{T}} + {\mathbf{v}}\), where x, f, and v are observed vectors of total output, final demand, and valueadded. In empirical applications, aggregation is such that all these vectors are typically positive. The nonnegative matrix Z = (z_{ij}) shows all intermediate transactions taking place between sectors i and j.
Property 2
Let us consider an empirical input–output table that presents positive levels of total output and final demand; if we define the technology by \({\mathbf{A}} = {\mathbf{Z}} \cdot {\hat{\mathbf{X}}}^{  1}\) and assume constant returns to scale, we obtain a balanced empirical economy.
Proof
We first observe that \(\vec{1} = {\hat{\mathbf{X}}}^{  1} \cdot {\mathbf{x}}\) (since all observed outputs are assumed to be positive). Hence \(\,{\mathbf{x}} = {\mathbf{Z}} \cdot \vec{1} + {\mathbf{f}} = {\mathbf{Z}} \cdot {\hat{\mathbf{X}}}^{  1} \cdot {\mathbf{x}} + {\mathbf{f}} = {\mathbf{A}} \cdot {\mathbf{x}} + {\mathbf{f}}.\)
Notice that in this empirical economy, all technical coefficients a_{ij} = z_{ij}/x_{j} in matrix A are well defined since x_{j} > 0 for all j.
Property 3
If A is the technology matrix of a balanced empirical economy, then A is productive.
Proof
Postmultiply \({\mathbf{x}} = {\mathbf{A}} \cdot {\mathbf{x}} + {\mathbf{f}}\) by \({\hat{\mathbf{X}}}^{  1}\) and obtain \(\vec{1} = {\mathbf{A}} \cdot \vec{1} + {\mathbf{f}} \cdot {\hat{\mathbf{X}}}^{  1} > {\mathbf{A}} \cdot \vec{1}\) since both f and x are assumed positive. From the inequality \(\vec{1} > {\mathbf{A}} \cdot \vec{1}\) we verify that the Brauer–Solow sufficient condition (Solow 1952) holds, which implies \(\lambda ({\mathbf{A}}) < 1\).
The Armington–Leontief model
In this section, we assume we can perform all the required matrix operations and algebra. The standard linear model outlined above corresponds to a fully closed (no trade) economy. We now introduce the empirically relevant distinction that there are two sources of output, domestic output x^{d} and imports x^{m}. Total output satisfies
The technology matrix A must now capture the domestic production function and for this we need to define A in relation to domestic output x^{d}, not total output x. We now have
Alternatively
From expressions (1) and (2) we find
If the inverse of (I − A) exists we obtain
We introduce the Leontief inverse L = (I − A)^{−1} and we use it to solve for domestic output:
We now invoke the Armington (1969) assumption in a fixed coefficients setting. Domestic and imported output will now be linear functions of total output:
with the proportionality factors being the shares of domestic and imported output over total output. If we write the shares in two diagonal matrices \(\hat{\alpha }^{d} ,\;\;\hat{\alpha }^{m}\) we obtain
Let us assume for the time being that all shares are positive, i.e., \(\hat{\alpha }^{d} ,\;\;\hat{\alpha }^{m} > 0\). We now substitute the first equation in (6) into the second one:
Here in expression (7) \(\hat{\beta }\) is a positive diagonal matrix with entries \(\beta_{jj} = \alpha_{j}^{m} /\alpha_{j}^{d}\). We now use (7) to transform expression (4):
Solving for domestic output we find
We will refer to the matrix given by
as the Armington–Leontief multiplier matrix. It links the vector of final demand f with the vector of domestic output x^{d}.
We now explore the relationship between M and the standard multiplier matrix as captured by the Leontief inverse \({\kern 1pt} {\mathbf{L}} = ({\mathbf{I}}  {\mathbf{A}})^{  1}\). We consider two polar cases to provide limit bounds for M; first we go to one extreme and make all \(\beta_{jj} \to 0\), and then we consider the other extreme case with all \(\beta_{jj} \to \infty\).
Property 4
Limit bounds for M:

i.
if β_{jj} → 0 for all j then M → L,

ii.
if \(\beta_{jj} \to \infty\) for all j then M → 0.
Proof
The first statement turns out to be trivial and follows directly from expression (10). To check statement (ii) we will assume for the sake of the current argument that the inverse matrix \(({\mathbf{I}} + {\mathbf{L}} \cdot \hat{\beta })^{  1} \,\) exists. Since the identity I is (trivially) invertible and the matrix L. \(\hat{\beta }\) is invertible (provided A is productive, Property 1, and \(\,\hat{\beta } > 0\)), we can use a version of the matrix inversion lemma of Henderson and Searle (1981) that states that the inverse of a sum of invertible matrices can be written as
\(({\mathbf{I}} + {\mathbf{L}} \cdot \hat{\beta })^{  1} = ({\mathbf{L}} \cdot \hat{\beta })^{  1} \cdot ({\mathbf{I}}^{  1} + ({\mathbf{L}} \cdot \hat{\beta })^{  1} )^{  1} \cdot {\mathbf{I}}^{  1}.\)
We reorder and simplify a little bit:
Notice that \(\beta_{jj} \to \infty\) implies \(\hat{\beta }^{  1}\) → 0 and then M → 0.
The economic interpretation is straightforward. In a fully closed (no trade) economy, i.e., \(\hat{\beta } \to {\mathbf{0}}\), the Armington–Leontief multiplier matrix M coincides with the standard Leontief inverse L. Should all the domestic production be progressively eliminated and imports be increasingly dominant, \(\hat{\beta }^{  1}\) → 0, then there would be no domestic multiplier effect whatsoever as a result of changes in final demand. All impulses from final demand would leak outside the economy.
The main analytical result
Property 5
If the nonnegative matrix A is productive and the shares satisfy \(\hat{\beta } > {\mathbf{0}}\), then the multiplier matrix M exists and is nonnegative.
Proof
Recall first that if A is productive the inverse \({\mathbf{L}} = ({\mathbf{I}}  {\mathbf{A}})^{  1}\) exists and is nonnegative. Hence, trivially L^{−1} also exists and is equal to \(\,({\mathbf{I}}  {\mathbf{A}})\). Additionally, it is always the case that
If \({\mathbf{I}}  {\mathbf{A}} + \hat{\beta }\) should happen to be invertible, then we would have
and the multiplier matrix M would be recovered. We therefore need to see that matrix \({\mathbf{I}}  {\mathbf{A}} + \hat{\beta }\) is indeed invertible. Notice that \(\hat{\beta } > {\mathbf{0}}\) implies the diagonal matrix \(\hat{\rho }\) defined by \(\rho_{ii} = 1 + \beta_{ii}\) satisfies \(\hat{\rho } > {\mathbf{I}}\). From here we can write
Since the diagonal matrix \(\hat{\rho }\) is clearly invertible, nonnegative, and \({\mathbf{0}} < \hat{\rho }^{  1} < {\mathbf{I}}\) all that remains to check is that matrix \(({\mathbf{I}}  \hat{\rho }^{  1} \cdot {\mathbf{A}})\) is invertible too. For this, we invoke the property that eigenvalues for nonnegative matrices are a nondecreasing function of the matrix coefficients (Nikaido 1972, thm. 17.1). In this case from \({\mathbf{0}} < \hat{\rho }^{  1} < {\mathbf{I}}\) we verify that \({\mathbf{0}} < \hat{\rho }^{  1} \cdot {\mathbf{A}} < {\mathbf{A}}\). From this result and the fact that A is productive it follows that
Property 1(iii) now implies that the inverse of matrix \(({\mathbf{I}}  \hat{\rho }^{  1} \cdot {\mathbf{A}})\) exists and is nonnegative. Therefore the multiplier matrix M exists, and is nonnegative and equal to
Remark 1
The result that M exists and is nonnegative ensures that the inverse of \(({\mathbf{I}} + {\mathbf{L}} \cdot \hat{\beta })\,\) also exists.
Remark 2
Notice that without loss of generality, we can relax the restriction that \(\hat{\beta } > {\mathbf{0}}\) to \({\kern 1pt} \hat{\beta } \ge {\mathbf{0}}\). The only change would be that now \(\hat{\rho }^{  1} \cdot {\mathbf{A}} \le {\mathbf{A}}\) but the maximal eigenvalue of matrix \({\kern 1pt} \hat{\rho }^{  1} \cdot {\mathbf{A}}{\kern 1pt}\) would still be less than 1. Hence, productivity of matrix \(\hat{\rho }^{  1} \cdot {\mathbf{A}}{\kern 1pt}\) is guaranteed. The relaxation is relevant for empirical analysis since in these cases sectoral aggregation is selected such that for all i \(x_{i}^{d} > 0\) , whereas \(\,x_{i}^{m} \ge 0\).
Remark 3
If matrices \(\hat{\rho }^{  1} \cdot {\mathbf{A}}\) and A are both productive they can be expanded in convergent matrix series. Since \(\hat{\rho }^{  1} \cdot {\mathbf{A}} \le {\mathbf{A}}\) it follows that
Now \(({\mathbf{I}}  \hat{\rho } \cdot {\mathbf{A}})^{  1} \le {\mathbf{L}}\) and \({\mathbf{0}} < \hat{\rho }^{  1} < {\mathbf{I}}\) imply that M ≤ L. Thus matrix L is effectively an upper bound for matrix M. If in empirical analysis we use L when M is in fact called for, an evaluation error will ensue for we would be upward biasing the multiplier estimates and any results that derive from them. The size of the error will depend on the degree of openness of the economy. The more open to trade the economy, the larger the evaluation bias.
Remark 4
The standard Leontief system \(({\mathbf{I}}  {\mathbf{A}}) \cdot {\mathbf{x}} = {\mathbf{f}}\) is a particular case of the more general equation \((\rho \cdot {\mathbf{I}}  {\mathbf{A}}) \cdot {\mathbf{x}} = {\mathbf{f}}\) when the real number \(\rho\) satisfies \({\kern 1pt} \rho = 1\). This more general system is said to be solvable if for any nonnegative vector f there is a nonnegative vector x such that \((\rho \cdot {\mathbf{I}}  {\mathbf{A}}) \cdot {\mathbf{x}} = {\mathbf{f}}\) holds. A wellknown theorem (Nikaido 1972, thm. 15.3) establishes that solvability is equivalent to the matrix \((\rho \cdot {\mathbf{I}}  {\mathbf{A}})\) satisfying the Hawkins and Simon (1948) conditions. This property is readily extended to our Armington–Leontief model. Indeed, Eq. (9) can be rewritten as \({\mathbf{x}}^{d} = (\hat{\rho }  {\mathbf{A}})^{  1} \cdot {\mathbf{f}}\) and invertibility of \((\hat{\rho }  {\mathbf{A}})\) produces the linear system \({\kern 1pt} (\hat{\rho }  {\mathbf{A}}) \cdot {\mathbf{x}}^{d} = {\mathbf{f}}\). This system is solvable if and only if matrix \((\hat{\rho }  {\mathbf{A}})\) satisfies the Hawkins–Simon condition.
Remark 5
Thm. 6.4 in Nikaido (1968) can also be painstakingly extended to the more general case we study here. Let us define the matrix sequence
If matrix \((\hat{\rho }  {\mathbf{A}})\) has a nonnegative inverse, then the diagonal matrix satisfies \(\hat{\rho } > {\mathbf{0}}\) and the sequence \(\left\{ {{\mathbf{T}}_{k} } \right\}\) converges to \((\hat{\rho }  {\mathbf{A}})^{  1}\). And reciprocally, if \(\hat{\rho } > {\mathbf{0}}\) and the sequence \(\left\{ {{\mathbf{T}}_{k} } \right\}\) is convergent, then \((\hat{\rho }  {\mathbf{A}})\) has a nonnegative inverse and the limit of the sequence is the inverse \((\hat{\rho }  {\mathbf{A}})^{  1}\).
Remark 6
For the general linear equilibrium case of the Armington–Leontief model, with equation \((\hat{\rho }  {\mathbf{A}}) \cdot {\mathbf{x}}^{d} = {\mathbf{f}}\), no condition relating the maximal eigenvalue of A in relation to the eigenvalues of matrix \(\hat{\rho }\) seems to arise (or we have not been able to find). In the standard linear system case, it is known that solvability of the system is equivalent to the maximal eigenvalue of matrix A satisfying the condition \(\lambda ({\mathbf{A}}) < \rho\) (Nikaido 1972, thm. 17.1). This provision is clearly and trivially the same as \(\,\lambda ({\mathbf{A}}) < \lambda (\rho \cdot {\mathbf{I)}}\). However, the conjecture that \(\lambda ({\mathbf{A}}) < \lambda (\hat{\rho })\) would also suffice for solvability in the new setup does not hold. See the counterexample in Sect. 6.
Examples
We have constructed these examples using Smath Studio (2018)—a wonderfully simple but amazingly powerful and free piece of software.
Example 1
Consider a simple input–output (IOT) table with two sectors (“iron” and “wheat”) and total output comprising domestic and imported outputs:
IOT  Iron  Wheat  Demand  Output 

Iron  40  10  50  100 
Wheat  30  50  20  100 
Labor  20  10  
Imports  10  30  
Output  100  100 
For this linear economy we have total output, imports, domestic output, and final demand and intermediate flows equal to
The technology matrices A, \(\hat{\beta }\) and \(\hat{\rho }\) are given by
Matrix A verifies the eigenvalue productivity condition (\(\lambda ({\mathbf{A}}) = 0.836 < 1\)). From here we can calculate the Leontief multiplier matrix L and the new multiplier matrix M:
Notice that the total multiplier for good 1 under L is 5.571, whereas it only gets to be 2.445 under M. The difference indicates the upward bias that would ensue in practical calculations if matrix L is used when M is the correct modeling choice.
Example 2
We verify that matrix M exists and is nonnegative should the above economy stop trading in either wheat or iron. In the first case, β_{11} = 0 and β_{22} = 3/7, whereas in the second, β_{11} = 1/9 and β_{22} = 0. We obtain respectively
In both calculations, we still have L ≥ M but notice that as the economy restricts trade flows, the internal multiplier effect magnifies and gets closer to L.
Example 3
Take matrices A and \(\hat{\rho }\) defined by
We can check that \(\,\lambda ({\mathbf{A}}) = 0.8\), \(\lambda (\hat{\rho }) = 0.9\) , and so \({\kern 1pt} \lambda ({\mathbf{A}}) < \lambda (\hat{\rho })\). When we calculate M, however, we find a nonpositive matrix:
Hence, the system \({\kern 1pt} (\hat{\rho }  {\mathbf{A}}) \cdot {\mathbf{x}}^{d} = {\mathbf{f}}\) would not be solvable. In light of Remark 4, this system would not satisfy the Hawkins and Simon (1948) conditions. This is indeed the case as we can easily check. The eigenvalue condition is not sufficient for solvability. For our empirically based matrices, this possible negativity problem of the generalized multiplier matrix does not arise as Property 5 demonstrates.
Conclusions
The model we develop in this technical note includes the standard linear model as a special case. The relevance for its applicability rests in the empirical fact that all actual economies undertake trade with other countries or other regions, broadly defined. Therefore, it enhances the toolkit of the interindustry researcher by adding a complementary modeling tool. A recent empirical example of the use of this novel approach in input–output analysis is the work of Guerra and Sancho (2018) who study emission multipliers in the European Union.
The implementation of the Armington principle in the present text adopts the admittedly strong assumption of fixed proportions. In other words, the corresponding Armington elasticity is set to zero. Nonetheless, this value is numerically coherent with the perfect complements assumption implicit in the whole structure of the input–output model. This model, as is well known, does not allow for substitution in intermediate goods or primary factors either. We usually interpret this fixed proportion property in the light of the very short run. However, when we move to other modeling platforms, such as applied general equilibrium, or consider longer than shortrun periods, then the elasticity values become extremely relevant. Among the econometrics studies that estimate Armington elasticities, we highlight those of Reinert and RolandHolst (1992) for the US and Welsh (2008) for the UE. From the perspective of international economics, Balistreri and McDaniel (2003) point out that shortrun Armington elasticities are quite lower than longrun estimates while Ruhl (2008) emphasizes that small Armington elasticity values belong to the shortrun variations that describe the business cycle fluctuations. Hence, the zero elasticity assumption in the text, even if strong, could be somewhat defensible.
We have also shown the internal consistency of the model in terms of the existence and nonnegativity of the equilibrium solution. Finally, we have shown how to extend some of the main linear algebra results for nonnegative square matrices.
Availability of data and materials
Data sharing does not apply to this article as no dataset was used or generated during the current study
Notes
 1.
Notational conventions: For two vectors x and y, x < y means x_{i} < y_{i} for all i; \({\mathbf{x}} \le {\mathbf{y}}\) means x_{i} ≤ y_{i} for all i with x_{k} ≠ y_{k} for some k. The same types of notational considerations are extended to matrices. The vector \(\vec{1}\) represents the column unit vector. Any vector x can be rewritten in the format of a diagonal matrix \({\hat{\mathbf{X}}}\). Given a square matrix S, its inverse is given by S^{−1} (if it exists). If x > 0, then \({\hat{\mathbf{X}}}^{  1}\) exists. I denotes the identity matrix and S^{T} the transpose of S.
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Acknowledgements
The author is grateful to the comments of professors C. Pinyol and A.I. Guerra to an earlier draft and to the suggestions of two anonymous referees for improving the text.
Funding
The author is a participating member of the project “Social Choice, networks and welfare: Theory and applications” funded by the Ministry of Economy and Competitiveness of Spain, Grant number MICINNECO201783534P.
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Correspondence to Ferran Sancho.
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Sancho, F. An Armington–Leontief model. Economic Structures 8, 25 (2019). https://doi.org/10.1186/s400080190158y
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Keywords
 Armington principle
 Extended linear equilibrium model
 Solvability
 Technological productivity
JEL Classification
 C62
 C67
 D57