Leontief Model Research Articles

Hierarchical open Leontief models

We study the concept of a nonnegative matrix inducing hierarchies and apply it to the nonnegative input–output matrix of the open Leontief model, leading to properties which are stronger than the feasibility of the model. We show some examples of these matrices. The characterization of nonnegative matrices inducing hierarchies for any permutation of indices leads to certain matrices of class M(x).

Transcribing the Tableau Économique: Input-Output Analysis à la Quesnay

Since its publication in 1955, Almarin Phillips' article “The Tableau Économique as a Simple Leontief Model” has inspired an interesting line of interpretation of the arithmetical schemes presented two centuries earlier by François Quesnay. Subsequent attempts to represent the Tableaux économiques in the form of Input-Output transcriptions by Shlomo Maital (1972), Bernhard Korte (1972), Tibor Barna (1975), and Paul Samuelson (1986), among others, have underlined the brilliance of the French doctor's formal conception of the economy as a reproductive system.

Productive structure and income distribution: The Brazilian case

The objective of the paper is to assess the impacts of some of the important changes occurred in the Brazilian economy in the 90s on income distribution. A Pure Leontief Model and a Leontief-Miazawa Model were utilized to portray the structure of the economy in both years, and to perform counterfactual simulations on some important changes occurring during the period. The methodology allowed for the identification of the high- and low-inequality sectors in both years, and to assess their contribution to the increasing inequality during the period. It is interesting to notice that some sectors with low internal inequality ended-up provoking increased global inequality through their interaction pattern with other sectors in the economy, and through the consumption structure. The results also indicate that the change in sectoral shares in the period contributed to diminishing inequality. Therefore, the causes for increasing inequality remain within the distribution of wages within the sectors.

A NOTE ON IMPOSING LOCAL CURVATURE IN GENERALIZED LEONTIEF MODELS

In this paper, we build on Ryan and Wales (1998) and Moschini (1999) and impose curvature conditions locally on the generalized Leontief model, introduced by Diewert (1974). In doing so, we exploit the Hessian matrix of second order derivatives of the reciprocal indirect utility function, unlike Ryan and Wales (1998) and Moschini (1999), who exploit the Slutsky matrix.

The resource conservation effect of recycling in a dynamic Leontief model

The aim of the paper is to investigate the impact of recycling on the use of non-renewable resources in the economy. The paper tries to generalize the classical dynamic input–output model. In this regard we extend the standard Leontief model with the balance equation of recycled products, and we establish some properties of this augmented model. We investigate how recycling extends the availability of non-renewable natural resources for the next generations in an inter-industry framework. Supposing a balanced growth both for production and consumption, we examine the existence of the balanced growth path of this model and compare the results to the classical Leontief model. We try to answer the question whether recycling/reuse increases the growth possibility of an economy. Finally, we illustrate our results with a simple numerical example. Thus, we analyze a possible sustainable development of the economy on the basis of the product recovery management of industries.

Ronald W. Jones and Two-Sector Growth: Ramsey Optimality in the RSS and Leontief Cases

In this essay, we interpret the value-loss line, familiar in models of undiscounted optimal growth, as delineating the cone of diversification, familiar in models of international trade. This interpretation allows an embedding of the 2-sector Robinson-Solow-Srinivasan (RSS) model in the 2-sector Leontief model, and clarifies situations when the Leontief case offers additional insights into transition dynamics. The geometry underlying this embedding extends an earlier construction due to Khan-Mitra, and it has independent interest.

A critical comment on Oosterhaven–Stelder net multipliers

Recently, Oosterhaven and Stelder (OS; J Reg Sci 42(3):533–543, 2002; Trade, network and hierarchies: modeling regional and interregional economies, pp. 119–133, 2002) have introduced the operational idea of a “compensated net multiplier” (CNM) to take into account the double counting that occurs when output replaces final demand as an exogenous entry when the Leontief model is used to evaluate operationally the impact of large investments. Each output gross multiplier is compensated by multiplying it by the ratio of final demand to output. Oosterhaven (Research Report 04C01, 2004) has demonstrated that CNM fulfills an axiom of “total output preservation,” namely that the sum of effects equals the total output. It is shown in the present paper that the original CNM is only a homogenous formula with no causal link between the cause (the exogenous output) and the effect (the total output), so that it cannot be a multiplier. The paper explores all the other possibilities for deriving CNM by considering an exogenous output: (1) in absolute value; (2) in variation; and finally (3) at final equilibrium. Possibilities (1) and (2) do not work while only the variation from zero is option (3) working, although imperfectly. The derivations yield two results: (1) the final demand ratio, and so CNM, cannot be stable even if the technical coefficient matrix is fixed, except for very small perturbations; (2) even if total output preservation derives from the definition of a multiplier, CNM cannot fulfill total output preservation when an exogenous output is taken as the initial shock. De Mesnard (J Reg Sci 42(3):545–548, 200) has proposed another solution, “iterative net multiplier” (INM), derived by considering the successive rounds after an exogenous impact of output. INM is stable and makes a link between cause and effect. In contrast to the assertions by OS (J Reg Sci 42(3):533–543, 2002; Trade, network and hierarchies: modeling regional and interregional economies, pp. 119–133, 2002), INM fulfills output preservation when an exogenous output is taken as the initial shock. Neither CNM nor INM must be confused with the so-called “output-to-output multipliers.”

Coefficient stability and predictability in input‐output models: a comparative analysis for the Netherlands

Following the work of Bon, this article compares the stability of two types of input‐output models. These are, the Leontief (or demand‐driven) model and the Ghosh (or supply‐driven) model. The empirical results are based on a times series of annual input‐output tables for the Netherlands, covering the period 1948–84. Three types of analysis are carried out. The first to evaluate the stability of the models' underlying coefficients over time. The second to investigate the forecasting behaviour when there is perfect foresight with respect to the models' exogenous variables. The third to examine the predictability of the models, absent of perfect foresight.

Leontief and Samuelson on the Non-Substitution Theorem

The non-substitution theorem was first demonstrated by Samuelson and Georgescu-Roegen for a generalized model of Leontief. The theorem raised the question to know if the “deep-seated causes of [economic] development are to be found in the variations of the basic structural relationships themselves_ that is in modification of changes in consumers’ tastes_ and changes in the structure of productive processes” (Leontief). Without advanced empirical evidence, this episode brought at the core of linear economics the substitution principle and consider Leontief’s assumption of non-substitutions among production factors as simplistic and unsatisfactory. This can be viewed as an anomaly for the philosophy of science: a change in a research program without empirical anomalies but only a certain epistemological and theoretical context.

A Dynamic Leontief Model with Non-renewable Resources

In this paper we study a generalization of the dynamic Leontief input–output model. We extend the standard dynamic Leontief model with the balance equation of non-renewable resources. Obviously, the non-renewable stocks will decrease, exploiting primary resources. In this study we examine the controllability of this extended model by taking the consumption as the control parameter. Assuming balanced growth for both consumption and production, we investigate how long these scarce resources will cover the input needs of production and how the lifetime of the system depends on the balanced growth rate and on the consumption. In doing so, we apply classic results from control theory and on eigenvalue problems in linear algebra.

A world trade model based on comparative advantage with m regions, n goods, and k factors

This paper describes the World Trade Model, a linear program that determines world prices, scarcity rents, and international trade flows based on comparative advantage in a world economy with m regions, n goods, and k factors. The new model generalizes the World Model of Leontief et al. (1977) in ways that make it particularly useful for analyzing scenarios about sustainable development. Major properties of the model are demonstrated, and sources of the gains from trade are identified for the world as a whole and for individual regions. Illustrative results are reported for a 10-region, 8-good, 3-factor model of the world economy.

A New Matrix Theorem: Interpretation in Terms of Internal Trade Structure and Implications for Dynamic Systems

Economic systems often are described in matrix form as x = Mx. We present a new theorem for systems of this type where M is square, nonnegative and indecomposable. The theorem discloses the existence of additional economic relations that have not been discussed in the literature up to now, and gives further insight in the economic processes described by these systems. As examples of the relevance of the theorem we focus on static and dynamic closed Input-Output (I-O) models. We show that the theorem is directly relevant for I-O models formulated in terms of difference or differential equations. In the special case of the dynamic Leontief model the system’s behavior is shown to depend on the properties of matrix M = A + C where A and C are the matrices of intermediate and capital coefficients, respectively. In this case, C is small relative to A and a perturbation result can be employed which leads directly to a statement on the system’s eigenvalues. This immediately suggests a solution to the well-known problem of the instability of the dynamic Leontief model.

Pace, Pattern and Phase: A Morphology of Cycles

A closed dynamic Leontief model is implemented for the Business, Government and Household sectors of the Hungarian and the US economies at the end of the past century. It is built on rough estimates of its flow and stock coefficients. The growth rate and the length and pattern of the four most familiar cycles are reasonably well approximated by the model and are similar for the two countries in spite of the large discrepancy in the proportions of the equilibrium quantity vectors. The mathematics of cycles (for discrete or continuous time) is also discussed. Fluctuations are shown to be the elementary, basic and unavoidable consequence of all productive activities. They will emerge even in the absence of any price mechanism.

Some results on continuous type conditional input-output equation—Fixed point and surjectivity methods

Based on the classical (matrix type) input-output analysis, a type of nonlinear (continuous type) conditional Leontief model, input-output equation were introduced, as well as three corresponding questions, namely, solvability, continuity and surjectivity, and some fixed point and surjectivity methods in nonlinear analysis were used to deal with these questions. As a result, the main theorems are obtained, which provide some sufficient criterions to solve above questions described by the boundary properties of the enterprise’s consuming operator.

The Structure of the Sri Lankan Economy

Sri Lanka is a small island economy without much of natural resources. In spite of a troubled domestic environment in recent times, the country has recorded an impressive peiformance in the areas of Average Life Expectancy, Adult Literacy rates, Infant Mortality Rates and similar indices of develapment. However, economic growth, sustained employment generation, reasonable price stability, balance in the external sector, and reduction in income inequality have not been adequate though to proper economy into the league of Developed Nations. The objective of this paper is to study the inter-sectoral linkages of this economy using Multiplier Analysis to understand his structure by measuring the degree of interconnectedness of its different sectors, since the effectiveness of policy measures is expected to depend on the degree of interconnectedness of an economy. For this purpose, the Extended Input-Output Methodology combining the Leontief (1941) and Ghosh (1958) Models has been used. The results show that the various sectors of the economy are weakly integrated, which is perhaps the reason why the effects of growth and development oriented policies have not percolated down throughout the economy. The development of strong inter-sectoral linkages is therefore a necessary prerequisite for the country's overall economic development.

Studying a Basic Price Equation as a Periodic System

This paper is concerned with the study of the basic price equation of the dynamic Leontief model described by F(k)p(k+ 1) = [1 +r(k)] [F(k) - I] p(k) + Gp0(k+ 1)a0, k = 0, 1, …, L - 1, where F(k) = I + (I-AT)1 BT(k) and G = (I-AT)-1 considering both time-invariant and periodic coefficients, with particular emphasis on establishing some results in the periodic case where F(k) are commuting normal matrices.

A Turnpike Theorem in the Closed Dynamic Leontief Model with a Singular Matrix of Capital Coefficients

The paper refers to the well-known Tsukui turnpike theorem on convergence of optimal growth trajectories in the closed dynamic Leontief model to the maximum balanced growth trajectory, called turnpike. In the original proof of this theorem, the assumption that the matrix B of capital coefficients is non-singular plays an essential role. For many reasons this assumption, very convenient for theoretical analysis, is not always satisfied in input-output systems built for empirical purposes. This paper fills the gap between theory and empirical studies, presenting a proof that convergence of optimal trajectories towards the turnpike is also a characteristic feature of the closed Leontief model in the case when matrix B is singular. The general idea of the proof is based on the approximation of a singular matrix B by an infinite sequence of non-singular matrices.

Leontief-Based Model of Risk in Complex Interconnected Infrastructures

Wassily Leontief received the 1973 Nobel Price in Economics for developing what came to be known as the Leontief input-output model of the economy. Leontief's model enables understanding the interconnectedness among the various sectors of an economy and forecasting the effect on one segment of a change in another. A Leontief-based infrastructure input-output model is developed here to enable an accounting of the intraconnectedness within each critical infrastructure as well as the interconnectedness among them. The linear input/output model is then generalized into a generic risk model with the former as the first-order approximation. A preliminary study of the dynamics of risk of inoperability is discussed, using a Leontief-based dynamic model. Several examples are presented to illustrate the theory and its applications.

Optimal control for dynamic versions of the leontief and other matrix models*

In this paper dynamic versions of the Leontief model are formulated as two kinds of optimal control problems where the control regions are given by linear inequalities. Applying Pontryagins maximum principle a practicable form of an optimality test is presented. Finally, an extension to the case of multistage models is developed.

A Subsidy-Surplus Model and the Skorokhod Problem in an Orthant

We consider the deterministic Skorokhod problem in an orthant of the form [Formula: see text] with (Yw)i(·) nondecreasing, and (Yw)i(·) not increasing while (Zw)i(·) > 0. This can be viewed as a subsidy-surplus model in an interdependent economy. Existence of a unique solution is established under fairly general conditions (viz. with R(·, ·, ·) satisfying a uniform spectral radius condition). Comparison result for (SP) vis-a-vis the usual partial order on the orthant is studied; we show that the more “inward looking” the reflection vectors and the drift, the larger the values of Yw will be but the values of Zw will be smaller. In addition to showing that the Leontief-type output is a feasible subsidy, connection between (SP) and “minimality” of feasible subsidies is discussed (consequently it is suggested that (SP) may be taken as a continuous time feedback-form analogue of open Leontief model). In the stochastic case, (Y(t), Z(t)) turns out to be a strong Markov process if w(·) arises from aLevy process. Relevance of the comparison result to recurrence/positive recurrence of Z(·) process is pointed out.