Iterative Descent Method for Generalized Leontief Model
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RESEARCH ARTICLE
Iterative Descent Method for Generalized Leontief Model R. Jana1 • A. K. Das1 • Vishnu Narayan Mishra2
Received: 10 May 2019 / Revised: 16 July 2020 / Accepted: 28 August 2020 Ó The National Academy of Sciences, India 2020
Abstract In this article, we show that under certain condition the generalized Leontief model is solvable by iterative descent method based on infeasible interior point algorithm. We prove the convergence of the method from strictly positive starting point. A numerical example is presented to demonstrate the performance of the algorithm. Keywords Leontief model Generalized Leontief model Linear complementarity problem Vertical linear complementarity problem Infeasible interior point algorithm
1 Introduction The purpose of Leontief model [1] is to find the interrelationship among goods and services for different sectors of the economy. Leontief model considers production of items within some industries where number of industries and number of products are equal. In other words, the model indicates a balance between demand and supply. The model is very useful to analyze the national economy of
& Vishnu Narayan Mishra [email protected] R. Jana [email protected] A. K. Das [email protected] 1
SQC & OR Unit, Indian Statistical Institute, Kolkata 700108, India
2
Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak, Anuppur, Madhya Pradesh 484 887, India
any sector as each of the industries uses input from itself and other industries to produce a particular product. Leontief model is classified as open model and closed model (see [2]). Open model deals with finding the production level based on external demand, whereas the closed model deals only with internal demand. The input–output model has wide applications in the area of regional economics [3], international trade [4], multi-facility inventory systems [5]. The Leontief model describes a facilitated view of an economical situation. The target of this model is to state the exact level of production for each of various types of services or goods. Suppose bj denotes the units available or required at industry j, and ajk be the technical coefficients representing units of output of sector j required per unit output of sector k. The net output bj is normally called the final demand of the jth good. Suppose the basic input–output equations are given as n X xj ¼ ajk xk þ bj ; ð1:1Þ k¼1
then we can find xj which is the output from industry j. There is an assumption that each industry or sector produces only one output. In case if an industry produces more than one output, then the analysis is done by aggregation. Several variants of the input–output model are available in the literature [6]. Leontief model considers single technology. Ebiefung et al. [7] introduced generalized Leontief model by considering multiple technologies. In this paper, we consider an infeasible interior point method in line with Kojima et al. [8] and show that generalized Leontief model can be so
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