Some applications of linear algebra in economics and game theory

Linearna algebra se primjenjuje u mnogim matematičkim disciplinama te prirodnim i društvenim znanostima. U ovom radu želimo pokazati neke primjene linearne algebre u ekonomiji i teoriji igara. U ekonomiji veliki značaj ima input-output analiza. Analiza se temelji na pretpostavci da se proizvodni sustav privrede jedne zemlje može podijeliti na određeni broj međusobno povezanih proizvodnih sektora. Pritom svaki sektor u proizvodnji svojih proizvoda koristi proizvode ostalih sektora ili čak svoje proizvode. Osim što svaki sektor isporučuje svoje proizvode ostalim proizvodnim sektorima, isporučuje ih i finalnim potrošačima. Stoga ukupna razina proizvodnje određenog sektora ovisi o zahtjevima ostalih proizvodnih sektora i finalnih potrošača za njegovim proizvodima. Nabave i isporuke proizvoda između sektora prikazujemo input-output tablicama. Pitanje koje se postavlja je koju razinu proizvodnje svaki od sektora mora imati kako bi se zadovoljila ukupna potražnja za njihovim proizvodima. Odgovor na to pitanje nam daje Leontiefov input-output model. U proteklom razdoblju razvile su se tri osnovne verzije input-output modela: otvoreni statički model, zatvoreni statički model i dinamički model. U ovom radu naglasak je na otvorenom statičkom input-output modelu. Osim što smo detaljnije objasnili metodu input-output analize, u prvom poglavlju opisali smo ukratko i povijesni razvoj same analize. Autor koji je najviše pridonio njenom razvoju je Wassily Leontief. U drugom poglavlju smo predstavili Leontiefov otvoreni input-output model i naveli pretpostavke koje mora zadovoljavati. Model se temelji na rješavanju sustava n linearnih jednadžbi s n nepoznanica, stoga ovdje veliku ulogu ima linearna algebra i poznavanje matrica. Dokazali smo teoreme koji nam pod određenim uvjetima garantiraju egzistenciju i jedinstvenost nenegativnog rješenja modela i egzistenciju ekonomski značajnog rješenja. Na kraju poglavlja dali smo i primjer Leontiefovog otvorenog input-output modela. U trećem poglavlju smo pokazali primjene linearne algebre u teoriji igara. Naslasak je na matričnim igrama te smo, prije samih primjera matričnih igara, definirali osnovne pojmove na kojima se teorija igara temelji.Linear algebra is used in many mathematical disciplines and in natural and social sciences. In this thesis we want to show some applications of linear algebra in economics and game theory. Input-output analysis is of great importance in economics. It is based on assumption that the production system of the economy can be divided into certain number of interrelated production sectors. Each sector in a production of its products uses products of other sectors or even of its own sector. Besides giving the products to other sectors, each sector gives its products to final consumers. Therefore, the total level of production of a particular sector depends on the requirements of other production sectors and final consumers for its products. The interdependence between sectors is shown in input-output table. The question is what level of production each of the sectors must have in order to meet the total demand for their products. The answer is given by Leontief’s input-output model. In the past period, three basic versions of the input-output model have been developed: the open static model, the closed static model, and the dynamic model. In this thesis, the focus is on the open static input-output model. Besides explaining the method of input-output analysis, in the first chapter we have briefly described the historical development of the analysis. The author who contributed the most to its development is Wassily Leontief. In the second chapter, we have presented Leontief’s open input-output model and have given the assumptions which model must satisfy. The model is based on solving a system of n linear equations with n unknowns, so linear algebra and knowledge of matrices play an important role here. We have proved theorems that guarantee the existence and uniqueness of a non-negative model solution and the existence of an economically meaningful solution under certain conditions. At the end of the chapter, we have given the example of Leontief’s open input-output model. In the third chapter, we have shown the applications of linear algebra in game theory. The focus is on matrix games and before the examples of matrix games, we have defined the fundamental concepts on which game theory is based