Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Matematică Computaţională - în limba maghiară
Limba de predare: Maghiară

SYLLABUS

I. Általános adatok
Kód	Tantárgy neve	
MME1010	Modelarea proceselor economice	

Félév	Órák: C+S+L+P	Kategória	Típus	
4	2+1+0+1	specialitate	opţională	

II. Foállásban az  Egyetem  oktatói
Név	Tudományos	Oktatói	Tanszék	Tevékenység	
	fokozat	státus		E	Sz	L	
PETRUSEL Adrian Olimpiu	Dr. 	Prof.	Matematică aplicată	*			

Társult oktatók
Név	Tudományos	Intézmény	Állás	Tevékenység 	
	fokozat		típusa	E	Sz	L	

III. A tantárgy célkituzései
Scopul acestui curs este de a prezenta studentilor unele modele si probleme dim teoria economiilor matematice. Studiul unor procese economice modelate matematic se trateaza prin instrumente ale analizei neliniare, în special a teoriei punctului fix, si a ecuatiilor diferentiale.	

IV. A tantárgy tartalma
1) Modele matematice in economie: modelul Arrow-Debreu (problema consumatorului, elemente maximale, echilibru walras, jocuri generalizate, puncte echilibru de tip Nash)
2) Concepte si rezultate fundamentale din teoria analizei operatorilor multivoci (functionale pe spatiul partilor unui spatiu metric, continuitatea operatorilor multivoci, teoreme de selectie, teoreme de punct fix)
3) Lema KKM si teorema lui von Neumann. Aplicatii in modelarea matematica a proceselor economice
4) Inegalitati variationale
5) Elemente maximale si puncte echilibru pentru economii abstracte	

V. Könyvészet
1)  G. Mot, A. Petrusel, G. Petrusel: Topics in Nonlinear Analysis and Applications to Mathematical Economics, House of the Book of Science, Cluj-Napoca, 2007. 
2)  J.P. Aubin: Optima and Eqilibria, Springer, Berlin, 1993. 
3)  G.X.Z. Yuan: KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, 1999. 
4)  K. Border: Fixed Point Theorems with Applications to Economic and Game Theory, Cambridge University Press, London, 1985. 
5)  J. Dugundji, A. Granas.: Fixed Point Theory,.  Springer-Verlag, Berlin, 2003.

VI. Az oktatói tevékenység heti tematikája
Course 1. Arrow-Debreu model. References: [1]-pages v-xvii and [5]-pages 1-17. Seminar 1. The functionals D, Delta, Rho and H (exercises and examples) Reference: [1]-pages 3-16.
Course 2. The concept of generalized game and abstract economy. References: [1]-pages xviii-xx and [5]-pages 1-22. Seminar 2. Hausdorff-Pompeiu metric (exercises and properties) Reference: [1]-pages 3-16.
Course 3. Continuity concepts for multivalued operators. Reference: [1]-pages 27-36. Seminar 3. Examples and exercises in connection to the continuity of multivalued operators. Reference: [1]-pages 27-36.
Course 4. Selection results for multivalued operators. The continuous case. Reference: [1]-pages 39-46. Seminar 4. Lipschitz-type multivalued operators. Basic properties. Reference: [1]-pages 24-25.
Course 5. Topological fixed point theorems for multivalued operators. References: [1]-pages 57-63., [4]-pages 533-540. Seminar 5. Examples and counterexamples for topological fixed point theorems. References: [4]-pages 533-540, [6]-pages 99-104.
Course 6. Properties of the fixed point set. Reference: [1]-pages 65-71. Seminar 6. Middle Term Written Test
Course 7. KKM lemma. References: [1]-pages 103-105 and [6]-pages 96-98. Seminar 7. Examples and exercises, particular cases of KKM lemma. References: [1]-pages 103-105 and [6]-pages 96-98.
Course 8. Applications of KKM lemma to variational inequalities and game theory. References: [1]-pages 107-114. Seminar 8. Caristi’s theorem. Applications and extensions. Reference: [1]-pages 79-88.
Course 9. Stability results for the KKM set. References: [1]-pages 115-117. Seminar 9. (Report): Maximal elements. Reference: [1]-pages121-123 and [5]-pages 24-31.
Course 10. Coincidence theorems and applications to Nash equilibrium points. Reference: [1]-pages 89-99. Seminar 10 (Report): Coincidence theorems for singlevalued and multivalued operators. Reference: [1]-pages 89-99. 
Course 11. Strict fixed point theorems. Reference: [1]-pages 73-78. Seminar 11 (Report): Strict fixed point theorems. Reference: [1]-pages 73-78.
Course 12. Existence results for equilibrium points. Seminar 12. (Report) Existence for Walras-type equilibrium. Reference: [1]-pages 125-126.
Course 13. The excess-demand multifunction.  References: [1]-pages 127-128. and [4]-pages102-107. Seminar 13. (Report) Equilibrium price for the excess-demand multifunction. References: [1]-pages 127-128. and  [4]-pages 102-107.
Course 14. New research directions in mathematical economics. References: [1]-[5].  Seminar 14. Models of written tests. Presentation of the grades obtained during the semester evaluation.	

VII. Alkalmazott didaktikai módszerek
Expositions: description, explanation, class lectures, dialog-based lectures, lectures with demonstrations, introductive lectures,  synthesis lectures. 
Conversations: debate, dialog, introductive conversations, conversations for knowledge consolidation, conversations to systematize and synthesize knowledge 
Use of problems: use of problem questions, problems and problem situations.	

VIII. Értékelés
Nota finala se compune din:

50% examen in sesiune
25% test scris in timpul semestrului
25% referat in timpul semestrului	

IX. Kiegészíto könyvészet
1)	A. Petrusel: Multifunctii si aplicatii, Presa Univ. Clujeana, Cluj-Napoca, 2001.
2)	J.W.S.Cassels, Economics for Mathematicians, Cambridge Univ.Press, 1989.
3)	J. P. Aubin, A. Cellina: Differential Inclusions, Springer-Verlag, Berlin, 1993.
4)	A. Petrusel, G. Mot: Multivalued Analysis and Mathematical Economics, House of the Book of Science, Cluj-Napoca, 2004.
5)	C. D. Aliprantis, D. J. Brown, O. Burkinshaw: Existence and Optimality of Competitive Equilibria, Springer-Verlag, Berlin, 1990.
A. Muntean: Fixed Point Principles and Applications to Mathematical Economics, Cluj University Press, 2002.	

	Dékán,						Tanszékvezeto,
Prof.dr. Leon Tambulea				Prof.dr. O. Agratini

	Dátum,						Eloadás felelose,
								Prof. Dr.  PETRUSEL Adrian Olimpiu