Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Informatica
Programul de studii: Modelare si simulare - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MME1010	Modeling of Economical Processes	

Semester	Hours: C+S+L+P	Category	Status	
4	2+1+0+1	speciality	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
PETRUSEL Adrian Olimpiu	Ph.D.	Prof.	Applied Mathematics	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The goal of this course is to provide students with some models and problems of mathematical economics. After completing the course, the students will have basic concepts and tools of mathematical economics and they should be able to analyse and diagnose the behavior of some actual economical process.	

IV. Course contents
Course 1. Arrow-Debreu model. 
Seminar 1. The functionals D, Delta, Rho and H (exercises and examples)
Course 2. The concept of generalized game and abstract economy.
Seminar 2. Hausdorff-Pompeiu metric (exercises and properties) 
Course 3. Continuity concepts for multivalued operators. 
Seminar 3. Examples and exercises in connection to the continuity of multivalued operators. 
Course 4. Selection results for multivalued operators. The continuous case. 
Seminar 4. Lipschitz-type multivalued operators. Basic properties. 
Course 5. Topological fixed point theorems for multivalued operators. 
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. 
Seminar 6. Middle Term Written Test
Course 7. KKM lemma.
Seminar 7. Examples and exercises, particular cases of KKM lemma. 
Course 8. Applications of KKM lemma to variational inequalities and game theory.
Seminar 8. Caristi&#8217;s theorem. Applications and extensions. 
Course 9. Stability results for the KKM set.
Seminar 9. (Report): Maximal elements. 
Course 10. Coincidence theorems and applications to Nash equilibrium points. 
Seminar 10 Coincidence theorems for singlevalued and multivalued operators. 
Course 11. Strict fixed point theorems.
Seminar 11 Strict fixed point theorems. 
Course 12. Existence results for equilibrium points. 
Seminar 12. Existence for Walras-type equilibrium. 
Course 13. The excess-demand multifunction.  
Seminar 13. Equilibrium price for the excess-demand multifunction. 
Course 14. New research directions in mathematical economics. 
Seminar 14. Models of written tests. Presentation of the grades obtained during the semester evaluation.	

V. Bibliography
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. Thematic of didactic activities per weeks
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]-pag. 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. Didactic methods used
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. Assessment
The final grade is the weighted mean of the following three grades. 
The final grade = 50% Final Written Test + 25% Middle Term Written Test + 25% Report.  Successful passing of the exam is conditioned by the final grade that has to be at least 5. 
All university official rules with respect to students attendance of academic activities, as well as to cheating and plagiarism, are valid and enforced. 	

IX. Additional references
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.	

	Dean,							Chair head,
Prof.dr. Leon Tambulea				         Prof.dr. O. Agratini

	Date,							Course responsible,
							Prof. PETRUSEL Adrian Olimpiu, Ph.D.