Università Roma Tre

Part I: Functions of several variables – Constrained optimization
Vectorial functions and the Jacobian matrix. Chain rule. Implicit function theorem. Secon property of the gradient (w.p.). Constrained optimization: equality constraints and inequality constraints. Lagrange multiplier theorem NC (w.p. geometrical). Second order conditions for constrained local problems (bordered Hessian matrix). Hint on Khun-Tucker NC. Global case with compact constraint. Geometric representation of the constrained problem. Economic applications. The problem of the consumer.
Part II: Ordinary differential equations and systems
Definitions and examples. Malthusian growth model. Cauchy problem. General existence theorem and uniqueness of the solution (hint). First order linear differential equations: structure of solutions, the case with constant coefficients, the general formula for solutions. Separable variable equations. Second-order linear differential equations: structure of solutions, the constant coefficient case, the homogeneous case and the similarity principle. Logistic growth model. Economic applications. Systems of two-dimensional first-order differential equations. Systems of linear first order differential equations with constant coefficients: solving by eigenvalues, steady states and their stability.
(w.p. = with proof)

Textbook:
• Simon & Blume: “Matematica per le scienze economiche” ed. Egea.

Other materials will be available in the course Moodle class.

Other Textbooks:
• Mastroeni L. and Mazzoccoli A.: “Matematica per le applicazioni economiche” ed. Pearson.