Summary of real functions of a real variable. Main properties and definitions, elementary functions, derivatives and derivation rules. Second order derivatives, chain rule, monotonicity and first derivative, second order derivative and convexity NSC for the existence of local maxima and minima. Global max and min. Weierstrass theorem.
Topology and metrics of the n-dimensional real space. Sequences. Open sets. Closed sets. Compact sets.
Functions of several variables. Definition and examples. Level curves. Linear functions and quadratic forms. Continuous functions and Weierstrass theorem.
Differential calculation for several variables functions. Partial derivatives. Differential and tangent plane. Gradient and Jacobian matrix. Approximations by differentials. Darivative along a curve. Directional derivative. Chain rule. Second order derivatives and Hessian matrix.
Implicit function theorem: Case of a function in two variables. Geometric interpretation. Case of a function in several variables. Linear case: m equations, n+m variables. General case of systems of m equations and n+m variables. Inverse function theorem.
Static optimization. Free optimization. NC for the existence of local maxima and minima. Stationary points. Definite and semidefinite quadratic forms. Criterion for 2 or 3 variables. Second order SC for the existence of local free maxima and minima. Constrained optimization. Equality constraints. NC for the existence of local maxima and minima. Inequality constraints: NC for the existence of local maxima and minima. Kuhn-Tucker conditions for non-negative variables. Interpretation of the Lagrange multiplier.
Homogeneous functions. Geometric definition and properties. Euler theorem. Homothetical functions
Concave and convex functions. Definitions and geometrical properties. Properties and characterization. Concavity and sign of the Hessian matrix. Quasiconcave and quasiconvex functions. Cobb-Douglas functions. Pseudoconcave functions Free and constrained optimization in hypothesis of quasiconvexity and quasiconcavity.
Dynamical models: Malthus model in discrete and continuous time. Dynamical models in continuous and discrete time. Definition of differential equation and difference equations. Existence and uniqueness of solutions for first order difference equations.
Calculus of variations. Definition of a problem of calculation of variations. Euler equation in integral form (w.p.). Euler-Lagrange equation. Transversality conditions. Sufficient conditions in hypothesis of concavity or convexity.
Optimal control theory: Optimal control problem (in discrete and continuous time). Pontryagin maximum principle. Sufficient conditions in hypothesis of concavity.
Dynamic programming. Finite horizon and discrete time: Bellman’s principle. Dynamic programming principle. Bellman's equation and optimization conditions.
(w.p.)=”with proof”.

Textbooks
• Simon & Blume “Matematica per le scienze economiche” ed Egea.
• Salsa & Squellati “Modelli dinamici e controllo ottimo” ed. Egea: chapter 1 (pp.1-6, 17-26), chapter 3 (pp.69-73, 78-79) , chapter 9 (pp. 211-232), chapter 10 (pp. 245-249, 277-288); chapter 11 (pp. 293-296, 304-310).

1. The notion of Surplus
2. The capitalist mode of production
3. Sraffa's theory of prices
4. The integrated wage-commodity sector and the surplus equation
5. The marginalist theory of distribution
6. Capital as a factor of production

- Teaching material written by the teacher and available online.
- Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge University Press: Cambridge. Chapters I-IV and VI.

1. Theory of firm
1.1 Technology
1.2 Profit maximization
1.3 Profit function
1.4 Cost minimization
1.5 Cost function
1.6 Duality

2. Consumer theory
2.1 Preferences
2.2 Utility function
2.3 Utility maximization
2.4 Demand function and demand and correspodence
2.5 Minimization of consumre’s expenditures
2.6 Slutsky equation: substitution and income effects
2.7 Consumption with endowements

3. Consumer’s Surplus
3.1 Compensative variation
3.2 Equivalent variation
3.3 Quasi-linear utility
3.4 Gorman indirect utility function and aggregate demand

4. Choice under uncertainty
4.1 Lotteries
4.2 Expected utility
4.3 Risk aversion
4.4 Measures of absolute and relative risk aversion
4.5 The CAPM

5 Perfect competition
5.1 The firm in perfect competition
5.2 Aggregate supply function
5.3 Short and long run equilibrium
5.4 Pareto efficiency
5.5 Efficiency and welfare

6 Monopoly
6.1 Monopolistic firm
6.2 Profit maximization
6.3 First, second, and third degree price discrimination.

VARIAN, HAL R. MICROECONOMIC ANALYSIS.3RD ED. NEW YORK: W.W. NORTON & COMPANY, 1992.

THE MULTIPLE LINEAR REGRESSION MODEL. INTERPRETATION AND COMPARISON OF REGRESSION MODELS. HETEROSCHEDASTICITY AND AUTOCORRELATION. ENDOGENITY AND INSTRUMENTAL VARIABLES.

MODELS WITH LIMITED DEPENDENT VARIABLES. BINARY CHOICE MODELS: PROBIT AND LOGIT MODELS. MULTIPLE-CHOICE MODELS: PROBIT AND LOGIT MODELS ORDERED, LOGIT MODEL MULTINOMIAL. MODELS FOR COUNTING DATA. TOBIT MODELS.
Seemingly Unrelated Regression (SUR): ESTIMATION AND TESTING METHODS.
SIMULTANEOUS EQUATIONS Model (SEM).
MODELS FOR PANEL DATA. THE STATIC LINEAR MODEL: FIXED EFFECT MODELS; RANDOM EFFECT MODELS. ESTIMATORS WITHIN AND BETWEEN. HETEROSCHEDASTICITY AND AUTOCORRELATION TESTS. DYNAMIC MODELS FOR PANEL DATA: THE ARELLANO-BOND ESTIMATOR.
MODELS WITH LAGGED VARIABLES: DYNAMIC REGRESSION MODELS, MODELS WITH DISTRIBUTED LAGS (DL, ARDL).

For the course program students can consult the following texts (the first three can be considered alternative among them)
1) Introduzione all'Econometria, J. H. Stock, M. W. Watson, Ed. Pearson
2) Econometrica, J. Johnston, Ed. Franco Angeli
3) Econometria, M. Verbeek, Ed. Zanichelli
4) Lectures on advanced econometrics, L. Pieraccini, Ed. Aracne
5) Introduction to Time Series Analysis and Forecasting, D. C. Montgomery, C. L. Jennings, M. Kulahci, Ed. Wiley

Some of the topics in the course can also be found in
Introductory Econometrics for Finance, C. Brooks, Cambridge University Press

For introductory references of statistical inference and algebra of matrices students can consult:
Fondamenti di Inferenza Statistica, L. Pieraccini, Ed. Giappichelli
Matrix Differential Calculus in Statistics and Econometrics, J. R. Magnus, H. Neudecker, Ed. Wiley Series in Probability and Statistics

For the laboratory course with R, students can consult one of the following texts:
Introductory Statistics with R, P. Dalgaard, Ed. Springer
An Introduction to Applied Multivariate Analysis with R, B. Everitt, T. Hothorn, Ed. Springer

THE MULTIPLE LINEAR REGRESSION MODEL. INTERPRETATION AND COMPARISON OF REGRESSION MODELS. HETEROSCHEDASTICITY AND AUTOCORRELATION. ENDOGENITY AND INSTRUMENTAL VARIABLES.

MODELS WITH LIMITED DEPENDENT VARIABLES. BINARY CHOICE MODELS: PROBIT AND LOGIT MODELS. MULTIPLE-CHOICE MODELS: PROBIT AND LOGIT MODELS ORDERED, LOGIT MODEL MULTINOMIAL. MODELS FOR COUNTING DATA. TOBIT MODELS.
Seemingly Unrelated Regression (SUR): ESTIMATION AND TESTING METHODS.
SIMULTANEOUS EQUATIONS Model (SEM).
MODELS FOR PANEL DATA. THE STATIC LINEAR MODEL: FIXED EFFECT MODELS; RANDOM EFFECT MODELS. ESTIMATORS WITHIN AND BETWEEN. HETEROSCHEDASTICITY AND AUTOCORRELATION TESTS. DYNAMIC MODELS FOR PANEL DATA: THE ARELLANO-BOND ESTIMATOR.
MODELS WITH LAGGED VARIABLES: DYNAMIC REGRESSION MODELS, MODELS WITH DISTRIBUTED LAGS (DL, ARDL).

For the course program students can consult the following texts (the first three can be considered alternative among them)
1) Introduzione all'Econometria, J. H. Stock, M. W. Watson, Ed. Pearson
2) Econometrica, J. Johnston, Ed. Franco Angeli
3) Econometria, M. Verbeek, Ed. Zanichelli
4) Lectures on advanced econometrics, L. Pieraccini, Ed. Aracne
5) Introduction to Time Series Analysis and Forecasting, D. C. Montgomery, C. L. Jennings, M. Kulahci, Ed. Wiley

Some of the topics in the course can also be found in
Introductory Econometrics for Finance, C. Brooks, Cambridge University Press

For introductory references of statistical inference and algebra of matrices students can consult:
Fondamenti di Inferenza Statistica, L. Pieraccini, Ed. Giappichelli
Matrix Differential Calculus in Statistics and Econometrics, J. R. Magnus, H. Neudecker, Ed. Wiley Series in Probability and Statistics

For the laboratory course with R, students can consult one of the following texts:
Introductory Statistics with R, P. Dalgaard, Ed. Springer
An Introduction to Applied Multivariate Analysis with R, B. Everitt, T. Hothorn, Ed. Springer

THE MULTIPLE LINEAR REGRESSION MODEL. INTERPRETATION AND COMPARISON OF REGRESSION MODELS. HETEROSCHEDASTICITY AND AUTOCORRELATION. ENDOGENITY AND INSTRUMENTAL VARIABLES.

MODELS WITH LIMITED DEPENDENT VARIABLES. BINARY CHOICE MODELS: PROBIT AND LOGIT MODELS. MULTIPLE-CHOICE MODELS: PROBIT AND LOGIT MODELS ORDERED, LOGIT MODEL MULTINOMIAL. MODELS FOR COUNTING DATA. TOBIT MODELS.
Seemingly Unrelated Regression (SUR): ESTIMATION AND TESTING METHODS.
SIMULTANEOUS EQUATIONS Model (SEM).
MODELS FOR PANEL DATA. THE STATIC LINEAR MODEL: FIXED EFFECT MODELS; RANDOM EFFECT MODELS. ESTIMATORS WITHIN AND BETWEEN. HETEROSCHEDASTICITY AND AUTOCORRELATION TESTS. DYNAMIC MODELS FOR PANEL DATA: THE ARELLANO-BOND ESTIMATOR.
MODELS WITH LAGGED VARIABLES: DYNAMIC REGRESSION MODELS, MODELS WITH DISTRIBUTED LAGS (DL, ARDL).

For the course program students can consult the following texts (the first three can be considered alternative among them)
1) Introduzione all'Econometria, J. H. Stock, M. W. Watson, Ed. Pearson
2) Econometrica, J. Johnston, Ed. Franco Angeli
3) Econometria, M. Verbeek, Ed. Zanichelli
4) Lectures on advanced econometrics, L. Pieraccini, Ed. Aracne
5) Introduction to Time Series Analysis and Forecasting, D. C. Montgomery, C. L. Jennings, M. Kulahci, Ed. Wiley

Some of the topics in the course can also be found in
Introductory Econometrics for Finance, C. Brooks, Cambridge University Press

For introductory references of statistical inference and algebra of matrices students can consult:
Fondamenti di Inferenza Statistica, L. Pieraccini, Ed. Giappichelli
Matrix Differential Calculus in Statistics and Econometrics, J. R. Magnus, H. Neudecker, Ed. Wiley Series in Probability and Statistics

For the laboratory course with R, students can consult one of the following texts:
Introductory Statistics with R, P. Dalgaard, Ed. Springer
An Introduction to Applied Multivariate Analysis with R, B. Everitt, T. Hothorn, Ed. Springer

1. The notion of Surplus
2. The capitalist mode of production
3. Sraffa's theory of prices
4. The integrated wage-commodity sector and the surplus equation
5. The marginalist theory of distribution
6. Capital as a factor of production

- Teaching material written by the teacher and available online.
- Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge University Press: Cambridge. Chapters I-IV and VI.

1. The notion of Surplus
2. The capitalist mode of production
3. Sraffa's theory of prices
4. The integrated wage-commodity sector and the surplus equation
5. The marginalist theory of distribution
6. Capital as a factor of production

- Teaching material written by the teacher and available online.
- Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge University Press: Cambridge. Chapters I-IV and VI.

-The completeness and incompleteness of the markets
-The assumptions underlying the Black-Scholes-Merton model and the alternative valuation models
-Arbitrage and hedging techniques
-The Greeks
-Value at Risk and its generalizations
-Volatility smiles
-Insurance derivatives
-Risk management in energy markets
-Atmospheric derivatives
-Swaps
-Interest rate derivatives (standard models, short-term rate models, advanced models)
-Real Options
-Exotic Options
-Credit derivatives

HULL Opzioni , Futures e altri derivati. Ediz. Mylab. 2018

The course will present alternative theories of primary income distribution and employment and draw their implications for
- the analysis of growth,
- the causes and effects of technical change
- the impact of labour market reforms and macroeconomic policies
- the interpretation of the changes in income distribution and their impact on employment and growth
During the course, data and results of empirical literature will lso be presented.

Various papers and essays collected around four main subjects areas and available at the library. Upon request English alternatives to the Italian readings can be suggestd to foreign students. Below the detail:

INTRODUCTORY:
“Teorie del valore e della distribuzione – un confronto tra classici e neoclassici” di Fabio Petri, da pag. 87 a 116; e da pag. 249 a 305
“Appunti sulla teoria della distribuzione e del valore” di Garegnani e Palumbo, parte introduttiva, da pag. 1 a 3
“Unemployment, institutions and the living standard in the classical theory of wages” di Antonella Stirati, da pag. 42 a 66
“Emplyoment theory in the history of economic thought: an overview” di Antonella Stirati

PART I PRODUCTIVITY EMPLOYMENT AND GROWTH:
“Teorie dello sviluppo economico” di Terenzio Cozzi, da pag. 189 a 249
“Technical change, effective demand and emplyment” di Cesaratto, Serrano e Stirati da pag. 33 a 52
Lezioni sviluppo 2010-11 (fotocopie con tabelle di dati e grafici)
“Equilibrio, distribuzione e crescita” di Kaldor, da pag. 219 a 246
“L’industria – rivista di economia politica” da pag. 45 a 53
“Potential GNP: its measurement and significance” di Okun
“Puzzles, paradoxes, and regularities: cyclical and structural productivity in the United States (1950-2005)” di Jeon e Vernengo, da pag. 237 a 243
“L’approccio Kaldor-Verdoorn: una verifica empirica per il Centro-Nord e il Mezzogiorno d’Italia (anni 1951-2006)”, di Ferdinando Ofria, da pag. 179 a 203
“intervento alla discussione del seminario ASTRIL del 6/3/13” di Antonella Stirati

PART II: INCOME DISTRIBUTION
“La riduzione del prodotto che va al lavoro” di Antonella Stirati del 16 Novembre 2010
“Towards a sustainable recovery: the case for wage-led policies”, da pag. 161 a 186
“Wages, employment, distribution and growth” di Levrero e Stirati in Hein, Heise a Truger da pag. 93 a 114
“Il problema salariale in Italia” di Maurizio Zenezini, da pag. 2 a 31

PART III: LABOUR MARKET FLEXIBILITY AND EMPLOYMENT
“Manuale di economia del lavoro”, capitolo 9 e 10 da pag. 215 a 249
“Are protective labor market institutions at the root of unemplOyment? A critical review of the evidence” di Howel, Baker, Glyn and Smchitt, da pag. 53 a 71
“Employment growth in Italy in the 1990s: institutional arrangements and market forces” di Brandolini, Casadio, Cipollone, Magnani, Rosolia e Torrini, da pag. 31 a 68
“La flessibilità del mercato del lavoro e il mito del conflitto tra generazioni” di Antonella Stirati, da pag. 181 a 191
“Oltre l’austerità” di Cesaratto e Pivetti, da pag. 133 a 144 (articolo di Antonella Stirati)

1. The notion of Surplus
2. The capitalist mode of production
3. Sraffa's theory of prices
4. The integrated wage-commodity sector and the surplus equation
5. The marginalist theory of distribution
6. Capital as a factor of production

- Teaching material written by the teacher and available online.
- Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge University Press: Cambridge. Chapters I-IV and VI.

1. The notion of Surplus
2. The capitalist mode of production
3. Sraffa's theory of prices
4. The integrated wage-commodity sector and the surplus equation
5. The marginalist theory of distribution
6. Capital as a factor of production

- Teaching material written by the teacher and available online.
- Sraffa, P. (1960) Production of Commodities by Means of Commodities. Cambridge University Press: Cambridge. Chapters I-IV and VI.