Mathematical Economics
Lecture 1
Introduction to the issues in Mathematical Economics: generalized
demand and producer theory; the significance of concavity and
convexity; gambling and insurance; Kuhn-Tucker theorem; general
equilibrium analysis and decentralization; forms of markets with
symmetric and with asymmetric information; the economics of
uncertainty; classical vs keynesian macro models, the role of the state;
optimal growth models.
Lecture 2
Advanced demand theory; dual problems; the case of n goods; indirect
utility function; expenditure (minimum cost function); proof of
characterization of demand functions from the indirect functions;
marginal utility and marginal cost; the Slutsky equation without total
differentials.
Lecture 3
Concavity and convexity of functions; the concavity of the expenditure
function; the resulting prediction of demand theory: negative definiteness
of the slutsky matrix; not all goods can be substitutes.
Lecture 4
The profit function; proof of characterization of demand functions for
inputs and supply function of output through the (maximum) profit
function; convexity (but not strict convexity) of the profit function and
implications for comparative statics results. Con
Lecture 5
Optimal taxation; how to collect optimally an amount of money by taxing
goods; choosing between taxing all goods and taxing some goods; all
goods are taxed in the same proportion; Ramsey rule, all quantities are
reduced by the same proportion.
Lecture 6
The relation between the short run and the long run cost functiions; the
envelope property; deriving the long run cost functions through
minimization of the short run minimum cost function with respect to the
fixed inputs.
Lecture 7
General equilibrium analysis; exchange economies; noncooperative
(competitive solution) and cooperative, game theoretic (core) concepts;
the possibility of corner solution and Pareto efficiency; the tatonnement
process; replica economies and the limiting relation between competitive
allocation and the core.
Lecture 8
Convex sets and separating hyperplanes, alternative characterizations of
convex and concave functions; concave programming, the Kuhn-Tucker
theorem and its significance and wide applicability in economics; its
relation to linear programming; proof of global optimality of a solution;
applications such as a welfare function or a total revenuemaximization
subject to a transformation frontier.
Lecture 9
The wide applicability of the separating hyperplane theorem in
economics; the economic interpretation of the Kuhn-Tucker multipliers
and their significance in determining the coefficients of the separating
hyperplane in decentralizing the Kuhn-Tucker solution; decentralization
in the context of a production economy.
Lecture 10
Revision of the various forms of markets; revisiting the duopoly probem;
the case of symmetric and asymmetric information; the effect of
incomplete information on the plans of the firms; inefficiency of the
Cournot-Nash solution.
Lecture 11
The Stackelberg solution and its inefficiency; choosing between being a
Stackelberg leader and a Cournot-Nash duopolists; the decisions in the
Stackelberg model in terms of a tree and how it folds up through
backward induction.
Lecture 12
The economics of uncertainty; discrete and continuous random variables
in economics; von Neumann-Morgenstern expected utility hypothesis; the
significance of concave and convex utility functions of wealth for the
theory of gambling and insurance; Berenouilli's St. Petersburg paradox;
the relation between average loss and the maximum insurance premium;
measures of risk aversion; second derivative an unsafe measure;absolute
risk aversion, relative risk aversion.
Lecture 13
Maximization of expected utility of wealth; comparison with the actuarial
value criterion; a safe and a risky asset; implications of a constant relative
risk aversion and of a constant absolute risk aversion function; counter
intuitive implications; the significance of the quadratic utility function of
wealth; the normal distribution and the quadratic utility function the
mean-variance hypothesis; the expected utility function and the allocation
of wealth in terms of mean-variance.
Lecture 14
Introduction to game theory; its significance in economics;
interdependence of decicion; the normal form game and the extensive
form game; pure and mixed strategies; the concept of the Nash
equilibrium and the subgame perfect equilibrium (Stackelberg solution);
applications to economics (industrial economics and location theory).
Lecture 15
Analysis of the prisoner's dilemma problem; its normative significance in
economics; reaction functions; unique Nash equilibrium in mixed
strategies; applications to duopoly theory with strategies to cooperate and
cheat.
Lecture 16
Cooperative game theory; the generalized Nash bargaining solution
(GNBS) and the Nash bargaining solution (NBS); the Nash-Binmore
axioms; proof that the (GNBS) satisfies the axioms; the axiom of
symmetry requires equal bargaining powers (NBS); intuitive idea of the
Nash programme; the significance of the status-quo payoffs and of the
attitude to risk in the (NBS); dividing a cake of size one, and dividing the
joint profit of a collusion solution to a duopoly; intuitive idea of the Nash
programme.
.
Lecture 17
Assymmetric information models; moral hazard and adverse selection
problems; the principal-agent model; used cars; a model with a warranty;
auctions.
Lecture 18
Revision of the classical and the keynesian macroeconomic models; the
idea of rational expectations; the dichotomization of the classical model
into a real subsystem and a money subsystem with result that the money
variables do not affect the real variables in the long run; the possibility of
a trade off between inflation and unemployment in the keynesian model.
Lecture 19
The IS-LM curves and comparative statics results, comparing the effect
of the monetary policy and fiscal policy; the significance of the ratio of
the responsiveness of the investment function to the rate of interest over
the responsiveness of the liquidity preference function to the rate of
interest.
Lecture 20
The analysis of the complete models and formal proof of the
independence of the real variables from the money variables in the
classical model; the role of the state in the keynesian model;
Enlarged models in which the government imposes fiscal constraints and
an exports-imports sector is added; comparisons bertween the effects of
changes through money variables and exports.
Lecture 21
Optimal growth models and their normative significance in relation to the
golden rule capital-labour ratio; finite vs infinite horizon models; the
Euler-lagrange equation, the Pontryagin maximum principle, the
transversality condition and their interpretation; proof of global
optimality of the optimal paths.
Lecture 22 and 23
Revision of the above, emphasizing main points and taking questions.
There will be no new material covered in the last term.
A lecture is a two hour session. Participation of the students is actiovely
encouraged. For all lectures and classes there will be a number of exam
level examples and their solutions. In particular the actual two past years
exam papers will be considered in detail in the classes. On a number of
topics there will be lecture notes.
There will be two closed-book tests, one at the end of the first term and
one at the end of the second term. They count towards the coursework
mark and they are important for the final exam preparation.