Main questions (each question 8 points)
Input-output (Leontief) model: main assumptions and construction. Definition of productivity.
Inverse matrix (E-A)-1 in input-output model. How it is calculated by infinite series of Ak and what
does it mean? What is the meaning of rows and columns of matrix (E-A)-1 and why?
Difference equations, their application. The order of difference equation. Solution of difference
equations of the second order. Homogenous and non-homogenous case (proof of relevant
formulae).
Samuelson’s model of economic growth (multiplier and accelerator interaction). Corresponding
difference equation. Analysis of the model. Proof of convergence. Diagram for different pairs of α
(accelerator) and γ (propensity to consume).
Problem of mathematical programming. Local and global minima. Convex set. Convex and concave
function: definition and necessary and sufficient conditions of convexity. Necessary and sufficient
condition of optimal solution of convex optimization problem. Cases when a set of feasible
solutions is Rn , Rn+
Problem of mathematical programming.. Necessary and sufficient condition of optimal solution of
convex optimization problem when a set of feasible solutions is defined by constraints g(x)≤b.
Lagrangian function, saddle point as sufficient condition of optimal solution (proof). Kuhn-Tucker
theorem. Interpretation of dual variables (Lagrangian multipliers).
Optimal control theory: the main problem in discrete time. R. Bellman’s principle of optimality.
Recursive function. Ways of calculation of recursive functions. Examples of recursive functions: the
problem of distribution of units of equipment, the problem of equipment replacement.
Recursive function in the model of optimal choise between consumption and investment (CobbeDouglas production function and logarithmic utility). Solution in the infinite time (steady state
regime). Apply two approaches to the solution.
McCall’s intertemporal job search model. Reservation wage.
Problem of stochastic optimization (stochastic programming). One and two stage approach.
Deterministic equivalent of the simple one stage stochastic problem.
Problem of reservation of commodities under the uncertain demand.
Literature
K.Wainwright, A.S.Chiang. Fundamental Methods of Mathematical Economics.
L.Ljungqvist, T.J.Sargent. Recursive Macroeconomic Theory.
R.Bellman. Dynamic programming.
Simplified questions (4 questions for 6 points)
Input-output model: main assumptions and construction.
Definition of productivity in input-output model. Sufficient condition of productivity.
Inverse matrix (E-A)-1 in input-output model. How it is calculated and what does it mean?
What is the meaning of the rows and columns of input-output inverse matrix (E-A)-1
Difference equations, their application. The order of difference equation.
Analysis of the first order difference equation. Homogenous ad non-homogenous cases.
Solution of the homogenous difference equation of the second order. Cases of real roots.
Samuelson’s model of economic growth. Formulation and corresponding difference equation
Problem of optimization. Classic approach. Necessary and sufficient conditions of minimum
(maximum) in one – dimensional, n – dimensional constrained and non-constrained optimization
cases. Application of the bordered Hessian.
Optimization approach to the example of two enterprises: independent competition; leadership of
one; cartel agreement.
Non-classic problem of optimization (mathematical programming). The main terms and a
corresponding chart in R2. Local and global minima in optimization problem.
Convex set and convex function. Formal definitions and illustration by charts.
First and second order necessary and sufficient conditions of convexity of function.
Why convex optimization is more simple than general non-linear problem?
Necessary and sufficient conditions of optimal solution of convex optimization problem. Cases
with no constraints: Rn and Rn+.
Necessary and sufficient conditions (Kuhn-Tucker) of optimal solution of convex optimization
problem. Case with constraints g(x)≤b (saddle point).
Meaning of dual variables (Lagrangian multipliers) in convex optimization problem.
Optimal control theory: the main problem in discrete time .
R. Bellman’s principle of optimality. Recursive function.
Examples of recursive functions: the problem of distribution of units of equipment, the problem
of equipment replacement.
Recursive function in the model of optimal choise between consumption and investment (CobbeDouglas production function and logarithmic utility).
Recursive function in McCall' intertemporal job search model.
The problem of stochastic optimization. One and two stages approach.
Formulation of stochastic optimization problems in economics (what examples you know).
Exercise (2 points)
To solve the difference equation. To solve the convex optimization problem.
To solve the dynamic optimization problem (distribution of units of equipment).