ECON6003_Mathematical Methods of Economic Analysis_2009 Semester 1_ECON6003 S11

Kuhn-Tucker conditions
The Kuhn-Tucker conditions for the problem maxx f (x) subject to gj(x) ≤ cj for j = 1, ..., m are Li'(x) = 0 for i = 1 ,..., n λj ≥ 0, gj(x) ≤ cj and λj[gj(x) − cj] = 0 for j = 1, ..., m, where L(x) = f (x) − ∑jλj(gj(x) − cj).


Binding or not binding


Consider a single case with m = 1: maxx f (x) subject to g(x) ≤ c. There are two possibilities for the solution of this problem. In the following figures, the black closed curves are contours (level curves) of f; values of the function increase in the direction shown by the blue arrows. The downward-sloping red line is the set of points x satisfying g(x) = c; the set of points x satisfying g(x) ≤ c lie below and the the left of the line, and those satisfying g(x) ≥ c lie above and to the right of the line. In each panel the solution of the problem is the point x*. In the left panel the constraint binds at the solution: a change in c changes the solution. In the right panel, the constraint is slack at the solution: small changes in c have no effect on the solution.
Optimization with inequality constraints

Many models in economics are naturally formulated as optimization problems with inequality constraints. Take the consumer's choice problem as an example. There is no reason to insist that a consumer spend all her wealth, so that her optimization problem should be formulated with inequality constraints: maxx u(x) subject to p· x ≤ w and x ≥ 0. Depending on the character of the function u and the values of p and w, we may have p· x < w or p· x = w at a solution of this problem. This is the most general optimization problem so far.
The Final

Comprehensive with focus on optimization Format similar to the midterm Cheat-sheet: an one-sided A4 Time: 15/06/2009 6:00 - 8:00PM Location: Quad Room S421 A Guide to Examination Rooms Practice exam will be posted in the last week Outline topic guide will be updated too

Hale Waihona Puke General optimization problem

Consider a problem of the form maxx f (x) subject to gj(x) ≤ cj for j = 1, ..., m, where f and gj for j = 1, ..., m are functions of n variables, x = (x1, ..., xn), and cj for j = 1, ..., m are constants. All the problems we have studied so far may be put into this form. No constraints


m = 0. maxx f (x) subject to h(x) = 0 may be written as maxx f (x) subject to h(x) ≤ 0 and −h(x) ≤ 0. For a problem with a constraint xk ≥ 0 we let gj(x) = −xk and cj = 0 for some j. minx h(x) subject to gj(x) ≤ cj for j = 1,...,m is the same as maxx f (x) subject to gj(x) ≤ cj for j = 1,...,m, where f (x) = −h(x).
Optimization
Optimization with Inequality Constraints
Roadmap


Basic definitions and the existence of an optimum. Interior optima, focuses on conditions for solutions that are strictly inside the constraint set S --"interior" solutions. Optimization with equality constraints, discusses the key technique developed by Lagrange for finding the solutions of problems in which S is the set of points that satisfy some equations. Optimization with equality constraints, on the Kuhn-Tucker conditions for problems with inequality constraints, discusses a set of conditions that may be used to find the solutions of any problem in which S is the set of points that satisfy some inequalities.
ECON6003 Mathematical Methods of Economic Analysis
Semester 1, 2009, USYD
Topics Today
18/5/09 Group Assignment The Final Exam New Topics

Difference Equations

Equality constraints


Nonnegativity constraints


Minimization problems

How to solve it?


For the consumer's choice problem maxx u(x) subject to p· x≤w and x ≥ 0. Depending on the character of the function u and the values of p and w, at a solution of this problem we may have p· x < w or p· x = w. If p· x < w, the constraint is not binding and we effectively have a unconstrained optimization problem. What to do then? If p· x = w, the constraint is not binding and we effectively have a optimization problem with equality constraints. What to do then? Can we combine these two cases together?
When Lagrange meets Kuhn & Tucker

We have two cases and from our previous analysis of problems with equality constraints and with no constraints, if g(x*) = c, define the Lagrangean function L by L(x) = f (x) − λ(g(x) − c), then L’i(x*) = 0 for all i. if g(x*) < c, then fi'(x*) = 0 for all i. First we show that if g(x*) = c, we have λ ≥ 0. Suppose, to the contrary, that λ < 0. Then we know that a small decrease in c raises the maximal value of f (why?). That is, moving x* inside the constraint raises the value of f, contradicting the fact that x* is the solution of the problem. In the second case, the value of λ does not enter the conditions, so we can choose any value for it. Given the interpretation of λ, setting λ = 0 makes sense. Under this assumption we have fi'(x) = L'i(x) for all x, so that L'i(x*) = 0 for all i. Thus in both cases we have L'i(x*) = 0 for all i, λ ≥ 0, and g(x*) ≤ c. In the first case we have g(x*) = c and in the second case λ = 0. Or λ[g(x*) − c] = 0. We may combine the two cases by writing the conditions as Li'(x*) = 0 for j = 1, ..., n λ ≥ 0, g(x*) ≤ c, and λ[g(x*) − c] = 0.