E 600 Chapter 4: Optimization

Transcription

1 E 600 Chapter 4: Optimization Simona Helmsmueller August 8, 2018

2 Goals of this lecture: Every theorem in these slides is important! You should understand, remember and be able to apply each and every one of them! The best way to remember a theorem is to apply it over and over again - luckily, that is exactly what you will be doing in your econ master courses. Some theorems (implicit function, Lagrange) have a lot of conditions and scary formulas. I do not expect you to exactly write them down. However, you should know and actively remember the intuitive meaning behind each condition and you should be able to check them in a given (economic) context. The latter often requires being able to write down the economic problem in a form which fits the optimization framework as discussed in this lecture. You should be comfortable doing that.

3 Introduction Economics is the study of optimal (efficient) allocations given limited resources.

4 Introduction Economics is the study of optimal (efficient) allocations given limited resources. Examples include: Choice of optimal ratio of input factors in production given input and output prices Optimal division of time between labor and leisure given a time budget Optimal allocation of tax money to investments and subsidies given a federal budget

5 Definition Optimization Problem (P min ) minimize x dom(f ) f (x) subject to g i (x) = 0, i = 1,..., m. h i (x) 0, i = 1,..., k.

6 Definition Optimization Problem (P min ) minimize x dom(f ) f (x) subject to g i (x) = 0, i = 1,..., m. h i (x) 0, i = 1,..., k. (P max ) maximize x dom(f ) f (x) subject to g i (x) = 0, i = 1,..., m. h i (x) 0, i = 1,..., k.

7 Example 1: Utility maximization Consumer theory: agents maximize their utility by choosing an optimal consumption bundle of the n goods in the economy x = (x 1,...x n ) Objective function: f (x) = u(x 1,..., x n ) Constraints arise from the prices of goods p = (p 1,..., p n ) and the available income I : p x I The maximization problem then reads: max u(x) subject to p x I. x:x i 0

8 Example 2: Expenditure minimization Dual problem of utility maximization: minimize expenditures given a certain utility level The minimization problem reads: min p x subject to u(x) u(x). x:x i 0

9 Example 3: Profit maximization Single output producer (y) n input factors: x = x 1,..., x n, with prices w = w 1,..., w n Production function y = g(x) Price function p(y) (inverse demand function) Maximization problem: max p(g(x))g(x) x w subject to x 0 x

10 Example 4: Cost minimization As an exercise, formulate the dual problem of the firm.

11 Definition: (Maximum and Minimum) Let X be a subset of R. An element x in X is called a maximum in X if, for all x in X, x x. An element x in X is called a minimum in X if, for all x in X, x x. If the above inequalities are strict, one speaks of strict maximum (resp. strict minimum). A maximum or a minimum is often simply referred to as an extremum.

12 Definition: (Local and Global Maximizers) Let f be a real-valued function defined on X R n. A point x in X is: A global maximizer for f on X if and only if: x X, f ( x) f (x) A strict global maximizer for f on X if and only if: x X, x x, f ( x) > f (x) A local maximizer for f on X ε > 0 such that: x X B ε ( x), f ( x) f (x) A strict local maximizer for f on X ε > 0 s.t.: x X B ε ( x), x x, f ( x) > f (x)

13 Fact: (Equivalence Between Minimizers and Maximizers) Consider a problem of the form P min. x is a local (resp. global) extremizer for P min if and only if it is a local (resp. global) extremizer for a problem of the form P max with identical constraints but f (x) as an objective function.

14 Fact: (Equivalence Between Minimizers and Maximizers) Consider a problem of the form P min. x is a local (resp. global) extremizer for P min if and only if it is a local (resp. global) extremizer for a problem of the form P max with identical constraints but f (x) as an objective function. PLEASE CONSIDER P max AS A CANONICAL SHAPE AND ALWAYS TRY TO RESHAPE YOUR PROBLEM SO THAT IT FITS IT!

15 Objectives of Optimization Theory: 1. Identify conditions which guarantee existence of a solution. 2. Identify the set of optimal, feasible points: Necessary conditions, which every solution must fulfill Sufficient conditions, which guarantee that any point fulfilling them is a solution

16 Objectives of Optimization Theory: 1. Identify conditions which guarantee existence of a solution. 2. Identify the set of optimal, feasible points: Necessary conditions, which every solution must fulfill Sufficient conditions, which guarantee that any point fulfilling them is a solution 3. Identify conditions which guarantee uniqueness of the solution. 4. Analyze how the solution depends on parameters of the optimization problem

17 Theorem: (Weierstrass - Extreme Value Theorem) Let X be a nonempty closed and bounded subset of R n and f : X R be continuous. Then, f is bounded in X and attains both its maximum and its minimum in X. That is, there exist points x M and x m in X such that f (x m ) f (x) f (x M ) for all x X

18 Theorem: (Weierstrass - Extreme Value Theorem) Let X be a nonempty closed and bounded subset of R n and f : X R be continuous. Then, f is bounded in X and attains both its maximum and its minimum in X. That is, there exist points x M and x m in X such that What does this mean? f (x m ) f (x) f (x M ) for all x X

19 Exercise: what conditions would allow us to apply Weierstrass on the utility maximization and the cost minimization problems?

20 Consider the following optimization problem: (P) maximize x dom(f ) f (x) where dom(f ) R n and f is real-valued and continuously differentiable.

21 Theorem: (First Order Necessary Condition (FOC)) Consider (P). Let x be an element in the interior of dom(f ). If x is a local extremum of f, then: f ( x) = 0

22 Theorem: (First Order Necessary Condition (FOC)) Consider (P). Let x be an element in the interior of dom(f ). If x is a local extremum of f, then: f ( x) = 0 The reverse is not necessarily true! Any point x dom(f ) with f (x) = 0 is called a critical point of f.

23 Figure: f (x, y) = x 2 y 2, 0 is a Saddle Point

24 Theorem: (SOC: Necessity) Consider (P). Let x be an element in the interior of dom(f ) and B ε ( x) an open ε-ball around x. Assume f is in C 2 (B ε ( x)). If x is a maximizer of f, then: λ R n λ H f ( x)λ 0 i.e. the Hessian of f at x is negative semidefinite.

25 Theorem: (SOC: Necessity) Consider (P). Let x be an element in the interior of dom(f ) and B ε ( x) an open ε-ball around x. Assume f is in C 2 (B ε ( x)). If x is a maximizer of f, then: λ R n λ H f ( x)λ 0 i.e. the Hessian of f at x is negative semidefinite. Theorem: (SOC: Sufficiency) Consider (P). Let x be an element in the interior of dom(f ) and B ε ( x) an open ε-ball around x. Assume f is in C 3 (B ε ( x)). If f ( x) = 0 and H f ( x) is negative definite, then x is a local maximizer of f.

26 Consider the following optimization problem: (P min,c ) minimize x dom(f ) f (x) subject to g i (x) = 0, i = 1,..., m.

27 Definition: (Level Sets) Let X be a nonempty subset of R n, f : X R, and c be an element of R. The c-level set of f is the set L f c := {x x X, f (x) = c}. The c-lower level set of f is the set L f c := {x x X, f (x) c}. The strict c-lower level set of f is the set L f c := {x x X, f (x) < c}. The c-upper level set and the strict c-upper level set of f are defined symmetrically.

28 Figure: Level Sets in Geography (source: compass.php)

29 What are implicit functions? What do they have to do with constrained optimization?

30 Theorem: (Implicit Function Theorem) Let X R n and f : X R. Suppose also that f belongs to C 1 (A), where A is a neighborhood of x in X and that for some i in {1, 2,, n} f ( x) x i 0. Then φ i (x i ) defined on a neighborhood B of x i such that φ i ( x i ) = x i. Also, if x i B, then (x 1,..., x i 1, φ i (x i ), x i+1,..., x n ) A and f (x 1,..., x i 1, φ i (x i ), x i+1,..., x n ) = f (x). Finally, φ i is differentiable at x i and for j i φ i ( x i ) x j = f ( x) x j f ( x) x.

31 Theorem: (Lagrange optimization - several equality constraints) Let f, g 1,..., g m C 1 be functions of n variables. Consider the problem of maximizing f (x) on the constraint set C g = {x = (x 1,..., x n ) : g 1 (x) = a 1,..., g m (x) = a m }. Suppose that x C g and that x is a (local) max or min of f on C g. Suppose further that x satisfies the nondegenerate constraint qualification, i.e., the Jacobian matrix of the constraint functions has maximal rank (is invertible). Then, there exist λ 1,...λ m such that (x 1,..., x n, λ 1,..., λ m) = (x, λ ) is a critical point of the Langrangian L(x, λ) = f (x) λ 1 (g 1 (x) a 1 )... λ m (g m (x) a m ).

32 Consider the following optimization problem: (P min,c2 ) minimize x dom(f ) f (x) subject to g i (x) b i, i = 1,..., m.

33 Most of the economic problems are in this form It is more difficult to prove or illustrate graphically Justin Leduc did in excellent job in illustrating the idea homework: carefully read through his script on the Kuhn-Tucker theorem (uploaded on my webpage)

34 Theorem: (Lagrange optimization - one inequality constraints) Suppose that f and g are C 1 functions on R 2 and that (x, y ) maximizes f on the constraint set g(x, y) b. If g(x, y ) = b, suppose that g x (x, y ) 0 or g y (x, y ) 0. In any case, form the Lagrangian function L(x, y, λ) = f (x, y) λ(g(x, y) b). Then, there is a multiplier λ such that: L 1. x (x, y, λ ) = 0 L 2. y (x, y, λ ) = 0 3. λ (g(x, y ) b) = 0 4. λ 0 5. g(x, y ) b

35 Theorem: (Lagrange optimization - several inequality constraints) Suppose that f and g 1,..., g k are C 1 functions of n variables and that x R n maximizes f on the constraint set defined by the k inequalities g i (x) b i, i = 1,..., k. Suppose that nondegenerate constraint qualification is satisfied at x, i.e., the rank at x of the Jacobian matrix of the binding constraints is maximal. Form the Lagrangian function L(x, λ 1,..., λ k ) = f (x) λ 1 (g 1 (x) b 1 )... λ k (g k (x) b k ). Then, there is multipliers λ 1,..., λ k such that: L 1. (x, λ ) = 0,..., L (x, λ ) = 0 x 1 x n 2. λ i (g i (x ) b i ) = 0 i = 1,..., k 3. λ i 0 i = 1,.., k 4. g i (x ) b i i = 1,..., k