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[PDF] BEEM103 Mathematics for Economists Unconstrained Optimization

are satisfied, i e , either the k-Lagrange multiplier is zero or the k-th constraint binds for 1 ≤ k ≤ K Then (x∗ ,y ∗) is a maximum for the constrained maximization

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Constrained Optimization

An Example

Utility maximization

Summary

BEEM103 Mathematics for Economists

Constrained Optimization 1

Dieter Balkenborg

Department of Economics, University of Exeter

Department of Economics, University of Exeter

Week 3Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Unconstrained Optimization

-4 -2 -4 -40 -50-2 y x -30 z -20-100 00 2 4 2

4Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Constrained Optimization:

y≥1 -4 -2 -4 2000
-10 -20 z -30 xy-2 -40 -50 4 2

4Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Constrained Optimization

Examples:

1 A consumer maximizes his utility subject to his budget constraint. 2 A producer minimizes costs subject to the constraint that a certain amount is produced. 3 Moral hazard: An insurer tries to select an insurance contract that maximizes profits subject to the constraints that it is valuable to the consumer (“Participation Constraint") and that the consumer has an incentive to be careful (“Incentive Constraint"). Basic result: Full insurance is not optimal because it would make consumer act careless.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Constrained Optimization Problem

Objective:Find the (absolute) maximum of the function z=f(x,y) subject to the inequality constraints g

1(x,y)≥0

g

2(x,y)≥0

g

K(x,y)≥0

Thus find pair

(x ?,y ?)satisfying the constraints such that we have for all other pairs (x,y)satisfying the constraints:f(x ?,y ?)≥f(x,y). f (x,y)is called the “objective function" and I call g

1(x,y),...,gK(x,y)the “constraining functions" of the problem.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The constraints carve out a region of the plane.

g5(x,y)<0 g

1(x,y)>0g

1(x,y)<0g

5(x,y)=0

g

5(x,y)>0

Region where

all constraints satisfiedg

1(x,y)=0

g

2(x,y)=0

g

2(x,y)>0

g2(x,y)<0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Example 1: Consumer Optimization

A consumer wants to maximize his utility

u (x,y)=(x+1)(y+1) subject to his budget constraint b-pxx-pyy≥0 and the non-negativity constraints x≥0y≥0 0 1 2 3 0123
xy

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Example 2: Cost Minimization

A producer with production functionQ(K,L)=K

16L

12in a

perfectly competitive market wants tominimizecosts subject to producing at leastQ0units.

Maximize

(rK+wL) subject to Q (K,L)-Q0≥0

K≥0

L≥0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Example 3: Shortest Route

A swimmer who is currently at the coordinates(a,b)wants to swim along the shortest route to the square island with corner points (-1,1),(1,-1),(-1,1),(1,1). Instead of minimizing the distance we can maximize the negative of the square of the distance (x-a)2-(y-b)2 subject to the constraints x+1≥0

1-x≥0

y+1≥0

1-y≥0

x,y)is a point in the square carved out by the solutions to the four inequalities.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Island

-3-2-10123 y -3 -2 -1 1 2 3 x y+1=01-y=01-x=0x+1=0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Swimmer

y+1=01-y=01-x=0x+1=0 -3-2-10123 y -3 -2 -1 1 2 3 x (a,b)

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Binding constraints

A constraint isbinding at the optimumif it holds with equality in the optimum. In the above picture only one of the four the constraints is binding. All non-binding constraints can be ignored.

If they are left out the optimum does not change.

-3-2-10123 y -3 -2 -1 1 2 3 x y+1=01-y=01-x=0x+1=0 (a,b)

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Lagrangian Approach

The Lagrangian approach transfers a constrained optimization problem into 1 an unconstrained optimization problem and 2 a pricing problem. The new function to be optimized is called theLagrangian.For each constraint a shadow price is introduced, called aLagrange multiplier. In the new unconstrained optimization problem a constraint can be violated, but only at a cost. The pricing problem is to find shadow prices for the constraints such that the solutions to the new and the original optimization problem are identical.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Lagrangian Approach

The Lagrangian:

L

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

TheoremSuppose we are given numbersλ1,λ2,...,λK≥0and a pair of numbers (x ?,y ?)such that 1 λ1,λ2,...,λK≥0, i.e. Lagrange mutlipliers are non-negative, 2 (x ?,y ?)satisfies all the constraints, i.e., gk(x ?,y ?)≥0for 3 (x ?,y ?)is an unconstrained maximum of the Lagrangian L (x,y). 4

The complementarity conditions

kgk(x ?,y ?)=0 are satisfied, i.e., either the k-Lagrange multiplier is zero or Then (x ?,y ?)is a maximum for the constrained maximization problem.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Lagrangian:

L

(x,y)=f(x,y)+λ1g1(x,y)+λ2g2(x,y)+···+λKgK(x,y)Proof.Because of the complementarity conditions we have

L (x ?,y ?)=f(x ?,y ?,y ?)for any x,y).If(x,y)satisfies all constraints thenλkgk(x,y)≥0 for each constraint since theλkare non-negative. Hence f ?,y ?)for any point (x,y)satisfying the constraints.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Method

1 Make an informed guess about which constraints are binding at the optimum. (Suppose there arek ?such constraints.) 2 Set the Lagrange multipliers for all other constraints zero, i.e. ignore these constraints. 3

Solve the two first -order conditions∂

L∂x=0,∂

L∂y=0 together

with the conditions that thek ?constraints are binding. (Notice that we have 2+k ?constraints and equations, namelyx,yandk ?Lagrange multipliers.) 4 Check whether the solution is indeed an unconstrained optimum of the Lagrangian. (May be difficult.) 5 Check that the Lagrange multipliers are all non-negative and that the solution(x ?,y ?)satisfies all constraints. 6 If 4. and 5. are violated, start again at 1. with a new guess.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

The Swimmer"s Problem

The Lagrangian is

L (x,y)=-(x-a)2-(y-b)2 FOC: ∂L ∂x=-2(x-a)+λ1-λ2=0 ∂L∂y=-2(y-b)+λ3-λ4=0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 1:

Suppose-1 ?<1,-1None of the four constraints is “binding"

In this case the optimum has to be a stationary point of the objective function. This gives the conditions ∂f ∂x=-2(x-a)=0 ∂f∂y=-2(y-b)=0

UNIQUE SOLUTION:

(x ?,y ?)=(a,b).Must have:

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 1

y+1=01-y=01-x=0x+1=0 -3-2-10123 y -3 -2 -1 1 2 3 x

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 2

Optimum is on the upper side of the square, but not a cornerpoint, exceptλ4must be zero. So the Lagrangian is L (x,y)=f(x,y)+λ4(1-y) FOC: ∂L ∂x=-2(x-a)=0 ∂L∂y=-2(y-b)-λ4=0

Add that the constrainty=1 is binding, i.e.

y=1. unique solutionx?=a,y ?=1,λ4=-2(1-b).

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 2

For our candidate(x

?,y ?)=(a,1)to satisfy all constraints we nonnegative. True only ifb-1≥0 orb≥1.

Cases 3,4,5:Optimum is on a different side.

Solved symmetricallyBalkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 2

-3-2-10123 y -3 -2 -1 1 2 3 x y+1=01-y=01-x=0x+1=0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 6

Complementarity conditions implyλ1=λ3=0. Lagrangian is L (x,y)=f(x,y)-λ2(x-1)-λ4(y-1) FOC: ∂L ∂x=-2(x-a)-λ2=0 ∂L∂y=-2(y-b)-λ4=0 cornerpoint: x=1 y=1.

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 6

-3-2-10123 y -3 -2 -1 1 2 3 x y+1=01-y=01-x=0x+1=0

Balkenborg

Constrained Optimization 1

Constrained Optimization

An Example

Utility maximization

Summary

Case 6

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