Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel...

21
Econ 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel

Transcript of Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel...

Page 1: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Econ 205 - Slides from Lecture 14

Joel Sobel

September 10, 2010

Econ 205 Sobel

Page 2: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Theorem (“Lagrange Multipliers”)

TheoremIf x∗ solves max f (x) subject to G (x) = 0 then there exists λ suchthat Df (x∗) = λDG (x∗).

It is standard to write the equation

∇f (x∗) =m∑i=1

λi∇gi (x∗).

The λi are often called Lagrange Multipliers.

Econ 205 Sobel

Page 3: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Comments

I FOC replace a max problem.

I λ have economic interpretation.

I Write the constraints of the original problem are G (x) = b.

I What happens to the solution of the problem as b changes?

I Let V (b) ≡ f (x∗(b)) where x∗(b) solves: max f (x) subject toG (x) = b.

I Assume that you can solve problem at a point (say b = 0).

Econ 205 Sobel

Page 4: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

I If regularity condition holds, then IFT guarantees that we cansolve G (x) = b for W as a function of u and b in aneighborhood of (u, b) = (u∗, 0).

I If x∗(b) = (u∗(b),w∗(b)) satisfies the FOC, then we candifferentiate V (b) to obtain: DV (0) =

((Duf (x∗) + Dw f (x∗)) (DuW (x∗, 0)) Du∗(0)+Dw f (x∗)DbW (x∗, 0).

I The first two terms on RHS are 0 by FOC.

I . The final term on the right is the contribution that comesfrom the fact that W may change with b.

I

Dw f (x∗)DbW (x∗, 0) = Dw f (x∗) (DwG (x∗))−1 I = λ,

where the last equation is just the definition of λ. Hence wehave the following equality constrained envelope theorem:

DV (0) = λ. (1)

Econ 205 Sobel

Page 5: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Interpretation

How does the value of the problem change if the right-hand side ofthe ith constraint increases from 0?The answer is λi .

Econ 205 Sobel

Page 6: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

The Kuhn-Tucker Theorem

Inequality constraints: S = {x : gi (x) ≤ bi , i ∈ I}.I Includes equalities as a special case (because gi (x) = bi is

equivalent to −gi (x) ≤ −bi and gi (x) ≤ bi .

I We know how to solve problems without constraints.

I We know how to solve problems with equality constraints.

I If you have inequality constraints and you think that x∗ is anoptimum, then (locally) the constraints will divide into thegroup that are satisfied as equations (binding or activeconstraints) and the others.

I The first group we treat as we would in an equalityconstrained problem.

I The second group we treat as in an inequality constrainedproblem.

Econ 205 Sobel

Page 7: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

The First-Order Condition

I Expect a first-order condition like:

∇f (x∗) =∑i∈J

λi∇gi (x∗) (2)

where J is the set of binding constraints.

I Alternative form:

∇f (x∗) =∑i∈I

λi∇gi (x∗) (3)

and

λi (bi − gi (x∗)) = 0 for all i ∈ I . (4)

Econ 205 Sobel

Page 8: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Interpretation

I Equation (3) differs from equation (2) because it includes allconstraints (not just the binding ones).

I Equation (3) and (4) are equivalent to (2).

I Equation (4) can only be satisfied (for a given i) if eitherλi = 0 or gi (x∗) = 0.

I It says a multiplier associated with non-binding constraints iszero.

Econ 205 Sobel

Page 9: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Complementary Slackness

I Equation (4) is called a complementary slackness condition.(Constraints that do not bind are said to have slack.)

I Lagrange multipliers are “prices” or values of resourcesrepresented in each constraint.

I It is useful to think of multiplier λi as the amount that valuewould increase if the ith constraint were relaxed by a unit or,alternatively, how much the person solving the problem wouldpay to have an “extra” unit to allocate on the ith constraint.

I If a constraint does not bind, it should be the case that thisvalue is zero (why would you pay to relax a constraint that isalready relaxed?).

I If you are willing to pay to gain more resource (λi = 0) itmust be the case that the constraint is binding. It is possible(but rare) to have both λi = 0 and gi (x∗) = 0. This happenswhen you have “just enough” of the ith resource for yourpurposes, but cannot use anymore.

Econ 205 Sobel

Page 10: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Summary

TheoremSuppose x∗ solves: max f (x) subject to x ∈ S when S is defined byinequalities S = {x : gi (x) ≤ bi} and f and gi are differentiable.There exists λ∗k such that

∇f (x∗) +∑i∈I

λ∗i∇ (gi (x∗)− bi ) = 0 (5)

andλ∗i (gi (x∗)− bi ) = 0 for all i . (6)

1. Appropriate generalization of unconstrained and equalityconstrained cases.

2. Multipliers have interpretation: Di f (x∗) = λi

3. We need an additional “regularity” condition to get this resultin general. (Supplementary notes have details.)

Econ 205 Sobel

Page 11: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Second-Order Conditions

I First-order conditions for optimality do not distinguish localmaxima from local minima from points that are neither.

I Second-order conditions do provide useful information.

I Equality constraints: you derive first-order conditions byrequiring that the objective function be maximized in alldirections that will satisfy the constraints of the problem. Thesecond-order conditions must hold for exactly these directions.

Econ 205 Sobel

Page 12: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

I If x∗ is a maximum and x∗ + tv satisfies the constraints of theproblem for t near zero, then h(t) = f (x∗ + tv) has a criticalpoint at t = 0 and h

′′(0) ≤ 0.

I h′′

(0) = v tD2f (x∗)v , so h′′

(0) < 0 if and only if a quadraticform is negative.

I If the problem is unconstrained, then the second-orderconditions require that the matrix of second derivatives benegative semi-definite.

I In general, this condition need only apply in the directionsconsistent with the constraints.

I There is a theory of “Boardered Hessians” that allows you touse some insights from the theory of quadratic forms toclassify when quadratic form restricted to a set of directionswill be positive definite.

Econ 205 Sobel

Page 13: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

maxx1,x2

x21 − x2

subject to x1 − x2 = 0

The constraint says x1 = x2 so the problem is

maxx1

x21 − x1

and we can solve this as a one-variable unconstrained problem.

Econ 205 Sobel

Page 14: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

max log(x1 + x2) subject to: x1 + x2 ≤ 5, x1, x2 ≥ 0.

I log is a strictly increasing function, so the problem isequivalent to solving max(x1 + x2) subject to the sameconstraints.

I Since the objective function is increasing in both variables, theconstraint x1 + x2 ≤ 5 must bind.

I Hence the problem is equivalent to maximizing x1 + x2 subjectto x1 + x2 = 5 and x1, x2 ≥ 0.

I If x1 ∈ [0, 5] any pair (x1, 5− x1) solves the problem.

Solving this using KT is possible, but not easy.

Econ 205 Sobel

Page 15: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

max log(1 + x1)− x22 subject to: x1 + 2x2 ≤ 3, x1, x2 ≥ 0.

The first-order conditions are

1

1 + x1= λ1 − λ2 (7)

−2x22 = 2λ1 − λ3 (8)

λ1 (3− x1 − 2x2) = 0 (9)

λ2x1 = 0 (10)

λ3x2 = 0 (11)

3 = x1 + 2x2 and x1 > 0 in the solution. [Why?]Hence:

I the third constraint is satisfied,I λ2 = 0 by the fourth constraint,I and the first constraint gives: λ1 = 1/(1 + x1)I and the second constraint implies that λ3 > 0.I by the final constraint that x2 = 0I x1 = 3 and λ1 = 1/4.

Econ 205 Sobel

Page 16: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Simple Allocation

max x1 · · · xn subject tok∑

i=1

xi = K .

This problem has an equality constraint. If v = x1 · · · xn, then wecan write the first-order conditions as v = λxi for all i . Hencekv = λK and therefore each xi = K/k .

Econ 205 Sobel

Page 17: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

A Famous InequalitySolve:

max x21x2

2 · · · x2n subject to

n∑i=1

x2i = 1

is to set xn = (1/n)n for all n.Solution must satisfy

2f (x)

xi= 2λxi ,

which implies that the xi are independent of i .[Why is this a maximum and not a minimum?]Given any n positive numbers a1, . . . , an, let

xi =a1/2i

(a1 + · · ·+ an)1/2, for i = 1, . . . , n.

It follows that∑n

i=1 x2i = 1 and so(

a1 · · · an(a1 + · · ·+ an)n

)1/n

≤ 1

n

and so

(a1 + · · ·+ an)1/n ≤ (a1 + · · ·+ an)1/2

n.

The interpretation of the last inequality is that the geometric meanof n positive numbers is no greater than their arithmetic mean.

Econ 205 Sobel

Page 18: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Hype

It is possible to use techniques from constrained optimization todeduce other important results (the triangle inequality, theoptimality of least squares, . . . ).

Econ 205 Sobel

Page 19: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

Cobb-Douglas

Canonical consumer theory example.Cobb-Douglas Utility function:

f (x) = xa11 · · · x

ann ,

where the coefficients are nonnegative and sum to one.Consumer problem:

max f (x) subject to p · x ≤ w

where p ≥ 0, p 6= 0 is the vector of prices and w > 0 is wealth.Since the function f is increasing in its arguments, the budgetconstraint must hold as an equation.

Econ 205 Sobel

Page 20: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

The first-order conditions can be written

ai f (x)/xi = λpi

orai f (x) = λpixi .

Summing both sides of this equation and using the fact that the aisum to one yields:

f (x) = λp · x = λw .

It follows that

xi =waipi

and λ =

(a1p1

)a1

· · · (anpn)an .

Econ 205 Sobel

Page 21: Econ 205 - Slides from Lecture 14jsobel/205f10/notes14.pdfEcon 205 - Slides from Lecture 14 Joel Sobel September 10, 2010 Econ 205 Sobel Theorem (\Lagrange Multipliers") Theorem If

In Cobb-Douglas example, you have explicit formulas for solutionand value function.You can manipulate these to confirm generate properties(homogeneity and envelope theorem).

Econ 205 Sobel