Week 4: Calculus and Optimization (Jehle and Reny, Chapter...

Week 4: Calculus and Optimization (Jehle andReny, Chapter A2)

Tsun-Feng Chiang

*School of Economics, Henan University, Kaifeng, China

September 27, 2015

Microeconomic Theory Week 4: Calculus and Optimization (Jehle and Reny, Chapter A2)September 27, 2015 1 / 31

Alternatively, to see whether a multivariate function f is concave, wecan check if the Hessian matrix of the function is negativesemidefinite.

Definition: DefinitenessLet A be an n × n symmetric matrix, then A is:

positive definite if zT Az > 0 for all z 6= 0 in Rn,positive semidefinite if zT Az ≥ 0 for all z 6= 0 in Rn,negative definite if zT Az < 0 for all z 6= 0 in Rn,negative semidefinite if zT Az ≤ 0 for all z 6= 0 in Rn, andindefinite if zT Az > 0 for some z in Rn, and zT Az < 0 for someother z in Rn


Figure: (left) Negative definite function x3 = −x21 − x2

2 and (right) negative

semidefinite function x3 = −(x1 + x2)2 (Simon and Blume 1994, page 378-379)


With this in mind, the analog of a nonpositive second derivative of afunction of one variable would be a negative semidefinite matrix ofsecond-order partial derivatives (i.e., the Hessian matrix) of a functionof many variables. We can now put together Theorem A2.1 and A2.3to confirm this.

Theorem A2.4 Slope, Curvature, and Concavity in ManyVariablesLet D be a convex subset of Rn with a nonempty interior on which f istwice continuously differentiable. The following statements 1 to 3 areequivalent:1. f is concave.2. H(x) is negative semidefinite for all x in D.3. For all x0 ∈ D : f (x) ≤ f (x0) +5f (x0)(x− x0)Moreover,4. If H(x) is negative definite for all x in D, then f is strictly concave(the converse is not true).


In a single-variable case, a necessary and sufficient condition for afunction to be concave (convex) is that its second derivative not berising (falling). In the multivariable case, we can note a necessary, butnot a sufficient condition for concavity or convexity in terms of the signsof all "own" second partial derivative.

Theorem A2.5 Concavity, Convexity, and Second-Order OwnPartial DerivativesLet f : D → R be a twice differentiable function.1. If f is concave, then ∀x, fii(x) ≤ 0, i = 1, · · · ,n.2. If f is convex, then ∀x, fii(x) ≥ 0, i = 1, · · · ,n.


Homogeneous Functions

Homogeneous real-valued function arise quite often in microeconomicapplications. In this section, we briefly consider functions of this typeand use our calculus tools to establish some of their importantproperties.

Definition A2.2 Homogeneous FunctionsA real-valued function f (x) is called homogeneous of degree k if

f (tx) ≡ tk f (x) for all t > 0.

Two special cases are worthy of note: f (x) is homogeneous of degree1, or linear homogeneous, if f (tx) ≡ tf (x) for all t > 0; it ishomogeneous of degree zero if f (tx) ≡ f (x) for all t > 0.

Homogeneous function display very regular behavior as all variablesare increased simultaneously and in the same proportion. When a


function is homogeneous of degree 1, for example, doubling or triplingall variables doubles or triples the value of the function. Whenhomogeneous of degree zero, equiproportionate changes in allvariables leave the value of the function unchanged.

Example A2.3

The function f (x1, x2) ≡ Axα1 xβ2 , A > 0, α > 0, β > 0, is known as theCobb-Douglas function. We now multiply all variables by the samefactor t ,

f (tx1, tx2) ≡ A(tx1)α(tx2)

β ≡ tαtβAxα1 xβ2= tα+βf (x1, x2).

According to the definition, the Cobb-Douglas is homogeneous ofdegree α+ β > 0. If the coefficients are chosen so that α+ β = 1, it islinear homogeneous.


The partial derivative of homogeneous functions are alsohomogeneous. The following theorem makes this clear.

Theorem A2.6 Partial Derivative of Homogeneous FunctionsIf f (x) is homogeneous of degree k , its partial derivative arehomogeneous of degree k − 1.Proof: Assume f (x) is homogeneous of degree k , Then

f (tx) ≡ tk f (x) for all t > 0.

Differentiate the left-hand side and right-hand side with respective toxi . The resulting derivatives are equal,

∂f (tx)∂xi

= tk−1∂f (x)∂xi

,

for i = 1, · · · ,n, and t > 0, as we sought to show.

Let verify this theorem for the Cobb-Douglas form.


Example A2.4

Let f (x1, x2) ≡ Axα1 xβ2 , and suppose α+ β = 1 so that it is linearhomogeneous. The partial derivative with respect to x1 is

∂f (x1, x2)

∂x1= αAxα−1

1 xβ2

Multiply both x1 and x2 by the factor t , and evaluate the partialderivative at (tx1, tx2). We obtain

∂f (tx1, tx2)

∂x1= αA(tx1)

α−1(tx2)β = tα+β−1αAxα−1

1 xβ2 =∂f (x1, x2)

∂x1,

as required, because α+ β = 1 and tα+β−1 = t0 = 1.

Finally, Euler’s theorem, sometimes called the adding-up theorem,gives us an interesting way to completely characterize homogeneousfunctions.


Theorem A2.7 Euler’s Theoremf (x) is homogeneous of degree k if and only if

kf (x) =n∑

i=1

∂f (x)∂xi

xi for all x

This theorem says a function is homogeneous if and only if it canalways be written in terms of its own partial derivatives and the degreeof homogeneity.

Example A2.5

Let f (x1, x2) ≡ Axα1 xβ2 , and again suppose α+ β = 1. The partialderivatives are

∂f (x1, x2)

∂x1= αAxα−1

1 xβ2 ,∂f (x1, x2)

∂x2= βAxα1 xβ−1

2

Multiply the first by x1, the second by x2, add, and use the fact that


Example A2.5(Continued)α+ β = 1 to get

∂f (x1, x2)

∂x1x1 +

∂f (x1, x2)

∂x2x2 = αAxα−1

1 xβ2 x1 + βAxα1 xβ−12 x2

= (α+ β)Axα1 xβ2= 1× f (x1, x2).

just as we were promised by Euler’s theorem.


A2.2 Unconstrained Optimization

This section is devoted to the calculus approach to optimizationproblems, the most common form of problem in microeconomic theory.First we consider the function of a single variable, y = f (x), andassume it is differentiable. If we can find a point x∗ or x̃ , then

x∗ is a (unique) local maximum if f (x∗) ≥ f (x) (f (x∗) > f (x)) forall x 6= x∗ in some neighborhood of x∗.x∗ is a (unique) global maximum if f (x∗) ≥ f (x) (f (x∗) > f (x)) forall x 6= x∗ in the domain of the function.x̃ is a (unique) local minimum if f (x̃) ≤ f (x) (f (x̃) < f (x)) for allx 6= x̃ in some neighborhood of x̃ .x̃ is a (unique) global minimum if f (x̃) ≤ f (x) (f (x̃) < f (x)) for allx 6= x̃ in the domain of the function.

Various types of optima are illustrated in Figure A2.4 (see the nextslide).


Figure A2.4: Local and global optima

In Figure A2.5 (see the next slide), we can see the characters of themaximum and the minimum, which can be presented by the first-ordernecessary condition (FONC) and the second-order necessarycondition (SONC).


Figure A2.5: Necessary conditions for local optima

Theorem A2.8 Necessary Conditions for Local (Interior) Optimain the Single-Variable CaseLet f (x) be a twice continuously differentiable function of one variable.Then f (x) reaches a local1. maximum at x∗ ⇒ f ′(x∗) = 0 (FONC)

⇒ f ′′(x∗) ≤ 0 (SONC)2. minimum at x̃ ⇒ f ′(x̃) = 0 (FONC)

⇒ f ′′(x̃) ≥ 0 (SONC)


Real-Valued Functions of Several Variables

Suppose that D ⊂ Rn, and let f : D → R be a twice continuouslydifferentiable real-valued function of n variables. Similar to thesingle-variable case, if we can find a point x∗ or x̃, then

x∗ is a local maximum whenever there exists some ε > 0 suchthat f (x∗) ≥ f (x) for all x ∈ Bε(x∗).x∗ is a global maximum if f (x∗) ≥ f (x) for all x in the domain.x̃ is a local minimum whenever there exists some ε > 0 such thatf (x̃) ≤ f (x) for all x ∈ Bε(x̃).x̃ is a global minimum whenever f (x̃) ≤ f (x) for all x in thedomain.

These optima are unique if the inequalities hold strictly. As one mightexpect, the analogy to the derivative being zero at the optimum forfunctions of one variable will be the gradient vector must be zero at anoptimum of a function of many variables. Thus, the single first-orderequation f ′(x∗) = 0 characterizing optima for functions of one variable


generalizes to the first-order system of n simultaneous equations,5f (x∗) = 0 characterizing the optima of functions of n variables. Thisgives us our first-order necessary condition for any (interior) optima ofreal-valued functions.

Theorem A2.9 First-Order Necessary Condition for Local(Interior) Optima of Real-Valued FunctionsIf the differentiable functions f (x) reaches a local (interior) maximum orminimum at x∗, then x∗ solves the system of simultaneous equations,

∂f (x∗)∂x1

= 0,∂f (x∗)∂x2

= 0, · · · ∂f (x∗)∂xn

= 0.

Example A2.6

Let y = x2 − 4x21 + 3x1x2 − x2

2 . To find a critical point of this function,take each of its partial derivatives:


Example A2.6 (Continued)∂f (x1, x2)

∂x1= −8x1 + 3x2,

∂f (x1, x2)

∂x2= 1 + 3x1 − 2x2

We will have a critical point at a vector (x∗1 , x∗2 ) where both of these

equal zero simultaneously. To find x∗1 and x∗2 , set each partial equal tozero:

∂f (x∗1 , x∗2 )

∂x1= −8x∗1 + 3x∗2 = 0,

∂f (x∗1 , x∗2 )

∂x2= 1 + 3x∗1 − 2x∗2 = 0

and solve this system for x∗1 and x∗2 . We can see the critical point atx∗1 = 3/7 and x∗2 = 8/7. We do not yet know whether we have found amaximum or a minimum, though. For that we have to look at thesecond-order conditions.


Second-Order Conditions

Once we have found a point where 5f (x∗) = 0, we know we have amaximum (minimum) if the function is "locally concave" ("locallyconvex") there. Theorem A2.4 pointed out that curvature depends onthe definiteness property of the Hessian of f . Intuitively, it appears thatthe function will be locally concave around x if H(x) is negativesemidefinite, and will be locally convex if it is positive semidefinite.Intuition thus suggests the following second-order necessary conditionfor local (interior) optima.

Theorem A2.10 Second-Order Necessary Condition for Local(Interior) Optima of Real-Valued FunctionsLet f (x) be twice continuously differentiable.1. If f (x) reaches a local interior maximum at x∗, then H(x∗) is negativesemidefinite.2. If f (x) reaches a local interior minimum at x̃, then H(x̃) is positivesemidefinite.


Theorem A2.9 and A2.10 are both necessary conditions which help inlocating potential maxima (or minima) of specific functions, but to verifythat they actually maximize (or minimize) the function, we needsufficient conditions. Sufficient conditions for optima are more stringentthan necessary conditions. Simply stated, sufficient conditions forinterior optima are as follows:

If fi(x∗) = 0 for i = 1, · · · ,n and H(x∗) is negative definite at x∗,then f (x) reaches a local maximum at x∗.If fi(x̃) = 0 for i = 1, · · · ,n and H(x̃) is positive definite at x̃, thenf (x) reaches a local minimum at x̃.

The sufficient conditions require the point to be a critical point, andrequire the curvature conditions to hold in their strict forms. Forexample, when H(x∗) is negative definite, the function will be strictlyconcave in some ball around x∗.Locating a critical point is easy. We simply set all first-order partialderivatives equal to zero and solve the system of n equations.


Determining whether the Hessian is negative or positive definite therewill generally less easy.Various tests for determining the definiteness property of the Hessiankey on the sign pattern displayed by the determinants of certainsubmatrices formed from it at the point (or region) in question. Thesedeterminants are called the principal minors of the Hessian. By thefirst through nth principal minors of H(x) at the point x, we mean thedeterminants

D1(x) ≡∣∣f11∣∣ = f11, D2(x) ≡

∣∣∣∣f11 f12f21 f22

∣∣∣∣ · · · Di(x) ≡

∣∣∣∣∣∣∣f11 · · · f1i...

. . ....

fi1 · · · fii

∣∣∣∣∣∣∣ · · ·Dn(x) ≡

∣∣∣∣∣∣∣f11 · · · f1n...

. . ....

fn1 · · · fnn

∣∣∣∣∣∣∣ ,where it is understood that fi1 is evaluated at x. Each is thedeterminant of a matrix resulting when the last (n − i) rows and


columns of the Hessian H(x) are deleted, for i = 1, · · · ,n.The following theorem gives requirements on its principal minorssufficient to ensure definiteness of the Hessian.

Theorem A2.11Let f(x) be twice continuously differentiable, and let Di(x) be thei th-order principal minor of the Hessian matrix H(x).1. If (−1)iDi(x) > 0, i = 1, · · · ,n, then H(x) is negative definite.2. If Di(x) > 0, i = 1, · · · ,n, then H(x) is positive definite.If condition 1 holds for all x in the domain, then f is strictly concave. Ifcondition 2 holds for all x in the domain, then f is strictly convex.

From Theorem A2.9 and A2.11, we can state the first- andsecond-order sufficient conditions for local interior optima.

Theorem A2.12 Sufficient Conditions for Local Interior Optima ofReal-Valued FunctionsLet f(x) be twice continuously differentiable,


Theorem A2.12 (Continued)

1. If fi(x∗) = 0 and (−1)iDi(x∗) > 0, i = 1,2, · · · ,n, then f (x) reaches alocal maximum at x∗.2 If fi(x̃) = 0 and Di(x̃) > 0, i = 1,2, · · · ,n, then f (x) reaches a localminimum at x̃.

Example A2.7Check whether the critical point for the functiony = x2 − 4x2

1 + 3x1x2 − x22 was a maximum or a minimum. We

compute the second-order partials,

∂2f∂x2

1= −8;

∂2f∂x1∂x2

=∂2f

∂x2∂x1= 3;

∂2f∂x2

2= −2

and form the Hessian, H(x) =[−8 33 −2

]. In Example A2.6, we found

a critical point at x = (3/7,8/7). Checking the principal minors,


Example A2.7 (Continued)

D1(x) ≡∣∣−8

∣∣ = −8 < 0, D2(x) ≡∣∣∣∣−8 3

3 −2

∣∣∣∣ = 16− 9 = 7 > 0,

Because these principal minors alternate in sign, beginning withnegative, Theorem A2.12 tells us that x∗ = (3/7,8/7) is a localmaximum.

In this example the Hessian matrix was completely independent of x.We would therefore obtain the same alternating sign pattern on theprincipal minors regardless of where we evaluated them. In TheoremA2.11, we observed that this is sufficient to ensure that the functioninvolved is strictly concave. In other words, it means there is only a hillin the graph and it is the only highest point.Indeed, from Figure A2.5, it is intuitively clear that any local maximum(minimum) of a concave (convex) function must also be a globalmaximum (minimum).


Theorem A2.13 (Unconstrained) Local-Global TheoremLet f be a twice continuously differentiable real-valued concavefunction on D. The following statements are equivalent, where x∗ is aninterior point of D:1. 5f (x∗) = 0.2. f achieves a local maximum at x∗.3. f achieves a global maximum at x∗.Proof: Clearly, 3⇒ 2, and by Theorem A2.9, 2⇒ 1. Hence, it remainsonly to show that 1⇒ 3.Because f is concave, Theorem A2.4 implies that for all x in thedomain,

f (x) ≤ f (x∗) +5f (x∗)(x− x∗).

Then given 1, the inequality above implies that

f (x) ≤ f (x∗)

Therefore, f reaches a global maximum at x∗.


Theorem A2.13 says that under convexity or concavity, any localoptimum is a global optimum. Notice, however, that it is still possiblethat the lowest (highest) value is reached at more than one point in thedomain. If we want the highest or lowest value of the function to beachieved at a unique point, we have to impose strict concavity or strictconvexity.

Theorem A2.14 Strict Concavity/Convexity and the Uniquenessof Global Optima1. If x∗ maximizes the strictly concave function f , then x∗ is the uniqueglobal maximizer, i.e., f (x∗) > f (x) ∀x ∈ D, x 6= x∗.2. If x̃ minimizes the strictly convex function f , then x̃ is the uniqueglobal minimizer, i.e., f (x̃) < f (x) ∀x ∈ D, x 6= x̃.Proof: We will prove 1. If x∗ is a global maximizer of f but it is notunique, then there exists some other points x′ 6= x∗ such thatf (x′) = f (x∗). If we let xt = tx′ + (1− t)x∗, then strict concavityrequires that


Theorem A2.14 (Continued)

f (xt) > tf (x′) + (1− t)f (x∗) ∀t ∈ (0,1)

Because f (x′) = f (x∗), the requires that

f (xt) > tf (x′) + (1− t)f (x′)

or, simplyf (xt) > f (x′)

This, however, contradicts the assumption that x′ is a global maximizerof f . Thus, any global maximizer of a strictly concave function must beunique.

From Theorem A2.9 and A2.14, we have the sufficient conditions forunique global optima.


Theorem A2.15 Sufficient Conditions for Unique Global OptimaLet f (x) be twice continuously differentiable, 1. If f (x) is strictlyconcave and fi(x∗) = 0, i = 1, · · · ,n, then x∗ is the unique globalmaximizer of f (x).2. If f (x) is strictly convex and fi(x̃) = 0, i = 1, · · · ,n, then x̃ is theunique global minimizer of f (x).


Constrained Optimization

Scarcity is a pervasive fact of economic life. It is most commonlyexpressed as constraints on permissible value of economic variables.Agents, often consumers and firms, are then represented as seekingto do the best they can within the constraints they face. This is the typeof problem we will regularly encounter. We need to modify ourtechniques of optimization and the terms that characterize the optimain such cases, accordingly. There are three basic types of constraintswe will encounter. They are equality constraints, nonnegativityconstraints, and, amore generally, any form of inequality constraint.We will derive methods for solving problems that involve each of themin turn. We will confine discussion to problems of maximization, simplynoting the modifications (if any) for minimization problems.


Equality Constraints

Consider choosing x1 and x2 to maximize f (x1, x2), when x1 and x2must satisfy some particular relation to each other that we write inimplicit form as g(x1, x2) = 0. Formally

maxx1,x2 f (x1, x2) subject to g(x1, x2) = 0

Here f (x1, x2) is called the objective function. The x1 and x2 arecalled choice variables. The function g(x1, x2) is called theconstraint. The set of all x1 and x2 that satisfy the constraint aresometimes called the constraint set or the feasible set. One way tosolve this problem is by substitution. For example, suppose thatg(x1, x2) = 0 can be written to isolate x2 on one side as

x2 = g̃(x1)

Substitute the equation above into the objective function to replace x2.This way, the two-variable constrained maximization problem can berephrased as the single-variable problem with no constraints:


maxx1 f (x1, g̃(x1))

The usual first-order conditions require that we set the total derivative,∂f/∂x1, equal to zero and solve for the optimal x∗1 . That is,

∂f (x∗1 , g̃(x∗1 ))

∂x1+∂f (x∗1 , g̃(x

∗1 ))

∂x2

dg̃(x∗1 ))dx1

= 0 (by chain rule)

When we have found x∗1 , we plug it back to the constraint and findx∗2 = g̃(x∗1 ). The pair (x∗1 , x

∗2 ) then solves the constrained problem,

provided the appropriate second-order condition is also fulfilled.However, there more economic problems which involve more than twochoice variables and more than one constrains. The substitutionmethod is not well suited to these more complicated problems. Thebetter way is using Lagrange’s method.


First Midterm Exam

Date: Scheduled on Monday, October 26th, 2015

Time: 9:00 am ∼ 11:30 am

Location: To be Announced

Coverage: Week 1 - Week 5 and a small part of Week 6

Rules: The exam is closed-book, closed-notes and in-class.Taking the in-class exam is only way for earning grades in this

course fro any student. No scratch paper (will be provided by theproctor), calculator, cellphone and other electronic devices areallowed. Pens, correction tools and basic requirements for life

(ex. water, tissue paper, and medicine) are allowed.


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