mathematical economics: lecture 16 · the graph of f lies below the graph. a function f of n...

Mathematical Economics:Lecture 16

Yu Ren

WISE, Xiamen University

November 26, 2012

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Chapter 21: Concave and Quasiconcave Functions

Outline

1 Chapter 21: Concave and QuasiconcaveFunctions

Yu Ren Mathematical Economics: Lecture 16

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New Section

Chapter 21: Concaveand Quasiconcave

Functions


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Concave and convex functions

DefinitionA real-valued function f defined on aconvex subset U of Rn is concave if for allX ,Y in U and for all t between 0 and 1,f (tX + (1− t)Y ) ≥ tf (X ) + (1− t)f (Y ).Figure 21.2A real-valued function f defined on aconvex subset U of Rn is convex if for allX ,Y in U and for all t between 0 and 1,f (tX + (1− t)Y ) ≤ tf (X ) + (1− t)f (Y ).Figure 21.3


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f is concave if and only if −f is convex


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Different with convex set: whenever X and Yare points in U, the line segment joining X to Yl(X ,Y ) ≡ {tX + (1− t)Y : 0 ≤ t ≤ 1} is also inU. Figure 21.1


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A function f of n variables is concave if andonly if any secant line connecting two points onthe graph of f lies below the graph. A function fof n variables is convex if and only if any secantline connecting two points on the graph of f liesabove the graph.


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Theorem 21.1 Let f be a function defined on aconvex subset U of Rn. Then, f is concave(convex) if and only if its restriction to very linesegment in U is a concave (convex) function ofone variable.


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Calculus Criteria for Concavity:Theorem 21.2 Let f be a C1 function on aninterval I in R. Then, f is concave on I if andonly if f (y)− f (x) ≤ f ′(x)(y − x) for all x , y ∈ I.The function f is convex on I if and only iff (y)− f (x) ≥ f ′(x)(y − x) for all x , y ∈ I.


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Theorem 21.3 let f be a C1 function on aconvex subset U of Rn. Then, f is concave on Uif and only if for all X ,Y in U:f (Y )− f (X ) ≤ Df (X )(Y − X ). Similarly, f isconvex on U if and only if for all X ,Y in U:f (Y )− f (X ) ≥ Df (X )(Y − X ) for all X ,Y in U.


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Corollary 21.4 If f is a C1 concave function on aconvex set U and if X0 ∈ U, thenDf (X0)(Y − X0) ≤ 0 implies f (Y ) ≤ f (X0). Inparticular, if Df (X0)(Y − X0) ≤ 0 for all Y ∈ U,then X0 is a global max of f .


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Theorem 21.5 Let f be a C2 function on anopen convex subset U of Rn. Then, f is aconcave function on U if and only if the HessianD2f (X ) is negative semidefinite for all X in U.The function f is a convex function on U if andonly if D2f (X ) is positive semidefinite for all X inU.


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Example 21.1

Example 21.1 Let us apply the test of Theorem21.3 to show that f (x1, x2) = x2

1 + x22 is convex

on Rn. The function f is convex if and only if

(y21 + y2

1 )− (x21 + x2

2 ) ≥ (2x1 2x2)

(y1 − x1

y2 − x2

)= 2x1y1 − 2x2

1 + 2x2y2 − 2x22


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Example 21.1

if and only if

y21 + y2

1 + x21 + x2

2 − 2x1y1 − 2x2y2 ≥ 0

if and only if

(y1 − x1)2 + (y2 − x2)2 ≥ 0

which is true for all (x1, x2) and (y1, y2) in R2.


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Example 21.2

Example 21.2 The Hessian of the functionf (x , y) = x4 + x2y2 + y4 − 3x − 8y is

D2f (x , y) =

(12x2 + 2y2 4xy

4xy 12x2 + 2y2

)For (x , y) 6= (0,0), the two leading principalminors, 12x2 + 2y2 and 24x4 + 132x2y2 + 24y4,are both positive, so f is a convex function on allRn.


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Example 21.3

Example 21.3 A commonly used simple utilityor production function is F (x , y) = xy . ItsHessian is

D2F (x , y) =

(0 11 0

)whose second order principal minor isdetD2F (x , y) = −1. Since this second principalminor is negative, D2F is indefinite and F isneither concave nor convex.


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Example 21.4

Example 21.4 Consider the monotonictransformation of the function F in the previousexample by the functiong(z) = z1/4 : G(x , y) = x1/4y1/4, defined only onthe positive quadrant R2

+. The Hessian of G is

D2G(x , y) =

(− 3

16x−7/4y1/4 116x3/4y−3/4

116x−3/4y−3/4 − 3

16x1/4y−7/4

)


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Example 21.4

For x > 0, y > 0, the first order leading principalminor is negative and the second order leadingprincipal minor, x−3/2y−3/2/128, is positive.Therefore, D2G(x , y) is negative definite on R2

+

and G is a concave function on R2+.


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Example 21.5

Example 21.5 Now, consider the generalCobb-Douglas function on R2

+ : U(x , y) = xayb.Its Hessian is

D2U(x , y) =

(a(a− 1)xa−2yb abxa−1yb−1

abxa−1yb−1 b(b − 1)xayb−2

)


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Example 21.5

whose determinant is

detD2U(x , y) = ab(1− a− b)x2a−2y2b−2.

In order for U to be concave on R2+, we need

a(a− 1) < 0 and ab(1− a− b) > 0; that is, weneed 0 < a < 1,0 < b < 1, and a + b ≤ 1. Insummary, a Cobb-Douglas production functionon R2

+ is concave if and only if it exhibitsconstant or decreasing returns to scale.


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Properties of Concave functions

Theorem 21.6 Let f be a concave (convex)function on an open convex subset U of Rn. If x0is a critical point of f , that is, Df (x0) = 0, thenx0 ∈ U is a global maximizer (minimizer) of f onU.


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Theorem 21.7 Let f be a C1 function defined ona convex subset U of Rn. If f is a concavefunction and if x0 is a point in U which satisfiesDf (x0)(y − x0) ≤ 0 for all y ∈ U, then x0 is aglobal maximizer of f on U.


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Example 21.6

Example 21.6 If f is a C1 increasing, concavefunction of one variable on the interval [a,b],then f ′(b)(x − b) ≤ 0 for all x ∈ [a,b]. ByTheorem 21.7, b is the global maximizer of f on[a,b].


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Example 21.7

Example 21.7 Consider the concave functionU(x , y) = x1/4y1/4 on the (concave) triangle

B = {(x , y) : x ≥ 0, y ≥ 0, x + y ≤ 2}.

By symmetry, we would expect that(x0, y0) = (1,1) is the maximizer of U on B. Toprove this, use Theorem 21.7. Let (x,y) be anarbitrary point in B.


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Example 21.7

∂U∂x

(1,1)(x − 1) +∂U∂y

(1,1)(y − 1)

=14

(x − 1) +14

(y − 1)

=14

(x + y − 2)

≤ 0

since x + y − 2 ≤ 0 for (x,y) in the constraint setB. By Theorem 21.7, (1,1) is the globalmaximizer of U on B.


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Example 21.8

Example 21.8 Consider the problem ofmaximizing profit for a firm whose productionfunction is y = g(x), where y denotes outputand x denotes the input bundle. If p denotes theprice of the output and wi is the cost per unit ofinput i , then the firm’s profit function is

Π(x) = pg(x)− (w1x1 + · · ·+ wnxn)


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Example 21.8

As can easily be checked, Π will be a concavefunction provided that the production function isa concave function. In this case, the first ordercondition

p∂g∂xi

= wi for i = 1,2, . . . ,n,

which says marginal revenue product equals thefactor price for each point, is both necessaryand sufficient for an interior profit maximizer.


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Example 21.8

If one wants to study the effect of changes in wior p on the optimal input bundle, one wouldapply the comparative statics analysis tosystem. Since profit is concave for all p and w ,the solution to system will automatically be theoptimal input for all choices of p and w .


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Theorem 21.8 Let f1. · · · , fk be concave(convex) functions. each defined on the sameconvex subset U of Rn. Let a1, · · · ,ak bepositive numbers. Then, a1f1 + · · ·+ ak fk is aconcave (convex) function on U.


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Theorem 21.9 let f be a function defined on aconvex set U in Rn. If f is concave, then forevery x0 in U, the setC+

x0≡ {x ∈ U : f (x) ≥ f (x0)} is a convex set. If f

is convex, then for every x0 in U, the setC−x0≡ {x ∈ U : f (x) ≤ f (x0)} is a convex set.


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Quasiconcave and Quasiconvex

Definition: a function f defined on a convexsubset U of Rn is quasiconcave if for everyreal number a, C+

a ≡ {x ∈ U : f (x) ≥ a} is aconvex set. Similarly, f is quasiconvex if forevery real number a,C−a ≡ {x ∈ U : f (x) ≤ f (x0)} is a convex set.Figure 21.9


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Theorem 21.12 Let f be a function defined on aconvex set U in Rn. Then, the followingstatements are equivalent to each other:(a) f is a quasiconcave function on U(b) For all X ,Y ∈ U and all t ∈ [0,1] f (X ) ≥ f (Y )implies f (tX + (1− t)Y ) ≥ f (Y )(c) For all X ,Y ∈ U and all t ∈ [0,1]f (tX + (1− t)Y ) ≥ min{f (Y ), f (X )}.


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Example 21.9

Example 21.9 Consider the Leontief orfixed-coefficient production functionQ(x , y) = min{ax ,by} with a,b > 0. The levelsets of Q are drawn in Figure 21.7. Certainly,the region above and to the right of any of thisfunction’s L-shaped level sets is a convex set. Qis quasiconcave.


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Extra Theorem

Theorem 21.* Any monotonic transformation ofa concave function is quasiconcave.


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Theorem 21.13 Every Cobb-Douglas functionF (x , y) = Axayb with A,a and b all positive isquasiconcave.


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Example 21.10

Example 21.10 Consider the constant elasticityof substitution (CES) production function

Q(x , y) = (a1x r1 + a2x r

2)1/r , where 0 < r < 1.

By Theorem 21.8 and Exercise 21.4,(a1x r

1 + a2x r2) is concave. Since g(z) = z1/r is a

monotonic transformation, Q is a monotonictransformation of a concave function andtherefore is quasiconcave.


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Example 21.11

Example 21.11 Let y = f (x) be any increasingfunction on R1, as in Figure 21.8. For any x∗,{x : f (x) ≥ f (x∗)} is just the interval [x∗,∞), aconvex subset of R1. So, f is quasiconcave. Onthe other hand, {x : f (x) ≤ f (x∗)} is the concaveset (∞, x∗]. Therefore, an increasing function onR1 is both quasiconcave and quasiconvex. Thesame argument applies to decreasing function.


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Example 21.12

Example 21.12 Any function on R1 which risesmonotonically until it reaches a global maximumand then monotonically falls, such as y = −x2

or the bell-shaped probability density functiony = ke−x2, is a quasiconcave function, as Figure21.9 indicates. For any x1 as in Figure 21.9,there is a x2 such that f (x1) = f (x2). Then,{x : f (x) ≥ f (x1)} is the convex interval [x1, x2].


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Quasiconcave and quasiconvex

Calculus Criteria: Theorem 21.14 Suppose thatF is a C1 function on an open convex subset Uof Rn. Then, F is quasiconcave on U if and onlyif F (y) ≥ F (x) implies that DF (x)(y − x) ≥ 0; Fis quasiconvex on U if and only if F (y) ≤ F (x)implies that DF (x)(y − x) ≤ 0;


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Quasiconcave and quasiconvex

Theorem 21.15 Suppose that F is a real-valuedpositive function defined on a convex cone C inRn. If F is homogeneous of degree one andquasiconcave on C, it is concave on C.


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Pseudoconcave and Pseudoconvex

Definition : Let U be an open convex subset ofRn. A C1 function F : U → R is pseudoconcaveat x∗ ∈ U if DF (x∗)(y − x∗) ≤ 0 impliesF (y) ≤ F (x∗) for all y ∈ U. The function F ispseudoconcave on U if (15) holds for all x∗ ∈ U.To define a pseudoconvex function on U, onesimply reverses all inequalities.


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Theorem 21.16 Let U be a convex subset of Rn,and let F : U → R be a C1 pseudoconcavefunction. If x∗ ∈ U has the propertyDF (x∗)(y − x∗) ≤ 0 for all y ∈ U, for example,DF (x∗) = 0, then x∗ is a global max of F on U.An analogous result holds for pseudoconvexfunctions.


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Theorem 21.17 Let U be a convex subset of Rn.Let F : U → R be a C1 function. Then, (a) if F ispseudoconcave on U, F is quasiconcave on U,and (b) if U is open and if OF (x) 6= 0 for allx ∈ U, then F is pseudoconcave on U if andonly if F is quasiconcave on U.


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Theorem 21.18 Let U be an open convexsubset of Rn. Let F : U → R be a C1 function onU. Then, F is pseudoconcave on U if and only iffor each x∗ in U, x∗ is the solution to theconstrained maximization problem max F (x) s.tCx∗ ≡ {y ∈ U : DF (x∗)(y − x∗) ≤ 0}.


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Theorem 21.19 Let F be a C2 function on anopen convex subset W in Rn. Consider thebordered Hessian H (a) If the largest (n-1)leading principal minors of H alternate in sign,for all x ∈W , with the smallest of these positive,then F is pseudoconcave, and thereforequasiconcave, on W . (b) If these largest (n-1)leading principal minors are all negative for allx ∈W , then F is pseudoconvex, and thereforequasiconvex, on W .


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Theorem 21.20 Let F be a C2 function on aconvex set W in R2. Suppose that F ismonotone in that F ′x > 0 and F ′y > 0 on W. If thedeterminant (18) is positive for all (x , y) ∈W ,then F is quasiconcave on W . If thedeterminant (19) is negative for all (x , y) ∈W ,then F is quasiconvex on W . Conversely, if F isquasiconcave on W , then the determinant (19)is positive; if F is quasiconvex on W , then thedeterminant (19) is ≤ 0 for all (x , y) ∈W .


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Example 21.13

Example 21.13 Theorem 21.13 implies that theCobb-Douglas function U(x , y) = xayb isquasiconcave on R2

+ for a,b > 0 since it is amonotone transformation of a concave function.Let’s use Theorem 21.20 to prove thequasiconcavity of U. The bordered Hessian is 0 axa−1yb bxayb−1

axa−1yb a(a− 1)xa−1yb abxa−1yb−1

bxayb−1 abxa−1yb−1 b(b − 1)xayb−2


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Example 21.13

whose determinant is

(ab + ab2 + a2b)x3a−2y3b−2,

which is always positive forx > 0, y > 0,a > 0,b > 0. By Theorem 21.20, Uis pseudoconcave, and therefore quasiconcave.


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Concave Programming

Unconstrained Problems : Theorem 21.21 LetU be a convex subset of Rn. Let f : U → R be aC1 concave (convex) function on U. Then, x∗ isa global max of f on U if and only ifDf (x∗)(x − x∗) ≤ 0 for all x ∈ U. In particular, ifU is open, or if x∗ is an interior point of U, thenx∗ is a global max(min) of f on U if and only ifDf (x∗) = 0


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Concave Programming

Constrained Problem : Theorem 21.22 Let U bea convex open subset of Rn. Let f : U → R be aC1 pseudoconcave function on U. Letg1, · · · ,gk : U → R be C1 quasiconvexfunctions. If (x∗, λ∗) satisfy the Lagrangianconditions, x∗ is a global max of f on theconstraint set.


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Concave Programming

Theorem 21.23 let f ,g1, · · · ,gk be as in thehypothesis of Theorem 21.22. (a) For any fixedb = (b1, · · · ,bk) ∈ Rk , let Z (b) denote the set ofx ∈ Cb that are global maximizers of f on Cb.Then, Z (b) is a convex set. (b) For any b ∈ Rk ,let V (b) denote the maximal value of theobjective function f in problem (20). If f isconcave and the gi are convex, then b → V (b)is a concave function of b.


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Concave Programming

Saddle Point: Definition: Let U be a convexsubset of Rn. Consider the Lagrangian function(21) for the programming problem (20), as afunction of x and λ. Then, (x∗, λ) is saddle pointof L if L(x , λ∗) ≤ L(x∗, λ∗) ≤ L(x∗, λ) for all λ ≥ 0and all x ∈ U. Usually, U = Rn or U = Rn

+, thepositive orthant of Rn. In the latter case, we saythat (x∗, λ∗) is a nonnegative saddle point of L


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Concave Programming

Theorem 21.24 If (x∗, λ∗) is a (nonnegative)saddle point for L in Problem (20), then x∗maximizes f on Cb(Cb ∩ Rn

+).


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Concave Programming

Theorem 21.25 Suppose that U = Rn+ or that U

is an open convex subset of Rn. Suppose that fis a C1 concave function and that g1, · · · ,gk areC1 convex functions on U. Suppose that x∗maximizes f on the constraint set Cb as definedin (20). Suppose further that one of theconstraint qualifications in Theorem 19.12holds. Then, there exists λ∗ > 0 such that(X ∗, λ∗) is a saddle point of the Lagrangian (21)


mathematical economics: lecture 16 · the graph of f lies below the graph. a function f of n...

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