7. Concavity and convexityEcon 494

Spring 2013

Why are we doing this?

• Desirable properties for a production function:• Positive marginal product • Diminishing marginal product • Isoquants should have

• Negative rate of technical substitution • Diminishing RTS

• We want to link these desired properties to the shape of the production function.• This will also apply to the utility function when we discuss consumer

theory

2

Typo corrected

Where are we going with this?Why do we care?• Production function defines the transformation of inputs into

outputs• Postulates of firm behavior

• Profit maximization• Cost minimization

• Results: shape of production fctn is key to FONC and SOSC• Especially in evaluating comparative statics.

3

Math review:Shape of functions• Concavity and convexity• Quasi-concavity and quasi-convexity• Determinant tests

4

Concavity

• Strict concavity

5

0 1

0 1

0 1

( ) is if:

ˆ( ) ( ) (1 ) ( )

ˆwhere: (1 )

and (0,1).

ˆThis implies th

strictly conc

a

ave

t:

f x

f x f x f x

x x x

x x x

Concavity is a weaker condition than strict concavity Concavity allows linear segments

0 1

( ) is if:

ˆ( ) (

conc

) (1 ) )

a

(

vef x

f x f x f x

f(x0)

f(x1)

qf(x0) + (1- ) qf(x1)

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1 2

1 2

Let 0.5 and ( ) ln( ) :

ˆ2 20 11

ˆ( ) 0.69 ( ) 3 ( ) 2.40

0.5ln(2) 0.5ln(20) 1.84

f x x

x x x

f x f x f x

Concave Function: y = ln(x)

2.40

3.00

0.69

1.84

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28

x

ln(x

)

Convexity

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x2

x1

q2

q1

q0

0 1

( ) is if:

ˆ( ) (

con

) (1 ) )

v

(

exf x

f x f x f x

0 1

( ) is ifst :

ˆ( ) ( ) (

rictly co v

1

n

)

x

(

e

)

f x

f x f x f x

qf(x0) + (1- ) q f(x1)

• Strict convexity and convexity• Reverse direction of inequality

Shape of production function

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f(x)

convex concave

Concavity/convexity is usually defined for some region of f(x).

• Is this production function concave? Convex? Both?

Quasi-concavity

• Production functions are also quasi-concave

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0 1

0 1

0 1

( ) is if:

ˆ( ) min ( ), ( )

ˆ

qu

where: (1 )

and (0,1

asi-c

).

ˆThis

onc

implies that:

avef x

f x f x f x

x x x

x x x

^

f(x0)

f(x)^

f(x1)

x0 x x1

Quasi-concavity

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A quasi-concave function cannot have a “U” shaped portion

f(x0)f(x)̂

f(x1)

x0 x̂ x1

f(x)f(x) is not quasi-concave over its whole domain.

Strict quasi-concavity

• No linear segments (or “U” shaped portions)

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0 1

strictly quasi-conca( ) is if:

ˆ( ) min ( )

v

)

e

, (

f x

f x f x f x

0 1

( ) is if:

ˆ( )

quasi-

min

concave

( ), ( )

f x

f x f x f x

Replace weak inequality with strict inequality

x̂

f(x1)

x0 x1

f(x)

f(x0) = f(x)̂

Quasi-convexity• For quasi-convex function, change direction of inequality and

change “min” to “max”

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0 1

strictly quasi-conv( ) is if:

ˆ( ) (

ex

max ), ( )

f x

f x f x f x 0 1

( ) is if:

ˆ( ) ( ),

quas

( )

i-convex

max

f x

f x f x f x

f(x0)

f(x1)

x0x1

quasiconvex can’texceed this

quasiconcave can’tgo below here

Recap

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0 1

0 1

0 1

0 1

0 1

ˆconcave ( ) ( ) (1 ) ( )

ˆstrictly concave ( ) ( ) (1 ) ( )

ˆconvex ( ) ( ) (1 ) ( )

ˆstrictly convex ( ) ( ) (1 ) ( )

ˆquasi-concave ( ) min ( ), ( )

ˆstrictly quasi-concave ( )

f x f x f x

f x f x f x

f x f x f x

f x f x f x

f x f x f x

f x

0 1

0 1

0 1

min ( ), ( )

ˆquasi-convex ( ) max ( ), ( )

ˆstrictly quasi-convex ( ) max ( ), ( )

f x f x

f x f x f x

f x f x f x

Principal minors

• Best illustrated with an example:

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11 12 13

21 22 23

31 32 33

11 12 1311 12

1 11 2 3 21 22 2321 22

31 32 33

leading principal mi

For the 3 3 matrix:

The nors are:

; ;

f f f

f f f

f f f

f f ff f

f f f ff f

f f f

H

H H H H

This pattern applies for square matrices of any dimension

Using Hessian matrix

• Hessian matrix• Contains all 2nd partial derivatives

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11 12 1

21 22

1

1 2

1 11 12 1

2 21 22 2

1 2

2

2

1 2

the bordered Hessian is

For

the Hessian matrix is:

the function ( , , , ),

:

0 n

n

n

n

n

n

n nn n nn

n

nn

f f f

f

f f f

f

y f

f f

f f f

f f f

f f f f

f

x x

f f f

x

BHH

Remember Young’s theorem: fij = fji

Strict Concavity

• The function is strictly concave if its Hessian matrix is negative definite (ND).

• A negative definite matrix has leading principal minors with determinants that alternate in sign, starting with negative:• etc.

• Alternatively:

• Diagonal elements of H are all • This is a sufficient condition for ND

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Recall SOSC for maximum(2 variables)• and

• Note that is implied by the above

• Note sign reversal because .• Because and

• The above meet the conditions for a negative definite matrix: • (for all )• Let and

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Concavity

• The function is concave if its Hessian matrix is negative semi-definite (NSD).

• For NSD, replace strict inequality with weak inequality• Determinants still alternate in sign:

• etc.• Alternatively:

• This is a necessary condition for NSD

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Strict Convexity

• The function is strictly convex if its Hessian matrix is positive definite (PD).

• A positive definite matrix has leading principal minors with determinants that are all strictly positive:• etc.

• Alternatively:

• Diagonal elements of H are • This is a sufficient condition for PD

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Convexity

• The function is convex if its Hessian matrix is positive semi-definite (PSD).

• For PSD, replace strict inequality with weak inequality.• Determinants all non-negative:

• etc.• Alternatively:

• This is a necessary condition for PSD

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Quasi-concavity

• The function is quasi-concave if its bordered Hessian matrix is negative definite (ND).• ND is sufficient for quasi-concavity

• Strict quasi-concavity• No convenient determinant conditions for distinguishing quasi-concavity

from strict quasi-concavity.

• The bordered Hessian being NSD is a necessary condition for quasi-concavity.

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Example

22

1 21 2 2

1 11 12 1 11 1 11 11

2 12 22

1 22 2

2 1 11 12 1 22 1 2 12 2 11

2 12 22

bordered Hessian: 2 bordered principal minors

00

0 0

0

2 0

f ff

BH f f f BH f f ff f

f f f

f f

BH f f f f f f f f f f

f f f

1 2

21 1

2 22 1 22 1 2 12 2 11

( , ) is quasi-concave if:

0

2 0

f x x

BH f

BH f f f f f f f

Concavity and quasi-concavity

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Every concave function is quasi-c

Theorem:

oncave.

0 1 0 1Let ( ) ( ) ( ) ( ) (0,

Proo

)

:

1

f

f x f x f x f x 1 10 1 ( ) (1 ) (( ) (1 ) ( ))f x f x f x f x

0 1 0 1ˆD ( ) (1efinition of concavity: ) where:ˆ (1 )( ) ( )f x f xf x x x x

0 1

1

1

0If ( ) ( ), then it must follow that, for co

ˆ

ncave function

( )

s:

,(( ) (1 ) ( ) )f x f x f x

f f x

x

x

f

1ˆ(

The

)

refo

(

r :

.)

e

f xf x

10 (( ) )f f xx

1 0 1mi( ) n ( ), ( )ˆ( ) f x ff x f x x

0 1min ( ), ( )

which is the definition of Quasi-concavity

ˆ( )

. Q.E.D.

f x ff xx

1 0 1min ( ), (( ) )f xx ff x

1( )f x

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RecapConcavity H is NSD (–1)n |Hn| ³ 0 necessary

Strict concavity

H is ND (–1)n |Hn| > 0 sufficient

Convexity H is PSD (+1)n |Hn| ³ 0 necessary

Strict convexity

H is PD (+1)n |Hn| > 0 sufficient

Quasi-concavity

BH is NSD (–1)n |BHn| ³ 0 necessary

Quasi-concavity

BH is ND (–1)n |BHn| > 0 sufficient