7. concavity and convexity econ 494 spring 2013. why are we doing this? 2 typo corrected
TRANSCRIPT
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)
6
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
9
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
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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