Transcript Document
7. Concavity and convexity
Econ 494
Spring 2013
2
Why are we doing this?
• Desirable properties for a production function:
• Positive marginal product
𝑓𝑖 > 0
• Diminishing marginal product
𝑓𝑖𝑖 < 0
• Isoquants should have
• Negative rate of technical substitution
• Diminishing RTS
𝑑𝑥2 𝑑𝑥1 < 0
𝑑 2 𝑥2 𝑑𝑥1 2 > 0
Typo
corrected
• 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
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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.
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Math review:
Shape of functions
• Concavity and convexity
• Quasi-concavity and quasi-convexity
• Determinant tests
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Concavity
f(x1)
• Strict concavity
f ( x ) is strictly conc ave if:
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
f(x0) + (1-) f(x1)
w here: xˆ x 0 (1 ) x1
(0,1).
and
f(x0)
T his im plies th a t: x 0 xˆ x1
Concavity is a weaker
condition than strict
concavity
Concavity allows linear segments
f ( x ) is conc a ve if:
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
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L et 0.5 and f ( x ) ln( x ) :
x1 2
x 2 20
xˆ 11
f ( x1 ) 0.69
f ( x2 ) 3
f ( xˆ ) 2.40
0.5 ln(2) 0.5 ln(20) 1.84
Concave Function: y = ln(x)
3.50
3.00
3.00
ln(x)
2.50
2.40
2.00
1.84
1.50
1.00
0.69
0.50
0.00
0
2
4
6
8
10
12 14
x
16
18
20
22
24
26
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Convexity
• Strict convexity and convexity
• Reverse direction of inequality f(x ) + (1-) f(x )
0
1
f ( x ) is st rictly co n v e x if :
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
f ( x ) is con v ex if:
x2
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
q2
q1
q0
x1
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Shape of production function
• Is this production function concave? Convex? Both?
f(x )
Concavity/convexity is usually
defined for some region of f(x).
convex
c o n c av e
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Quasi-concavity
• Production functions are also quasi-concave
f ( x ) is qu asi-c onc ave if:
f ( xˆ ) m in f ( x 0 ), f ( x1 )
f(x 1 )
w here: xˆ x 0 (1 ) x1
and
(0,1).
T his im plies that: x 0 xˆ x1
^
f(x)
f(x 0 )
x0
^x
x1
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Quasi-concavity
A quasi-concave function cannot have a
“U” shaped portion
f(x)
f(x 1 )
f(x 0 )
^
f(x)
x0
^x
x1
f(x) is not quasiconcave over its
whole domain.
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Strict quasi-concavity
• No linear segments (or “U” shaped portions)
Replace weak inequality with
strict inequality
f(x)
f ( x ) is quasi- concave if:
f ( xˆ ) m in f ( x 0 ), f ( x1 )
f(x 1 )
f ( x ) is strictly quasi-conca v e if:
f ( xˆ ) m in f ( x 0 ) , f ( x1 )
f(x 0 ) = f(x^ )
x0
^x
x1
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Quasi-convexity
• For quasi-convex function, change direction of inequality and
change “min” to “max”
f ( x ) is quas i-convex if:
f ( x ) is strictly quasi-conv ex if:
f ( xˆ ) m ax f ( x 0 ), f ( x1 )
f ( xˆ ) m ax f ( x 0 ), f ( x1 )
f(x 1 )
qu asic on vex c an’t
ex cee d thi s
f(x 0 )
qu asic on cav e can ’t
go be low her e
x0
x1
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Recap
concave
strictly concave
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
convex
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
strictly convex
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 )
quasi-concave
f ( xˆ ) m in f ( x 0 ), f ( x1 )
strictly quasi-concave
f ( xˆ ) m in f ( x 0 ), f ( x1 )
quasi-convex
f ( xˆ ) m ax f ( x 0 ), f ( x1 )
strictly quasi-convex
f ( xˆ ) m ax f ( x 0 ), f ( x1 )
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Principal minors
• Best illustrated with an example:
For the 3 3 m atrix:
f 11
H f 21
f 31
f 12
f 22
f 32
f 13
f 23
f 33
T he leading principal m i nors are:
H 1 f 11 ;
f 11
H2
f 21
f 12
;
f 22
f 11
H 3 f 21
f 31
f 12
f 22
f 32
f 13
f 23 H
f 33
This pattern applies for square matrices of any dimension
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Using Hessian matrix
• Hessian matrix
• Contains all 2nd partial derivatives
For the function y f ( x1 , x 2 ,
, x n ),
the H essian m atrix is:
the bordered H essian is :
f 11
f 21
H
f n1
BH
f 12
f 22
fn2
f1 n
f2n
f nn
0
f1
f2
f1
f 11
f 12
f2
f 21
f 22
fn
f n1
fn2
Remember Young’s theorem: fij = fji
fn
f1 n
f2n
f nn
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Strict Concavity
• The function 𝑦 = 𝑓 𝑥1 , 𝑥2 , … , 𝑥𝑛 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:
• 𝐇1 < 0
𝐇2 > 0
𝐇3 < 0
• Alternatively: −1 𝑛 ∙ 𝐇𝑛 > 0
• Diagonal elements of H are all < 0
• This is a sufficient condition for ND
etc.
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Recall SOSC for maximum
(2 variables)
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• 𝑓11 < 0 and 𝑓11 𝑓22 − 𝑓12 > 0
• Note that 𝑓22 < 0 is implied by the above
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• 𝑓11 𝑓22 − 𝑓12 > 0
• 𝑓11 𝑓22 > 𝑓12
• 𝑓22 < 𝑓12
2
• 𝑓22 < 𝑓12
2
𝑓11
2
𝑓11 < 0
Note sign reversal because 𝑓11 < 0.
Because 𝑓12 2 > 0 and 𝑓11 < 0
• The above meet the conditions for a negative definite matrix:
• −1 𝑛 ∙ 𝐇𝑛 > 0 (for all 𝑛)
2
• Let 𝐇1 = 𝑓11 < 0 and 𝐇2 = 𝑓11 𝑓22 − 𝑓12 > 0
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Concavity
• The function 𝑦 = 𝑓 𝑥1 , 𝑥2 , … , 𝑥𝑛 is concave if its Hessian
matrix is negative semi-definite (NSD).
• For NSD, replace strict inequality with weak inequality
• Determinants still alternate in sign:
• 𝐇1 ≤ 0
𝐇2 ≥ 0
𝐇3 ≤ 0
• Alternatively: −1 𝑛 ∙ 𝐇𝑛 ≥ 0
• This is a necessary condition for NSD
etc.
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Strict Convexity
• The function 𝑦 = 𝑓 𝑥1 , 𝑥2 , … , 𝑥𝑛 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:
• 𝐇1 > 0
𝐇2 > 0
𝐇3 > 0
• Alternatively: +1 𝑛 ∙ 𝐇𝑛 > 0
• Diagonal elements of H are > 0
• This is a sufficient condition for PD
etc.
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Convexity
• The function 𝑦 = 𝑓 𝑥1 , 𝑥2 , … , 𝑥𝑛 is convex if its Hessian matrix
is positive semi-definite (PSD).
• For PSD, replace strict inequality with weak inequality.
• Determinants all non-negative:
• 𝐇1 ≥ 0
𝐇2 ≥ 0
𝐇3 ≥ 0
• Alternatively: +1 𝑛 ∙ 𝐇𝑛 ≥ 0
• This is a necessary condition for PSD
etc.
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Quasi-concavity
• The function 𝑦 = 𝑓 𝑥1 , 𝑥2 , … , 𝑥𝑛 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 𝑦 = 𝑓 𝑥1, 𝑥2
bordered H essian:
0
B H f1
f 2
f1
f 11
f 12
2 bordered principal m i nors
f2
f 12
f 22
BH 1
BH 2
0
f1
f1
f 11
0
f1
f2
f1
f 11
f 12 f 1 f 22 2 f 1 f 2 f 12 f 2 f 11 0
f2
f 12
f 22
0 f 11 f 1 f 1 0
2
2
f ( x1 , x 2 ) is quasi-concave if:
B H 1 f1 0
2
B H 2 f 1 f 22 2 f 1 f 2 f 12 f 2 f 11 0
2
2
2
2
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Concavity and quasi-concavity
T heorem :
E very concave function is quasi-c oncave.
f ( x1 )
P roo f :
f ( x 0 ) f ( x1 ) (0, 1)
f ( x 0 ) (1 ) f ( x1 ) f ( x1 ) (1 ) f ( x1 )
L et f ( x 0 ) f ( x1 )
D efinition of concavity:
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 ) w here: xˆ x 0 (1 ) x1
If f ( x 0 ) f ( x1 ), then it m ust follow that, for c o ncave function s:
f ( xˆ ) f ( x 0 ) (1 ) f ( x1 ) f ( x1 ) ,
T he refo r e :
f ( xˆ ) f ( x1 ) .
f ( x 0 ) f ( x1 )
f ( x1 ) m in f ( x 0 ), f ( x1 )
f ( xˆ ) f ( x1 ) m i n f ( x 0 ), f ( x1 )
f ( xˆ ) m in f ( x 0 ), f ( x1 )
w hich is the definition of Q uasi-concavi ty . Q .E .D .
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Recap
Concavity H is NSD
(–1)n |Hn| 0
necessary
H is ND
Strict
concavity
Convexity H is PSD
(–1)n |Hn| > 0
sufficient
(+1)n |Hn| 0
necessary
H is PD
(+1)n |Hn| > 0
sufficient
BH is NSD
(–1)n |BHn| 0 necessary
BH is ND
(–1)n |BHn| > 0 sufficient
Strict
convexity
Quasiconcavity
Quasiconcavity