30C00300 Mathematical Methods for Economists (6 cr)
7) Important properties of functions:
homogeneity, homotheticity, convexity
and quasi-convexity
Abolfazl Keshvari
Ph.D. Aalto University School of Business
Slides originally by: Timo Kuosmanen
Updated by: Abolfazl Keshvari
1
Outline
•
•
•
•
•
Concave and convex functions
Quasiconcave and quasiconvex functions
Definition of homogeneity
Homogenizing a function
Definition of homotheticity
• Examples of concave and quasi-concave functions in
microeconomics
• Examples of homogeneous functions in microeconomics
2
Concavity and convexity
Definition: Let U be a convex subset of Rn. A real valued
function f : U  R is concave if
f ( x  (1   )y )   f ( x)  (1   ) f ( y )   [0,1], x, y  R n
A real valued function g : U  R is convex if
g ( x  (1   )y )   g (x)  (1   ) g ( y )   [0,1], x, y  R n
3
Concavity and convexity
Note 1: If function f is concave, then its additive inverse –f is
convex. Thus, all properties of concave functions carry over to
convex functions, and vice versa.
Note 2: Convexity of a function and convexity of a set should
not be confused. A notion of concave set does not exist.
Definition: set S is convex if
x, y  S   x  (1   )y  S   [0,1]
4
Concave function vs convex set
Recall that production function f and production possibility set
T are equivalent representations of technology:
y  f (x)  (x, y )  T
Proposition: Production function f is a concave function if
and only if production possibility set T is a convex set.
5
Concavity of cost function
Recall the cost function
C : R k m  R  ,
C (w, y )  min w  x ( x, y )  T
x
Theorem: The cost function is concave in input prices w:
C ( w1  (1   )w 2 , y )   C ( w1 , y )  (1   )C ( w 2 , y )
  [0,1], w1 , w 2  R k , y  R m
Note: concavity of C does not depend on T.
6
Concavity of profit function
Recall the profit function
 : R k m  R  ,
 (w, p)  max p  y  w  x (x, y )  T
x,y
Theorem: The profit function is concave in prices w, p
 ( w1  (1   )w 2 , p1  (1   )p 2 )   (w1 , p1 )  (1   ) ( w 2 , p 2 )
  [0,1], w1 , w 2  R k , p1 , p 2  R m
Note: concavity of π does not depend on T.
7
Concavity and convexity – univariate case
Recall that the following inequality holds for all convex
functions f : R  R
f ( x  h)  f ( x)  f ( x)h x, h  R
where f ’ is the first derivative (or the sub-derivative).
Analogously, the following inequality holds for all concave
functions
f ( x  h)  f ( x)  f ( x)h x, h  R
8
Derivative test for univariate functions
Assume function f is twice continuously differentiable.
Then f is convex if and only if
f ( x)  0 x  R
Analogously, f is concave if and only if
f ( x)  0 x  R
9
Concavity and convexity – multivariate case
The following inequality holds for all convex functions
f : Rn  R
f (x  h)  f (x)  f (x)  h x, h  R n
Analogously, the following inequality holds for all concave
functions
f (x  h)  f (x)  f (x)  h x, h  R n
Note: inequalities apply to the gradient vectors of differentiable
functions as well as all subgradients in the subdifferential.
10
Derivative test for multivariate functions
Assume function f is twice continuously differentiable. The Hessian
matrix is defined as
 f11 (x)
 f  (x)
 2 f (x)   21


 f n1 (x)
f12 (x)
f 22 (x)
f n2 (x)
f1n (x) 
f 2n (x) 


f nn (x) 
Function f is convex if and only if the Hessian matrix is positive
semidefinite for all x in the domain of f.
Analogously, f is concave if and only if the Hessian matrix is
negative semidefinite for all x in the domain of f.
Note: We will examine the specific criteria for positive/negative
semidefiniteness later in the context of optimization.
11
Quasi-concavity and quasi-convexity
Definition: a function f defined on a convex subset U of Rn is
quasiconcave if the upper level set
Ca  x  U f (x)  a
is a convex set for every real number a.
Similarly, f is quasiconvex if the lower level set
Ca  x  U f (x)  a
is a convex set for every real number a.
Note: Concavity implies quasiconcavity, but the converse does
not hold. Similarly, convexity implies quasiconvexity, but the
converse does not hold.
12
Quasiconcavity of production function
Definition: Input correspondence L is the mapping

L : R  2 , L( y )  x  R k (x, y )  T
m

R k

• The input set for a given output vector y, L(y), is the set of all
input vectors x that can produce output y.
Theorem: production function f is quasiconcave if and only if
the input sets L(y) are convex for all non-negative y.
13
Concavity and quasiconcavity
of the Cobb-Douglas function
Consider the Cobb-Douglas (CD) production function
f CD (x)   x11 x22 ...xkk
Proposition: The CD function is quasiconcave at all nonnegative parameter values α, β1, β2, …, βk.
Proposition: The CD function is concave if all parameters are
non-negative and
k

i 1
i
1
14
Homogeneity
Definition: For any scalar k, a real valued function f : R n  R
is homogeneous of degree k if
f ( x)   k f (x)   0, x  R n
Although k can be any scalar, in economics, we are typically
interested in cases where
k = -1
k = 0 (zero homogeneity)
k = 1 (linear homogeneity)
15
Constant returns to scale
Consider production function f : R k  R  . f exhibits constant
returns to scale if and only if it is homogeneous of degree 1:
f ( x)   f (x)   0, x  R n
Equivalently, the production possibility set T satisfies
T   T   0
16
Cobb-Douglas function
Consider the CD production (or utility) function
f CD (x)   x11 x22 ...xnn
Proposition: The f CD function is homogeneous of degree
n
k   i
i 1
17
Euler’s homogenous function theorem
Theorem 20.4. Let f be a continuously differentiable
homogeneous function of degree k. Then for all x
x f (x)  kf (x)
18
Homogeneity of cost function
Consider again the cost function
C : R k m  R  ,
C (w, y )  min w  x ( x, y )  T
x
Theorem: The cost function is homogeneous of degree one in
input prices w:
C ( w, y )   C ( w, y )   0, w  R k , y  R m
Interpretation: if the input prices are doubled, the cost doubles as
well.
19
Homogeneity of output distance function
Recall Shephard’s output distance function
DO (x, y)  inf   R (x, y /  )  T
Theorem: The output distance function is homogenous of
degree one in outputs y:
D O (x,  y )   D O (x, y )
20
Homogenizing a function
Homogeneity property allows us to estimate the distance
function from empirical data. Starting from
D O (x,  y )   D O (x, y )
choose one of the outputs, say output 1, and set   1 / y1
y
1 O
O
Then
D (x, )  D ( x, y )
y1
y1
Taking logs of both sides, we have
ln D O (x,
y
)   ln y1  ln D O ( x, y )
y1
ln y1   ln D O (x,
y
)  ln D O ( x, y )
y1
21
Homogenizing a function
Suppose the outputs are subject to random disturbance
according to
y  y * exp( )
such that
D O (x, y * )  1
Then ln DO (x, y )  
This yields the regression equation
ln y1   ln D O (x,
y
)
y1
Note: normalization by output 1 ensures that linear
homogeneity is satisfied.
22
Homogenizing a function
Suppose we want to estimated a CD function from data,
k
subject to the linear homogeneity constraint (CRS):  i  1
i 1
We can use the homogeneity property to normalize the inputs
by dividing all inputs one of the inputs, say input 1:
1
2
 x1   x2 
 xk 
y
      ...  
x1
 x1   x1 
 x1 
 y
 x2
ln    ln    2 ln 
 x1 
 x1
k

 xk 
  ...   k ln  

 x1 
Having estimated the parameters β2,…, βk, we use
k
1  1   i
i 2
23
Homothetic functions
Definition: A function v : R n  R is homothetic if it is a
monotone transformation of a homogeneous function,
that is, if there exist a monotonic increasing function
g : R   R and a homogeneous function u : R n  R
such that
v(x) g (u ( x)) x  R n
Note: the level sets of a homothetic function are radial
expansions and contractions of each other.
A homogeneous function is also homothetic, but the converse
24
does not hold.
Next week
Unconstrained optimization
• Simon & Blume, Ch. 17
25