Download Homogeneity

AAE 635 Discussions
09/11/2006
Homogeneous and Homothetic functions,
Monotonic Transformation, Level Curves and Discounting
Homogeneity
A function f ( x1 ,...xn ) is said to be homogenous of degree r (HODr) if multiplication of each of
its independent variables by a constant t will alter the value of the function by the proportion tr .
I.e. f (tx1 ,...txn )  t r f ( x1 ,...xn ) , or f (tx)  t r f ( x) .
Note: r may be negative.
Properties of homogeneous functions:
Now take the derivatives of the above equation with respect to t and xi. These yield respectively:
f (txi )
xi  rt r 1 f ( x)
(1) 
txi
i
f
f ( x)
t  tr
(2)
txi
xi
f ( x)
x i  rf ( x) . I.e. Equation (1) at t=1.
1. Euler’s Theorem: If f(x) is HODr, then 
x i
i
2. The derivative of a function that is HODr is HOD(r-1). This is simply the interpretation of (2),
f
f ( x )
 t r 1
which may be rewritten as:
.
txi
xi
Exercise1: 1.Determine whether or not the following functions is homogeneous. 2. If so, of what
degree? 3. Verify the Euler’s Theorem for the homogenous functions.
(1) f ( x)  x 2
(2) f ( L, K )  L K 1
x1 x2 2
 2 x1 x3
(3) f ( x1 , x2 , x3 ) 
x3
(4) f ( x, y)  x3  xy  y 3
(x  x )
(5) f ( x1 , x2 )  21 2 2
x1  2 x2
Homotheticity
A homothetic function is a monotonic transformation of a homogeneous function. Let
y  f ( x1 ,...xn ) be HODr, and let z  F ( y ) , where F ( y ) is a monotonic transformation of y. The
function z ( x1 ,...xn ) is called a homothetic function.
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(Monotonic transformation: A function g(x), g: XX is a positive monotonic transformation if
it is strictly increasing at all values of x. It is called a negative monotonic transformation if it is
strictly decreasing at all values of x. Positive monotonic transformations preserve the ordering of
elements of X. So, if g(x) is a positive monotonic transformation, and x1>x2, then g(x1)> g(x2).
Negative monotonic transformations reverse the ordering of the elements of X. So, if g(x) is a
negative monotonic transformation, and x1>x2, then g(x1)< g(x2).)
Homogeneity and homotheticity:
1. A homogeneous function is always a homothetic function.
2. A homothetic function may not be a homogeneous function.
Exercise 2: Verify that function y(a, b)  Aa b is a homogeneous function, but its monotonic
transformation F ( y(a, b))  e y ( a ,b ) is not homogeneous.
Level Curves
Consider a function y  f ( x1 , x2 ) , the set of points ( x1 , x2 ) needed to reach a given level y is
called a level curve (or level set).
In production theory, the level curves of the production function are often called isoquants, while
in consumption theory the level curves of the utility function are usually called indifference
curves.
For simplicity, we assume X2, f(x)C1, and the level curves of f(x) corresponding to any real
value k can be expressed as the function x2(x1). (Note: All the arguments presented in this
handout follow through without the above restrictions under fairly general conditions too tedious
and notation intensive to worry about for now. The goal today is to develop a somewhat
rigorous intuition.)
The slope of a level curve:
1. The calculation:
(1) Direct Approach: Solve for x2 ( x1 ) , and then calculate x2 ' ( x1 ) .
(2) Indirect Approach:
The slope of the level set x2(x1) is given by:
dx 2 ( x1 )
f x1 , x 2  x1

dx1
f x1 , x 2  x 2
Proof: Start at a point (x1,x2). Now move along the level set through (x1,x2) to a nearby point on
the same level set (x1+dx1,x2+dx2). Note that:
f  x1 , x 2 
f x1 , x 2 
f(x1+dx1,x2+dx2)-f(x1,x2)  f(x1,x2)+
dx1 
dx 2 - f(x1,x2)
x1
x 2
f  x1 , x 2 
f x1 , x 2 
=
dx1 
dx 2
x1
x 2
However, as both points are on the same level set, we have f(x1+dx1,x2+dx2)-f(x1,x2) = 0.
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Putting these two equations together we have:
dx 2
f x1 , x 2  x1
.

dx1
f x1 , x 2  x 2
f  x1 , x 2 
f x1 , x 2 
dx1 
dx 2 = 0, or:
x1
x 2
Exercise3: Using two approaches to calculate the slope of the level curve for function y=f(x1,x2)
= x12 x 2 when y=k, and find out whether the answers are the same.
2. The slopes of the level curves of a homogenous function are unchanged along any ray through
the origin.
Proof: The slope of the level set of a function at point (tx1, tx2):
f  tx1 , tx2  tx1
f  x1 , x2  x1
dx2 ( x1 )
t r 1 f  x1 , x2  x1

  r 1

dx1 x (tx ,tx )
f  tx1 , tx2  tx2
t f  x1 , x2  x2
f  x1 , x2  x2
1
2
3. The slopes of the level curves of a homothetic function are unchanged along any ray through
F
F '( y ) f1
f
 1 
the origin too. We see that slope of the function z level curve   1  
F2
F '( y ) f 2
f2
slope of the function f level curve.
Some Financial Mathematics
Discounting:
Given a nominal periodic interest rate compounded k times equals to i ( k ) , the present value (PV)
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of $1 due in t periods 
i(k )
(1  )k t
k
Note: In Jean-Paul’s math review, “a periodic interest rate” refers to a nominal periodic interest
rate.
(Accumulation Function: Imagine a fund growing at interest. It would be very convenient to
have a function representing the amount in the fund at any time t. The function a(t ) is defined as
the accumulated value (AV) of the fund at time t of an initial investment of $1 at time 0. a(t ) is
called “Accumulation function”.
a(t )  a(t  1)
Effective rate of interest: it 
a(t  1)
Simple interest: Under simple interest, the accumulation function is linear. a (t )  1  it
Compound interest: Under compound interest, previous interest earns interest. a(t )  (1  i)t
Nominal rate of interest: In general, if i ( k ) is the nominal annual rate compounded k times a year,
i(k )
i(k ) k 1
) .)
is the effective rate for an kth of a year and the AV of $1 in one year is (1 
k
k
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Execise4: Two banks offer the following rates:
Bank#1 – An effective annual rate of interest of 6%.
Bank#2 – A nominal annual rate of interest of 6% compounded monthly.
Which bank would you rather deposit your money in?
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