Breaz_A_comparison. 109KB Jun 04 2011 12:08:27 AM

ACTA UNIVERSITATIS APULENSIS

A COMPARISON BETWEEN TWO INDICES
OBTAINED BY MDF
by
Nicoleta Breaz and Daniel Breaz

Abstract. A comparison between two indices obtained by FPM. It is well known that the price
index obtained by FPM (factors path method) depends on the path. In this paper, we consider
two particular paths namely the linear path and the exponential path. Then we make a
comparison between the two indices from an axiomatically theory point of view.

A statistical index can be understood either by the algorithm used for the calculation
of its values or by the related function proprieties.
For the index of integral variation of a variable z = z (t ) we have the formula

I zk / j =

z (k )
where j and k are two values of t. Unlike of this, for the index of the
z( j)

factorial variation related to a variable z (t ) = f ( x1 (t ), x 2 (t ),..., x n (t )) it have been
propose more methods of calculus.
In this paper we focused on the factors path method (MDF).
Definition 1. The factorial index of a variable z = z (t ) = f ( x1 (t ), x 2 (t ),...x n (t )) with

respect to the xi factor, measured in time interval ( j , k ) obtained by MDF method is
given by the formula:

(

Ι

k/ j
z / xi

f x´i ( x1 , x2 ,...xn )
dx
= exp ∫
(
)

f
x
x
x
,
,...
1
2
n
(Pj Pk )

) represents the curve described by the factors evolution between the
(1)

points Pj ( x1 ( j )...x n ( j )) , Pk ( x1 (k )...x n (k )) from R n that is parameterized by the
where Pj , Pk

equation x1 = x1 (t ),...x n = x n (t ).
Remark 2. The factorial index MDF depends on the factors path.
Remark 3. If we want to measure the variation with t (time, space or

economical category) of a variable z, that is the value of a panel with n goods

z (t ) = ∑ pi (t )qi (t ) ,
n

i =1

39

socio-

ACTA UNIVERSITATIS APULENSIS

p(t ) = ( p1 (t ), p 2 (t ),..., p n (t )), q(t ) = (q1 (t ), q 2 (t ),..., q n (t )) then the factorial index
with respect to p is called the price index. Here p and q are the prices and the
quantities vectors.
Definition 4. An index given by the formulas

Ι kz // pj = Ι vk // pj = ∏ Ι kz // pj i
n

Ι kz // pj i = exp ∫

Pj Pk

v ′pi
v

dpi = exp ∫

Pj Pk

i =1

∑pq
qi

n

i =1

or

Ι

k/ j
z/ p

= exp ∫

i

dpi = exp ∫

(2)
t1

t0

∑ q (t ) p (t )

n

i =1

i

∑ q (t ) p′ (t )

q1 (t ) pi′ (t )
i

dt

(2’)

i

n

t1

t0

i =1
n

i

1

∑ q (t ) p (t )
i =1

i

dt

(2’’)

i

is called the MDF price index.
Remark 5. The curve which links the two situations of price-quantity Pj , Pk is in

general parametrised with respect to parameter t ∈ [t 0 ,t1 ] , but it has an unknown
form.
Case of a linear path-definition of the linear MDF price index

xi (t ) = xi ( j ) + t [xi (k ) − xi ( j )] = xi ( j ) + t∆xi , i = 1, n, t ∈ [0,1] the index from (1)

In the particullar case of a linear path given by the parametrised equation
become:

Ι kz // xji = exp ∫

1

0

f x′ (t )∆xi

dt
f (t )

For the calculation of a price index we use the parametrisation:

 pi (t ) = pi ( j ) + t ( pi (k ) − pi ( j )) = pi ( j ) + t∆pi

qi (t ) = qi ( j ) + t (qi (k ) − qi ( j )) = qi ( j ) + t∆qi

with i = 1, n , t ∈ [0,1]

In 2n dimensional price-quantity space the situation can be represented as:

40

ACTA UNIVERSITATIS APULENSIS

q

Pk

C

Pj

p
Then the index from the definition 4 become:

[qi ( j ) + t∆qi ]∆pi
dt
0
∑ [qi ( j ) + t∆qi ]( pi ( j ) + t∆pi )

Ι kz // pj i = exp ∫

1

n

i =1

or

Ι

= exp ∫

1

k/ j
z/ p

0

since dpi = ∆pi dt .

∑ [q ( j ) + t∆q (t )]∆p

(3)

n

i =1

i

i

i

∑ [q ( j ) + t∆q ]( p ( j ) + t∆p )
n

i =1

i

i

i

dt

(3’)

i

 x (k ) 
 , t ∈ [0,1], i = 1, n , the index
After we use the parametrisation xi (t ) = xi ( j ) i
 xi ( j ) 
Case of an exponential path-definition of the exponential MDF price index
t

f x' (t )xi ( j ) ln

from definition 4 becomes

I zk // xji = exp ∫
1

0

xi (k )
xi ( k )
1
f x' (t )xi ( j ) xi (k ) ln I xki / j
xi ( j )
dt
dt = exp ∫
f (t )
f (t )
0

In case of the value of a panel v =

(4)

∑ p q , we have the following representation in
n

i =1

i

i

2n dimensional price-quantity space:

41

ACTA UNIVERSITATIS APULENSIS

q

Pk
C

Pj
t

 qi (k ) 
t
 = qi ( j ) ⋅ Ι qk i/ j
qi (t ) = qi ( j ) ⋅ 

 qi ( j ) 

t
 pi (k ) 

 = pi ( j ) ⋅ Ι kpi/ j
 pi (t ) = pi ( j ) ⋅ 
 pi ( j ) 


p

( )

With the equations

with i = 1, n , t ∈ [0,1]

( )

t

the MDF price index becomes :

 p (k )   p (k ) 
 q (k ) 
q i ( j ) ⋅  i  ⋅ pi ( j ) ⋅  i  ln i 
∑
1 i =1
 pi ( j )   p i ( j ) dt =
 qi ( j ) 
= exp ∫
t
0
n
 qi (k ) ⋅ p i (k ) 

qi ( j ) ⋅ pi ( j ) ⋅ 
∑
i =1
 qi ( j ) ⋅ pi ( j ) 
n

Ι vk // pj

= exp ∫

∑
n

1 i =1

0

t

t

 q (k ) ⋅ pi (k )   pi (k ) 
 ln

q i ( j ) pi ( j ) i
 qi ( j ) ⋅ pi ( j )   pi ( j )  dt
t

 q (k ) ⋅ pi (k ) 

qi ( j ) ⋅ pi ( j ) ⋅  i
∑
i =1
 qi ( j ) ⋅ pi ( j ) 
n

Ι vk // pj = ∏ Ι vk // pj
n

or

i =1

42

t

(5)

ACTA UNIVERSITATIS APULENSIS

 q (k ) ⋅ pi (k )   pi (k ) 
 ln

qi ( j ) p i ( j ) i
1
qi ( j ) ⋅ pi ( j )   pi ( j ) 

= exp ∫
dt
t
0
n
 qi (k ) ⋅ pi (k ) 

q i ( j ) ⋅ pi ( j ) ⋅ 
∑
i =1
 qi ( j ) ⋅ pi ( j ) 
t

where

Ι vk // pj i

(5')

Remark 6. Is obviously that the factors path is neither a linear nor an exponential one
except the particular cases. The real path is unknown. However, on small time
intervals we can suppose that the path is one of a linear or an exponential path. The
problem which arising here is which of these two path are more appropiate for the
calculus of a price index. We can partialy solve such a problem if we use the
axiomatical approach for an index. In this approach, the index has the following
definition:
Definition 7. Is called the statistical index, a function f : D → R which maps an
economical object set D into the real number set R and which satisfies a system of
economically relevant conditions. The form of these conditions depends on the
information which one wants to obtain from the particular index. In 1911, I. Fisher
gave an axioms list suitable for a price index. Later, these axioms are grouped in
systems that satisfy natural requirements of independence and consistency. Thus, these
systems can be used as definitions for different types of price index. Here we work
with three systems of axioms that are used as comparison tool for the linear MDF
price index and the exponential MDF price index. Since the axioms (properties) are
desirable for a price index we call a better index that index which satisfies more
axioms.
Definition 8.A function P : R+4+n → R+ + , (q j , p j , q k , p k ) → P (q j , p j , q k , p k ) is
called a price index if P satisfies the following four properties (axioms): monotonicity,
proportionality, dimensionality and comensurability.
Then P (q j , p j , q k , p k ) represents the value of price index on price-quantity
situation (q j , p j , q k , p k ) .
Definition 9.(Eichhorn-Voeller’s axioms system)(1978).
A function P : R+4+n → R+ + , (q j , p j , q k , p k ) → P (q j , p j , q k , p k ) is called a price
index if P satisfies the next five properties: monotonicity, dimensionality,
comensurability, identity and linear homogeneity.
Remark 10. In 1979 the definition given by Eichhorn and Voeller is modified in sense
that the proportionality substitutes the identity and is required a weak linear
homogeneity: P (q j , p j , q j , λp k ) = λP (q j , p j , q j , p k ), λ ∈ R+ +
Definition
11.
(Olt’s
axioms
system)(1995).
A
function
4n
j
j
k
k
j
j
k
k
P : R+ + → R+ + , (q , p , q , p ) → P(q , p , q , p ) is called a price index if P
satisfies the following four properties (axioms):dimensionality, comensurability,

43

ACTA UNIVERSITATIS APULENSIS

simetry and the mean value test. Next we state these mentioned properties and other
that are necesary for our goal.
Proportionality axiom. The index equals the individual price relatives when they
agree with each other. P (q j , p j , q k , λp j ) = λ .
Comensurability axiom. A change in the units of measurement of comodities (goods)
does not change the value of the function P.

P(

q1j

λ1

,...,

qnj

λn

, λ1 p1j ,...,λn pnj ,

q1k

λ1

,...,

qnk

λn

, λ1 p1k ,...,λn pnk ) = P(q j , p j , q k , p k )whereλi ∈ R++

Time reversal test. When the two situations are interchanged, the price index yields
the reciprocal value.

P(q j , p j , q k , p k ) =

1
.
P(q , p , q j , p j )
k

k

Linear homogeneity. The value of the function P changes by the factors λ if all
prices of the observed situation change λ -fold.

P (q j , p j , q k , λp k ) = λP(q j , p j , q k , p k ), λ ∈ R+ +

k
Monotonicity. The function P is strictly increasing with respect to p and strictly
j
decreasing with respect to p .

P(q j , p j , q k , p k ) > P(q j , p j , q k , ~
p k ), if , p k > ~
pk
P(q j , p j , q k , p k ) < P (q j , ~
p j , q k , p k ), if , p j > ~
pj

Dimensionality. A dimensional change in the unit of currency in which all prices are
measured does not change the value of the function P.

P ( q j , λp j , q k , λpk ) = P (q j , p j , q k , p k ), λ ∈ R+ +

Identity. The value of the function P equals one if all prices remain constant.

P(q j , p j , q k , p j ) = 1 .

Mean value test. The value of the price index can be represented as a convex
combination of the smallest and the biggest price relative.

 pk
 pk
pk 
p k1 
min  1j ,..., nj  ≤ P (q j , p j , q k , p k ) ≤ max  1j ,..., nj 
pn 
pn 
 p1
 p1
 pk 
 pk 
P ( q j , p j , q k , p k ) = λ min  ij  + (1 − λ ) max  ij 
i
 pi 
 pi 

or

44

ACTA UNIVERSITATIS APULENSIS

Simetry. The same permutation of the components of the four vectors does not change
the value of the index.

P ( q j , p j , q k , p k ) = P ( q~ j , ~
p j , q~ k , ~
pk ).

Theorem 12. The MDF price index I

k/ j
z/ p

= exp ∫

z 'p
z

Pj Pk

dp on the linear path is a

price index in sense of definition (8).
Proof:
We will proove that the index satisfies the monotonicity, the proportionality, the
dimensionality and the comensurability.
Let
the

n
 q = qi ( j ) + t (q i (k ) − qi ( j ))
z = ∑ pi q i , z = z (t ), t ∈ [ j , k ], and  i
, t ∈ [0,1]
i =1
 p i = pi ( j ) + t ( p i (k ) − pi ( j ))

parametrisation

of

the

linear

path

between

the

points

Pj (q ( j ), p ( j )), Pk (q(k ), p(k )) .

We get I zk // pji = exp

∫

1

0

[qi ( j ) + t∆q i ]∆pi

∑ (q ( j ) + t∆q )( p ( j ) + t∆p )
n

i =1

= ∏I

and I

i =1

1

k/ j
z / pi

∑
n

= exp ∫

n

k/ j
z/ p

i

i =1

i

i

dt

i

[qi ( j ) + t∆qi ]∆pi

∑ (q ( j ) + t∆q )( p ( j ) + t∆p )
n

0

i =1

i

i

i

dt

i

-dimensionality

I zk // pj (q ( j ), λp ( j ), q (k ), λp (k )) = I zk // pj (q ( j ), p ( j ), q (k ), p (k ))
I zk // pj (q ( j ), λp ( j ), q (k ), λp (k )) = exp ∫
1

0

= exp ∫

1

0

∑ [q ( j ) + t∆q ]∆p

∑ [q ( j ) + t∆q ]λ∆p
n

i =1

i

i

i

∑ (q ( j ) + t∆q )(λp ( j ) + tλ∆p )
n

i =1

i

i

i

i

n

i =1

i

i

i

∑ [q ( j ) + t∆q ][ p ( j ) + t∆p ]
n

i =1

i

i

i

dt = I zk // pj (q( j ), p( j ), q (k ), p (k ))

i

45

dt

ACTA UNIVERSITATIS APULENSIS

-proportionality

I zk // pj (q ( j ), p( j ), q(k ), λp( j )) = λ

∑ [q ( j ) + t∆q ] p ( j )(λ − 1)
n

I zk // pj (q( j ), p( j ), q(k ), λp( j )) = exp ∫

i =1

1

0

i

i

∑ [q ( j ) + t∆q ][ p ( j ) + tp ( j )(λ − 1)]

1

n

0

= exp ∫

i

i =1

i

i

i

dt =

i

λ −1
dt = exp ln[1 + t (λ − 1)] / 10 = exp ln λ = λ
1 + t (λ − 1)

-commensurability

I zk// pj (

λ1

q1 ( j )

,...,

λn

qn ( j )

, λ1 p1 ( j ),...., λn pn ( j ),

λ1

q1 (k )

= I zk// pj (q( j ), p( j ), q(k ), p(k )).
I zk// pj (....) = exp ∫

1

0

∑ λ [q ( j) + t∆q ]λ ∆p
n

i =1

= exp ∫

0

i

∑ λ [ p ( j) + t∆p ] ⋅ λ

1

i

i

∑[q ( j) + t∆q ]∆p

i

λn

q n (k )

, λ1 p1 (k ),...., λn pn (k )) =

dt =
[qi ( j ) + t∆qi ]
i

i

n

i =1

1

1

,....,

i

i

i

n

dt = I zk// pj (q( j ), p( j ), q(k ), p(k ))
∑[ pi ( j) + t∆pi ][qi ( j) + t∆qi ]
i =1

i

i

i

n

i =1

-monotonicity

I zk // pj (q( j ), p( j ), q(k ), p(k )) > I zk / /pj (q ( j ' ), p( j ' ), q(k ' ), p(k ' ))
'

a)

if q( j ) = q( j ' ), p( j ) = p ( j ' ), q(k ) = q(k ' ) and p (k ) ≥ p(k ' )
I zk // pj (q( j ), p( j ), q(k ), p (k )) < I zk / /pj (q( j ' ), p( j ' ), q(k ' ), p(k ' ))
'

b)

'

'

if q( j ) = q ( j ' ), p ( j ) ≥ p( j ' ), q(k ) = q (k ' ) and p (k ) = p(k ' )

46

ACTA UNIVERSITATIS APULENSIS

First we prove the part b) of the property

I

k /j
z/ p

= exp ∫

1

0

∑ [q ( j ) + t∆q ]∆p
n

i =1

∑( p ( j
n

i =1

i

i

i

i

) + t∆pi )(qi ( j ) + t∆qi )

dt

∑ [q ( j ) + t∆q ]∆p

Denoting

n

F ( p1 ( j ), p 2 ( j ),..., p n ( j )) =

i =1

i

i

i

∑ [ p ( j ) + t∆p ][q ( j ) + t∆q ]
n

i =1

i

i

i

i

∂F
< 0, ∀i = 1, n ,
∂pi ( j )

we obtain
for any t ∈ [0,1] .

From ( p1 ( j ), p 2 ( j ),..., p n ( j )) ≥ ( p1 ( j ' ), p 2 ( j ' ),..., p n ( j ' )) follows

F(( p1 ( j), p2 ( j),...,pn ( j))) ≤ F(( p1 ( j ' ), p2 ( j ),...,pn ( j))) ≤ F(( p1 ( j ' ), p2 ( j ' ),...,pn ( j))) ≤

≤ ... ≤ F (( p1 ( j ' ), p 2 ( j ' ),..., p n ( j ' )))

and now we apply the integrand with respect to t and the exponential function and we
have I zk // pj < I zk /'/pj ' .
a)
According [1], the MDF index satisfies the time reversal test independently
of the path, so we have:

I zk// pj (q( j), p( j), q(k ), p(k )) =

I

= I zk/'/pj ' (q( j), p( j), q(k ), p(k ' ))

j/k
z/ p

1
1
> k '/ j '
=
(q(k ), p(k ), q( j), p( j)) I z / p (q(k ), p(k' ), q( j) p( j)

where we use the hypothesis p(k ) ≥ p(k ' ) and the part b) of the monotonicity.
Theorem 13. The exponential MDF price index is a price index in sense of the
definition (8).
Proof:
The most part of these axioms are verified in [3]. Here we prove the monotonicity.
Let the function

47

ACTA UNIVERSITATIS APULENSIS

F ( p1 ( j ), p 2 ( j ),..., p n ( j ) ) =

p (k )
∑ [q ( j ) p ( j )] [q (k ) p (k )] ln p ( j )
1−t

n

i =1

i

i

i

i

∑ [q ( j ) p ( j )] [q (k ) p (k )]
1−t

n

i =1

We consider pi , i = 1, n ,

t

i

i

i

.

t

i

i

i

p (k )
p1 (k ) p 2 (k )
(6)
≤
≤ ... ≤ n
p1 ( j ) p 2 ( j )
pn ( j)

without loss the generality. Moreover we make the assumption that

 p (k ) p (k )
p n (k )
p (k ) 
≤ min  1 ' , 2 ' ,..., n ' 
pn ( j)
pn ( j ) 
 p1 ( j ) p 2 ( j )

(7)

For fixed p 2 ( j ), p3 ( j ),..., p n ( j ) , the function becomes a function of single variable

p1 ( j ) .


p (k) 
∑[q ( j) p ( j)] [q (k) p (k)] q ( j)[q ( j) p ( j)] [q (k) p (k)] (1− t )ln p ( j) −1
−t

n

Fp'1 ( j ) =

i =1

i

i

1−t

t

i

i

1

i

t

i

i

i




1−t
t
∑[qi ( j) pi ( j)] [qi (k) pi (k)] 
 i=1

n


[qi ( j) pi ( j)]−t [qi (k) pi (k)]t q1 ( j)[qi ( j) pi ( j)]1−t [qi (k) pi (k)]t (1− t) ln pi (k) 
∑
pi ( j) 
i =1

−
2

n

n
1−t
t
∑[qi ( j) pi ( j)] [qi (k) pi (k)] 
 i=1


1

1

2

Since the prices and the quantities at different moments are positive real numbers if


p (k ) 
p1 (k )
− 1 − (1 − t ) ln i
(1 − t ) ln
≤0
p
(
j
)
p
(
j
)
1
i



for any i = 1, n then we have F p'1 ( j ) ≤ 0 .

From (6) the above inequality holds because

48



ACTA UNIVERSITATIS APULENSIS

p1 (k ) p 2 (k ) p1 (k ) p3 (k )
p (k ) p n (k )
,
,…, 1
.
≤
≤
≤
p1 ( j ) p 2 ( j ) p1 ( j ) p 3 ( j )
p1 ( j ) p n ( j )
So the function F p1 ( j ) is monoton decreasing.

From the hypothesis we have p ( j ) ≥ p ( j ' ) , and further p1 ( j ) ≥ p1 ( j ' )

(

)

so it results F ( p1 ( j ), p 2 ( j ),..., p n ( j ) ) ≤ F p1 ( j ' ), p 2 ( j ),..., p n ( j ) .
Now we fix
respect to p 2 ( j ) .
Similarly if

p1 ( j ' ), p3 ( j ),..., p n ( j ) and prove that

F is decreasing with

p 2 (k ) p1 (k ) p 2 (k ) p i (k )
≤
, i = 2, n ,
≤
,
p 2 ( j ) p1 ( j ' ) p 2 ( j ) pi ( j )

then we have
F p' 2 ( j ) ≤ 0 .
But according to (6) and (7) the above relations hold.
So if p 2 ( j ) ≥ p 2 ( j ' ) we can write

(

) (

F ( p1 ( j), p2 ( j),..., pn ( j)) ≤ F p1 ( j ' ), p2 ( j),..., pn ( j) ≤ F p1 ( j ' ), p2 ( j ' ), p3 ( j),..., pn ( j)
.

We repeat this method and if p ( j ) ≥ p ( j ' ) then results

(

)

F ( p1 ( j ), p 2 ( j ),..., p n ( j ) ) ≤ F p1 ( j ' ), p 2 ( j ' ),..., p n ( j ' ) .

According to the properties of integrand and exponential function the second
part of monotonicity is proved. For the first part one can be repeat the proof for the
linear case.
Proposition 14. Any function P : R+4+n → R+ + , that satisfies the proportionality and
the monotonicity satisfies the mean value test, too.
Proof:
Let

 p (k )
p (k ) 
a = min  1
,..., n 
pn ( j) 
 p1 ( j )
and

49

)

ACTA UNIVERSITATIS APULENSIS

 p (k )
p (k ) 
b = max  1
,..., n  .
pn ( j) 
 p1 ( j )
If the price index satisfies the proportionality then results
a = P(q ( j ), p ( j ), q (k ), ap ( j ) ) ,and according to the monotonicity we have

a = P(q ( j ), p ( j ), q (k ), ap ( j ) ) ≤ P(q ( j ), p ( j ), q (k ), p (k ) ) since
ap ( j ) ≤ p(k ) .The last inequality means that
ap1 ( j ) ≤ p1 (k ) , ap 2 ( j ) ≤ p 2 (k ) ,…, ap n ( j ) ≤ p n (k ) ,which is true since

a≤

pi (k )
∀ i = 1, n .
pi ( j )

Similarly, we have b = P(q ( j ), p ( j ), q ( k ), bp ( j ) ) ≤ P(q ( j ), p ( j ), q ( k ), p ( k ) ) , so
the mean value test is verified.
Corollary 15. The MDF price index both on the linear and the exponential path
satisfies the mean value test.
Remark 16. It can be proved that the MDF linear factorial index does not satisfy the
linear homogeneity.
Proposition 17. The MDF exponential price index satisfies the linear homogeneity.

Proof: P (q ( j ), p ( j ), q ( k ), λp ( k ) ) = λP(q ( j ), p ( j ), q (k ), p (k ) ), λ ∈ R+ + .

 q (k )λpi (k )   λpi (k ) 

 ln
q ( j ) pi ( j ) i
1 ∑ i
qi ( j ) pi ( j )   pi ( j ) 
i =1

dt =
P(q( j ), p ( j ), q (k ), λp(k ) ) = exp ∫
t
n


q (k ) pi (k )λ
0

qi ( j ) pi ( j ) i
∑
q
j
p
j
(
)
(
)
i =1
i

 i
t

n

 q (k ) pi (k )  
p (k ) 

  ln λ + ln i
λ ∑ qi ( j ) pi ( j ) i
1
qi ( j ) pi ( j )  
pi ( j ) 
i =1

dt =
exp ∫
t
n


q
k
p
k
(
)
(
)
0
i

λt ∑ qi ( j ) pi ( j ) i
q
j
p
(
)
i =1
i ( j) 
 i
n

t

t

50

ACTA UNIVERSITATIS APULENSIS

t
n

 pi (k)  


q
k
p
k
(
)
(
)
i
i



q
j
p
j
(
)
(
)
ln
∑
i
i
 q ( j) p ( j)   p ( j)  
1
1
i =1
i
  i 
 i

=
dt
exp∫ lnλ +
exp
lnλdt⋅ P(q( j), p( j), q(k), p(k)) =
∫
t
n




q (k) pi (k)
0
0

qi ( j) pi ( j) i


∑
i =1
 qi ( j) pi ( j) 


λP(q( j), p( j),q(k), p(k)).

Proposition 18. Any function f : R+4+n → R+ + , that satisfies the proportionality
satisfies also the identity.
Proof:
We have P (q ( j ), p ( j ), q (k ), λp ( j )) = λ .

P(q( j ), p( j ), q(k ), p( j )) = P(q ( j ), p( j ), q(k ),1 ⋅ p( j )) = 1.

Theorem 19.The MDF exponential price index is a price index in sense of definition
(9).
Proof: The theorem is a consequence of (17) and (18) and the theorem (13).
Remark 20. The MDF linear price index is not a price index by the definition (9)
since does not satisfy the linear homogeneity.
Proposition 21. The MDF price index both on linear and exponential index satisfy the
symmetry.

∑
n

Proof: This will results from the form of the two indices inside we have

i =1

.

Theorem 22.The MDF price index (linear or exponential) is a price index in sense of
the definition (11).
Proof: The theorem results since we have (12), (13), (15) and (21).
In conclusion, the MDF linear price index satisfies less axioms than the MDF
exponential price index. The first index satisfies just two of the three systems given
here. From the three filter definitions (axioms systems) point of view, the MDF
exponential price index is more appropiate than MDF linear price index.

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ACTA UNIVERSITATIS APULENSIS

REFERENCES
1. I. Florea, I. Parpucea, A. Buiga, “Statistică descriptivă-teorie şi aplicaţii”, Ed.
Continental, Aisteda, Alba-Iulia,1998.
2. A. Vogt, J.Barta, “The Making of Tests for Index Number-Mathematical Methods
of
Descriptive Statistics”, Physica-Verlag, Heidelberg 1997.
3. N. Breaz,D. Breaz “ Dimensionalitate, proporţionalitate şi comensurabilitateaxiome ale indicelui preţurilor MDF, pe drum exponenţial” , Annales
Universitates Apulensis, Series Economica, Nr.1/2001.

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