J a c e k Bia łek , A ndrzej C zajk o w sk i**
A P R O P O S I T I O N O F T H E S Y S T E M O F W E I G H T S F O R A G G R E G A T I V E I N D E X E S O N T H E E X A M P L E
O F T H E I N D E X O F W O R K E F F I C I E N C Y 1
ABSTRACT. In this paper we propose a construction of the aggregative index of work efficiency. The proposed system of weights is based on theoretical considerations over the situation in which the number of observations - coming from some of the con sidered enterprises - is insufficient. In the first part of this paper we consider a group of
N - enterprises and two periods of their activity. We propose a construction of index to
compare the periods taking into consideration the work efficiency. Next we consider the case when we intend to measure the average, one-period dynamics of the efficiency of work, having data from T >2 periods. We construct a new index which is a more general version of the previous index.
Key words: aggregative index, arithmetic mean, harmonic mean, work efficiency.
I. IN TR O D U CTIO N
The contemporary economy makes use o f lots o f statistical indexes to calcu late the dynamics o f prices, quantities, and in particular, work efficiency. For example: Laspeyres and Paasche indexes have been known since 19-th century (see Diewert (1976), Shell (1998)). Depending on the type o f an economic prob lem we may also use one o f the following indexes: Fisher ideal index (see Fisher (1972)), Tömqvist index (Tömqvist (1936)), Lexis index and other indexes (see Zając (1994), Domański (2001)). Indexes are also used to calculate national income (see Moutlon, Seskin (1999), Seskin, Parker (1998)). Balk (1995) wrote about axiomatic price index theory, Diewert (1978) showed that the Tömqvist index and Fisher ideal index approximate each other. But it is really hard to indi cate the best o f the statistical indexes (see Dumagan (2002)). The choice o f index
' Ph. D ., Chair o f Statistical Methods, University o f Łódź. " Ph. D ., Chair o f Statistical Methods, University o f Łódź.
depends on the information we want to get. Unfortunately, most o f indexes take into account no event from the inside o f the considered time interval. So if we want to consider also the omitted periods we should use a different formula.
A system o f weights for the index o f work efficiency (next we consider only this type o f indexes) should satisfy all economic postulates (see Gajek, Kałuszka (2000)). But the construction o f index, based on economic postulates, have to take into consideration the accidental noise of partial indexes. The partial in dexes o f work efficiency, based on small number o f observations, can lead to wrong conclusions about the global work efficiency. In this paper the proposed system o f weights is based on theoretical considerations over the situation in which the number o f observations - coming from some o f the considered enter prises - is insufficient. We are going to construct the aggregative index, which strongly limit the influence o f partial indexes o f work efficiency connected with the small number o f observations. In the first part o f this paper we consider a group o f N - enterprises and two periods o f their activity. Next, we consider the case when we want to measure the average, one-period dynamics o f the work efficiency, having data from T >2 periods.
II. CO N ST R U C TIO N O F INDEX IN T IIE CASE O F TW O PER IO D S Let us consider a group o f N — enterprises observed in discrete moments:
s (base period) and t (testing period). Let us signify:
W* - work efficiency o f i - th enterprise at time s , / e {1,2,...,TV}, W! - work efficiency o f i - th enterprise at time t , i e {1,2,..., A^},
WJ
I j (s , t ) = I j = j p j - partial index o f work efficiency o f i - th enterprise
comparing
periods s and t , where j e {1,2,....N } ,
n* - number o f employees o f i - th enterprise at time s , n\ - number o f employees of i - th enterprise at time t .
We are going to find the right vector o f weights {g| , g2,— Usi ng the above significations we can write the index of work efficiency as follows:
When we treat each I , as a random variable on some probabilistic space ( ii , F , P ) and each as a real number, we must treat also / as a random variable. We are interested in the differences among the calculated, noised by small number o f observations from some enterprises index / and its theoretical, expected value 702. We will calculate the influence o f accidental noise o f partial indexes on the global index 1 as:
d l = 1 - 1 0 = 1 - E l . (2) so we are going to minimize the value of dispersion o f random variable in (1):
a f = E ( d ľ f . (3)
Let us signify
Г,
-
№
Z S i /•= I We get from (1) and (4) that
/ = 2 > л . (5) M 70 = £ / = Z r / / r0, (6) i=i where 7 i 0 = £ / / ,ie { lA ...,J V } , (7) N Z * - 1- (8) i=i
Let us assume that í / / ( and d l j are independent random variables for each
i and j . Hence, we get the following consequence of this fact:
a 2j = E (d J)2 = ECl - I n)2 = - 70)2] = /=i
= 2 > , 2< (9)
I '- l
Now we have the optimization task where the aim function is
N
F = - ( i° )
i=i with the constraints specified in (8).
The essential and sufficient condition for the optimization task defined by (8) and (10) is formulated as follows:
d F d F
—— = —— , for each i and k. (11)
д / i d y k
The formula (11) leads to
= Y i v \ = - = У ы °1 , with = 1 (12) /=1
or equivalently
S i Ą = S i^ i2 = - = g ^ N- (13)
From (13) we get that
Under the additional assumption that variation coefficients o f work effi ciency at the moments s and t are similar, after some technical operations we get that
2 M 1 v
° 5) Using (14) and (15) we can calculate the weights as follows
2
f t — j--- p (16)
nj n\
From (16) we can get the following conclusion: each weight g : is a har monic mean o f number o f employees o f i - th enterprise at considered moments
í and t . It is easy to verify that if we assumed weights as an arithmetic mean o f these numbers:
- ni + ni
(17)
we would not solve the optimization task for function F under the constraints specified in (8). In our opinion, this fact recommends the definition g , over g , .
III. CO N ST R U C TIO N O F INDEX IN CASE O F M O R E THAN TW O PERIO DS
Let us consider a group o f N - enterprises observed in discrete moments: {1,2,.., Г } . We are going to measure the average, one-period work efficiency. As in the previous case we expect that the new index / strongly limits the influ ence o f partial indexes o f work efficiency connected with the small number o f observations. We propose a list o f postulates for this aggregative index:
Postulate 1
This postulate says that in case when partial indexes show no change of work efficiency o f given enterprises during the time interval, then the global index must absolutely inform us about no change o f work efficiency o f the group.
Postulate 2
The influence o f enterprises with relatively small number o f employees on the average one-period work efficiency is asymptotically negligible.
Postulate 3
If all partial indexes of work efficiency grew by about the same m% then the value o f global index / would increase by about the same m%.
Postulate 4
If we increased the number o f employees o f each enterprise by about the same m% the index I would not change.
Under the above assumptions and significations we propose the following index o f the average work efficiency on time interval:
N T / = Ё д £ « д о - / (о - 1 . о , (18) where T- 1 (19) and 2
The formulas (19) and (20) are the alternative proposition for results coming from the paper by Białek (2005). We have the following interpretation o f these coefficients: Д informs the producer how important is a share o f i - th enter prise taking into consideration the number o f employees and a " informs the producer how important is и - th moment in the case o f i — th enterprise.
We can also notice that the numerators and denominators o f formulas (19) and (20) are the harmonic mean o f right numbers o f employees.
It is easy to verify that
From (21) and (22) we get the index / as a weighted mean o f all
/ , ( / — 1,
t )
. Besides, the following theorems are true:Theorem 1
N
(21)
T
(22)
Index I , defined in (18), satisfies all postulates 1-4 (proof is omitted). Theorems 2
In the special case, when T = 2 (two periods), the formula / leads to I . It is an immediate consequence o f the fact that for T = 2 we have
a, = 1, (23) 2 (24) and finally / = £ д / (м = & , / ( = / . /-i /-i
Conclusion
Index / proposed in (18) is a more general version o f index /. Both in dexes have the required properties and strongly limit the influence of partial indexes of work efficiency connected with the small number o f observations.
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P R O P O Z Y C JA SYSTEM U W AG DLA IN D EK SÓ W A G R E G A T O W Y C H NA PR Z Y K Ł A D Z IE INDEKSU W Y D A JN O ŚCI PRA CY
W pracy zaproponow ano konstrukcję agregatow ego indeksu w yd ajn ości pracy. Pro p on ow an y system w ag w ynika z teoretycznych rozw ażań nad sytuacją, g d y liczba ob serw acji p och od zących od któregoś z analizow anych p rzedsiębiorstw je s t n iew ystarcza jąca.
W pierw szej c z ę śc i pracy rozw ażania d oty czą grupy N - p rzedsiębiorstw i dw óch ok resów ich funkcjonow ania. Podajem y konstrukcję indeksu dla porów nania tych okre só w z punktu w id zen ia w ydajności pracy. N astępnie rozw ażam y przypadek, gd y ch cem y zm ierzyć przeciętną, jed n o -o k reso w ą dynam ikę w ydajności pracy posiadając dane p o ch od zące z T >2 okresów . K onstruujem y n o w y indeks stan ow iący o g ó ln iejszą w ersję p oprzedniego indeksu.
