Indici statistici - Hummer9999 · 2010. 9. 27. · • pentru factorul calitativ: ∑ ∑ = 0 0 ( )...

INDICI INDIVIDUALI Indicii individuali VH� FDOFXOHD] � FD� LQGLFL� VLPSOL� IRORVLQG� GDWHOH� vQUHJLVWUDWH� SHQWUX� ILHFDUH�

YDULDELO �OD�QLYHOXO�XQLW LL�GH�REVHUYDUH�IRORVLW � • Pentru fenomenul complex y:

00

11

0

1

0/1 fx

fx

y

yi y == ,

unde:

y - fenomenul complex, de tipul fxy ⋅= x - factorul de tip calitativ f - factorul de tip cantitativ

• Pentru factorul calitativ x:

0

1

0/1 x

xi x =

Indici statistici

• Pentru factorul cantitativ f:

0

1

0/1 f

fi f =

&HL�WUHL�LQGLFL�YHULILF �vQWRWGHDXQD�UHOD LD�GH�VLVWHP�

fxy iii 0/10/10/1 ⋅= 5HOD LL�XWLOH�

( ) 1000/1%0/1 +=yy ri

( ) 1000/1%0/1 +=xx ri

( ) 1000/1%0/1 +=ff ri ,

unde:

yr 0/1 -�ULWPXO�GH�GLQDPLF �SHQWUX�IHQRPHQXO�FRPSOH[�y

xr 0/1 - ritmuO�GH�GLQDPLF �SHQWUX�IDFWRUXO�FDOLWDWLY�x

fr 0/1 -�ULWPXO�GH�GLQDPLF �SHQWUX�IDFWRUXO�FDOLWDWLY�f

INDICI DE GRUP Indicii de grup VH�FDOFXOHD] �OD�QLYHOXO�JUXSHL�VDX�SH�vQWUHJXO�DQVDPEOX� Sisteme de ponderare folosite la construirea indicilor de grup factoriali a) Sistemul de ponderare propus de E. Laspeyres, la care ponderile folosite sunt cele din

SHULRDGD�GH�ED] � • pentru factorul cantitativ:

∑∑=

00

10)(0/1 fx

fxI fy

• pentru factorul calitativ:

∑∑=

00

01)(0/1 fx

fxI xy

b) Sistemul de ponderare propus de H. Paasche, la care ponderile utilizate sunt cele din

SHULRDGD�FXUHQW � • pentru factorul cantitativ:

∑∑=

01

11)(0/1 fx

fxI fy


∑∑=

10

11)(0/1 fx

fxI xy

c) Sistemul propus de Edgeworth�VH�ED]HD] �SH�FXPXODUHD�FDQWLW LORU�GLQ�SHULRDGD�GH�ED] �FX�

FHOH�GLQ�SHULRDGD�FXUHQW � úL� IRORVLUHD�DFHVWRUD�FD�SRQGHUH� OD�P VXUDUHD�YDULD LHL� UHODWLYH�D�SUH XULORU��3UH XO�ILLQG�R�YDULDELO �FDOLWDWLY ��UHOD LD�JHQHUDO �GLQ�FDUH�UH]XOW �LQGLFHOH�HVWH�

∑∑

++

=)(

)(

010

011)(0/1 ffx

ffxI x

d) Sistemul de ponderare propus de I. Fisher presupune calcularea indicelui de grup al

SUH XULORU� FD� R� PHGLH� JHRPHWULF � D� FHORU� GRL� LQGLFL� DJUHJD L�� GH� WLS� /DVSH\UHV� úL� GH� WLS�Paasche.

Generalizând,


∑∑

∑∑

⋅=10

11

00

01)(

01 fx

fx

fx

fxI xy

• pentru factorul cantitativ:

∑∑

∑∑

⋅=01

11

00

10)(

01 fx

fx

fx

fxI fy

0RGDOLW L�GH�FDOFXO�DO�LQGLFLORU�GH�JUXS

∑∑

=00

11),(0/1

fx

fxI

fxy pentru fenomenul complex

∑∑

=00

10)(0/1

fx

fxI fy pentru factorul cantitativ

10

11)(0/1 ∑

∑=

fx

fxI

xy pentru factorul calitativ

Indicii de gruS� IDFWRULDOL� FDOFXOD L� FX� VLVWHPH� GLIHULWH� GH� SRQGHUDUH�� YHULILF � UHOD LD� GH� VLVWHP��

úL�DQXPH�

( ) ( ))(

0/1

)(

0/1

),(0/1

xy

P

fy

L

fxy III ⋅=

0RGLILFDUHD� DEVROXW � VH� FDOFXOHD] �� vQ� JHQHUDO�� FD� GLIHUHQ � vQWUH� QXP U WRUXO� úL� QXPLWRUXO�

indicelui:

∑∑ −=∆ 0011),( fxfxfxy pentru fenomenul complex

∑∑ −=∆ 0010)( fxfxfy pentru factorul cantitativ

∑∑ −=∆ 1011)( fxfxxy pentru factorul calitativ ùL�vQ�FD]XO�PRGLILF ULORU�DEVROXWH�VH�YHULILF �UHOD LD�GH�VLVWHP�

)()(),( xyfyfxy ∆+∆=∆ 3H�ED]D�DFHVWHL�UHOD LL��VH�FDOFXOHD] �FRQWULEX LD�UHODWLY �D�ILHF UXL�IDFWRU�OD�PRGLILFDUHD�DEVROXW ��

pe total:

100),(

)()( ⋅

∆

∆=fxy

fyfyK , respectiv 100

),(

)()( ⋅

∆

∆=fxy

xyxyK

Exemplificare pentru fond de salarii (FS)� FD� IHQRPHQ� FRPSOH[� FX� GRL� IDFWRUL� GH� LQIOXHQ ��

salariul mediu ( S ��úL�QXP UXO�GH�VDODULD L��7��

∑∑

=00

11),(0/1

TS

TSI

TSFS

∑∑

=00

10)(0/1

TS

TSI

TFS

10

11)(0/1 ∑

∑=

TS

TSI

SFS

)(

0/1)(

0/1),(

0/1SFSTFSTSFS III ⋅=

∑∑ −=∆ 0011),( TSTSTSFS

∑∑ −=∆ 0010)( TSTSTFS

∑∑ −=∆ 1011)( TSTSSFS

)()(),( SFSTFSTSFS ∆+∆=∆

100),(

)()( ⋅

∆

∆=TSFS

TFSTFSK

100),(

)()( ⋅

∆

∆=TSFS

SFSSFSK

,QGLFL�GH�JUXS�FDOFXOD L�FD�PHGLH�GH�LQGLFL�LQGLYLGXDOL� • pentru fenomenul complex:

∑∑ ⋅

=00

000/1),(0/1

fx

fxiI

y

fxy sau

∑

∑=

11

0/1

11),(

0/1 1fx

i

fxI

y

fxy

• pentru factorul cantitativ:

∑∑ ⋅

=00

000/1)(0/1

fx

fxiI

f

fy


1

11

0/1

11)(0/1

∑

∑=

fxi

fxI

x

xy

([HPSOLILFDUH�SHQWUX�LQGLFLL�YDORULL��YROXPXOXL�IL]LF�úL�DO�SUH XULORU:

∑∑ ⋅

=00

000/1),(0/1

qp

qpiI

vqpv

sau

∑∑

=

11

0/1

11),(0/1 1

qpi

qpI

v

qpv

∑∑ ⋅

=00

000/1)(0/1

qp

qpiI

q

qv

1

11

0/1

11)(0/1

∑∑

⋅=

qpi

qpI

p

pv

,QGLFLL�GH�JUXS�FDOFXOD L�FD�UDSRUW�D�GRX �PHGLL • SHQWUX�IHQRPHQXO�FRPSOH[��LQGLFH�FX�VWUXFWXU �YDULDELO ��

( ) ( )

100:

100:

*%00

*%11

0

00

1

11

0

1),( * ∑∑∑

∑∑

∑===

fxfx

f

fx

f

fx

x

xI fxxSV

∑∑

==*

00

*11

0

1),( *

fx

fx

x

xI fxxSV , unde ∑

=i

ii

f

ff *

• SHQWUX�IDFWRUXO�FDQWLWDWLY��LQGLFHOH�YDULD LHL�VWUXFWXULL��

∑∑

∑∑

==0

00

1

10

0

*)( :

*

f

fx

f

fx

x

xI fxVS sau ∑

∑==

*00

*10

0

*)( *

fx

fx

x

xI fxSV

• SHQWUX�IDFWRUXO�FDOLWDWLY��LQGLFH�FX�VWUXFWXU �IL[ ��

∑∑

∑∑

==1

10

1

11

*

1)( :f

fx

f

fx

x

xI xxSF sau ∑

∑==

*10

*11

*

1)(

fx

fx

x

xI xxSF

,QGLFHOH�FX�VWUXFWXU �YDULDELO ��FX�VWUXFWXU �IL[ �úL�DO�YDULD LHL�VWUXFWXULL� IRUPHD] �vPSUHXQ �XQ�

sistem de indici:

)()(),( ** fxVS

xxSF

fxxSV III ⋅=

MoGLILFDUHD� DEVROXW � D� PHGLHL� VXE� LQIOXHQ D� XQXLD� VDX� DOWXLD� GLQ� IDFWRUL� VH� FDOFXOHD] � FD�

GLIHUHQ �GLQWUH�QXP U WRUXO�úL�QXPLWRUXO�LQGLFHOXL�IDFWRULDO�FRUHVSXQ] WRU��5HOD LLOH�GH�FDOFXO�YRU�IL�

*1*10*

111

10

1

11)( xxfxfxf

fx

f

fxxx −=−=−=∆ ∑∑∑

∑∑

∑

0*

00*

10

0

00

1

10)( **

xxfxfxf

fx

f

fxfx −=−=−=∆ ∑∑∑

∑∑

∑

ùL� vQ� DFHVW� FD]�� PRGLILFDUHD� PHGLHL� HVWH� HJDO � FX� VXPD� PRGLILF ULORU� GDWRUDWH� IDFWRULORU� GH�

LQIOXHQ ��UHVSHFWLY�

)()(),( ** fxxxfxx ∆+∆=∆ ([HPSOLILFDUH�SHQWUX�LQGLFLL�SURGXFWLYLW LL�PXQFLL

∑∑

∑∑

∑∑

∑∑

===0

00

1

10

0

00

1

11

0

1),( ::*

T

qp

T

qp

T

Tw

T

Tw

w

wI TwwSV , unde ∑

=i

ii

T

TT *

∑∑

∑∑

==0

00

1

10

0

*)(

:*

T

Tw

T

Tw

w

wI

Tw

VS

∑∑

∑∑

==1

10

1

11

*

1)( :T

Tw

T

Tw

w

wI wwSF

)()(),( ** TwVS

wwSF

TwwSV III ⋅=

01

0

00

1

11*),( wwT

Tw

T

TwTww −=−=∆

∑∑

∑∑

*1

1

10

1

11)( wwT

Tw

T

Twww −=−=∆

∑∑

∑∑

0*

0

00

1

10*)( wwT

Tw

T

TwTw −=−=∆

∑∑

∑∑

)()(),( ** wwTwTww ∆+∆=∆

'(6&20381(5($�3(�)$&725,�$�9$5,$ ,(,�818,�)(120(1�&203/(;

Metoda indicilor

ÌQ�WHRULD�úL�SUDFWLFD�VWDWLVWLF �GHVFRPSXQHUHD�LQGLFHOXL�JHQHUDO�vQ�SURGXVXO�LQGLFLORU�IDFWRULDOL�VH�QXPHúWH� GHVFRPSXQHUH� JHRPHWULF �� LDU� VHSDUDUHD� PRGLILF ULL� DEVROXWH� WRWDOH� vQ� VXPD� PRGLILF ULORU�DEVROXWH��GDWRUDWH�IDFWRULORU��HVWH�GHQXPLW �GHVFRPSXQHUH�DQDOLWLF .

MetRGD�VXEVWLWX LHL�vQ�ODQ

3UHVXSXQH�DSOLFDUHD�XUP WRDUHORU�UHJXOL� • LQGLFHOH� LQIOXHQ HL� SULPXOXL� IDFWRU�� GH� UHJXO � FHO� FDQWLWDWLY�� VH� FRQVWUXLHúWH� IRORVLQG� GUHSW�

SRQGHUH�FHDODOW �VDX�FHOHODOWH�YDULDELOH�OD�QLYHOXO�SHULRDGHL�GH�ED] � • XQ�IDFWRU�RGDW �VXEVWLWXLW�U PkQH�GUHSW�SRQGHUH�OD�QLYHOXO�SHULRDGHL�FXUHQWH�SH�WRW�SDUFXUVXO�

GHVFRPSXQHULL�SHQWUX�FHLODO L�LQGLFL�IDFWRULDOL� ,QGLFLL�IDFWRULDOL�úL�PRGLILF ULOH�DEVROXWH�VH�FDOFXOHD] �SH�ED]D�UHOD LLORU� Varianta I (se substituie mai întâi factorul cantitativ)

;

00010)(

00

10)(

01

∑∑∑∑∑

∆=−=∆

=

ffy

fy

xfxfx

fx

fxI

∑∑∑∑∑

∆=−=∆

=

xxy

xy

ffxfx

fx

fxI

11011)(

10

11)(

01;

Varianta II (se substituie mai întâi factorul calitativ)

∑∑∑∑∑

∆=−=∆

=

xxy

xy

ffxfx

fx

fxI

00001)(

00

01)(

01;

∑∑∑∑∑

∆=−=∆

=

ffy

fy

xfxfx

fx

fxI

10111)(

01

11)(

01;

0HWRGD�LQIOXHQ HORU�L]RODWH�D�IDFWRULORU��PHWRGD�UHVWXOXL�QHGHVFRPSXV� Pentru cazul a doi facWRUL�� LQGLFLL� IDFWRULDOL� VH�RE LQ�DSOLFkQG�VLVWHPXO�GH�SRQGHUDUH�SURSXV�GH�

/DVSH\UHV��úL�DQXPH�

( ) ∑∑

=00

10)(

fx

fxI fyL

, respectiv ( ) ∑

∑=

00

01)(

fx

fxI xyL

LDU�PRGLILF ULOH�DEVROXWH�VH�FDOFXOHD] �SH�ED]D�UHOD LLORU�

∑∑∑ ∆=−=∆ ffy xfxfx 00010)(

∑∑∑ ∆=−=∆ xxy ffxfx 00001)( )RORVLQG� DFHODúL� VLVWHP�GH�SRQGHUDUH�SHQWUX� DPELL� IDFWRUL� QX� VH�PDL�YHULILF � UHOD LD�GH� VLVWHP�

GHRDUHFH�U PkQH�XQ�³UHVW�QHGHVFRPSXV´�FDUH�DUDW �LQIOXHQ D�VLPXOWDQ �D�DPELORU�IDFWRUL�

∑∑

∑∑

=∩

00

01

10

11)( :fx

fx

fx

fxI fxy

( ) ( ) ∑∑∑∑∑ ∆∆=−−−=∆ ∩ fxfxy fxfxfxfx 00011011)(

ModificarHD�DEVROXW �SH�WRWDO�

∑∑∑ ∆∆+∆+∆=∆ fxxffxy fx 00),( 5HSDUWL]DUHD�UHVWXOXL�QHGHVFRPSXV�SH�IDFWRUL�GH�LQIOXHQ �VH�SRDWH�IDFH�vQ�XUP WRDUHOH�YDULDQWH� a) VH�DWULEXLH�LQWHJUDO�XQXLD�GLQ�IDFWRUL��VLWXD LH�FDUH�FRQGXFH�OD�SURFHGHXO�VXEVWLWX LHL�vQ�ODQ �� b) se repartizHD] �vQ�PRG�HJDO�SH�IDFWRUL� c) VH� UHSDUWL]HD] �SURSRU LRQDO�FX� LQIOXHQ HOH� L]RODWH�DOH� IDFWRULORU� (kx, kf)�� FDUH� VH�GHWHUPLQ �

astfel:

∑∑∑

∆+∆∆

=xf

xx

fx

fk

00

0, respectiv

∑∑∑

∆+∆∆

=xf

ff

fx

xk

00

0

Sporul total al variabilei y care revine factorului f:

fxfffy kx ∆∆+∆=∆ ∑ ∑0)( Sporul total al variabilei y care revine factorului x:

fxxxxy kf ∆∆+∆=∆ ∑ ∑0)( &RQWULEX LD�UHODWLY �D�IDFWRULORU�OD�IRUPDUHD�VSRUXOXL�WRWDO�

100 respectiv , 100),(

)(

),(

)(

⋅∆∆⋅

∆∆

fxy

xy

fxy

fy

PROBLEME REZOLVATE

1. Într-R�VRFLHWDWH�FRPHUFLDO �V-au înregistUDW�XUP WRDUHOH�GDWH�FX�SULYLUH�OD�WUHL�P UIXUL��$��%�úL�&��

vQ�GRX �OXQL�FRQVHFXWLYH��YH]L�WDEHOXO��

6LWXD LD�FDQWLW LORU�úL�SUH XULORU�GH�YkQ]DUH�SHQWUX�WUHL�SURGXVH�SULQFLSDOH

Tabelul 7.1.

&DQWLW L�YkQGXWH 3UH XO�XQLWDU� (mii lei/U.M.) 0 UIXULOH

U/M

perioada GH�ED]

perioada FXUHQW

perioada GH�ED]

perioada FXUHQW

A B 1 2 3 4 I II III

buc. buc. Kg

500 800 300

600 1000

200

20 12 30

30 20 60

Total - - - - - Se cere: 1. V �VH�FDOFXOH]H�LQGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU�SHQWUX�FHOH�WUHL�

P UIXUL� 2. V � VH� FDOFXOH]H� LQGLFLL� GH� JUXS� DL� YDORULL�� YROXPXOXL� IL]LF� úL� DO� SUH XULORU� úL� V � VH� YHULILFH�

UHOD LD�GLQWUH�HL� 3. V � VH� FDOFXOH]H�� SH� WRWDO�� PRGLILF ULOH� DEVROXWH� SHQWUX� YDORDUH�� SUH XUL� úL� YROXP� IL]LF��

specificându-VH�FDUH�HVWH�SRQGHUHD�IDFWRULORU�GHWHUPLQDQ L� 4. V �VH�YHULILFH�HJDOLWDWHD�GLQWUH�LQGLFLL�DJUHJD L�úL�LQGLFLL�GH�JUXS�FDOFXOD L�FD�PHGLH�DULWPHWLF �

VDX�DUPRQLF �D�LQGLFLORU�LQGLYLGXDOL� 5. V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�YDORULL�vQ�IXQF LH�GH�YDORDUHD�FHORU�WUHL�P UIXUL�vQ�FHOH�GRX �

perioade.

Rezolvare 1. &DOFXOXO�LQGLFLORU�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU

00

11

0

1

0/1 qp

qp

v

vi v == ;

0

1

0/1 q

qi q = ;

0

10/1 p

pi p =

3HQWUX� LQGLFHOH� YDORULL� HVWH� QHFHVDU� V � VH� FDOFXOH]H� YDORULOH� GLQ� FHOH� GRX � SHULRDGH�

(vezi tabelul 7.2.).

&DOFXOXO�LQGLFLORU�LQGLYLGXDOL�úL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU

Tabelul 7.2.

&DQWLW LOH�YkQGXWH�

(U.M.) 3UH XO�XQLWDU (mii lei/ U.M.)

9DORDUHD�P UIXULORU (mii. lei)

0 UIXUL U.M. SHU��GH�ED]

(qo) per. crt.

(q1)

per.de ED] (po)

per. crt. (p1)

per. de ED] (q0p0)

perioada FXUHQW (q1p1)

în per. crt. în SUH XULOH�SHU�

GH�ED] ��

(q1p0) A B 1 2 3 4 5 6 7

I buc. 500 600 20 30 10000 18000 12000

II buc. 800 1000 12 20 9600 20000 12000

III Kg 300 200 30 60 9000 12000 6000

Total - - - - - 28600 50000 30000

=⋅

=⋅

=⋅

==

)IIImarfapentru(%67,66100300

200

)IImarfapentru(%125100800

1000

)Imarfapentru(%120100500

600

0

10/1 q

qi q

=⋅

=⋅

=⋅

==

)IIImarfapentru(%20010030

60

)IImarfapentru(%67,16610012

20


30

0

10/1 p

pi p

=⋅

=⋅

=⋅

==

)IIImarfapentru(%33,1331009000

12000

)IImarfapentru(%3,2081009600

20000


18000

00

110/1 qp

qpi v

2. Indicii de grup VH�SRW�FDOFXOD�vQ�DFHVW�FD]�VXE�IRUP �GH�LQGLFL�DJUHJD L�

%8,174sau748,128600

50000

00

11),(0/1 === ∑

∑qp

qpI qpv

%89,104sau0489,128600

30000

00

10)(0/1 === ∑

∑qp

qpI qv

%67,166sau6667,130000

50000

10

11)(0/1 === ∑

∑qp

qpI pv

5HOD LD�GLQWUH�FHL�WUHL�LQGLFL�

)(0/1

)(0/1

),(0/1

pvqvqpv III ⋅=

748,174816,16667,10489,1 ≅=⋅ 3. 0RGLILF ULOH�DEVROXWH�DOH�YDORULL��YROXPXOXL�IL]LF�úL�DOH�SUH XULORU:

2140028600500000011),(

0/1 =−=−=∆ ∑∑ qpqpqpv mii lei

140028600300000010)(

0/1 =−=−=∆ ∑∑ qpqpqv mii lei

2000030000500001011)(

0/1 =−=−=∆ ∑∑ qpqppv mii lei

21400200001400)(0/1)(

0/1),(

0/1 =+=∆+∆=∆pvqvqpv mii lei

Ponderea celor doi factori:

%54,610021400

1400100

),(0/1

)(0/1)( =⋅=⋅

∆

∆=

qpv

qv

qvK

%46,9310021400

2000100

),(0/1

)(0/1)( =⋅=⋅

∆

∆=

qpv

pv

pvK

4. Orice indice agregat se poate substitui printr-un indice mediu aritmetic sau armonic al indicilor individuali FRUHVSXQ] WRUL�

0489,128600

3,30000

28600

90006667,0960025,1100002,1

00

00)(0/1 ==

⋅+⋅+⋅==

∑∑

qp

qpiI

qqv

sau 104,89%

6667,16,29997

50000

2

12000

6667,1

20000

5,1

1800050000

111

11)(0/1 ==

++==

∑∑

qpi

qpI

p

pv sau 166,67%

5. 5HSUH]HQWDUHD�JUDILF �D�VWUXFWXULL�YDORULL�GLQ�FHOH�GRX �OXQL�FRQVHFXWLYH

Tabelul 7.3.

6WUXFWXUD�YDORULL�YkQ] ULORU�� 0 UIXUL

SHULRDGD�GH�ED] ��3%� perioada cXUHQW ��3&� A 1 2 I II III

35,0 33,6 31,4

36 40 24

Total 100% 100%

36,00

33,6040,00

31,4024,00

35,00

0

20

40

60

80

100

PB PC

%

III

II

I

)LJXUD��6WUXFWXUD�YDORULL�GHVIDFHULORU�GH�P UIXUL

2. Într-R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�XUP WRDUHOH�GDWH�FX�SULYLUH�OD�GHVIDFHULOH�GH�P UIXUi:

9DORDUHD�P UIXULORU�YkQGXWH�vQ�SHULRDGD�GH�ED] �úL�FXUHQW

Tabelul 7.4.

9DORDUHD�P UIXULORU� (mil. lei)

0 UIXUL

perioada GH�ED]

perioada FXUHQW

Modificarea volumului fizic

(%)

A 1 2 4 A B C

500 600 800

1500 900

1400

+7 + 10 - 20

Total 1900 3800

Se cere: 1. V �VH�FDOFXOH]H�LQGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL��SUH XULORU� 2. LQGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�DO�SUHWXULORU� 3. PRGLILF ULOH�DEVROXWH�DOH�YDORULL��YROXPXOXL�IL]LF�úL�DOH�SUH XULORU� 4. FRQWULEX LD�FHORU�GRL�IDFWRUL�DVXSUD�GLQDPLFLL�YDORULL�P UIXULORU�YkQGXWH� Rezolvare 1. Indicii individuali ai valorii se pot calcula pe baza datelor din tabel, iar indicii individuali ai

SUH XULORU�SH�ED]D�UHOD LHL�GLQWUH�FHL�WUHL�LQGLFL�

• indicii valorii�GHVIDFHULORU�GH�P UIXri:

=

=

=

==

C)marfa(pentru%175sau75,1800

1400

B)marfa(pentru%150sau5,1600

900

A)marfa(pentru%300sau3500

1500

00

110/1 qp

qpi v

• indicii volumului fizic

( )

=+−

=+

=+

=+=

)Cmarfapentru(%8010020

)Bmarfapentru(%11010010

)Amarfapentru(%1071007

100%0/1

qq ri

• LQGLFLL�SUH XULORU�VH�FDOFXOHD] �SH�ED]D�UHOD LHL�

pqv iii 0/10/10/1 ⋅= ,

de unde:

=

=

=

==

)Cmarfapentru(%75,218sau1875,28,0

75,1

)Bmarfapentru(%36,136sau3636,11,1

5,1

)Amarfapentru(%4,280sau804,207,1

3

0/1

0/10/1 q

vp

i

ii

2. &DOFXOXO�LQGLFLORU�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�DO�SUH XULORU

• indicele de grup al valorii�VH�RE LQH�GLUHFW�GLQ�GDWHOH�GLQ�WDEHO�

%200sau21900

3800

00

11),(0/1 === ∑

∑qp

qpI qpv

• indicele de grup al volumului fizic nu se poate calcula aici decât ca un indice mediu

DULWPHWLF� SRQGHUDW� FX� YDORDUHD� P UIXULORU� GLQ� SHULRDGD� GH� ED] �� 3Hntru aceasta se SRUQHúWH�GH�OD�UHOD LD�

9659,01900

1835

1900

8008,06001,150007,1

00

00)(0/1 ==

⋅+⋅+⋅=

⋅=

∑∑

qp

qpiI

qqv

• LQGLFHOH�GH�JUXS�DO�SUH XULORU�SRUQHúWH�FD�úL�vQ�FD]XO�LQGLFLORU�LQGLYLGXDOL�GH�OD�UHOD LD�

pqv III 0/10/10/1 ⋅= , de unde:

0706,29659,0

2

0/1

0/10/1 === q

vp

I

II sau 207,06%

3. 0RGLILF ULOH� DEVROXWH� DOH� YDORULL�� YROXPXOXL� IL]LF� úL� DOH� SUH XULORU� VH� FDOFXOHD] � SH� ED]D�

indicilor de grup respectivi:

1900190038000011),(

0/1 =−=−=∆ ∑∑ qpqpqpv mil. lei

65190018350000)(

0/1 −=−=−⋅=∆ ∑∑ qpqpi qqv mil. lei

)(0/1

)(0/1

),(0/1

pvqvqpv ∆+∆=∆ , de unde:

1965)65(1900)(0/1),(

0/1)(

0/1 =−−=∆−∆=∆qvqpvpv

mil. lei

4. &RQWULEX LD�FHOor doi factori

%42,31001900

65100

),(0/1

)(0/1)( −=⋅−=⋅

∆

∆=

qpv

qv

qvK

%42,1031001900

1965100

),(0/1

)(0/1)( =⋅=⋅

∆

∆=

qpv

pv

pvK

3. Se cunosc datele:

Tabelul 7.5.

9DORDUHD�vQFDVDW ��PLO�OHL� Categorii de

confort perioada de ED]

perioada FXUHQW

Modificarea SUH XULORU

(%)

A 1 2 4 1 2 3 4

1500 2000

400 300

2500 5500

520 380

+75 +125

- -25

Total 4200 8900

Se cere: 1. FDOFXOXO�LQGLFLORU�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU� 2. FDOFXOXO�LQGLFHOXL�GH�JUXS�DO�SUH XULORU��SUHFL]kQGX-VH�FH�IHO�GH�IRUPXO �VH�SRDWH�DSOLFD�úL�GH�

ce; 3. indicele de grup�DO�YDORULL�úL�DO�YROXPXOXL�IL]LF� 4. PRGLILF ULOH�DEVROXWH�DOH�YDORULL��YROXPXOXL�IL]LF�úL�SUH XULORU��JUDGXO�GH�LQIOXHQ �D�FHORU�GRL�

IDFWRUL�DVXSUD�GLQDPLFLL�vQFDV ULORU�

Rezolvare 1. &DOFXOXO�LQGLFLORU�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XUilor

• indicii individuali ai valorii:

=

=

=

=

==

)confortde4categoriapentru(%67,126sau2667,1300

380

)confortde3categoriapentru(%130sau3,1400

520

)confortde2categoriapentru(%275sau75,22000

5500

)confortde1categoriapentru(%67,166sau6667,11500

2500

00

110/1 qp

qpi v

• LQGLFHOH� SUH XULORU� VH� YD� FDOFXOD� SH� ED]D� UHOD LHL� GLQWUH� LQGLFH� úL� ULWP�� GHRDUHFH� vQ� WDEHOXO� �� vQ� FRORDQD� �� VH� G � SURFHQWXO� GH� PRGLILFDUH� D� SUH XULORU�� DGLF � ULWPXO� GH�modificare

=+−

=+

=+

=+

=+=

%75sau75,0100

10025

%100sau1100

1000

%225sau25,2100

100125

%175sau75,1100

10075

100

1000/1

qq ri

• indicele volumului fizic�VH�RE LQH�GLQ�UHOD LD�

pqv iii 0/10/10/1 ⋅= ,

de unde:

=

=

=

=

==

%89,168sau6889,175,0

2667,1

%130sau3,10,1

3,1

%2,122sau222,125,2

75,2

%2,95sau952,075,1

6667,1

0/1

0/10/1 p

vq

i

ii

2. &DOFXOXO�LQGLFHOXL�GH�JUXS�DO�SUH XULORU�VH�YD�IDFH�vQ�DFHVW�H[HPSOX�FD�PHGLH�DUPRQLF ��SH�

ED]D�UHOD LHL�

816,17,4899

8900

75,0

380

1

520

25,2

5500

75,1

2500

8900

111

11)(0/1 ==

+++=

⋅=

∑∑

qpi

qpI

p

pv sau 181,6%

1RW � Nu s-D� SXWXW� FDOFXOD� XQ� LQGLFH� DJUHJDW� DO� SUH XULORU�� GHRDUHFH� QX� VH� FXQRVF� VHSDUDW�

FDQWLW LOH�úL�SUH XULOH� 3. ,QGLFHOH�GH�JUXS�DO�YDORULL�úL�DO�YROXPXOXL�IL]LF:

119,24200

8900

00

11),(0/1 === ∑

∑qp

qpI qpv sau 211,9%

1667,1816,1

119,2

0/1

0/10/1 === p

vq

I

II sau 116,67%

4. 0RGLILF ULOH�DEVROXWH�DOH�YDORULL��YROXPXOXL�IL]LF�úL�DOH�SUH XULORU:

4700420089000011),(

0/1 =−=−=∆ ∑∑ qpqpqpv mil. lei

3,40007,489989001

1111)(

0/1 =−=−=∆ ∑∑ qpi

qpp

pv mil. lei

7,6993,40004700)(0/1),(

0/1)(

0/1 =−=∆−∆=∆pvqpvqv

mil. lei

&RQWULEX LD�FHORU�GRL�IDFWRUL

%9,141004700

7,699100

),(0/1

)(0/1)( =⋅=⋅

∆

∆=

qpv

qv

qvK

%1,851004700

3,4000100

),(0/1

)(0/1)( =⋅=⋅

∆

∆=

qpv

pvpvK

Deoarece )( pvK > )(qvK UH]XOW �F �vQ�SULQFLSDO�FUHúWHUHD�VXPHL�vQFDV ULORU�vQ�SHULRDGD�FXUHQW �ID �

GH�SHULRDGD�GH�ED] �FX��PLO��OHL�VH�GDWRUHD] �vQ�SULQFLSDO�FUHúWHULL�SUHWXULORU��úL�QXPDL��% pe VHDPD�FDQWLW LORU�

4. 3HQWUX�R�VRFLHWDWH�FRPHUFLDO �VH�FXQRVF�GDWHOH�

Tabelul 7.8.

0 UIXUL Valoarea

vQFDV ULORU�vQ�

PB (mil. lei)

Dinamica volumului fizic (%)

Modificarea SUH XULORU�

(%) 0 1 2 3

A 100 120 150 B 400 107 230 C 600 130 50 Total 1100 ... ...

Se cere: 1. LQGLFLL�LQGLYLGXDOL�DL�YDORULL�úL�DL�SUH XULORU� 2. LQGLFLL�GH�JUXS�DL�YDORULL��YROXPXOXL�IL]LF�úL�DO�SUH XULORU� 3. V �VH�GHVFRPSXQ �YDORDUHD�DEVROXW �SH�WRWDO

• IRORVLQG�PHWRGD�VXEVWLWX LHL�vQ�ODQ • folosind metoda restului nedescompus.

Rezolvare: 1. ,QGLFLL�LQGLYLGXDOL�DL�SUH XULORU�VH�GHWHUPLQ �XWLOL]kQG�UHOD LLOH�

( ) 10001%01 +=pp ri

De exemplu, pentru marfa „A”:

( ) %250100150%01 =+=pi

,QGLFLL�LQGLYLGXDOL�DL�YDORULL�VH�GHGXF�GLQ�UHOD LD�GH�VLVWHP�

pqv iii010101

⋅=

35,22,101

=⋅=vi pentru marfa „A”.

5H]XOWDWHOH�FDOFXOHORU�VXQW�SUH]HQWDWH�vQ�WDEHOXO��FRORDQHOH��úL��

Tabelul 7.9.

0 UIXUL

Valoarea vQFDV ULORU�

în PB

00 qp (mil. lei)

qi pi vi viqpqp ⋅= 0011 0010 qpiqp

q ⋅= qi

qpqp 1101 =

0 1 2 3 4 5 6 7 A 100 1,2 2.5 3 300 120 250 B 400 1,07 3,3 3,53 1412 428 1320 C 600 1,3 1,5 1,95 1170 780 900 Total 1100 1,2073 2,1702 2,62 2882 1328 2470

2. PeQWUX� FDOFXOXO� LQGLFLORU� GH� JUXS� VH� FDOFXOHD] � YDORDUHD� vQFDV ULORU� vQ� SHULRDGD� FXUHQW �( 11 qp ��úL�YDORDUHD�vQFDV ULORU�vQ�SUH XUL�FRPSDUDELOH�� 10 qp ��YH]L�FRORDQHOH��úL��

( ) 62,21100

2882

00

11,01

===∑∑

qp

qpI qpv sau 262%

( ) 2073,11100

1328

00

10

01===

∑∑

qp

qpI qv sau 120,73%

( ) 1702,21378

2882

10

11

01===

∑∑

qp

qpI pv sau 217,02%

3. a) 'HRDUHFH�SUH XULOH�DX�FUHVFXW�PDL�UDSLG�GHFkW�FDQWLW LOH��SHQWUX�GHVFRPSXQHUHD�SH�IDFWRUL�

D� PRGLILF ULL� DEVROXWH�� IRORVLQG� PHWRGD� VXEVWLWX LHL� vQ� ODQ , vom utiliza varianta I (se substituie mai întâi factorul cantitativ):

( ) 2280010 =−=∆ ∑∑ qpqpqv mil. lei

( ) 17821011 =−=∆ ∑∑ qpqppv mil. lei

0RGLILFDUHD�DEVROXW �SH�WRWDO�

( ) 17820011, =−=∆ ∑∑ qpqpqpv mil. lei

3H�ED]D�DFHVWHL� UHOD LL� VH�SRDWH�GHWHUPLQD�FRQWULEX LD� IDFWRULORU� OD�PRGLILFDUHD�DEVROXW �SH�total:

( )( )

( ) %79,12100, =⋅∆∆=

qpv

qvqvK

( )( )

( ) %21,87100, =⋅∆∆=

qpv

pvpvK

b) Metoda restului nedescompus 6H� GHWHUPLQ � PDL� vQWkL� LQIOXHQ D� L]RODW � D� IDFWRULORU� �YDORULOH� 10 qp sunt prezentate în tabelul 7.9., coloana 6).

( ) 2280010 =−=∆ ∑∑ qpqpqv ii mil. lei

( ) 13700001 =−=∆ ∑∑ qpqppv ii mil. lei

$SRL�GH�GHWHUPLQ �UHVWXO�QHGHVFRPSXV�

( ) ( ) 1841598178213702281782, =−=+−=∆−∆= v iiqpvRN mil lei ,

unde 1598=∆v ii 3HQWUX� UHSDUWL]DUHD� UHVWXOXL� QHGHVFRPSXV� SH� IDFWRUL� GH� LQIOXHQ � VH� SRDWH� XWLOL]D� XQD� GLQ�variantele: Varianta I: Se atribuie, în cazul nostru, tot restul nedescompus factorului calitativ:

( ) 228=∆ qv mil. lei

( ) ( ) 15541841370 =+=+∆=∆ RNpv iipv mil. lei

&RQWULEX LD�IDFWRULORU�HVWH�FHD�GH�OD�PHWRGD�VXEVWLWX LHL�vQ�ODQ .

Varianta a II-a: Se distribXLH��vQ�PRG�HJDO��UHVWXO�QHGHVFRPSXV�SH�FHL�GRL�IDFWRUL�GH�LQIOXHQ �

( ) ( ) 320922282

1 =+=+∆=∆ RNqv iiqv mil. lei

( ) ( ) 14629213702

1 =+=+∆=∆ RNpv iipv mil. lei

&RQWULEX LD�UHODWLY �D�IDFWRULORU�vQ�DFHVW�FD]�

( ) %96,17=qvK

( ) %04,82=pvK

Varianta a III-a: 6H� UHSDUWL]HD] � UHVWXO� QHGHVFRPSXV� SH� IDFWRULL� GH� LQIOXHQ � LQkQG� FRQW� GH� SRQGHUHD�IDFWRULORU�vQ�PRGLILFDUHD�GDWRUDW �LQIOXHQ HL�L]RODWH�D�DFHVWRUD�

( ) ( )( )

( ) 2568,254184%27,14228, =⋅+=⋅∆

∆+∆=∆ RN

qpvii

qviiqv

iiqv mil. lei

( ) ( )( )

( ) 7432,1527184%73,851370, =⋅+=⋅∆

∆+∆=∆ RN

qpvii

pviipv

iipv mil. lei

&RQWULEX LD�UHODWLY �D�IDFWRULORU�vQ�acest caz este:

( ) %25,14=qvK

( ) %75,85=pvK

2EVHUYD LH� Valorile cele mai realiste sunt cele de la varianta a III-a a metodei restului nedescompus deoarece PHWRGD� VXEVWLWX LHL� vQ� ODQ � DFFHQWXHD] � LQIOXHQ D� XOWLPXOXL� IDFWRU� VXEVWLWXLt, în FD]XO�QRVWUX�LQIOXHQ D�SUH XULORU�

5. 3HQWUX�GRL�DJHQ L�FDUH�GHVI úRDU �DFHHDúL�DFWLYLWDWH�VH�FXQRVF�GDWHOH�

Tabelul 7.10.

3URGXF LD��vQ�SUH XUL�FXUHQWH�PLO��OHL� 1U��VDODULD L Agent economic

PB PC PB PC Modificarea

SUH XULORU�� 0 1 2 3 4 5

A 260 300 23 17 +15 B 200 240 22 25 +20 Total 460 540 45 42

Se cere: 1. SURGXFWLYLWDWHD�PXQFLL�SH�ILHFDUH�DJHQW�HFRQRPLF�úL�SH�WRWDO�vQ�FHOH�GRX �SHULRDGH� 2. LQGLFLL�LQGLYLGXDOL�DL�SURGXFWLYLW LL�PXQFLL� 3. LQGLFLL�SURGXFWLYLW LL�PXQFLL�FX�VWUXFWXU �YDULDELO ��FX�VWUXFWXU �IL[ �úL�DO�YDULD LHL�VWUXFWXULL� 4. PRGLILFDUHD�DEVROXW �D�SURGXFWLYLW LL�PHGLL�D�PXQFLL�FX�HYLGHQ LHUHD�LQIOXHQ HL�IDFWRULORU� 5. PRGLILFDUHD�SURGXF LHL��SH�WRWDO��SH�VHDPD�PRGLILF ULL�QXP UXOXL�VDODULD LORU�

Rezolvare 1. Pentru calculul SURGXFWLYLW LL� PXQFLL� GLQ� SHULRDGD� FXUHQW � WUHEXLH� FDOFXODW � YDORDUHD�

SURGXF LHL� vQ� SUH XUL� FRPSDUDELOH� �FRQVWDQWH�� DGLF � vQ� SUH XULOH� SHULRDGHL� GH� ED] �

(pi

qpqp

01

1110 = ). Rezultatele calculului sunt prezentate în tabelul 7.11., coloana 2.

304,1123

260

0

000 === T

qpw �PLO��OHL�SHUVRDQ

0909,922

200

1

101 === T

qpw �PLO��OHL�SHUVRDQ

Tabelul 7.11.

9DORDUHD�SURGXF LHL�vQ�SUH XUL�

constante (mil. lei) Agent economic

PB PC 0

000

T

qpw =

1

101

T

qpw = 10 Tw

(mil. lei)

wi01

0 1 2 3 4 5 6 A 260 260,86956 11,304 15,345 192,168 1,3574 B 200 200 9,0909 8 227,2725 0,88 Total 460 460,86956 10,222 10,973 419,4405 1,0734

3H�WRWDO��DJUHJDUHD�YDORULORU�LQGLYLGXDOH�SRDWH�IL�UHDOL]DW �QXPDL�SULQ�LQWHUPHdiul mediei ( )w :

222,1045

460

0

00

0

000 ====

∑∑

∑∑

T

Tw

T

qpw �PLO��OHL��SHUVRDQ

973,1042

869,460

1

101 ===

∑∑

T

qpw �PLO��OHL��SHUVRDQ

2. ,QGLFLL�LQGLYLGXDOL�DL�SURGXFWLYLW LL�PXQFLL�VH�FDOFXOHD] �GXS �UHOD LD�

0

101 w

wi w = ,

úL�DQXPH�

3574,1304,11

345,1501

==wi sau 135,74% pentru agentul economic „A”

88,00909,9

801

==wi sau 88% pentru agentul economic „B”

3. ,QGLFLL�SURGXFWLYLW LL�PHGLL�D�PXQFLL�

a) FX�VWUXFWXU �YDULDELO

( )∑

∑∑

∑∑

∑∑

∑===

0

00

1

11

0

00

1

01

0

1, ::T

Tw

T

Tw

T

qp

T

pq

w

wI

Tgwwvs

( ) 0734,1

222,10

973,10, ==Tgww

vsI sau 107,34%

3HQWUX�FDOFXOXO� LQGLFLORU�GH�JUXS� IDFWRULDOL� WUHEXLH�FDOFXODW �SURGXFWLYLWDWHD�PXQFLL� *w IRORVLQG�XUP WRDUHD�UHOD LH�

∑∑

=1

10*

T

Tww

Valorile 10 Tw sunt prezentate în tabelul 7.11., coloana 5.

9867,942

4405,419* ==w

b) DO�YDULD LHL�VWUXFWXULL��GH�WLS�/DVSH\HUV�

( ) 977,0222,10

9877,9

0

* ===w

wI

Tgwvs sau 97,70%

5H]XOW � F � SURGXFWLYLWDWHD� PHGLH� D� PXQFLL� DU� IL� VF ]XW� FX� �� GDF � V-ar fi modificat QXPDL�VWUXFWXUD�VDODULD LORU�

c) FX�VWUXFWXU �IL[ ��GH�WLS�3DDVFKH�

( ) 0987,19867,9

973,10

*

1 ===w

wI wwSF sau 109,87%

'DF �V-DU�IL�PRGLILFDW�QXPDL�SURGXFWLYLWDWHD�PXQFLL�OD�QLYHOXO�ILHF UXL�DJHQW�HFRQRPLF��productivitatea medie a muncii ar fi crescut cu 9,87%. Pe ansamblu, productivitatea medie a muncii a crescut doar cu 7,34%, deoarece o parte GLQ� FUHúWHUHD� GDWRUDW � PRGLILF ULL� SURGXFWLYLW LL� PXQFLL� SDU LDOH� D� IRVW� FRPSHQVDW � GH�VF GHUHD�SURGXFWLYLW LL�PHGLL�D�PXQFLL�GDWRUDW �PXWD LLORU�vQ�VWUXFWXUD�VDODULD LORU�

4. 0RGLILFDUHD�DEVROXW �D�SURGXFWLYLW LL�PHGLL�D�PXQFLL��SH�WRWDO�

( ) 751,0222,10973,1001, =−=−=∆ ww

Tgww mil. lei / pers. = 751 mii lei / pers.

0RGLILFDUHD�DEVROXW �D�SURGXFWLYLW LL�PHGLL�SH�VHDPD�PXWD LLORU�vQ�VWUXFWXUD�VDODULD LORU�

( ) 235,0222,10987,90* −=−=−=∆ wwTgw mil. lei / pers. = - 235 mii lei / pers.

0RGLILFDUHD� DEVROXW � D� SURGXFWLYLW LL� PHGLL� SH� VHDPD� PRGLILF ULL� SURGXFWLYLW LL� PXQFLL�SDU LDOH��SH�ILHFDUH�DJHQW�HFRQRPLF��

( ) 986,0987,9973,10*1 =−=−=∆ wwww mil. lei = 986 mii lei

Verificare:

( ) ( ) ( ) 751986235, =+−=∆+∆=∆ wwTgwgww T mii lei

5. ( ) 9118,0460

4405,419

00

10 ===∑∑

TW

TWI TQ sau 91,18%

( ) 5595,400010 −=−=∆ ∑∑ TWTWTQ mil. lei

9DORDUHD�SURGXF LHL�SH�WRWDO�D�VF ]XW�FX��FHHD�FH�vQ�YDORDUHD�DEVROXW �UHSUH]LQW �R�VF GHUH�FX�–40,5595 mil. lei.

PROBLEME PROPUSE

1. 3HQWUX�GRX �SURGXVH�VH�FXQRVF�GDWHOH�

Tabelul 7.12.

9DORDUHD�YkQ] ULORU��PLO��OHL� Produs

PB PC

Modificarea SURFHQWXDO �D�

SUH XULORU 0 1 2 3

A 300 375 +20 B 600 900 +60

Se cere: 1. LQGLFLL�LQGLYLGXDOL�DL�YDORULL��YROXPXOXL�IL]LF�úL�DL�SUH XULORU� 2. PRGLILFDUHD�DEVROXW �úL�UHODWLY �D�YDORULL�GHVIDFHULORU�SH�WRWDO�úL�LQIOXHQ D�IDFWRULORU� 3. úWLLQG� F �QXP UXO�SHUVRQDOXOXL�� SH� WRWDO�� D� FUHVFXW� vQ�SHULRDGD�FXUHQW � ID �GH�SHULRDGD�GH�

ED] �FX��V �VH�GHWHUPLQH�PRGLILFDUHD�UHODWLY �D�SURGXFWLYLW LL�PXQFLL�

2. /D�R�VRFLHWDWH�FRPHUFLDO �V-au înregistrat datele:

Tabelul 7.13.

Structura valorii vâQ] ULORU�� Marfa

PB PC

Dinamica volumului fizic

(%)

Modificarea UHODWLY �D�

SUH XULORU�� 0 1 2 3 4

A 35,0 35,0 120 50 B 33,6 40,0 125 66,7 C 31,4 25,0 66,7 100

Se cere: 1. LQGLFLL�LQGLYLGXDOL�DL�SUH XULORU�úL�DL�YDORULL�YkQ] ULORU� 2. PRGLILFDUHD�UHODWLY �D�YDORULL��YROXPXOXL�IL]LF�úL�D�SUH XULORU��SH�WRWDO� 3. V �VH�UHSUH]LQWH�JUDILF�VWUXFWXUD�YDORULL�YkQ] ULORU�vQ�FHOH�GRX �SHULRDGH�

3. 3HQWUX�GRL�DJHQ L�HFRQRPLFL�VH�FXQRVF�GDWHOH�

Tabelul 7.14.

Salariul mediu (mii. lei/pers.) Structura�DQJDMD LORU�� Agent economic PB PC PB PC

0 1 2 3 4 A 2400 2880 50 45 B 2700 3200 50 55

Se cere:

1. dinamica salariului mediu pe fiecare agent economic; 2. VDODULXO�PHGLX�SH�WRWDO�vQ�FHOH�GRX �SHULRDGH� 3. LQGLFLL� GH� JUXS� DL� VDODULXOXL� PHGLX� FX� VWUXFWXU � YDULDELO �� FX� VWUXFWXU � IL[ � úL� DO� YDULD LHL�

structurii; 4. PRGLILF ULOH�DEVROXWH�FRUHVSXQ] WRDUH�SH�WRWDO�

4. Se cunosc datele:

Tabelul 7.15.

Produs Structura valorii

YkQ] ULORU�vQ�SHULRDGD�

FXUHQW ��

Dinamica volumului fizic (%)

Modificarea SUH XULORU�

(%) 0 1 2 3

A 35 120 50 B 40 125 66,7 C 25 66,7 100

Se cere:

1. LQGLFLL�LQGLYLGXDOL�DL�YDORULL�úL�DL�SUH XULORU� 2. LQGLFLL�GH�JUXS�SULYLQG�YDORDUHD��YROXPXO�IL]LF�úL�SUH XULOH� 3. úWLLQG�F �QXP UXO�SHUVRQDOXOXL�� SH� WRWDO�� V-D� UHGXV� vQ�SHULRDGD�FXUHQW � ID �GH�SHULRDGD de

ED] �FX��V �VH�GHWHUPLQH�PRGLILFDUHD�UHODWLY �D�SURGXFWLYLW LL�PXQFLL�

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