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Demand Estimation

 Demand Estimation

Year

2005

2006

2007

2008

2009

2010

Advertising Outlay

120

130

110

100

170

140

Sales Proceeds
(in ‘000 units)

200

500

900

700

1200

800

Analyse the least square method of deterioration with the data given in the tablet.

Solution

Least Square Method

Let us now scrutinise in some detail the deterioration method of least squares. To commence with we have schemed the data of sales proceeds and relative advertising outlay of several levels in the diagram. A deterioration line that apparently fits the data has been framed.

It may be noted that from this diagram that a point on the deterioration line relating to the advertising outlay of a year fluctuates or deviates from the observed or actual values of sales proceeds. It is this vertical divergence that depicts the blunder in the approximated value of the sales proceeds.

It is the sum of the square of these blunders refers that is reduced by the choice of parameters ‘a and b’ in this deterioration technique among advertising outlay and sales proceeds can be presented as below:
                        ^    ^
Yt        =          a + bXt + et

Illustration 29

Year

Advertising Outlay
Xt

Sales Proceeds
Yt

2005

120

200

2006

130

500

2007

110

900

2008

100

700

2009

170

1200

2010

140

800

Calculate the mean values of actual advertising outlay and sales proceeds and also show the estimated value of the intercept and β Co-efficient to show the association among advertising outlay and sales proceeds. What would be the estimated equation of the demand function.

Solution

Year

Advertising Outlay
Xt

Sales Proceeds
Yt

       _
Xt - X

       _
Yt - Y

_
(Xt – X)^2

          _             _
(Xt – X) (Yt – Y)

2005

120

200

-8.33

-516.67

69.39

4303.86

2006

130

500

1.67

-216.67

2.79

-361.84

2007

110

900

-18.33

183.33

335.99

-3360.44

2008

100

700

-28.33

-16.67

802.59

472.26

2009

170

1200

41.67

483.33

1736.39

1805.56

2010

140

800

11.67

83.33

136.19

972.46

Σ               

770

4300

0

0

3083.34

3831.86

                                                      _         _            _        _
Mean is ascertained by computing X and Y where X and Y are calculated as follows:

Number of years (n) = 6;
                        _                                              _
                        X         =          ΣXt     and     Y         =          ΣYt
                                                  n                                              n
                                                                              ^  ^
Estimated Equation of the demand function Y = a + bXt

The association of the outlay and proceeds are calculated by the following formula:
^                                                                    ^
a is the estimated value of the intercept and b is the estimated β Co-efficient.

To ascertain these estimates, the formulae used are as follows:

^                       _  ^_
a          =          Y - bX

                        ^                  t = last year_            _
                        b          =             Σ (Xt – X) (Yt – Y)
                                             t = first year                                   

                                             t = last year_
                                                     Σ (Xt – X)^2
                                            t = first year

                     _
                     X                        =          ΣXt      =          770      =          128.33
                                                              n                      6
                     _
                     Y                        =          ΣYt      =          4300    =          716.67
                                                              n                        6
                                 _
                     Σ(Xt – X)=        0
                                 _
                     Σ(Yt – Y)=        0

            ^                    t = 2010_          _
            b          =         Σ (Xt – X) (Yt – Y)    =          3831.86           =          1.24
                                   t = 2005                                   3083.34
                                   
                                  t = 2010_
                                    Σ (Xt – X)^2
                                  t = 2005

            ^              _    ^_
            a          =  Y – bX                        =          716.67 – 1.24*128.33 =          557.54
            _
            Y = a + bXt    =          557.54 + 1.24X

Illustration 30

Estimate the Demand Price Association with the given details in the below tablet.

Restaurant

1

2

3

4

5

Price

72

76

52

64

56

Items Served per day

360

340

480

420

400

Solution

Let us assume the price of items to be P and Items served per day to be I.

The estimated demand function will be I = a – bP

Numbers 1 to 6 represents number of hotels
Pi is the price meal and Ii is the number of meals per day in a restaurant.

Restaurant

Price P in $

Meals per day (I)

       _
Pi – P

     _
Ii - I

   _
(Pi – P)^2

          _          _
(Pi – P) (Ii – I)

1

72

360

8

-40

16

320

2

76

340

12

-60

144

-720

3

52

480

-12

80

144

-960

4

64

420

0

20

0

0

5

56

400

-8

0

16

0

Σ

320

2000

0

0

320

-1360

            n          =          number of restaurants
            _
            P          =          ΣP        =          320      =          64
                                     n                       5
            _
            I           =          ΣI        =          2000    =          400
                                     n                        5
          _                                            _       
Σ(Pi – P)          =          0 and   Σ(Ii – I)           =          0
         _
Σ(Pi – P)^2      =          320
       _          _
(Pi – P) (Ii – I) =          -1360

            ^                                 _         _
            b          =          Σ (Pi – P) (Ii – I)        
                                                                                                       
                                               _
                                    Σ (Pi – P)^2

                     =             -1360   =          - 4.25
                                      320

^                      _   ^ _
a          =          I – b P

            =          400 – 4.25 * 64

            =          400 – 272        =          128
                                                                                                                      _
Therefore the estimated demand function will be as follows:       I = 128 – 4.25P

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