    # 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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