Numericals

Chapter 4

MARKET AND DEMAND ANALYSIS

1. We have to estimate the parameters a and b in the linear relationship
Yt = a + bT
Using the least squares method.
According to the least squares method the parameters are:

∑TY–nTY
b=
∑T2–nT2

a = Y – bT
The parameters are calculated below:
Calculation in the Least Squares Method
T Y TY T2
1 2,000 2,000 1
2 2,200 4,400 4
3 2,100 6,300 9
4 2,300 9,200 16
5 2,500 12,500 25
6 3,200 19,200 36
7 3,600 25,200 49
8 4,000 32,000 64
9 3,900 35,100 81
10 4,000 40,000 100
11 4,200 46,200 121
12 4,300 51,600 144
13 4,900 63,700 169
14 5,300 74,200 196
∑ T = 105 ∑ Y = 48,500 ∑ TY = 421,600 ∑ T 2 = 1,015
T = 7.5 Y = 3,464

∑TY–nTY 421,600 – 14 x 7.5 x 3,464
b= =
∑T2–nT2 1,015 – 14 x 7.5 x 7.5

57,880
= = 254
227.5
a = Y – bT
= 3,464 – 254 (7.5)
= 1,559

, Thus linear regression is
Y = 1,559 + 254 T
2. In general, in exponential smoothing the forecast for t + 1 is
Ft + 1 = Ft + α et

Where Ft + 1 = forecast for year ) α = smoothing parameter
et = error in the forecast for year t = St = Ft
F1 is given to be 2100 and α is given to be 0.3
The forecasts for periods 2 to 14 are calculated below:
Period t Data (St) Forecast Error Forecast for t + 1
(Ft) (et St =Ft) (Ft + 1 = Ft + α et)

1 2,000 2100.0 -100 F2 = 2100 + 0.3 (-100) = 2070
2 2,200 2070 130 F3 = 2070 + 0.3(130) = 2109
3 2,100 2109.0 -9 F4 = 2109 + 0.3 (-9) = 2111.7
4 2,300 2111.7 188.3 F5 = 2111.7 + 0.3(188.3) = 2168.19
5 2,500 2168.19 331.81 F6 = 2168.19 + 0.3(331.81) = 2267.7
6 3,200 2267.7 932.3 F7 = 2267.7 + 0.3(9332.3) = 2547.4
7 3,600 2547.4 1052.6 F8 = 2547.4 + 0.3(1052.6) = 2863.2
8 4,000 2863.2 1136.8 F9 = 2863.2 + 0.3(1136.8) = 3204.24
9 3,900 3204.24 695.76 F10 = 33204.24 + 0.3(695.76) = 3413.0
10 4,000 3413 587.0 F11 = 3413.0 + 0.3(587) = 3589.1
11 4,200 3589.1 610.9 F12 = 3589.1 + 0.3(610.9) = 3773.4
12 4,300 3772.4 527.6 F13 = 3772.4 + 0.3(527.6) = 3930.7
13 4,900 3930.7 969.3 F14 = 3930.7 + 0.3(969.3) = 4221.5


3. According to the moving average method
St + S t – 1 +…+ S t – n +1
Ft + 1 =
n
where Ft + 1 = forecast for the next period
St = sales for the current period
n = period over which averaging is done

Given n = 3, the forecasts for the period 4 to 14 are given below:

,Period t Data (St) Forecast Forecast for t + 1
(Ft) Ft + 1 = (St+ S t – 1 + S t – 2)/ 3

1 2,000
2 2,200
3 2,100 F4 = (2000 + 2200 + 2100)/3 = 2100
4 2,300 2100 F5 =(2200 + 2100 + 2300)/3= 2200
5 2,500 2200 F6 = (2100 + 2300 + 2500)/3 = 2300
6 3,200 2300 F7 = (2300 + 2500 + 3200)/3= 2667
7 3,600 2667 F8 = (2500 + 3200 + 3600)/3 = 3100
8 4,000 3100 F9 = (3200 + 3600 + 4000)/3 = 3600
9 3,900 3600 F10 = (3600 + 4000 + 3900)/3 = 3833
10 4,000 3833 F11 = (4000 + 3900 + 4000)/3 =3967
11 4,200 3967 F12 =(3900 + 4000 + 4200)/3 = 4033
12 4,300 4033 F13 = (4000 + 4200 + 4300)/3 = 4167
13 4,900 4167 F14 = (4200 + 4300 + 4900) = 4467
14 5,300 4467


4.
Q1 = 60
Q2 = 70
I1 = 1000
I2 = 1200
Q1 – Q2 I1 + I2
Income Elasticity of Demand E1 = x
I2 - I1 Q2 – Q1
E1 = Income Elasticity of Demand
Q1 = Quantity demanded in the base year
Q2 = Quantity demanded in the following year
I1 = Income level in base year
I2 = Income level in the following year

70 – 60 1000 + 1200
E1 = x
1200 – 1000 70 + 60

22000
E1 = = 0.846
26000

, 5.
P1 = Rs.40
P2 = Rs.50
Q1 = 1,00,000
Q2 = 95,000
Q2 – Q1 P1 + P2
Price Elasticity of Demand = Ep = x
P2 –P1 Q2 + Q1
P1 , Q1 = Price per unit and quantity demanded in the base year
P2, Q2 = Price per unit and quantity demanded in the following year
Ep = Price Elasticity of Demand

95000 - 100000 40 + 50
Ep = x
50 - 40 95000 + 100000

- 45
Ep = = - 0.0231
1950