MARKETING (UNR471)
Dr. Mohamed Sameh
RESPONSE MODEL PHENOMENA
P1: Through Origin P2: Linear
Y Y
X X
P3: Decreasing Returns
P4: Saturation
(concave) —
Q
Y Y
X X
PHENOMENA
P5: Increasing Returns P6: S-shape
(convex)
Y Y
X X
P7: Threshold P8: Super-saturation
Y Y
X X
Aggregate Response Models:
Linear Model
Y = a + bX
• Handles Linear/through origin (P1, P2).
• Handles Saturation and threshold (P4, P7) (in ranges)
Aggregate Response Models:
Power Series Model
Y = a + b X + c X2+ d X3 + e X4 + …
• fits well within the range of the data but will.
• behaves badly (becoming unbounded) outside the data range.
• may be designed to handle phenomena P1, P2, P3, P5, P6, and P8
Aggregate Response Models:
Fractional Root Model
c
Y = a + bX
• When c=1/2 the model is called the square root model.
• When c= –1 it is called the reciprocal model.
• c can be interpreted as elasticity when a = 0.
• handles P1, P2, P3, P4, and P5.
Aggregate Response Models:
Semi-log Model
Y = a+ b ln(X)
• Handles situations in which constant increase in marketing result in constant
absolute increases in sales.
• handles P3 and P7.
Aggregate Response Models:
Exponential Model
Y = ae bx; x > 0
• Is widely used as a price-response function for b<0 (i.e., increasing returns to
decreases in price).
• handles phenomena P5 and, if b is negative, P4
Aggregate Response Models:
Modified Exponential Model
Y = a(1- e – bx)+ c
• has an upper bound at a+c and a lower bound of c
• shows decreasing returns to scale.
• handles phenomena P3 and P4 and
• can accommodate P1 when c=0
Aggregate Response Models:
Logistic Model
𝑎𝑎
𝑌𝑌 = −(𝑏𝑏+𝑐𝑐𝑐𝑐)
+ 𝑑𝑑
1 + 𝑒𝑒
• the most common algorithm to handle S- Shaped models.
• has a saturation level at a+d .
• has a region of increasing returns followed by decreasing return.
• Handles phenomena P4 and P6
Aggregate Response Models:
Adbudg Function
𝑋𝑋 𝑐𝑐
𝑌𝑌 = 𝑏𝑏 + (𝑎𝑎– 𝑏𝑏)
𝑑𝑑 + 𝑋𝑋 𝑐𝑐
• S-shaped for c>1 and concave for 0<c<1.
• bounded between b (lower bound) and a (upper bound).
• handles phenomena P1, P3, P4, and P6
Aggregate Response Models:
Multiple Instruments
• Additive model for handling multiple marketing instruments
Y = af (X1) + bg (X2)
Easy to estimate using linear regression.
Aggregate Response Models:
Multiple Instruments cont’d
• Multiplicative model for handling multiple marketing instruments
Y = aXb1 X2c
b and c are elasticities.
Widely used in marketing.
Can be estimated by linear regression.
Regression
The observed value yi corresponding to xi and the value α+βxi on the
regression line y = a+ bx.
CISE301_Topic4
METHOD OF LEAST SQUARES
We want to find a and b to minimize :
n
Φ ( a , b) = ∑ ( a + bxi − yi ) 2
i =1
How do we obtain a and b to minimize : Φ ( a, b) ?
CISE301_Topic4
DETERMINE THE UNKNOWNS
Necessary condition for the minimum :
∂ Φ ( a, b)
=0
∂a
∂ Φ ( a, b)
=0
∂b
DETERMINING THE UNKNOWNS
∂ Φ ( a , b) n
= ∑ 2(a + bxi − yi ) = 0
∂a i =1
∂ Φ ( a , b) n
= ∑ 2(a + bxi − yi )xi = 0
∂b i =1
Estimation
After some simple algebraic rearranging, we obtain:
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )
𝛽𝛽 = (slope)
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖2 − (∑ 𝑥𝑥𝑖𝑖 )2
𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽 𝑥𝑥̅𝑛𝑛 (intercept)
EXAMPLE
• A company wants to launch a new product. They want the marketing
department to make a proper estimation of the advertisement budget. Using
historical sales data of similar products and the accompanying advertisement
campaign sizes, estimate the budget if the company wants to sell at least 40000
units of the product.
Budget(1000 $) 11 12 11 15 8 10 11 12 17 11
Sales (1000 units) 25 33 22 41 18 28 32 24 53 26
FİRST STEP:
� 𝑥𝑥𝑖𝑖 = 118
� 𝑥𝑥𝑖𝑖2 = 1450
� 𝑦𝑦𝑖𝑖 = 302
� 𝑦𝑦𝑖𝑖2 = 10072
� 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 = 3779
SUM OF SQUARES:
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )
𝛽𝛽 = 2 2
= 3.74
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )
𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽𝑥𝑥̅𝑛𝑛 = −13.93
Therefore, 𝑦𝑦� = −13.93 + 3.74𝑥𝑥
SUM OF SQUARES:
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )(∑ 𝑦𝑦𝑖𝑖 )
𝛽𝛽 = 2 2
= 3.74
𝑛𝑛 ∑ 𝑥𝑥𝑖𝑖 − (∑ 𝑥𝑥𝑖𝑖 )
𝛼𝛼 = 𝑦𝑦�𝑛𝑛 − 𝛽𝛽𝑥𝑥̅𝑛𝑛 = −13.93
Therefore, 𝑦𝑦� = −13.93 + 3.74𝑥𝑥
LINEARIZATION OF NONLINEAR BEHAVIOR
Linear regression is useful to represent a linear relationship.
•If the relation is nonlinear either another technique can be used or the data can be
transformed so that linear regression can still be used.
The latter technique is frequently used to fit the following nonlinear equations to a set of
data.
• Exponential equation
• Power equation
• Saturation-growth rate equation
LINEARIZATION OF NONLINEAR BEHAVIOR
lnY = ln A + BX
Y = Ae BX
y= a + bx
LINEARIZATION OF NONLINEAR BEHAVIOR
•Calculate a= 6.25 and b= 0.841. Straight line is ln y = 6.25 + 0.841 x
•Switch back to the original equation. A= ea= 518, B= b= 0.841.
•Therefore, the exponential equation is y = 518 e0.841x.
LINEARIZATION OF NONLINEAR BEHAVIOR
logY = log A + B logX
Y = AX B
y= a + bx
