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Demand as a function of price variation
Let us define the sigmoid function:
This function is defined over 
, has values on 
and its derivative is:
Let 
with 
the price, 
the reference price and 
the price variation (uncorrelated from policy attributes). As the demand 
tends to 1 when the price is sufficiently low and to 0 when the price goes to infinity, we will write
with 
being the static component and 
being the dynamic component, which we will also refer to as the linear sensitivity. This is the equation that the demand module will fit to explain the observed conversion/retention rates.
The static model controls the value of the demand at the reference price, while the dynamic model relates to the slope of the conversion curve.
Note: When importing / exporting a dynamic model, the dynamic coefficients in the file integrate the minus sign in front of log(t) in the definition of d:
To sum it up:
In the app, positive dynamic coefficients correspond to decreasing demand with increasing price (what we expect).
In import and export files, negative dynamic coefficients correspond to decreasing demand with increasing price. You should therefore use the following formula to compute the predictions of a demand model coming from an exported file:
Elasticity and slope
The elasticity is defined in economics as the unitless ratio
Note: With our definition of demand and elasticity, a positive elasticity means that the demand decreases when the price increases, which is what we expect. We choose this definition as it's easier to deal with positive numbers, but you may find other definitions in other works.
An elasticity below 1 means that in our portfolio, a price increase will more than compensate the portfolio shrinkage, and we are therefore in a situation where we can increase both our margin and our GWP.
For convenience, let us also define the slope as
This is just the derivative of the demand function with respect to the variation in price.
Therefore
as we are computing at 
.
The slope allows us to easily compare the sensitivity to price of different segments.
From this we derive the value of the elasticity as a function of the dynamic model and the demand function:
Warning: In Akur8, when we quote values of elasticity or slope, we are always talking about the value at
so at price variation t = 1.
Summary: The elasticity conveys important information about business and how our portfolio will evolve when we change the price, whereas the slope makes it easier to compare different segments sensitivity. In Akur8, you will be able to switch from one view to the other to facilitate your decision making process.
A quick example
In the example below, the slope measured at 1% price decrease is equal to 3: the difference of absolute change in demand (3% = 23% - 20%) divided by the price change (1%).
Meanwhile, the elasticity is equal to 15: the ratio between relative change in demand (0.15 = (23%-20%)/20%) divided by the price change (1%).
