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I am attempting to model the decreasing price elasticity/response for a good. I need to control for place and time features and available alternatives. Besides this, I also need to add time and location fixed effects to control for seasonality. The outcome of this model would be to calculate a multiplier to adjust the price.

I have some sales data for a good g and other data such as availability of alternatives, time and place features.

The question:

How do I fit a model that estimates the decreasing price elasticity (image below) as an exponential function and estimate the parameters while controlling for alternatives, time and location features? What assumptions do I need to make?

P(g) ~ α ⋅ exp(− β ⋅ p) + loc + time + alt

where, p(g) is purchase / conversion calculated as a % (customers who purchased / total lead volume)

       α & β are parameters to be estimated

       loc - location features
       time - time features
       alt - availability of alternatives

enter image description here

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    $\begingroup$ You can fit a model of that form with a linear regression of $\ln(P(g)-loc-time-alt)$ against $p$. Alternatively, you could try a model of the form $P(g)=\alpha\exp(-\beta p)\ loc\ time\ alt$, where all the effects are multiplicative; then you can take logs of everything. $\endgroup$ Commented Nov 8, 2021 at 14:05

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