Here's the elegant proof cited in the report, which you can download here:
"where it q is the quantity sold for brand i in time t (in 000s), it p is the price of brand i at time t (in dollars per unit), the lower case advertising variables represent the flow or current period advertising (in $000’s), the upper case advertising variables represent the current stock of advertising up to time t (in $000’s), it Exper is the number of months that drug i was on the market prior to time t , and it cp is a weighted average of competitors’ prices at time t . We take logs of both price and quantity so that 1 β represents the price..."
And so on. I think you get the drift.
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