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I have data on

  • purchases
  • gross prices
  • discounts (like coupons)
  • net prices (i.e. gross price - discounts)
  • number of leads (potential customers at any given time)
  • conversion rates (# purchases / # leads) over time

A business wants to know how price elastic its consumers are and to use this information in part to set their net price by changing either gross price and/or discounts.

The problem is that there's some endogeneity (or recursion) in the system - when the business predicted higher conversion rates it raised prices. The business also varied the discount levels based on the conversion rates and gross price.

While the business is clearly selling a normal good, the endogeneity/recursion makes it looks like higher prices are associated with more sales and higher conversion rates. I need to disentangle these effects. Perhaps there's some great IV I'm not thinking of, but barring that, is there any other strategy to address this situation?

We can think of this as a single good case (I can add complexity later, but based on my subject matter knowledge it would be safe to ignore other goods and cross-price effects, etc).

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  • $\begingroup$ What do you know about the process by which prices are endogenized? You may be able to use a Heckman correction. Do you have any variables which alter prices and not quantities or quantities and not prices? You may be able to a simultaneous equations approach. $\endgroup$ – BKay Feb 15 '17 at 20:39
  • $\begingroup$ @BKay Good questions. In answer to the first question, the business doesn't make sales continuously, but at regular intervals (approximately once per month, several thousand sales occur on the same day). As time approaches the sale date the business becomes increasingly sure of their estimates of lead projections, conv. rates, and sales projections, and offers discounts and firms up prices accordingly. Interesting thoughts on the other questions; I will have to think more about it $\endgroup$ – Hack-R Feb 16 '17 at 14:28
  • $\begingroup$ Purchases at time $t$ over leads at time $t-1$ I guess? $\endgroup$ – Alecos Papadopoulos Feb 19 '17 at 21:52
  • $\begingroup$ @AlecosPapadopoulos interesting thought; I will give that a try tomorrow $\endgroup$ – Hack-R Feb 19 '17 at 21:54

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