Estimating Price Elasticity from sales

I'm looking at data for an online shop with clothing items listed at different prices. I'd like to use this information to do some kind of profit maximization (i.e. I'd like to have some understanding of the impact on sales if I raise/lower price of an individual item).

Unfortunately, I don't have information of the variety "Q sales for clothing item X at price P" for a variety of prices; this would be hard to estimate due to the small number of items in each category.

However, I do have a distribution of sales vs. price and the distribution of price of items in the shop. I'm thinking if I compute $$\frac{\text{% of all sales at this price}}{\text{% of available items at this price}}$$, that gives me something analogous to a "quantity demanded". For simplicity, I bucketed the prices and sales into $5 increments. Having done this and doing your traditional log-linear fit$ \ln(P) = a\ln(Q)+b $I get a pretty good fit and a reasonable number (around$-.9$). Question 1: Does my methodology make sense? If so, what did I just calculate? I feel like it's an "aggregate" price elasticity of all items in the shop. The clothing items for sale are mostly substitutable with each other, the price elasticity of an individual item must be a lot higher (since there are substitutes available). Question 2: Is there a way to figure the "average" price elasticity of each item from this (approximating that they all have about the same price elasticity)? EDIT: Based on the suggestion from @optimal-control and @NickJ, it appears the issue is that I don't really have a good model of what's going on. Here's what I'm imagining: Suppose you have$n$unique items, each with a fixed price$p$. Each item has a limited supply available$s$. I observe$q(p, t)$sales for each unique item, limited by$s$. Let's say$q(p, t)$is something like a homogeneous poisson process, for simplicity. The above is starting to look like a markdown optimization problem; I could imagine finding an optimal price for each item based on these factors and my cost structure (storage cost, time value of money, etc.), if I could, say, change prices and estimate the change in$q(p, t)$. Unfortunately, this function is so small, and$s$is so small (maybe 1 to 5 items) that this may be impossible to estimate. What I'm trying to figure out is some way to estimate the response of$q(p, t)$with respect to$p$without modifying prices, off of the assumption that the$n$items are partial substitutes for one another. If they were perfect substitutes, I would expect ALL of my lowest price items to sell out, and then my next lowest, etc. going on upward and I could use these results to estimate$q(p, t)$. But clearly this isn't the case, because some more expensive items sell first. The problem is,$q(p, t)$is dependent on the mix of items and prices available. Any thoughts on how to approach this? • No, what you're doing does not make sense to me. Can you be more specific or give a specific example? What data do you have, exactly and what are you computing? May 3, 2015 at 14:08 • Are you regressing$\ln P$on$\ln P$or is it just a typo? May 3, 2015 at 15:50 • Sorry, that's a typo. Should be$\ln(Q)\$! May 3, 2015 at 18:12
• Hi @NickJ, thanks for the comment. I updated the question; hopefully that answers what you were asking. May 3, 2015 at 20:38
• I sort of understand your problem now. I still don't understand your method. You're selling apples and oranges, and at fixed prices p1 and p2. You see the time at which apples were bought, the time at which oranges were bought, and you see at each purchase whether the buyer could choose between apples and oranges, or whether one was sold out and so they could only buy one type of fruit. Is this correct? If so, do you see people who came into the store and left without buying anything? May 8, 2015 at 14:59

1 Answer

In this kind of cases, I think you can not use a simple OLS estimation because of endogeneity of your variables.

Probably, there would be causality between prices and sales data, this means that sales are influenced by prices but also prices are influenced by sales. So, in this case, in order to do a non-biaised estimation, you should use instrumental variables in order to avoid endogeneity issues between your variables.

• Thanks! What might I choose as an instrumental variable in this case? I'm guessing your point is that prices are influenced by sales in the sense that we can only sell up to a maximum of a certain item (say, we might have 5 of them). Therefore quantity demanded is strictly less than or equal to amount available. Is this what you meant? May 3, 2015 at 20:05
• My point was a technical one. If you do some econometrics, you can not use simply linear regressions for this kind of question to which you try to answer. What I tried to say was that prices are influenced by quantities, as demand increases prices increases. But the reverse effect is always present, prices increase/decrease if quantity demand increases/decreases. I recommend you to search a little bit about some notions on causality, endogenous variables and instrumental variables. May 3, 2015 at 20:39