# Establish demand curve with infrequent sales

I sell variations of an item on eBay. It is an unusual commodity which I purchase in bulk lengths and then cut to various sizes. They come in about 20 types, which I then sell as 10 different lengths, giving a total of 200 variations.

My goal is to determine what price I should charge for each, to maximise monthly profit.

The problem is I only make an average of one sale per day. I have no idea how I can establish a demand curve for the products, given such limited data. (Some variations I only get one sale per 18 months!)

What techniques could I use for this problem?

How could I go about establishing the most profitable price for each of the variations, with such limited information? (Given known costs).

The only clear idea I have is this:

• Treat all 200 variations as a single product, and vary all their prices simultaneously, to derive a single demand curve. So for example, one month I might set all prices to generate 10% profit on top of the costs, then next month, set all prices to generate 20% profit, and so on, and this way establish what price generates the highest monthly profit.

But the variations tend to have wildly different applications, as well as, I expect, alternative products that can satisfy the customers' requirements, and so on - so having a single demand curve could leave significant profits untapped.

The only other idea I have is to profile the demand curve for each item in a different way:

• For each product, take an example: one that only sells approx every 6 months on average: On each sale, if the time since the last sale was 6 months, I divide the total profit made on that item by 6 - or for a product that sold twice in one month, multiply by two. And then each time there is a sale I can some how establish the demand curve for it.

But the snag is for a product that sells infrequently, the demand curve could be static, but obviously this does not mean it would sell, say, once every 6 months like clockwork. So it could take years to establish a demand curve for that variation.

I'm wondering if I should entirely forget about the demand curve and concentrate instead on a marketing-oriented approach, for example:

• Surveys: (How I would phrase this I'm not sure - I doubt customers would tell me the maximum they'd pay!)
• Market research: plot graphs of how much other retailers/distributors charge for each variation. (Although I have seen very few indeed that sell small lengths such as those that I offer)

If you are fairly confident that your current price is not too far from the real profit-maximising price then there is little loss in accuracy from assuming the demand curve to be locally linear (i.e. a straight line in the area around your current point). This is illustrated in the figure below: note how the red (linear) approximation to the curve much more closely follows the shape of the curve when the two endpoints are close together than when they are far apart. This observation can help you a little because it means you only need two data points to characterize the demand curve provided the price adjustments are small. Moreover, note that you can implement one large price change as a sequence of smaller changes. As soon as you have two datapoints, you can work out which direction you should be adjusting your price in. One procedure would be as follows:

1. reduce your price by a small amount—say, 1–5%. (note: a price decrease might work better than an increase because you have said that you do not sell often and increasing your price will just mean you end up with even fewer sales and hence even less data). Unless there is a kink in the demand curve, a sufficiently small price increase and a small price decrease should give you approximately the same estimate for the demand curve's slope.
2. observe demanded quantity at the new price (e.g. '$q_\text{new}$ sales per month').
3. calculate the slope of the inverse demand curve: $$\text{slope}=\frac{p_\text{new}-p_\text{old}}{q_\text{new}-q_\text{old}}$$ where $q$ means quantity and $p$ means price.
4. The profit maximising price, $p^*$, solves the equation $$p^*=c-(q\times \text{slope})$$ where $c$ is your unit cost, slope is calculated above, and $q$ is quantity sold at price $p^*$. If you find that neither $p_\text{new}$ nor $p_\text{old}$ is equal to $p^*$ then you need to make an adjustment to the price (see step 5.).
5. The linear approximation to demand introduces very little error so long as you don't venture far from the starting price. It therefore makes sense to incrementally adjust towards optimum. For example, suppose that you do steps 1–4 and find that, at your current $p$ and $q$, $p>c-(q\times \text{slope})$ (meaning that your price is above the profit-maximising level). Then it makes sense to reduce the price by a small amount and repeat steps 2–3 to get a new estimate for the slope of the demand function that is more accurate in the region surrounding the new lower price. Repeated application of this procedure should allow you to converge towards the optimal price.

Important caveat: You should be mindful that the quantity you sell will change not only because you reduced price (i.e. a movement along the demand curve), but also if the whole demand curve changes (e.g. demand for toys will be higher at Christmas independent of whether price changes or not). For this exercise to work, you need to be confident that the demand curve doesn't change too much over time (and to re-estimate the demand curve during periods when it is likely to be different).

What you're looking for is the price elasticity of demand of your product. That is: by how much will your sales rise or fall depending on how you manipulate price? It depends on what you're selling and, like you said, each variation of your product will have a different elasticity. So, say you increase prices by 5% and experience a %5 decrease in sales; your price elasticity of demand would be -1 (unit free).

The strategy you outlined to parse out the price elasticity sounds reasonable from a data collection standpoint, albeit a bit risky. I might change the price of just one of your variations (maybe one that sells somewhat more frequently) to assess the sensitivity your customers are to price changes. However, if your product's elasticity is high, your sales might drop dramatically or stop altogether.

While you only make 1 sale per day and you said that the market is limited; you should still find your competitors listings as well. I would rip that data from the internet, including the sellers name, buyers name, and the names of the bidders. Look to other sites for the same product, again getting data on seller, buyers, whatever you can find. This way you have an idea on what your market is. Break it down by type. What type has the most liquid market? Which type has the lowest turnover rate? These factors must go into the price.

You should also find information on substitutes. That is, if the buyer does not buy your product, how much does he have to spend to still complete his task. While the exact dollar amount matters, also think about non-tangible things, like enjoyment or time.

To use eBay's bidding system, let the most popular type and length go to auction for a few rounds so you get an idea on what the auction price would be. Note however that only bargain hunters are bidding looking for the lowest cost so expect them to be lower. Then find other products that are similar to yours(same market, same demand, or same price range, etc) and find out how much of a discount they bid for, then apply that rate to your product. Make sure that it is still in range of the competitors, and that your option is equal or less then substitutes. For your rarer variations, pro-rate them and add storage. For example, if A sells once a day and B sells once a week, and you could make 2 A's with one B, then B should be twice as much as A plus a week of storage. Price it a little high, and place a best offer on it. The fact that they are rarer should make them more expensive.