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I'm trying to support an argument being made with respect to camp fee changes, and I would need to know if anyone has ever come up with a "rule of thumb" for demand drop due to price increases for a standard 1-week sleep-away summer camp? No fancy expert training, or limited-access development, just standard summer camp activities.

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It seems not, judging for the results of this and this search (but you might want to take a look yourself).

There are some more informal analysis of the market for summer camps (e.g. here) and some organisations recollecting some form of data (e.g. here). The former is interesting because it shows, unsurprisingly, that the services proved by a summer camp are very heterogeneous. They are also targeted to different audiences. This means that the good itself is heterogeneous, so thinking of a unified "demand for summer camps" might be problematic.

Still, there are studies estimating demand for such heterogeneous goods. Here (subscription-wall) is an example which perhaps might be related to your interest (not the least because it uses the word camp too, which is how I found it, btw). The study estimates the demand for camping park's amenities. The key is that these amenities differ across camping sites (e.g. electricity access, natural beauty, etc). The authors estimate, among other things, the "price and income elasticity" for 48 camping parks in a region of Canada. The main finding with respect to elasticities are:

... it is immediately evident that there exists no definite pattern with regard to the price elasticity; there appear to be almost as many cases of inelastic as there are instances of elastic demand. [...] Since our results also indicate a negative income elasticity, higher user fees would represent a regressive form of taxation. The principle of user pay may still be justified on efficiency, not equity, grounds.

If you want to do some form of econometric analysis of the market for summer camps, I think this method is a possible approach.

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