The distinction between marginal and average values is a key point in economics. Last year I was one of a team of researchers who got into a robust debate with some other researchers – played out through a series of journal papers and letters to the editor – on matters related to average values, marginal values, and total values. 

The debate was about sharks. We had published a paper on the economic value of shark-related tourism in Palau (Vianna et al. 2012) (see PD207). People go there specially to dive with sharks, and they spend a lot of money in the process. We found that the sharks that interact with tourists in Palau would generate much less income if they were caught by fishers.

Catlin et al. (2013) subsequently published a paper in which they criticised several papers (including ours) for estimating the values for tourism of individual animals, rather than larger aggregations of animals.

They claimed that, ‘‘The fundamental characteristics of wildlife tourism activities do not lend themselves to valuation at the individual animal level.’’ Their stated concerns were that: the tourists might have spent the money anyway; it is hard to determine the number of animals in the population; it is easy to confuse annual values with lifetime values; the animals that interact with tourists may be atypical of the population, making extrapolation dangerous; and if comparing tourism to extractive uses, the relevant population sizes might be different.

They didn’t state whether they were concerned with the average value of a shark, or the marginal value of a shark, or both. They never used the terms “average value”, or “marginal value”, just the ambiguous term “individual value”.

The simplest, and most amusing, mistake in their paper was the claim that we presented individual values in our paper at all. All our results were presented at the level of the whole population of sharks that engage with the Palau tourism industry.

In an earlier report, which became the subject of an article published in the New York Times, we had converted the results into an average value per shark, based on high-quality information about the size of the population. I guess Catlin et al. must have seen that report and assumed we’d done the same thing in our journal article, without reading it properly.

Anyway, back to average and marginal values. In a Letter to the Editor, we pointed out that invidual values could be average or marginal, that average and marginal values are different, that Catlin et al. had not distinguished between them, that none of the concerns raised by Catlin et al. affected the conclusion of our study, and that, although we hadn’t done so in the journal article, it would have been meaningful and valid to present the average value of sharks for tourism and compare it with their average value for fishing.

That last point is simply demonstrated. If you know the size of the population, comparing tourism and fishing values for the whole population of sharks is no different to comparing them on average, per animal. The average values are just the population values divided by the population size. If the question is, ‘should we do all one or all the other?’, and you use averages correctly, you will never be misled about which option is more valuable.

The other way to get an “individual” value would be to calculate the marginal value of a shark: the change in value resulting from changing the number of sharks by one. For example, if the Palau authorities wanted to determine the optimal balance between sharks for fishing and sharks for tourism, they would need to estimate the marginal values for each use for different levels of fishing versus tourism. The optimal allocation would be where the marginal value for tourism equaled the marginal value for fishing (because, if the marginal value for one was smaller than the other, you could make more money by moving towards the other.) If they were never equal, the optimum would be either all fishing or all tourism. Once again, it is meaningful and valid to look at individual values – in this case marginal values.

Of course, it would be difficult to accurately estimate the marginal value of sharks for tourism, but no more difficult than estimating the aggregate values for different levels of fishing versus tourism. In fact, the marginal value curve is simply the slope of the aggregate value curve, so if you know one, you can derive the other. Once again, the distinction between individual (marginal) values and aggregate values is a false distinction. They are closely inter-related.

So, what happened in the robust debate? Catlin et al. published a Letter in reply to our Letter, but they just made matters worse, really.

Most outrageously, they said that “Our article is trying to point out that the two values [average and marginal] are often confused.” Talk about moving the goal posts! It does nothing of the sort. It doesn’t even mention average or marginal values. It just lumps them together as “individual” values.

They said that we had compared average values for tourism with marginal values for fishing “without explaining that they are not directly comparable”. Since we didn’t present average values for tourism in the paper, that’s clearly not strictly correct. But reinterpreting the comment, there is something to it.  It implies that catching more or fewer sharks could conceivably alter the market price of sharks, and it’s true that we didn’t allow for that. However, in the case of Palau, there are only about 100 sharks that hang around at tourist spots. If caught and sold, these sharks would be a tiny speck in the market. Clearly, market prices for the 100 sharks would hardly vary, if at all, if fewer or more of them were caught. That’s why we assumed a constant price (= a constant marginal value for fishing), which means that the marginal value equals the average value and can validly be compared with the average value for tourism.

Note that the price of sharks would have to be staggeringly sensitive to the number caught in order to change the result that the sharks are worth more alive than dead. Specifically, the market value of a caught shark would have to increase from $100 per shark to, I’d estimate, several hundred thousand dollars per shark. There is no chance that our assumption of a constant price had any material effect on the conclusion.

In any case, the criticism is irrelevant to the basic question of whether marginal or average values are valid and meaningful if used correctly. It is a criticism of using marginal and average values incorrectly.

To wrap up their reply, Catlin et al. repeated their claim that “valuation at an industry, location, or other higher scale is a more robust and reliable measure than is the valuing of individual animals.” Clearly, that’s wrong. Individual and aggregate values are too closely inter-related for that statement to make sense. For some management questions, the required information is marginal values, or, equivalently, aggregate values used in a way that reveals what the marginal values are. For other management questions, the required information is average values, or, equivalently, aggregate values that are equal to the average values times the population size.

We decided not to send another Letter to the Editor, thinking they’d be unlikely to publish a third letter on the topic. Instead we sent a letter directly to Catlin. He acknowledged receipt, but that’s all.

Further reading

Catlin, J., Hughes, M., Jones, T., Jones, R. and Campbell, R. (2013). Valuing individual animals through tourism: Science or speculation? Biological Conservation 157, 93–98. Abstract at journal web site

Catlin, J., Jones, T., Campbell, R., Hughes, M. and Jones, R. (2013). Letter to the Editor: Keeping perspective on using tourism values for conservation – Reply to Vianna, Biological Conservation 166, 303.  Journal web site

Vianna, G.M.S., Meekan, M.G., Pannell, D.J., Marsh, S.P. and Meeuwig, J.J. (2012). Socio-economic and community benefits from shark diving by tourists in Palau: a sustainable use of reef shark populations, Biological Conservation 145(1), 267-277. Abstract at journal web site

Vianna, G.M.S., Meekan, M.G., Pannell, D.J., Marsh, S.P. and Meeuwig, J.J. (2013). Letter to the Editor: Valuing individual animals through tourism: Science of speculation? – Reply to Catlin et al. (2013), Biological Conservation 166, 301-302. Journal web site

Kirillov, S. (1974). Marginal and average values in the theory of price formation, Problems of Economic Transition 16, 42-59. IDEAS page