## The Marginal Value of Parks

Environmental valuation studies should be clearer as to whether the value they estimate is marginal value or something else.

The idea of a marginal quantity is one of the most-used in the economist’s toolkit.  In the theory of the firm, profit is maximised when marginal revenue equals marginal cost.  In macroeconomics, the effect of a change in disposable income depends on the marginal propensity to consume.  In the literature on valuation of non-market environmental goods, however, this tool seems not to be used as often as it should.  Travel cost studies estimating the value of recreational sites often fail to consider whether what they have estimated is the marginal or some other value of the site, and this can lead to inappropriate policy recommendations.

The sense of marginal value with which I am concerned here is that which considers a whole recreational site as one unit.  Admittedly that could raise a difficulty in respect of open countryside with no clear subdivisions, but for urban parks surrounded by developed land and for designated national or country parks with well-defined boundaries it is usually clear enough.  So I am not concerned with the difference in value between a park as it is and the same park with one less square metre of land.  The margin I mean is that between the status quo in respect of all the parks in a region and a situation with one less park.  One might argue that incremental would be a more correct term, but I consider that marginal has connotations that are relevant here, such as the idea of diminishing marginal utility.

A couple of examples will illustrate why it matters whether or not the value estimated by a study is marginal value. Suppose there is a proposal to convert a recreational site for housing development, which we want to evaluate by cost-benefit analysis.  This involves, for both costs and benefits, comparing the situations with and without the project (1).  The recreational value that should be used in the analysis is therefore a marginal value, the value that would be lost if the site were no longer available for recreational visits.  If there are other recreational sites in the vicinity, then people who would have visited that particular site might visit other sites instead.  If so, then the numbers of visits to the particular site and the costs incurred by its visitors will not give a good guide to the value that would be lost.  It should of course be borne in mind, here and throughout this post, that recreational value is only one component of the total economic value of undeveloped land – others include the values of water and air purification, biodiversity and carbon sequestration -, and decisions on land use conversions should have regard to all such components.

Sometimes, however, marginal value is not the value we need.  Suppose we want to include the value of a recreational site in the national accounts, as part of an estimate of the aggregate value of the country’s environmental assets or natural capital.  In that case the marginal value is not appropriate. Suppose two sites A and B are not far apart.  If we value site A on a marginal basis, excluding the value attributable to visitors who would have visited site B if site A had not been available, and if we apply the same approach, vice versa, when valuing site B, then we will not capture the full combined value of the two sites.  In simple terms, the value attributable to visitors with no strong preference between the sites will be missed.

How then should we estimate the marginal recreational value of a park?  And where marginal value is not what we want, what other value concepts are available and how should they be estimated?  A key consideration here is that parks are not a homogeneous good: they differ in location, size and other characteristics.  This has several implications.  We should not expect a smooth curvilinear relation between the number of parks in a region and their combined value.  Obtaining the marginal value of a park is certainly not a matter of estimating such a curve and then applying differential calculus.  Each park will have its own marginal value.  Furthermore, the average value of a park, obtained by dividing the total value of the parks in a region by the number of parks, is unlikely to be a useful statistic.

To address the above questions, I shall consider the example below of a stylised small town with two parks.

The grey zone C2 is the commercial and shopping centre, with no residents.  The two green zones B2 and E1 are parks with free entry, also without residents.  The remaining seven zones are residential, each with the same number of residents: what the number is does not matter as all my calculations were per resident.

In the interests of simplicity I make the following assumptions:

1. All residents are alike in their behaviour in respect of park visits.  Their visit rates depend only on the travel costs of their visits to parks.
2. Residents perceive the two parks as different but equally attractive, except in so far as visits require different travel costs.
3. Distances are measured as straight lines between the centres of zones, the unit of distance being the side of one square zone.
4. The travel cost (TC) of a resident’s visit to a park is measured in monetary units such that it equals the park’s distance from the resident’s zone.
5. There are no complications arising from congestion or multi-purpose trips.

I pass over the important practical question, a key focus of attention in many published studies, of how the formulae modelling visit rates (trip-generating functions) might be estimated from observational data.  My focus here is on the determination of site values from the trip-generating functions, and I therefore start from plausible assumptions about those functions.  By plausible I mean that the functional forms are credible; the coefficient values are chosen so that all residents will make some visits to each park, but their visit rates will vary considerably depending on their zone of residence.

It is convenient to define the functions in two stages.  The first stage consists of formulae stating what the visit rate to a park would be if, hypothetically, the other park were not available (to indicate the hypothetical nature of these visit rates I write VR’).  These formulae are:

$VR'(B2)=18-3TC(B2)$

$VR'(E1)=18-3TC(E1)$

The formulae for actual visit rates, given the availability of both parks, are then expressed in terms of these hypothetical visit rates:

$VR(B2)=VR'(B2)-\Bigg(\dfrac{VR'(E1)}{VR'(B2)+VR'(E1)}\times\dfrac{\min{(VR'(B2),VR'(E1)}}{2}\Bigg)$

The formula for VR(E1) is as above but with B2 and E1 interchanged throughout.  These formulae may appear complicated, but are chosen for various desirable properties (further details are in the download at the end of this post).  Note that a simple linear functional form as below will not work:

$VR(B2)=a-bTC(B2)+cTC(E1)$

While it correctly indicates that a higher travel cost to park E1 will be associated with a higher visit rate to park B2, it has the very implausible implication that the relation between VR(B2) and TC(E1) is linear.  The higher TC(E1) is, the smaller we would expect the effect on VR(B2) of a unit increase  in TC(E1) to be.

To calculate values, starting from these trip-generating functions, I applied the standard method of deriving points on the demand curves by considering various price additions to the travel cost, then taking the area under the demand curve (the consumer surplus) to be the value (2).  Applying this method to the actual trip-generating functions, I obtained values per resident of 21.66 for park B2 and 16.06 for park E1, implying a total value of 37.72.  An appropriate description for these values would be “contribution of visits to the park to the total value of the two parks”.  This (generalised to all the parks in a country) is the value concept that would be relevant for inclusion in the national accounts as above.

The lower value for park E1 – the more distant park for more than half of the residents – is unsurprising, notwithstanding the equal attractiveness of the two parks.  There is much evidence that the recreational value of sites is lower when they are further from centres of population, other things being equal (3).

I also applied the method to the hypothetical trip-generating functions, obtaining values per resident (in each case in the absence of the other park) of 28.26 for park B2 and 22.26 for park E1.  This enabled the marginal values to be obtained as follows (calculations may not exactly agree due to rounding:

Marginal value of park B2

=  (Total value of two parks) less (Value of park E1 in absence of park B2)

=  37.72 – 22.26  =  15.45

Marginal value of park E1

=  (Total value of two parks) less (Value of park B2 in absence of park E1)

=  37.72 – 28.26  =  9.45

The table below summarises these results.

Total value is shown only for the “contribution” row: totals of the other rows would not be meaningful.

Once one has both the actual and the hypothetical trip-generating functions, the calculations of these distinct values do not present any special difficulty.  Why then do published travel cost valuation studies often fail to consider whether the values they obtain are marginal values or contributions to total value?

One possible reason is that researchers may not be entirely impartial.  They may be seeking results that would support a case for preservation of a site, and recognise that a marginal value, which could be low, would not be helpful.  Certainly, I have seen quite a few such studies that use their results in recommending preservation, perhaps with the help of government funding.  I cannot however recall a single study concluding that a site was not worth preserving.

Another reason is that many studies, perhaps because of resource limitations, focus on a single site.  They may, with the aim of avoiding omitted variable bias, estimate a trip-generating function that includes travel costs to alternative sites as independent variables.  But even then, it is impossible to obtain the marginal value of a site from a trip-generating function for that site only.  As we have seen above, that requires such functions for other sites too.

Suppose however that a researcher is both impartial and well-supplied with resources.  Suppose that they collect data on visit rates and travel costs for a number of sites in a region.  They could then estimate the actual trip-generating function for each site and hence calculate each site’s contribution to total value.  To obtain the marginal value of a site, however, they would still have to estimate the hypothetical trip-generating functions describing the unobservable visit rates to other sites that would prevail if that site were unavailable.  That seems, at best, statistically challenging.

It is understandable, therefore, that many studies lead to value estimates for a site that are a reasonable approximation to “contribution to total value”, but do not estimate, even approximately, the marginal value.  What is less defensible is when such values are used to justify policy recommendations, for example whether a site should be preserved or converted to an alternative land use, that really require the marginal value.  One such example is a paper by Bharali & Mazumder (2012) which estimates the recreational value of Kaziranga National Park, Assam, India (4).  Starting from sample data on visit rates, travel costs and other relevant variables, the authors obtain an estimate of consumer surplus which, they state, “signifies the value of the benefits that the visitors derived from visiting the park” (5).  But they do not consider whether those benefits are gross, or net of the benefits that visitors would have obtained at alternative sites if the park had not been present.  Since the paper does not consider alternative sites at all, we can take it that these benefits are gross and that the value it estimates is not marginal value.  Nevertheless, the author’s conclude that the government should allocate large funds to preserve the site (6).  That conclusion would be much better supported if it had been shown that the marginal value of the site were significant.

The calculations underlying the above results may be downloaded here: Marginal Park Value Calculation (MS Excel 2010 format).

Notes and References

1. See for example quote from Watkins T at  https://www.smartsheet.com/expert-guide-cost-benefit-analysis  (select Compare Aggregate Costs and Benefits)
2. See for example Perman R, Ma Y, McGilvray J & Common M (3rd ed’n 2003) Natural Resource and Environmental Economics  Pearson Addison Wesley  pp 413-4
3. See for example Bateman I et al (2013) Bringing Ecosystem Services into Economic Decision-Making: Land Use in the United Kingdom  Science Vol 341 Issue 6141 pp 45-50 (section headed National-Scale Implications)   http://science.sciencemag.org/content/341/6141/45
4. Bharali A & Mazumder R (2012)  Application of Travel Cost Method to Assess the Pricing Policy of Public Parks: the Case of Kaziranga National Park  Journal of Regional Development and Planning Vol 1(1) pp 41-50  http://www.jrdp.in/archive/1_1_4.pdf
5. Bharali & Mazumder, as 4 above, p 47.  An unusual feature of this paper (an error?) is that, having estimated the consumer surplus, it then adds on the actual travel costs to arrive at what it describes as total recreational value.
6. Bharali & Mazumder, as 4 above, p 48
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