2 Background to the VOI concept

Consider an individual plot of forested land controlled by a “seller” who may either preserve it, or convert it to other uses (such as agriculture), within a given period. An outside “buyer” has an interest in saving this forest plot, and makes a payment H to the seller when the plot is saved. V is the net, positive or negative, value to the seller of converting the forest to other uses.Footnote ² The buyer has a positive value B related to preserving the plot as forest.Footnote ³ The “seller” retains control of the plot beyond the period of analysis. If the plot is saved for this period, the seller is then free to manage the plot for possible future conversion. The buyer then does not actually “buy” the plot, but “rents” its protection, in the form of forest, forest for that period.

Consider first the perfect (and symmetric) information case where the buyer knows B precisely, and both buyer and seller know V precisely. The buyer could then pay V (+ ε; where ε is small) to the seller, given (a) V > 0 (since we assume that, otherwise, the land will not be converted even with no payments) and (b) B > V (since otherwise the opportunity value to the seller would be so high that the buyer could not profitably make the transaction). This would result in an efficient (and first-best) solution where the forest is preserved if and only if its protection value exceeds its opportunity value. All net rent associated with the transaction will then go to the buyer.

Two sets of problems with this solution may here arise. First, V may or may not be observed by the buyer, only its statistical distribution, while V is known by the seller, as his or her use value of a converted plot. Second, the buyer may not observe the true protection value of the particular resource to be preserved, B, but instead only its statistical distribution in a subjective (Bayesian) sense. At least two factors could be behind this uncertainty. First, the buyer may know the “true” distribution of these values across a large sample of plots, but not which plot has which value. Secondly, not even a “true” distribution across plots may be known; for rainforests there could be some more fundamental uncertainty related to biodiversity values that is common to all plots.

Under asymmetric information, when only the seller observes V, protecting or not protecting the resource is a mechanism design problem, similar to Myerson (Reference Myerson1979) and Myerson and Satterthwaite (Reference Myerson and Satterthwaite1980); in the context of forest protection, Montero (Reference Montero2008), Salas and Roe (Reference Salas and Roe2011), Benthem and Kerr (Reference Benthem and Kerr2013), and Mason and Plantinga (Reference Mason and Plantinga2013). We, here, do not go deeply into deriving the buyer’s optimal mechanism. We instead propose a simple mechanism where the buyer sets a fixed offer to the seller for protecting the resource, that maximizes the buyer’s expected value to protect or not protect a given forest, assuming risk neutrality of all parties.

When the distribution for B is a Bayesian prior distribution for the buyer, this is a Bayesian decision problem; see DeGroot (Reference DeGroot1970), Berger (Reference Berger1985), Pratt et al. (Reference Pratt, Raiffa and Schlaifer1995), Peck and Teisberg (Reference Peck and Teisberg1996), Gollier (Reference Gollier2001), Robert (Reference Robert2001), Bergland (Reference Bergland2005, Reference Bergland2014), and Eeckhoudt et al. (Reference Eeckhoudt, Thomas and Treich2011). Our aim is to study the effect of “precision” on the decision to protect or not protect the resource, following from reduced uncertainty; parameterized by the standard deviation on B. Our focus will be on the utility gains from avoiding or reducing the statistical prevalence of two types of mistakes that occur under imperfect information: convert the resource to other uses when it ought to be saved; and save it when it ought to be converted. Under perfect information for the buyer, there will be no such mistakes. Under improved but still imperfect information, the number and seriousness of mistakes will be reduced, but all mistakes will not be eliminated.

An interest to avoid or reduce such mistakes lies behind all work on benefit valuation, but little has been done to quantify the value of greater such precision. Some studies however exist.Footnote ⁴ The first theoretical paper on the topic is to our knowledge Crocker (Reference Crocker, Peskin and Seskin1975), who discusses “the BCAs of BCA.” A decision maker here seeks more information about an environmentally related decision problem from his or her staff whose time is limited, and asks whether staff time ought to be allocated to this task. Other early studies include Mendelsohn (Reference Mendelsohn1986) who studies precision in the measurement of environmental pollution; Yohe (Reference Yohe1996) and Peck and Teisberg (Reference Peck and Teisberg1996) on the social cost of carbon when mitigating climate change; and Polasky and Solow (Reference Polasky and Solow2001) who consider the VOI about biological reserves.Footnote ⁵ Costello et al. (Reference Costello, Polasky and Solow2001) and Kennedy and Barbier (Reference Kennedy and Barbier2013) have studied related renewable resource management problems; while Newbold and Marten (Reference Newbold and Marten2014) have analyzed integrated assessments of climate change.

More VOI studies are found in the epidemiology and medical risk literature. See Claxton and Posnett (Reference Claxton and Posnett1996), Claxton (Reference Claxton1999), Ades et al. (Reference Ades, Lu and Claxton2004), Yokota and Thompson (Reference Yokota and Thompson2004), and Eckermann and Willan (Reference Eckermann and Willan2007) for overviews; and prominent specific analyses by Tanner and Wong (Reference Tanner and Wong1987), Samson et al. (Reference Samson, Wirth and Rickard1980), Dakins et al. (Reference Dakins, Toll and Small1994), and Gold et al. (Reference Gold, Siegel, Russel and Weinstein1996). Phillips (Reference Phillips2001) presents an analytical framework for deriving the informational value of research for testing pharmaceutical medications or medical procedures, which appears as most closely related to our paper within this literature.

As noted, our paper also recognizes that practical decisions to save or protect a natural resource object are often not globally optimal, and that the value of additional information needs to be considered in that light.

3 The model

Assume that a buyer (“principal”) observes not his true value, B, of the forest plot to a seller (“agent”), but instead its statistical distribution F(B), with expectation E(B) and standard deviation σ_B. The seller knows the value of the resource, V, with certainty. For the buyer, V is uncertain with subjective and continuous distribution function G(V) with density g(V). Both buyer and seller are considered risk neutral.

Our model builds heavily on Mason and Plantinga (Reference Mason and Plantinga2013), and our solution is similar to theirs. Our model is however more general, as Mason and Plantinga assume B to be known for the buyer, while the buyer here knows only its distribution. B and V are assumed to be either uncorrelated, or positively correlated. In the presentation here and in Section 4, we assume for presentational simplicity zero correlation. Several of the simulations in Appendix 1, and mathematical derivations in Appendix 2, build on positive correlation. For tropical rainforests, low but positive correlation between B and V is reasonable when carbon and biodiversity dominate forest values for the buyer but not for the seller. When important protection values (nontimber forest products; ecotourism) depend on forest access, B and V are more likely positively correlated. For the Brazilian Amazon (our main example, studied in Section 5) the latter factors are relatively minor; see May et al. (Reference May, Soares-Filho and Strand2013). Any positive correlation is then likely to be weak. For ease of exposition, we first assume that B and V are uncorrelated. Section 4, and Appendix 2, develop and simulate a more general model which allows for positive correlation between B and V.

For the buyer, V has a continuous subjective prior distribution function G(V) defined on (−∞, ∞).Footnote ⁶ For a payment H from buyer to seller, the resource is saved with probability G(H).

The objective function for the buyer is

(1) $$ ER=G(H)\left( EB-H\right)+\alpha \left[G(H)H-\underset{0}{\overset{H}{\int }} Vg(V) dV\right]. $$

α $ \in \left[0,1\right] $ is a weight given by the buyer to the net expected gain for the seller.Footnote ⁷ The resource is purchased with probability G(H) < 1. H is offered to some sellers whose values of converting the resource exceed H (with probability 1 − G(H)); these will not accept the offer H.

Two useful special (limiting) cases are:

1. (a) α = 1: the welfare of the seller is perfectly internalized by the buyer.Footnote ⁸ The buyer then maximizes “social welfare” related to protecting the resource, considering seller and buyer as the only participants in this market with preferences for the resource.Footnote ⁹ For “realistic” rainforest values, a “typical” case should imply B > V so that the rainforest plot is (almost always) worthy of protecting. Thus in “by far the most” cases, a correct decision will be made by setting H below (but not too much below) EB. However, as long as V is unknown to the buyer, the probability that V > EB remains; an efficient mechanism, with α = 1, requires that H = EB; which will be chosen when the buyer perfectly incorporates the seller’s preferences.

2. (b) α = 0: the utility of the seller plays no role for the buyer.

Cases where the buyer considers at least some “development impact” of its forest-saving policy are likely to involve α > 0. α = 1 is however extreme, but could be relevant when the buyer is a donor (as often in a North–South context), or the government of the seller’s country.Footnote ¹⁰

Maximizing Equation (1) with respect to H yields

(2) $$ \frac{dER}{dH}=g(H)\left( EB-H\right)-\left(1-\alpha \right)G(H)=0. $$

The optimal solution depends on the weight α given by the buyer to the utility of the seller (or formally here, to the net utility of the seller associated with conserving the forested land). We focus on two cases considered above, namely α = 1 and α = 0.

In case (a), α = 1, the solution simplifies to

(3) $$ EB=H. $$

Equation (3) represents a constrained efficient solution, overall, the best feasible under the information constraints given. The reason for (constrained) efficiency is that when α = 1, the buyer maximizes “global” expected utility with respect to H. This is not the case when α = 0.

In case (b), α = 0, we have the solution

(4) $$ EB-H=\frac{G(H)}{g(H)}>0, $$

where the right-hand side of Equation (4) is the “inverse hazard” rate, G(H)/g(H). A necessary second-order condition for maximum of Equation (1) must here hold:

(5) $$ \frac{d^2 ER}{(dH)^2}=g\hbox{'}(H)\left( EB-H\right)-2g(H)<0. $$

A sufficient (but not necessary) condition for this to hold is g′(H)  $ \le $  0.

From Equation (4), the optimal solution for the buyer when α = 0 is to select H < EB. This buyer is averse to making payments to the seller, and acts as a monopsonist in purchasing “offsets” from the seller, under-paying its expected value. This is similar to recent analyses of offset markets such as Strand (Reference Strand2013) and Rosendahl and Strand (Reference Rosendahl and Strand2014).

Consider $ \alpha \in \left(0,1\right) $ , and an associated payment H from buyer to seller. The cumulative distribution function for V is in this case conditional on V  $ \le $  H; and given by

(6) $$ {G}_C(V)=\frac{G(V)}{G(H)},V\in \left(-\infty, H\right], $$

with probability density function given by g(V)/G(H) on the same domain.

Consider effects on welfare in this case. Using Equation (2), we can write Equation (1) as

(7) $$ ER=G(H)\left[\left(1-\alpha \right)\frac{G(H)}{g(H)}+\alpha \left( EB-{E}_CV\right)\right], $$

where E_CV is the conditional expectation of V. Simulations may help us identify the probabilities of incorrect decisions (when the resource is saved for B − V < 0, and converted for B − V > 0), and how these decision are impacted by “risk parameters” (standard deviations of B and V). In the Appendix we derive the mathematical distribution for B − V = Z, and simulate this distribution (see also Figures 10–13 in Appendix 1).

Improved decisions to protect the resource can occur in terms of either more precise B, or more precise V whenever B and V are uncorrelated. In either case, only the total variance on Z matters, and this is affected in symmetric fashion by each. It also follows that the most valuable information improvements are those that most reduce this total variance.

4 The VOPI about B when B and V are uncorrelated

4.1 Mathematical derivation

What is the additional VOPI about B to the buyer, in the context of our optimal forest protection problem, when information is asymmetric and only the seller knows the opportunity value of the resource, V, for certain? We will now derive the solution to this problem in the simplest case where B and V are uncorrelated, and later in the section discuss a number of simulations, with more simulations discussed in Appendix 1. The mathematical solution for the correlated case is derived in the Appendix 2.

Under imperfect information about B, when the buyer perfectly incorporates the seller’s utility (α = 1), two types of mistakes can be made: the resource can be converted when it should not be; and the resource can be saved when it should be converted.Footnote ¹¹ This leads to welfare losses in an ex ante expected sense. These losses are avoided when information is perfect ex post, and the buyer acts optimally upon this information.

Improving the information about B reduces the uncertainty about this value for the buyer, and reduces the ex-ante welfare loss due to incorrect decisions. An equivalent way to represent this welfare improvement for the buyer is his maximum willingness to pay to acquire the additional information. We assume that informational improvements for the buyer are due solely to reductions in the standard deviation of B, which represents the uncertainty of B for the buyer.

Define the standard deviations of B and V by σ_B, and σ_V, respectively. The perfect information case with σ_B = 0 and thus no uncertainty about B, is useful as it provides us with an appropriate benchmark for the study of informational values in intermediate and more realistic, and more complex, cases where the initial uncertainty is only partly resolved.

In the objective function (1) one now needs to replace EB by B which is now known, and the optimal solution is Equation (2) replacing EB by B. Define D = B − H = the net realized surplus for the buyer related to this particular unit. We can then express the expected gain to the buyer, for given B and α, as follows:

(8) $$ ER\left(B;\alpha \right)=G\left(B-D\right)D+\alpha \left[G\left(B-D\right)\left(B-D\right)-\underset{0}{\overset{B-D}{\int }} Vg(V) dV\right]. $$

This solution is simple when α = 1. The buyer then always sets H = B, so that D = 0, giving all the surplus to the seller, and making sure that the resource is always saved when B > V. Since the seller knows V perfectly, the solution is then always Pareto optimal.

When α < 1, H < B. H will vary with B, and B − H will also generally vary. From Equation (5), only when the hazard rate G(H)/g(H) is constant will B − H be constant, independent of B.

Consider first a general value of α. Define M = B − V. The expected value of a random “project” (across the entire distribution of possible projects) is then, for the simplified case where D = B − H only depends on α Footnote ¹²:

(9) $$ ER\left(\alpha \right)=\left[1-M\right(D\left(\alpha \right)\Big]D\left(\alpha \right)+\alpha \left[\underset{B-V=D\left(\alpha \right)}{\overset{\infty }{\int }}\left(B-V-D\left(\alpha \right)\right)m\left(B-V-D\left(\alpha \right)\right)d\left(B-V\right)\right]. $$

The resource will here be protected with probability 1 − M((B − H)(α)), which increases in α.

When α = 1, the solution implements an optimal (first-best) allocation in terms of saving the resource, subject to the given informational constraints. From the modified relation (5), H then always equals B, and D(1) = 0. Equation (9) then simplifies to:

(10) $$ ER(1)=\underset{B-V=0}{\overset{\infty }{\int }}\left(B-V\right)m\left(B-V\right)d\left(B-V\right), $$

which expresses the (maximized) ex-ante expected social surplus of a random unit of the resource in this case (drawn randomly from the universe of possible B − V values), evaluated before perfect information about B is achieved, but such that the actual decision about H will be taken under perfect information. Note that when B and V are independent normal random variables, M = B − V is a normal random variable with mean E(B) − E(V).

Rewriting Equation (7) by setting α = 1 and EB = H, we get the value of imperfect information in this case which we call ER(1;I):

(11) $$ ER\left(1;I\right)=\left[G(H)H-\underset{0}{\overset{H}{\int }} Vg(V) dV\right]. $$

VOPI, defined the ex-ante VOPI about B achieved ex post, over the ex-ante value of imperfect information about B, is given by

(12) $$ ER(1)- ER\left(1;I\right)=\left[\underset{B-V=0}{\overset{\infty }{\int }}\left(B-V\right)m\left(B-V\right)d\left(B-V\right)\right]-\left[G(H)H-\underset{0}{\overset{H}{\int }} Vg(V) dV\right]. $$

This is equivalent to the buyer’s willingness to pay ex ante to have all uncertainty associated with B removed ex post; or put otherwise, to have σ_B reduced to zero ex post.

This solution is still extreme as the buyer, in implementing a first-best solution, collects no net benefit, as the purchase price H equals his surplus.

The VOPI in the case of α = 0 is similarly given by Equation (9) minus Equation (7), inserted for α = 0, which implies $ D=\frac{G(H)}{g(H)} $ . The closed-form solution can be obtained but involves limits that are dependent on the functional forms that define the problem.

4.2 Simulations for α = 1 and zero correlation

Figure 1 illustrates how changing EV influences the additional VOPI about B, with both B and V being normal, and independent, and with α = 1. We find that the VOPI is lower the more EV differs from EB. Intuitively, when 0 < < EV < < EB, the resource is always protected by the buyer in most cases, and having more precise information about B makes little difference for this decision. Similarly, when 0 < < EB < < EV, the resource will almost always be converted. By contrast, when EV and EB are close together, the decision on how to optimally manage the resource is highly sensitive to more precise information about the resource value. It then matters greatly for such decisions whether there is perfect or imperfect information, and this increases the VOPI.

Figure 1 The value of perfect information as function of EV; α = 1, zero correlation.

Figure 1 also illustrates that a higher σ_B leads to increased value of fully resolving the uncertainty (reducing σ_B to zero). This is intuitive: a higher σ_B leads to more and greater mistakes in saving or not saving the resource B. Resolving this uncertainty completely will, in the case of α = 1, eliminate all such mistakes.

When σ_V is increased simultaneously with σ_B, as pictured in Figure 2, the VOPI about B is reduced, as compared to the higher standard deviation imperfect information case for EV = EB. Intuitively, when V is more dispersed, the agent’s decision to convert or not convert the resource is less affected by a more precise assessment of B by the buyer. Most of the decision is left to the seller; the buyer can do little to improve the information. A lower standard deviation of V leaves more of the decision to the buyer, and increases the usefulness of perfect information. From Figure 2, this holds when EV = EB (=10).

Figure 2 The value of perfect information as function of σ_B (=σ_V); α = 1, zero correlation.

The functional shape of the VOPI in Figure 2 (the unbroken red line) is specific to equal expectations for V and B (EV = EB = 10); with normal and independent distributions. Consider instead different EV and EB; in Figure 2, EV = 10, EB = 11. The VOPI is then still positive, but changes nonmonotonically with the standard deviation. Two factors work in opposite directions: When σ_B and σ_V are small, protection is likely to be chosen regardless of V. The decision to protect the resource is then hardly affected by additional information about B. When standard deviations increase, this decision is affected more by more information about B. This increases the VOPI, which approaches its value for EV = EB as standard deviations grow.

In Figure 3 we increase σ_B keeping σ_V, EB and EV fixed, which increases the VOPI. We find a strictly convex relationship with σ_B (converging to linear as σ_B increases). Perfect information is now more valuable for the buyer as mistakes are reduced by more. Similarly, a partial increase in σ_V reduces the VOPI about B (albeit only slightly). The same holds when EB and EV differ. (The two cases considered, EV = 11, EB = 10; and EV = 10, EB = 11, are similar).

Figure 3 The value of perfect information as function of σ_B with σ_V = 1; α = 1, zero correlation.

5 Empirical magnitudes of the VOPI in a simple example of rainforest valuation

How significant is the VOPI; is it likely to matter economically? We discuss this issue by considering further parameterized examples, where the object valued is an area of a rainforest. We focus on the case of an altruistic donor (α = 1), and uncorrelated B and V. From Figures 1 and 3, gains from perfect information depend strongly on two magnitudes: higher when the relative expectations of B and V are more similar; and higher when the standard deviation on the prior distribution of B is large. Intuitively, when expectations are similar, the resource is “highly contested,” subject to a high level of competition, and the decision to save or not save the resource is highly sensitive to detailed information about B. More information about B is then highly valuable, and more so the more uncertain B is at the outset.

Figures 1–3 provide us with some notion of VOPI values. Consider first EV = EB = 1. The forest is then highly contested as the expected opportunity value equals the prior expected protection value. Assume that the standard deviation for the prior distribution of B is ½ EB: high but not unreasonable; while the standard deviations of B and V are equal (σ_B = σ_V). VOPI values are found to be basically proportional to these standard deviations when their levels are identical. The VOPI is then approximately 0.08 times EB. Footnote ¹³

To get a notion of this value, assume that 10% of the entire forest will be “contested” over the period in question (for the scenario depicted in Strand et al. (Reference Strand, Carson, Navrud, Ortiz–Bobea and Vincent2017), up to 2050, see also Strand et al. (Reference Strand, Soares-Filho, Costa, Oliveira, Ribeiro, Pires and Oliveira2018)). Protection and opportunity values are similar, and optimality of the decision to save or not save this part of the forest depends highly on its assessed protection value in relation to its opportunity value.Footnote ¹⁴ Assume that the parameter values are otherwise as in the last example; in particular EB = EV. Then the VOPI for this forest area would be 8% of its gross protection value.

These numbers could be large for the Amazon rainforest. Consider a gross (noncarbon; conservative) protection stock value of $5,000 per hectare.Footnote ¹⁵ 8% of this value is $400 per hectare, which will be the VOPI for this forest area when managed optimally for given information. With more than 500 million hectares of Amazon rainforest, 10% of the area means more than 50 million hectares of contested rainforest, with a VOPI of $20 billion. This corresponds to the (present discounted) value of providing perfect and comprehensive valuation data for this extent of the forest, when decisions about saving and not saving the forest are always optimal subject to given available information (improved in the VOPI case); and expected protection values and opportunity values are identical. Even if only ⅒ of the area considered above is subject to explicit evaluation, and using the lower σ_B value, the VOPI would still be $500 million.

For several reasons, the realistic VOI related to forest protection values could however be smaller. First, information may be extracted and used nonoptimally. Deforestation may take place with no detailed analysis of its value for any one particular forest plot. Uncertainties could also be smaller. If instead the standard deviation for the prior distribution of B was ¼ EB, instead of ½ EB, this would by itself reduce the VOPI value by half, though the value, $10 billion (in the most expansive case considered above), would still be very high. No study resolves all uncertainty; there is always residual uncertainty. Protection and opportunity values are also often less similar. From Figure 1, the more dissimilar EB and EV are, the lower is the VOPI. To exemplify, when EB and EV differ by one (two standard deviations on either of the variables), the VOPI from Figure 3 is reduced by a factor of about three. If EB − EV exceeds four standard deviations in absolute value, the VOPI virtually vanishes.

Consider a more limited case where we only include actual deforestation taking place in the Amazon in any given year, today about 1 million ha.Footnote ¹⁶ Assume that 10% of this area is contested in the sense that ex ante protection and opportunity values are similar, and the decision to deforest or not can be affected by precision of information. In this case, and with parameter values otherwise as above, the VOPI would be $4 million per year.

The other extreme parametric case is α = 0, where the buyer has no concern for the seller. From Figures 1 and 7, the VOPI values for α = 0 are approximately ⅔ of the VOPI values for α = 1. If decisions to save or not save contested rainforest areas are driven by preferences of the donor/principal (while preferences of the forest-managing country have no weight in this decision), the VOPI values would be scaled down by about ⅓ of the values for α = 1.

Our analysis might also provide clues to the VOI from studies that do not resolve all uncertainty, from Figure 3 (α = 1) and Figure 9 (α = 0), showing the VOPI as functions of the standard deviation of B for given standard deviation of V. A conjecture (so far not formally substantiated) is that the value gain from achieving a lower uncertainty estimate can be found by considering the difference in the VOPI value gains for different initial standard deviations on B. These gains seem to be close to proportional to reductions in the standard deviations for small risk changes; but less than proportional as the remaining risk drops toward zero. If so, a valuation study that reduces σ_B to half of its prestudy value will provide (slightly more than) half of the VOPI value. We intend to further analyze, and simulate, such cases in follow-up work.

Figures 3 and 9 indicate that eliminating the last bit of uncertainty (going from σ_B = 0.5 to σ_B = 0 for σ_V = 1) is likely not worthwhile: the additional utility value is low, while the cost of eliminating the last remaining uncertainty is likely to be high.

6 Discussion and final comments

This paper has analyzed a class of VOI problems, in the context of public-goods valuation with emphasis on environment and natural resources, using surveys or other data-acquisition methods, when it is costly to achieve a high degree of certainty about the true values. We use tropical rainforests as a canonical example: decisions to deforest or save a given rainforest area are made by local “sellers,” who are incentivized or influenced by “buyers” with preference for preserving the rainforest.

Our analysis also defines and represents a first attempt to systematically define, and analyze, the “cost–benefit analysis of public-good valuation projects or procedures,” where the costs of such valuation work are compared to the benefits of achieving more precise values (often, very difficult to measure). A key objective of this paper has been to make precise what is meant by the “value of valuation studies,” and how such a value or values can be measured. When an environmental valuation study is carried out, the expected benefits from the study are typically not compared to the costs of carrying out the study: benefits are usually not specified, and much less monetized. Thus, environmental and other public-good valuation project funders or executors have usually little idea about the VOI provided by the valuation study. In our view, such analysis ought to become a much more standard tool and procedure in the science and practice of valuation of public goods in a wide range of sectors including the environment, natural resources, health, transportation, and education.

Another novel aspect of our analysis is to relate the VOI problem to a mechanism design problem under asymmetric information, involving constrained optimal decisions in response to improved information. The buyer with interest to protect the resource seeks to extract information about the resource’s opportunity value known by the seller. In our canonical case, a rainforest area is either protected or converted to other uses (deforested) by the seller, and the buyer influences this decision through a payment to the seller conditional on saving or converting the resource. The rainforest has true preservation value B, and true opportunity value V. In Section 3, the seller is assumed to know V perfectly, while the buyer only knows a prior (Bayesian) probability distribution over B and V. For most of our simulations (in Section 4, and in Appendix), these prior distributions of B and V are assumed to be normal and independent. In deriving a (monetized) VOI value related to providing more information about B, we also focus on cases where uncertainty is eliminated completely by the valuation project considered. We compare a preinformation state where the buyer is uncertain about B; with the ex-ante expected value of obtaining a postinformation state where all uncertainty about B is resolved for the buyer. The resulting VOI is the “VOPI,” which we call the VOPI, which indicates the potential informational value. In realistic cases some uncertainty about B will always remain after a valuation study has been carried out, so that the VOI will be smaller than the VOPI. This also often happens in practice because such additional information is not applied optimally in the actual decision to protect or not protect the resource.

We show how the VOPI value depends on uncertainties about the true value of the resource for the buyer, B, and about its opportunity value to the seller, V; and on the relation between their expected values, EB and EV. More information about B is particularly helpful when EB and EV are similar, as the resource is then highly contested and the decision to save or not save the resource is affected by more precise information in “many” cases. The value of more precise information is also higher when B is more uncertain before a valuation study is carried out. In fact, for given uncertainty about V, the VOPI value increases more than proportionately to the initial standard deviation on B. This is reasonable as the buyer then makes many and perhaps grave mistakes when acting upon only prior information; such mistakes are avoided when this uncertainty is resolved through the valuation studies. The informational value is also higher when the standard deviation of V is smaller (within certain bounds). When the standard deviation of V is large, the decision to save the resource is less affected by B’s precision.

We show in the simulations that the VOPI as share of the gross protection value of the resource depends heavily on the prior standard deviation of B relative to the gross protection value. Some simple back-of-the-envelope calculations, discussed in Section 5, illustrate the potential values of providing more precise resource value information, and show that the VOPI can be quite large in seemingly reasonable cases.

While we have not provided a complete proof, our results (in particular, the simulations in Figures 3 and 9) indicate that eliminating only part of the uncertainty through valuation work more than proportionately reduces the value loss due to imperfect information. This indicates a falling marginal return to more precise information, when a certain precision has been reached.

Our analysis does not consider nor discuss the particular costs of carrying out the valuation work to provide the required information. But an implication of our analysis is that very significant expenditures could in many cases be justified for providing the added information. Further details have not been pursued vigorously here, and must be left for future research.

Our presentation focuses on two extreme types of “buyer” preferences: the buyer cares fully about the outcome for the seller; or the buyer is completely selfish. Much of our analysis is based on the first assumption, which is often unrealistic. Section 4.3 and Appendix 1 consider the case of a selfish buyer. The solution is then generally inefficient as the resource is protected in fewer cases; the payment from buyer to seller is lower; and overall efficiency is reduced; and VOI and VOPI are both reduced by between ½ and ⅓; and saving the resource becomes less frequent. This demonstrates a novel, but intuitive, result that the value of information depends on the way in which information is used: when decisions made upon new information are less efficient, informational values tend to be lower.Footnote ¹⁷

Our analysis has several limitations. All examples are based on standard normal distributions for B and V, which can take (unbounded) positive and negative values. Negative gross rainforest values are of course unrealistic. On the other hand, only the difference Z = B − V matters in terms of efficiency; this difference could in principle be (even large) negative.

For most of our analysis we assume that protection and opportunity values (in our specified case for rainforests) are independent. This is probably not the case for most tropical rainforests, in particular if values (both preservation values, and exploitation values) depend heavily on easy access to the forest area. But some key aspects of rainforest value, such as carbon content and biodiversity, could have low positive (or even negative) correlations with opportunity values. The paper includes, in Figures 10 and 11, simulations of the VOPI when EB = EV, and B and V are positively (although not very highly) correlated. We show that the VOPI may then be either increased or reduced; but in the cases we study they are not substantially changed.

We have assumed that V, the resource opportunity value, is uncertain for the buyer but certain for the seller. In practice, there is hardly ever perfect information about V for the seller, and such uncertainty could also be reduced through valuation work. In a BCA context, it is uncertainty about B-V which is relevant. In our model context, a more precise assessment of V is useful only for a buyer who is not fully altruistic. When V is not known for certain by any party, additional effort to find a more precise V would be useful.

Our set-up to address VOI issues is constrained to natural resource values where a particular game is constructed and solved. Different public-good valuation problems take different forms. A common case is where a valuation study is done as basis for a public decision to take an environmental action or protect a natural resource, where there is no game structure like the one in our paper. With an unconstrained, “benevolent,” government, the problem would then correspond closely to our simplest case, α = 1. Financially constrained governments could then be modeled as in Mason and Plantinga’s (Reference Mason and Plantinga2013), with marginal costs of public funds exceeding unity. But further complications can arise when the information is not used optimally by the implementing government (the “buyer” in our terminology). The VOI will then be below its maximum. These are among issues that will need to be addressed in future research on this topic, which we consider as highly desirable.