Pay for the Option to Pay? The Impact of Improved Scientific Information on Payments for Ecosystem Services

Abstract

The scientific information needed to precisely link land use changes to changes in ecosystem service provision is often unavailable, particularly for hydrological ecosystem services. Potential buyers of ecosystem services must weigh the risk of paying to induce conservation that fails to deliver valuable ecosystem services against the risk that land is developed before its value is known. This paper uses a two period model to assess when expected future improvements in scientific information should substantially impact payments for conservation today. Optimal current payments increase, often substantially, as the quality of expected future information increases, but the gain from increasing payments to account for the expected degree of improved information is small in many cases. Larger values occur when the buyer believes that the land whose private development value is the highest also provides the highest ecosystem service value, when the buyer faces relatively more uncertainty about ecosystem service provision than about the cost of inducing conservation, and when the buyer believes land is highly likely to be developed absent incentive payments.

Appendices

Appendix A

1.1 Simplification of Buyer’s Payoff

The buyer’s single period payoff as a function of the payment offered is given by

$$\begin{aligned} W\left( \phi ,s\right) =\intop _{-\infty }^{\phi }\intop _{-\infty }^{\infty }\left( e-\phi \right) f_{e|s,p}\left( e|s,p\right) def_{p|s}\left( p|s\right) dp \end{aligned}$$

The multivariate normal assumption allows us to simplify this expression considerably. First, note that

$$\begin{aligned} \mu _{e|s,p}\left( s,p\right) =\frac{\sigma _{e}\left( \rho _{ep}-\rho _{ep}\rho _{sp}\right) }{\sigma _{p}\left( 1-\rho _{sp}^{2}\right) }p+\mu _{e}+\frac{\sigma _{e}\left( \rho _{es}-\rho _{ep}\rho _{sp}\right) }{\left( 1-\rho _{sp}^{2}\right) }s-\frac{\sigma _{e}\left( \rho _{ep}-\rho _{ep}\rho _{sp}\right) }{\sigma _{p}\left( 1-\rho _{sp}^{2}\right) }\mu _{p} \end{aligned}$$

so the inner integral can be rewritten as

$$\begin{aligned} \intop _{-\infty }^{\infty }\left( e-\phi \right) f_{e|s,p}\left( e|s,p\right) de= & {} \intop _{-\infty }^{\infty }ef_{e|s,p}\left( e|s,p\right) de-\intop _{-\infty }^{\infty }\phi f_{e|s,p}\left( e|s,p\right) de\\= & {} \mu _{e|s,p}\left( s,p\right) -\phi \\= & {} \frac{\sigma _{e}\left( \rho _{ep}-\rho _{es}\rho _{sp}\right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }p+\mu _{e}+\frac{\sigma _{e}\left( \rho _{es}-\rho _{ep}\rho _{sp}\right) }{\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }s\\&-\frac{\sigma _{e}\left( \rho _{ep}-\rho _{ep}\rho _{sp}\right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\mu _{p}-\phi \end{aligned}$$

To simplify derivations, let $\alpha =\mu _{e}+\frac{\sigma _{e}\left( \rho _{es}-\rho _{ep}\rho _{sp}\right) }{\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }s-\frac{\sigma _{e}\left( \rho _{ep}-\rho _{ep}\rho _{sp}\right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\mu _{p}-\phi$ and let $\beta =\frac{\sigma _{e}\left( \rho _{ep}\left( 1-\rho _{es}^{2}\right) \right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }$. The full first integral is thus

$$\begin{aligned} \intop _{-\infty }^{\phi }\left( \alpha +\beta p\right) f_{p|s}\left( p|s\right) dp \end{aligned}$$

or

$$\begin{aligned} \alpha F_{p|s}\left( \phi |s\right) +\beta \intop _{-\infty }^{\phi }pf_{p|s}\left( p|s\right) dp. \end{aligned}$$

The latter integral can also be simplified under the normality assumption to

$$\begin{aligned} \sigma _{p|s}^{2}\left( -f_{p|s}\left( \phi |s\right) \right) +\mu _{p|s}F_{p|s}\left( \phi |s\right) . \end{aligned}$$

Combining elements, we have the full first integral as

$$\begin{aligned} \alpha F_{p|s}\left( \phi |s\right) +\beta \left( \sigma _{p|s}^{2}\left( -f_{p|s}\left( \phi |s\right) \right) +\mu _{p|s}F_{p|s}\left( \phi |s\right) \right) \end{aligned}$$

or

$$\begin{aligned} \left( \alpha +\beta \mu _{p|s}\right) F_{p|s}\left( \phi |s\right) -\beta \sigma _{p|s}^{2}f_{p|s}\left( \phi |s\right) . \end{aligned}$$

We also know that

$$\begin{aligned} \mu _{p|s}=\mu _{p}+\sigma _{p}\rho _{sp}s\end{aligned}$$

and

$$\begin{aligned} \beta \mu _{p|s}= & {} \frac{\sigma _{e}\left( \rho _{ep}\left( 1-\rho _{es}^{2}\right) \right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\left( \mu _{p}+\sigma _{p}\rho _{sp}s\right) \\= & {} \frac{\sigma _{e}\left( \rho _{ep}\left( 1-\rho _{es}^{2}\right) \right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\mu _{p}+\frac{\rho _{sp}\left( \rho _{ep}\left( 1-\rho _{es}^{2}\right) \right) }{\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\sigma _{e}s\\ \alpha +\beta \mu _{p|s}= & {} \mu _{e}+\frac{\sigma _{e}\rho _{es}\left( 1-\rho _{ep}^{2}\right) }{\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }s-\frac{\sigma _{e}\rho _{ep}\left( 1-\rho _{es}^{2}\right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\mu _{p}-\phi \\&+\,\frac{\sigma _{e}\rho _{ep}\left( 1-\rho _{es}^{2}\right) }{\sigma _{p}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\mu _{p}+\frac{\rho _{es}\rho _{ep}\rho _{ep}\left( 1-\rho _{es}^{2}\right) }{\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\sigma _{e}s\\= & {} \mu _{e}+\rho _{es}\sigma _{e}s-\phi \end{aligned}$$

Finally, note that

$$\begin{aligned} \sigma _{p|s}^{2}=\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) \end{aligned}$$

so

$$\begin{aligned} \beta \sigma _{p|s}^{2}= & {} \sigma _{e}\sigma _{p}\rho _{ep}\left( 1-\rho _{es}^{2}\right) . \end{aligned}$$

Putting all these pieces together, we have

$$\begin{aligned} W\left( \phi ,s\right) =-\,\sigma _{e}\sigma _{p}\rho _{ep}\left( 1-\rho _{es}^{2}\right) f_{p|s}\left( \phi |s\right) +\left( \mu _{e}+\rho _{es}\sigma _{e}s-\phi \right) F_{p|s}\left( \phi |s\right) \end{aligned}$$

Appendix B

1.1 Proof of Proposition 1

Case1$\bar{p}$ is finite.

If $\bar{p}$ is finite, then the interval $\left[ 0,\bar{p}\right]$ is closed and bounded and since $W\left( \phi ,s\right)$ is continuous, it attains a maximum on this interval.

Case2 $\bar{p}=\infty$

In this case, the buyer believes the landowner will choose to conserve in period 1 regardless of the value of $p$. The $\lim _{\phi \rightarrow \infty }W\left( \phi ,s\right) =-\infty$ so there exists a value $\hat{\phi }$ such that $W\left( \phi ,s\right) <W\left( \hat{\phi },s\right)$ for any $\phi >\hat{\phi }$. Again, continuity of $W\left( \phi ,s\right)$ implies that it attains a maximum on the closed and bounded interval $\left[ 0,\hat{\phi }\right]$, with the maximized value greater than or equal to $W\left( \hat{\phi },s\right)$. Thus, this value also maximized $W\left( \phi ,s\right)$ on $\phi \ge 0$.

Appendix C

1.1 Proof of Proposition 2

A local interior maximum requires $\frac{\partial W}{\partial \phi }>0$ and $\frac{\partial ^{2}W}{\partial \phi ^{2}}\le 0$.

$$\begin{aligned} \frac{\partial W}{\partial \phi }\left( \phi ,s\right) =-\,\sigma _{e}\sigma _{p}\rho _{ep}\left( 1-\rho _{es}^{2}\right) \frac{df_{p|s}}{dp}\left( \phi |s\right) +\left( \mu _{e}+\rho _{es}\sigma _{e}s-\phi \right) f_{p|s}\left( \phi |s\right) -F_{p|s}\left( \phi |s\right) \end{aligned}$$

Since $p|s\sim N\left( \mu _{p}+\sigma _{p}\rho _{es}\rho _{ep}s,\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) \right)$, we know that

$$\begin{aligned} \frac{df_{p|s}}{dp}\left( \phi |s\right) =-\frac{\phi -\mu _{p}-\sigma _{p}\rho _{es}\rho _{ep}s}{\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }f_{p|s}\left( \phi |s\right) . \end{aligned}$$

giving

$$\begin{aligned} \frac{\partial W}{\partial \phi }\left( \phi ,s\right)= & {} f_{p|s}\left( \phi |s\right) \left[ \left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right) \phi +\sigma _{e}\rho _{es}\left( 1-\gamma \rho _{ep}^{2}\right) s+\mu _{e}-\frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}\gamma \mu _{p}\right] \nonumber \\&-F_{p|s}\left( \phi |s\right) \end{aligned}$$

(10)

where $\gamma =\frac{1-\rho _{es}^{2}}{1-\rho _{es}^{2}\rho _{ep}^{2}}$. The second derivative is

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)= & {} f_{p|s}\left( \phi |s\right) \left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right) +\frac{df_{p|s}}{dp}\left( \phi |s\right) \left[ \left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right) \phi \right. \\&\left. +\,\sigma _{e}\rho _{es}\left( 1-\gamma \rho _{ep}^{2}\right) s+\,\mu _{e}-\frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}\gamma \mu _{p}\right] -f_{p|s}\left( \phi |s\right) . \end{aligned}$$

Using the same substitution as above, we get

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \phi ^{2}}= & {} f_{p|s}\left( \phi |s\right) \left[ \left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right) +\left( \frac{\mu _{p}+\sigma _{p} \rho _{es}\rho _{ep}s-\phi }{\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\right) \left[ \left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right) \phi \right. \right. \\&\left. \left. +\,\sigma _{e}\rho _{es}\left( 1-\gamma \rho _{ep}^{2}\right) s+\,\mu _{e}-\frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}\gamma \mu _{p}\right] -1\phantom {\left( \frac{\mu _{p}+\sigma _{p} \rho _{es}\rho _{ep}s-\phi }{\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }\right) }\right] . \end{aligned}$$

Letting $a=\left( \gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}-1\right)$, $b=\frac{1}{\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }$, $c=\frac{\mu _{p}+\sigma _{p}\rho _{es}\rho _{ep}s}{\sigma _{p}^{2}\left( 1-\rho _{es}^{2}\rho _{ep}^{2}\right) }$, and $d=\sigma _{e}\rho _{es}\left( 1-\gamma \rho _{ep}^{2}\right) s+\mu _{e}-\frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}\gamma \mu _{p}$, we have

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \phi ^{2}}=f_{p|s}\left( \phi |s\right) \left[ a+\left( c-b\phi \right) \left( a\phi +d\right) -1\right] \end{aligned}$$

or

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \phi ^{2}}=f_{p|s}\left( \phi |s\right) \left[ a+\left( ca\phi +dc-ba\phi ^{2}-bd\phi \right) -1\right] \end{aligned}$$

or

$$\begin{aligned} \frac{\partial ^{2}W}{\partial \phi ^{2}}=f_{p|s}\left( \phi |s\right) \left[ -ba\phi ^{2}+\left( ca-bd\right) \phi +a+dc-1\right] . \end{aligned}$$

Since $f_{p|s}\left( \phi |s\right) >0$ for all $\phi$, $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) =0$ only when the term in brackets equals 0. This expression is quadratic in $\phi$, implying that $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)$ has at most two zeros. If there are multiple interior local maxima of $W\left( \phi ,s\right)$, they must both have $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) \le 0$ and be separated by an interior minimum where $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) \ge 0$. This implies that there must be two zeros of $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)$ between the local maxima. Since $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)$ has at most two zeros, this requires that the sign of $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)$ is the same for any value of $\phi$ above the largest value of $\phi$ that results in a local maximum. Since we must have$\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) <0$ and $\frac{\partial W}{\partial \phi }\left( \phi ,s\right) =0$ at the largest local maximum, this requires that $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) <0$ and $\frac{\partial W}{\partial \phi }\left( \phi ,s\right) =0$ for any value of $\phi$ above the largest value of $\phi$ that results in a local maximum. Moreover, it requires that $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) <0$ and $\frac{\partial W}{\partial \phi }\left( \phi ,s\right) >0$ for any value of $\phi$ below the smallest value of $\phi$ that results in a local maximum. Note that the $\lim _{\phi \rightarrow \infty }-\,ba\phi ^{2}+\left( ca-bd\right) \phi +a+dc-1=-\,ba\left( \infty \right)$ and $b>0$, suggesting that the sign of $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right)$ at very high values of $\phi$ is the opposite of the sign of a. Since we must have $\frac{\partial ^{2}W}{\partial \phi ^{2}}\left( \phi ,s\right) <0$ at high values of $\phi$, we can only have two local maxima of $W\left( \phi ,s\right)$ if $a>0$. The $\lim _{\phi \rightarrow -\infty }\frac{\partial W}{\partial \phi }\left( \phi ,s\right) =\left( 1-\gamma \frac{\sigma _{e}\rho _{ep}}{\sigma _{p}}\right) \infty =-a\infty$. Thus, at very low values of $\phi$, the sign of $\frac{\partial W}{\partial \phi }\left( \phi ,s\right)$ is opposite the sign of a. So if $a>0$, then $\frac{\partial W}{\partial \phi }\left( \phi ,s\right) <0$ at very low values of $\phi$, which contradicts the requirement that $\frac{\partial W}{\partial \phi }\left( \phi ,s\right) >0$ for values of $\phi$ below the smallest value of $\phi$ that results in a local maximum. Since $W\left( \phi ,s\right)$ has at most one interior local maximum on $\left( -\infty ,\infty \right)$, it has at most one interior local maximum on $\left[ 0,\bar{p}\right]$.

Appendix D

1.1 Numerical Simulation Details

As described in Sect. 3, neither the first nor the second period buyer optimization problems have closed form solutions. To explore the implications of the model, I use numerical simulations programmed in Matlab. The second period problem is a fairly straightforward maximization problem. There is a closed form expression for both the buyer’s objective function and the first order conditions for an interior maximum and we know that there is at most one interior local maximum. In contrast, the first period problem is fairly complex because the buyer’s objective function involves taking expectations over possible future signals, recognizing that the second period offer will be optimized in response to the observed signal. I use Matlab’s built-in fmincon solver to solve the first period maximization problem. Within the buyer’s objective function, I use a vectorized global adapative quadrature algorithm from the Mathworks File Exchange (quadvgk) to calculate the integral in Eq. 2. The quadvgk algorithm works by picking sets of values for the variable of integration (the observed signal) and computing the integrand simultaneously for each of these possible values to economize on computations.

Since the value of the integrand is the optimized second period payoff conditional on a particular signal times the probability of observing that signal in period 2, the function called by quadvgk must compute the solution to the second period optimization problem for many observed signal values. To solve this maximization problem efficiently for multiple observed signal values simultaneously I use the nonlinear complementarity problem solver ncpsolve from the CompEcon 2016 toolbox, which looks for solutions to the Kuhn-Tucker conditions of the individual maximization problems simultaneously.¹⁶

For most parameter sets, the second period problem is well behaved and solves quickly and easily, but three issues can arise. First, as described in Sect. 2.1, there is a single local interior maximum to the buyer’s problem, but the true maximum may occur at one of the endpoints and there may be local minima within the interval. This can lead the ncpsolve algorithm to get stuck in a region away from the solution. I use a coarse grid search on possible values to identify good starting points for the solver algorithm to avoid this problem. Second, for some parameter sets, the buyer’s objective function is extremely flat. These examples represent cases with a very high expected private value relative to the expected conservation value. The buyer will receive a payoff of 0 by making no offer but can increase the payoff a very small amount by making an offer arbitrarily close to the conditional expected benefit of conservation. The offer is virtually certain to be rejected and hence yields the buyer an expected payoff of almost but not quite zero. When this gain gets too small, the algorithm may fail to identify this true maximum. I supplement the ncpsolve algorithm with checks to identify these cases and manually set the optimal offer in these cases. Finally, for some parameter cases, the ncpsolve algorithm is unable to solve the period 2 problem simultaneously for all of the different signal values due to large scaling differences between the cases. When the algorithm fails in these cases, I revert to solving the remaining individual problems case by case using Matlab’s built-in nonlinear constrained optimization algorithm (fmincon). The simulation codes are present in the electronic supplementary material of this article.

Appendix E

1.1 Supplemental Graphs

1.1.1 E.1 Benefit of Improved Information

In Fig. 6, the impact of information on optimal values is shown by the different lines. It can be difficult to see how the size of these differences varies from this figure. Figure 9 emphasizes these difference by subtracting the value for $\rho _{es}=0$ (e.g. the lightest line in Fig.  6) from the lines for each remaining value of $\rho _{es}$ and plotting the result.

1.1.2 E.2 Impact of Correlation

Figure 10 replicates the left-hand columns of Fig. 5 with negative and positive correlation. Two effects are apparent from this figure. First, in the left hand column, we see that with negative correlation, the optimal offer first increases and then decreases as the observed signal of conservation value rises. At low signal value values, the increase in the expected conservation value dominates and leads to higher offers. When a high signal is observed, this causes the buyer to believe the development is likely to be low, suggesting that the parcel can be induced to conserve with a lower offer. Second, we see that when there is correlation, the signal reveals information about the expected development value, so the probability the parcel is in each location in the figure changes as the signal increases. With negative correlation, the probability of observing high p and low s or low s and high p increases, while with negative correlation, the probability of observing a high p and a high s or a low p and a low s increases.

Fig. 9

Change in optimal outcomes due to improved information

Figures 11, 12, and 13 illustrate the impact of correlation on optimal offers, the probability of first period conservation, and the optimized expected buyer payoff.

Fig. 10

Impact of correlation on parcel categorization

Fig. 11

Impact of correlation on the the increase in optimal first period offer for various levels of future information across different parameter cases

Fig. 12

Impact of correlation on the the increased probability of first period conservation for various levels of future information across different parameter cases

Fig. 13

Impact of correlation on the increase in buyers’ optimized payoff for various levels of future information across different parameter cases