• Open Access

Optimal contract lengths for voluntary ecosystem service provision with varied dynamic benefit functions

Authors


  • Editor
    Stephen Polasky

Correspondence
Amy W. Ando, Department of Agricultural and Consumer Economics,
University of Illinois, 326 Mumford Hall,
1301 W. Gregory Dr., Urbana, IL 61801, USA.
Tel: +(217) 333-5130; fax: +(217) 333-5538. E-mail: amyando@illinois.edu

Abstract: 

Programs to stimulate provision of terrestrial environmental services yield benefits that result from complex dynamic functions; we study optimal contract lengths for such programs. We show that the optimal contract length depends on a tradeoff between an ecological effect—long contracts increase the environmental benefits from one landowner—and an enrollment effect—long contracts decrease the number of landowners enrolled. In a simplified model of the contracting environment, we find that optimal contracts are longer when environmental benefits mature slowly, but it may not be best to offer permanent easements or even to offer contracts as long as the maturation period for the benefits in question. Nonecological characteristics matter to contract design as well; optimal contracts are longer where the turnover rate of parcels enrolled in conservation programs is high and where the average private land income is low.

Introduction

Programs exist worldwide that pay landowners to change land use activities in support of a wide range of ecosystem services and goods such as wildlife habitat, runoff filtration, and nutrient recycling. Such programs include government policies to promote conservation practices on agricultural lands (USDA 2007). Contract lengths offered for U.S. conservation programs vary from a few years to indefinite, while private conservation groups rely primarily on permanent purchase or easements. However, no research has informed those choices. The benefits from conservation practices result from complex and varied dynamic functions; hence, contract length may affect the total benefits that these programs produce. We provide a novel analysis of the optimal lengths of contracts for such conservation activities that helps conservation groups and agencies think about how contract lengths can be varied across situations to promote more conservation benefits for the same cost.

Optimal contract length has been explored in labor (Rey & Salanie 1990) and transaction-cost economics (Shelanski & Klein 1995), but that literature does not study how contracts should vary with the dynamic processes through which benefits mature. Gulati & Vercammen (2005) study optimal contracts for carbon sequestration assuming land returns to pre-contract use after a single contract expires. However, experience with the U.S. Conservation Reserve Program shows that many landowners choose to renew their conservation contracts when the contracts expire (Skaggs et al. 1994; USDA 2007).

We use theory and simulation to investigate the optimal contract length of land conservation programs. We show how optimal contract lengths vary with characteristics of economic conditions and the ecological processes that generate environmental benefits over time, and we discuss the possible implications of extensions of our basic model.

Methods

We constructed a simple theoretical economic model to find the land-conservation contract length and payment that maximize environmental benefits when total spending has a limit and when landowner participation is voluntary; details of the model are in the Appendix. The model is in an agricultural setting where landowners are farmers, but we did not attempt to model the precise details of any particular program, and the intuition of the results is not limited to agricultural programs. We modeled the government's goal as one of maximizing environmental benefits rather than social welfare (the sum of farmer and taxpayer well-being and environmental benefits) subject to a budget constraint because the former better matches the goals of most conservation groups and agencies.

In our model, a farmer must choose whether or not to accept a conservation contract. If the farmer does not accept, he receives benefits from farming (denoted Bf) each year. If the farmer accepts the contract, he receives a lump sum payment P. At the end of the L year contract, the farmer chooses whether to enter a new contract with the same payment and length.

For simplicity, we assumed the farmer to be risk neutral (indifferent between a certain payment and an uncertain payment with the same expected value) and we abstracted from the fact that many farmers enjoy nonmonetary benefits both from farming and from engaging in environmental stewardship (Smith & Shogren 2002). We also assumed that Bf follows a process such that the expected value of a farmer's benefit from farming does not change over time; that allows us to abstract from phenomena such as option values to isolate the extent to which optimal contract lengths and payments might vary just as a result of differences in the ecological processes that determine benefits. In this stylized world, a given farmer will never change his mind about whether or not to enroll in the program because nothing about the problem he faces will have changed when the contract ends. We discuss later the implications of relaxing some of these assumptions.

We solved for the relationship between contract length L and payment P that will make a farmer with benefits Bf indifferent between farming and entering land conservation programs. Intuitively, the lump-sum payment must compensate for all the years of benefits the farmer could reap from the land if it were not in a conservation contract. Because Bf varies across farmers, the smallest payment needed to induce farmers to enroll land in the program varies across farmers for a given contract length L. For simplicity we assumed that the government offers only a single price and contract length to all farmers. Our model can be extended to permit menus of contract lengths and payments; some of the qualitative results presented here regarding the relationships between ecological benefit functions and optimal contract lengths still apply (Chen 2006).

We assumed that farmers vary in the net incomes they derive from farming, with average farm income equal to μ, and quantified the fraction of the population of farmers that will enroll in the program for given values of P and L. Holding other variables constant, a higher payment leads to more enrollments, as that payment is enough to compensate more high-farm-income farmers for retiring their land. For a given payment, a longer contract length results in fewer enrollments because more years in the contract forces a farmer to give up more years of farm income. For any given payment and contract length, total program enrollment decreases with average farm income in the population because high farm income makes it more profitable for farmers to keep farming and forego the conservation payment.

When a parcel is enrolled in the conservation program, annual environmental benefits from that parcel change in a manner dictated by ecological processes, represented here by the function E(t). We assumed that once the parcel drops out of enrollment the benefit flow falls to zero. If the parcel is later re-enrolled, benefits again flow but with the time counter re-set to zero. Because ecological heterogeneity is not a primary focus of this research, we assumed that all farms in the region have the same environmental benefits function so that these functions can be aggregated simply across farms. When lands have heterogeneous benefits, the optimal set of payments and contract lengths would tend to have higher payments for lands with higher benefits; future work could explore this further.

Based on research on ecological restoration (e.g., MacArthur & MacArthur 1961; Bosch & Hewlett 1982; Forman & Godron 1986; Hobbs & Harris 2001), we specified the environmental benefits function as logistic: E(t) = 1/(1 +keat). A typical logistic function rises up to and levels off at a constant horizontal limit. By varying parameters a and k, our model can simulate a variety of stylized ecological situations. Some important ecosystem benefits might accrue over time according to qualitatively different functional forms so our model may not apply to all land conservation problems. However, other researchers can apply our framework to alternative environmental benefit function types.

With a stationary stochastic process governing income from each farm over time, the number of farmers that enroll in the program is constant over time for fixed payment and contract length. If there were a trend in the returns to farming, payments would need to move with the trend if the program manager wanted to keep enrollment steady; future work could explore the exact effect of a trend on the optimal path of payments and contract lengths over time. In the basic model described thus far, the identity of the parcels enrolled in the program remains fixed in perpetuity. However, many conservation groups worry that unforeseen factors might prevent lands from being re-enrolled if short-term contracts are used. Therefore, we allowed for that phenomenon in our model by assuming that at each new contracting period, some lands in conservation programs are not re-enrolled while some lands outside the program are enrolled for the first time. Such churning might very well occur because of factors like changes in land ownership (either through sale or inheritance); a new landowner may have different preferences over conservation and farming than the old one. We used α to indicate the percentage of enrollments that are new at each contracting period—the churning factor.

We developed one version of the model in which the government seeks to maximize the present discounted value of environmental benefits from all enrolled parcels given an infinite number of rounds of contracting where total spending on the program over time is limited by a total budget. The environmental benefits from each contracting period equal the benefits from all new enrollments plus benefits from all the lands that are already in the program. Benefits in a given period are higher for lands that are not newly enrolled because environmental processes there have had time to develop and mature.

Some conservation programs might be designed only for one contracting cycle. Thus, we also modeled cases in which the policymaker seeks to maximize the cost-effectiveness of a conservation program: the difference between the benefits of a single round of contracts and its associated cost.

The problems described above (and represented in Equations 7 and 8 of the Appendix) are too complex to derive analytical results. Hence, we used Matlab to simulate optimal program design. To illustrate our model in an agricultural setting, we used the following parameter values for our simulations: average net farm income per acre equals $100; the standard deviation of farm income equals $100; the discount rate equals 5%; and the churning rate equals 20%. The effects of changes in some of the parameters were explored in analyses reported below.

In many of the simulations, we used two sample environmental benefits functions illustrated in Figure 1. The solid line represents a grassland case with k=a= 1 such that environmental benefits start out high and level off after very short periods of time (e.g., erosion control benefits from perennial grass) while the dashed line corresponds to a forest case (k= 10,000 and a= 0.25) in which forest benefits are zero on the day the new trees are planted and it takes much longer for the benefits to reach their maximum. The maximum benefits were just normalized to equal 100 for both cases; readers should focus on relative values of payment and contract length between scenarios rather than absolute values of the results.

Figure 1.

Environmental benefits functions for two baseline ecological processes. The solid line shows annual benefits over time for a grassland case with E(t) = 100/(1 + et). The dashed line shows annual benefits over time for a forest case with E(t) = 100/(1 + 10000e−.25t). For both processes, maximum annual benefits are normalized to be equal to 100.

Results

We found the total environmental benefits that can be achieved by varied combinations of payments and contract lengths (Figure 2). For a benefit that matures quickly, total program benefits are decreasing in contract length because of a strong negative enrollment effect of farmers refusing to enroll in long contracts (panel a). However, for an environmental benefit that matures slowly, there is a range of contract lengths over which total environment benefits can increase with contract length (panel b). The ecological effect of longer contracts increases benefits by reducing the frequency with which some parcels can be taken out of conservation status; even if withdrawn parcels are replaced by new enrollments, newly enrolled parcels have lower benefit levels than the parcels on which benefits had been maturing. However, the negative enrollment effect eventually dominates the ecological effect of having even longer contracts. Hence, the total program benefits of a given lump-sum payment are first increasing and then decreasing in contract length.

Figure 2.

Contour lines of total programmatic environmental benefits. The curves in the two panels show combinations of contract length and payment that yield the same total environmental benefits from the program. Panel (A) uses the grassland benefit function, and panel (B) uses the forest benefit function.

Intuitively, the optimal contract length (L*) is found by overlaying the contour line for the budget constraint (Figure 3) on the contour map for environmental benefits and choosing the combination of payment and contract length that lies on the budget constraint and that yields the greatest benefits. We did simulations (Figure 4) that explored how L* varies with economic factors. Panel (a) shows the impact of the churning factor, α, on L*. The optimal contract length is longer when parcels are more likely to be pulled out of the program instead of being re-enrolled because that churning interrupts the maturation of benefits on the land. While this effect exists for both types of ecosystem benefits, the effect is larger in absolute value for benefits that mature slowly because those contracts are generally longer, and because the harm done by frequent non-re-enrollment in the program is greater when maturation is slow. Panel (b) shows the impact of mean farm income (μ) on L*. There is a negative relationship between μ and L* because high farm income makes it difficult to entice farmers to enroll their lands in a conservation program, and one way to increase total enrollment for a given payment is to reduce the length of the contract.

Figure 3.

Contour lines of total budget outlays. The curves show combinations of contract length and payment that give the same present discounted value of the sum of all payments made for the program.

Figure 4.

Effects of economic variables on optimal contract length. Panel (A) shows how optimal contract length for a program with many rounds of contracts varies with the churning factor for both the grassland and the forest case. Panel (B) shows how optimal contract length varies with average farm income for both the grassland and the forest case.

We investigated how L* varies with characteristics of the environmental benefit function. One important feature of that function is its Y-intercept, which indicates the level of annual benefits at the beginning of the contract as a fraction of the maximum benefits that will accrue each year once the benefit matures. Another important feature of the benefit function is the rate of benefit increase, as measured by the parameter a. We found (Figure 5) that shorter contracts should be offered when conservation aims to provide environmental benefits that mature quickly. Long contracts are useful for preventing turnover that hinders maturation of benefits on the land, but if benefits mature quickly, contracts do not need to be very long in order for significant benefits to accrue.

Figure 5.

Effects of ecological variables on optimal contract length. Panel (A) shows how optimal contract length for a program with many rounds of contracts varies with the level of annual benefits at the beginning of the contract as a fraction of the maximum benefits that will accrue each year once the benefit matures. Panel (B) shows how optimal contract length varies with the rate of benefit increase.

To further explore this effect, we constructed a scatter plot of L* against benefit maturation age for all of all the simulations conducted for Figure 5 (see Figure 6). This graph shows that L* tends to be shorter than the amount of time needed for an environmental benefit to mature when farmers have repeated options to re-enroll in the program even when the churning rate is high. The optimal contract length may fall short of maturation time in our model because long contracts are expensive and much benefit growth occurs in the early part of a logistic process, so it is worth the risk of non-re-enrollment to allow the contract to end a little before maturation in order to save on payment costs.

Figure 6.

Relationship between optimal contract length and maturation age. This graph plots the optimal contract length against the maturation age for all the simulations that were done to produce the results of Figure 5. Maturation is defined as the point in time where 99% of the maximum annual benefits are accruing in a given year.

Finally, we simulated a scenario in which the policymaker maximizes the cost-effectiveness of one round of contracts. Figures 7 and 8 show how L* varies with mean farm income μ and characteristics of the environmental benefit function for two types of ecosystems. Some of the qualitative results are similar to the multiple-round scenario. Optimal contracts are shorter when the average farmer makes more money from farming and when conservation aims to provide environmental benefits that mature quickly, and L* falls with both initial benefits and the rate of benefit increase. The optimal contract is much longer if the policymaker only has one round of contracting in mind then if repeated rounds will occur. However, an optimal single-round contract is still not necessarily as long as the maturation age of the benefit at hand.

Figure 7.

Effect of mean farm income on optimal length of single contract. This figure shows how optimal contract length varies with average farm income when the program has only a single round of contracts.

Figure 8.

Effects of ecological variables on optimal length of single contract. Panel (A) shows how optimal contract length for a program with a single round of contracts varies with the level of annual benefits at the beginning of the contract as a fraction of the maximum benefits that will accrue each year once the benefit matures. Panel (B) shows how optimal contract length varies with the rate of benefit increase.

Discussion

We found that the optimal contract length depends on a tradeoff between an ecological effect—long contracts increase the environmental benefits from one landowner—and an enrollment effect—long contracts decrease the number of landowners enrolled. Optimal contracts are longer when the environmental benefits in question take time to develop, as in cases like filtration or flood-control services from restored wetlands or biodiversity in restored woodlands. However, at least in the particular model of this article, it is not typically optimal to have the indefinitely lived contracts favored by some conservation groups; high payments must be offered to induce landowners to accept such contracts, which would limit the number of acres enrolled given a budget constraint. Contracts should be longer if the program might not extend past a single round of enrollments, but even then this particular model does not find much support for using permanent easements.

When ecological benefits mature quickly, optimal contracts in our model are short. This scenario might fit cases like the ecosystem services from fast-growing plants: prairie restoration, or riparian vegetation planted to cool streams. Factors such as transaction costs might dominate ecological and economic concerns as administrators choose exactly how long to set contracts for such activities.

We found that nonecological factors could affect the optimal length of conservation programs. Optimal contracts are longer in regions where the turnover rate of parcels enrolled in conservation programs is high, although the effect of turnover on optimal contract length is small for ecological services that mature quickly. Also, average private land income is inversely related to optimal contract length, as more incentives are needed to attract higher-productivity parcels into conservation.

Our model is fairly deterministic. If there is uncertainty about future environmental benefits, optimal contract lengths will be even shorter than in our model because of what economists refer to as option value to the conservation agent: that agent can gain by preserving flexibility to not re-enroll parcels that turn out not to have high environmental value in the future. Also, if there is real uncertainty about returns to farming that will partially resolve over time, then long contracts may be even more costly because programs must compensate farmers for losing flexibility to return to farming if crop prices turn out to be consistently high. On the other hand, if returns to farming are highly uncertain and farmers do not like risk, then farmers might be more willing to accept conservation contracts because those payments are not uncertain. These are promising lines for future research.

Our model does not apply directly to cases of payments for avoided deforestation because the model assumes that if a parcel enrolls, drops out, and then is re-enrolled, it starts generating benefits again at the same level as the original enrollment. The benefits of avoided deforestation are very high in the first round of enrollment, but if a parcel drops out of the program and is deforested, re-enrollment of that parcel would yield zero initial benefits of avoided deforestation. Future research would do well to extend our framework to this kind of situation.

Such work is likely to be worthwhile, because even our simple model makes clear that conservation programs can be made more cost-effective if conservation agents refine their understanding of the dynamic benefit functions that govern the flow of services such programs provide, and if they use that understanding to inform choices of contract payments and lengths.

Appendix

This technical appendix contains the detailed theoretical model that is described in the article.

Given contract length L and lump-sum payment P, a landowner chooses whether to enter into the land conservation contract. The farmer's individual benefit from farming, Bf, follows a stationary process (e.g., Brownian motion without drift) such that this year's benefit does not help the farmer to know if it will be high or low next year. V denotes the discounted value of payoffs to the farmer from making his/her optimal choice between (a) entering the conservation program and (b) staying out. If staying out of the program is optimal, then

image(1)

since staying out will yield a payoff of Bf in every period from now until the next contracting opportunity and will in expectation lead to exactly the same choice between options (a) and (b) in that next enrollment period. Defining inline image,

image(2)

If enrolling in the program is optimal, then

image(3)

because enrolling will yield a fixed lump-sum payment P at the beginning of the contract period and will in expectation lead to exactly the same choice between options (a) and (b) once the contract expires at the end of contract length L. Therefore,

image(4)

We set inline image=inline image and solve for the relationship between contract length L and payment P that will make a farmer indifferent between farming and entering the conservation program. That gives K(r, L)Bf=P. Thus, the fraction of farmers that will enter the program is the fraction of farmers for whom BfP/K(r, L). Net incomes from farming were assumed to be normally distributed across farmers according to Bf∼ N (μ, σ2). Thus, the quantity (Bf−μ)/σ has a standard normal distribution and the fraction of farmers that enroll is inline image.

We assumed there are n farmers in total, so N = nϕ (L,P) is the number of farmers that enroll for a given contract design. The number of farmers that enroll in the program is constant. The parameter α indicates the percentage of enrollments that are new at each contracting period. The environmental benefits function, E(t) is logistic: E(t) = 1/(1 +keat) .

The government seeks to maximize the present discounted value of environmental benefits from all enrolled parcels given an infinite number of rounds of contracting, which is given by

image(5)

The first term of Equation (5) shows the environmental benefits derived from the first contracting period, the next two terms equal those from the second period, and so on. The last element of each set of terms gives the benefits that period from lands that have been enrolled for the entire duration of the program, but the quantity of such lands decreases in each contracting period as a fraction of those lands is not re-enrolled due to churning. The total environmental benefits are derived by adding up the discounted environmental benefits of each period. Using the identity of infinite geometric series, inline imageEquation 5 becomes

image(6)

Therefore the government's optimization problem is written as

image

subject to

image(7)

where the budget constraint is M.

Where the policymaker seeks to maximize the cost-effectiveness of a conservation program, the government's problem is

image(8)

Acknowledgments

We received useful comments from C. Nelson, R. Brazee, S. Polasky, many seminar participants, and three anonymous referees. This work was supported in part by the USDA NIFA, Hatch project number #ILLU-470-316, and by the National Science Foundation Grant No. SES-0314445.

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