Implementing the Optimal Provision of Ecosystem Services under Climate Change

Transcription

1 Implementing the Optimal Provision of Ecosystem Services under Climate Change David J. Lewis Department of Applied Economics Oregon State University 200A Ballard Ext. Hall Corvallis, OR Stephen Polasky Department of Applied Economics University of Minnesota 337e Ruttan Hall St. Paul, MN Date of Draft: 3/11/16 Preliminary Draft, Please Do Not Cite Without Permission Abstract: Climate change is widely expected to alter future ecosystem services through mechanisms such as range shifts for wildlife. The provision of many ecosystem services depends on the spatial pattern of land use across multiple independent landowners with private information regarding their opportunity costs of conservation. When land-use decisions are irreversible and climate change causes uncertain changes in the ecosystem service production function, then the dynamically optimal spatial pattern of land conservation and development critically depends on knowledge of current and future opportunity costs of conservation across a landscape. This paper develops an auction that sets payments between landowners and the regulator to the time path of the optimized increased value of ecosystem services with conservation. Landowners have a dominant strategy to truthfully reveal their time path of conservation costs under the auction mechanism, which allows the regulator to implement the optimal provision of ecosystem services through conservation planning in the case of spatially dependent benefits, asymmetric information, uncertainty, and irreversibility. We show how the mechanism can be used as a subsidy or a tax depending on whether the government or individual landowners hold the property rights to land. Keywords: payments for ecosystem services, spatial modeling, land use, auctions, conservation planning, climate change, biodiversity conservation 1

2 1. Introduction This paper describes a method for internalizing an externality associated with conserving land under dynamically changing environmental conditions. Land conservation placing one s land-use in some type of natural state generates ecosystem services that contribute to human well-being. Many ecosystem services flowing from conserved land such as climate regulation, water quality, and habitat for wildlife are not traded in markets and so provide benefits to others that are external to the landowner that chooses to conserve. Therefore, under-provision of ecosystem services occurs in lieu of a policy mechanism that internalizes the external benefits to the landowner. Appropriately internalizing an environmental externality requires knowledge of the underlying environmental production function. The production function describing many nonmarket ecosystem services is subject to spatial dependencies a parcel of land s provision of an ecosystem service depends on the land-use choices arising on spatially proximate land (Armsworth et al. 2004). For example, the number of successful breeding birds on a piece of forestland depends on the fragmentation of nearby forestland (Robinson et al. 1995). The Where to Put Things approach developed in Polasky et al. (2008) shows how spatial optimization tools can be used to optimize landscape pattern to efficiently provide spatiallydependent habitat benefits and market returns on a landscape. However, the opportunity cost of conserving a piece of land a necessary piece of information to implement the Where to Put Things approach is private information. Lewis et al. (2011) show that voluntary incentivebased policies that do not attempt to truthfully reveal each landowner s costs will provide a small fraction of the efficient provision of spatially-dependent wildlife habitat. In response, Polasky, Lewis, Plantinga, and Nelson (2014) hereafter PLPN develop an auction mechanism to 2

3 truthfully reveal private landowner conservation costs and generate a spatial pattern of conservation that optimally provides ecosystem services on landscapes with many independent landowners. An important result from PLPN is that under a spatially-dependent benefit function, the positive externality generated from conserving a parcel of land depends on the conservation status and hence, the conservation costs of neighboring parcels, and so internalizing the externality requires a mechanism that truthfully reveals landowner conservation costs. This paper s primary contribution is to develop a dynamic extension of the PLPN auction mechanism to the problem of conservation planning under climate change. The PLPN mechanism is static and not well-suited to dealing with two key characteristics of conservation planning under climate change. First, the spatial dependencies that affect ecosystem service provision from land are likely to change over time since the suitable range of many species is expected to shift under a changing climate (Thuiller et al. 2005; Araujo et al. 2006; Staudinger et al. 2013). For example, high latitude regions are projected by ecologists to experience a large influx of new species (Lawler et al. 2009), with the constraint that spatial landscape patterns that are unsuitable to a potential new species (or to species movement) will be unlikely to be colonized even under climate change (Opdam and Wascher 2004; Lawler et al. 2013). Second, many land-use changes (e.g. development to urban uses, cutting old-growth forest, etc.) are irreversible, and the failure to prevent irreversible land-use changes today reduces the number of feasible spatial patterns available to manage under an uncertain future (Albers 1996). The economics of resource allocation under uncertainty and irreversibility is centered on the concept of option value (Arrow and Fisher 1974), but has not been applied to the problem of optimal spatial-dynamic management of landscapes. 3

4 This paper uses a spatial-dynamic applied theoretic framework to develop the key features of an auction mechanism that can truthfully reveal landowner development values (which are conservation costs) in a manner that allows for the dynamically optimal provision of spatially-dependent ecosystem services under uncertainty and irreversibility. Briefly, the mechanism works as follows. First, each landowner simultaneously submits a bid for their values to developing land today and in the future, which is equivalent to bidding the sale price of land. A landowner develops today if and only if their development value is greater than the incremental current value of ecosystem services from conserving their land plus a value of retaining the option to conserve or develop in the future. Second, if the bid is accepted, the payment between the regulator and the landowner is set equal to the current value of their parcel s contribution to ecosystem services, plus they retain an option to receive a future conservation payment conditional on the realized state of climate change. The optional future payment is made when the expected net gain in social net benefits from conserving a parcel under future climate change is positive. If the bid is accepted today, the landowner conserves today and grants the regulator the right to make the conservation decision in the future. If the landowner is optimally conserved once climate uncertainty is revealed in the future, they receive a payment equal to their parcel s contribution to ecosystem service provision. The truth-revealing property of this paper s auction mechanism arises because the payments to the landowner are independent of their bid and set to maximize the landowner s private stream of returns from land, and so it is a dominant strategy for landowners to set their bid equal to their development value. With knowledge of this stream of expected development values over time, the regulator can identify the set of parcels that maximizes the social benefits from the landscape today, accounting for the value of maintaining the option to conserve or 4

5 develop some parcels in the future depending on the realizations of climate change. With spatially-dependent benefits, the current and option value generated by an individual parcel, and hence the optimal payment between a landowner and the regulator, is a function of land uses on all parcels and so can only be determined once all bids have been submitted. Finally, in the spirit of the Coase Theorem (Coase 1960), we show how this subsidy mechanism can easily be reframed as a dynamic auction tax mechanism that can be used when governments retain the property right to land and charge a fee for development. The practical relevance of the tax auction mechanism cannot be understated, as a recent study estimates that 86% of the world s forests are owned by governments (Siry et al. 2009), and forests provide a wide array of ecosystem services. The mechanism developed in this paper is a dynamic extension of the static PLPN auction mechanism, which was developed in the spirit of the public good auction mechanisms introduced by Vickrey (1961), Clarke (1971), and Groves (1973), and more recently in a pollution control context by Montero (2008). By setting landowner payments equivalent to the marginal social benefit of conserving their land, the PLPN approach can be viewed as an integration of Pigouvian subsidies, Vickrey-Clarke-Groves auctions, and Where-to-Put-Things spatial optimizations. By considering a dynamic model under uncertainty and irreversibility, this paper presents an integration of Pigouvian incentives, Vickrey-Clarke-Groves auctions, and Where-to-Put-Things spatial optimizations with the literature on real options in resource economics (Arrow and Fisher 1974; Albers 1996). The application to conservation planning under climate change is an important practical policy problem regarding the ability of society to adapt to climate change and make ecosystems more resilient. The analysis in this paper clarifies how option payments for conservation planning under spatial dependencies critically depend on 5

6 the private information regarding development values across a landscape. Insights from this research provide a direct method for internalizing dynamic-spatial externalities from conserving land, and thereby contributes to work that examines the optimal design of spatial landscape conservation policies under climate change. The paper s organization is as follows. Section 2 introduces the basic setup and notation used in our spatial dynamic model. Section 3 develops a simple example of a three-parcel landscape over two time-periods to set ideas regarding optimal dynamic-spatial conservation. Section 4 introduces the auction mechanism as a subsidy and develops a proof that the mechanism truthfully-reveals landowners development values and generates the dynamically optimal landscape. Section 5 shows how the auction mechanism can be re-framed as a tax on development to be used when governments own the property rights to land. Section 6 revisits the simple example and illustrates how the auction mechanism would apply. Section 7 offers concluding thoughts. 2. Setup and notation for the spatial dynamic model We extend the model in PLPN to a two-period multiple-state model. There are i=1, 2,, N individually-owned parcels in a landscape that can contribute to the provision of an ecosystem service that is a public good (e.g., water quality or wildlife habitat) and a private good (e.g., agricultural crops). We assume that each risk-neutral landowner only cares about the private return on their parcel and not about their contribution to the public good. We assume there is a regulator whose objective is to maximize net social returns (the sum of the value of public and private goods). The regulator sets policy, which may involve subsidy payments to or taxes from landowners. Given policy, each landowner makes a binary land-use decision to develop or conserve their parcel. Let xx iiii = 1 if parcel i is developed in time period t and let xx iiii = 0 if 6

7 parcel i is conserved in time period t. Development is irreversible so if xx ii1 = 1, then xx ii2 = 1. The pattern of development and conservation in the landscape at time t is XX tt = (xx 1tt, xx 2tt,, xx NNNN ). With development of parcel i, the landowner of parcel i earns a development value of dd iiii in profit from production of the private good. The development value for parcel i is known by the landowner of parcel i but is not known by the regulator or by other landowners. If parcel i is conserved, the parcel contributes to provision of the ecosystem service but does not earn the landowner any private return. The value of the ecosystem service at time t, BB ss tt (XX tt ), depends on the landscape pattern (XX tt ), and on the realization of the climate state ss SS, where S is the set of possible climate states. We assume the regulator knows the function BB ss tt (XX tt ) so that the regulator will know the value of the ecosystem service if the landscape pattern and the climate state are known. The climate state for period 1 is known when land-use decisions for period 1 are made. The climate state in period 2 is not known in period 1, only the probability distribution over the possible states is known. The climate state in period 2 is known when period 2 land-use decisions are made. In our model, it does not matter whether landowners know BB ss tt (XX tt ). For completeness, however, we assume that they do not know BB ss tt (XX tt ) but have some prior over what this function might be. The regulator wishes to implement the land-use pattern over the two time periods that maximizes the present value of social benefits, which is the sum of the value of the ecosystem service over the landscape plus the value of the private good produced. If the regulator knew the development value of each parcel i, the regulator's problem would be to maximize the following expression: NN 2 tt=1 δδ tt 1 [BB ss tt (XX tt ) + ii=1 xx iiii dd iiii ] (1) 7

8 where δ is the discount factor between periods. Solving for the optimal land-use pattern over periods 1 and 2 involves solving a stochastic dynamic programming problem because the optimal period 2 choice depends on the climate state in period 2, which is not revealed until period 2. In period 2, the optimal land-use pattern for a given climate state, s, is given by: NN XX ss 2 = aaaaaaaaaaaa[bb ss (XX 2 ) + ii=1 xx ii2 dd ii2 ] (2) ss. tt. xx ii2 xx ii1 for all ii Let VV ss 2 (XX 1 ) represent the value of social benefits in period 2 given climate state s and the choice of X 1 in period 1. Note that period 1 choices only show up in the period 2 problem via the constraint that development is irreversible. Without this constraint, the period 2 problem can be solved independently of the period 1 problem. The optimal land use choice in period 1 can then be found by solving NN XX 1 = aaaaaaaaaaaa[bb(xx 1 ) + ii=1 xx ii1 dd ii1 ] + δδδδ[vv ss 2 (XX 1 )] (3) where the expectation is taken over potential climate states in period 2. We discuss how to solve this problem optimally given decentralized decision-making among N landowners who have private information about development value (d it ) in section 4 below. First, however, we provide a simple example to illustrate ideas and demonstrate the challenge of finding the dynamically optimal landscape pattern with changing climate, spatial dependencies, and asymmetric information. 3. A simple example 8

9 Consider the landscape shown in Figure 1 with three adjacent parcels and two time periods. Benefits of development (top line) and conservation (bottom line) in period 1 are shown in figure 1.a. The ecosystem service production function incorporates spatial dependency, increasing in value with more neighboring parcels conserved (moving from left to right). The present value of the benefits of development for period 2 are identical to development benefits in period 1. The benefits of conservation in period 2 are uncertain and will take one of two values: a low value where the present value of conservation remains the same as in period 1, and a high value where the present value of ecosystem services from conserving parcels (1) and (3) are much greater when each parcel is adjacent to a conserved parcel (shown in figure 1b). The high value climate state could represent a range shift in wildlife species that increases the value of contiguous habitat. The probability of the high value climate state is q, and the probability of the low value climate state is 1-q. Consider first the static version of the problem with only period 1 values. Note that parcel (1) is always optimally conserved regardless of the conservation status of neighboring parcel (2) because the benefit of conserving the parcel with no neighboring conserved parcels (12) outweighs the benefits of development (10). Next note that it is never optimal to conserve parcel (2) because the high value of development (25) outweighs the maximum possible benefit from conservation. The maximum benefit from conserving parcel (2) is 24 (15 for conserving parcel 2 with both neighbors conserved, an additional 3 on parcel 1 and 6 on parcel 3 for having a conserved neighboring parcel). Given that it is not optimal to conserve parcel (2), it is then not optimal to conserve parcel (3) as the benefits of development (10) outweigh the benefits of conservation (9). Optimal choices require information about both the benefits of conservation and development. Without both of these pieces of information it is not, in general, possible to 9

10 solve for an optimal solution, a point we return to below when we consider the problem of finding an optimal solution given asymmetric information. Now consider the dynamic version of the problem and the solution to the stochastic dynamic programming problem. An important aspect of the dynamic problem is irreversible development if a parcel is developed in period 1, it is not eligible for conservation in period 2. Following the backwards induction logic of stochastic dynamic programming, consider the conservation decision in period 2 if all parcels are eligible for conservation. Under the low climate state (ss = SS ll ), all benefits and costs are identical to period 1 and so the conservation decision would remain as described in the static case above. Under the high climate scenario (ss = SS h ), the benefits of ecosystem services are higher for parcels (1) and (3) when they have one conserved neighbor, thus the optimal conservation decision for parcels (2) and (3) need to be reconsidered. Note that it is optimal to conserve parcel (3) when parcel (2) is conserved as the conservation benefits (20) outweigh the development benefits (10). We can then check whether it is optimal to conserve parcels (2) and (3) by comparing the value with all three parcels conserved with the value of conserving parcel (1) and developing parcels (2) and (3). Since the value of conserving all three parcels (55) is greater than the value of conserving parcel (1) while developing parcels (2) and (3) ( = 47), it is optimal to conserve all three parcels in the high climate scenario. Should parcels (2) and (3) be conserved in period 1? Conditional on parcels 2 and 3 being conserved in period 1, we have shown that they should be conserved in period 2 if ss = SS h and developed in period 2 if ss = SS ll. Therefore, the present value of conserving all parcels in period 1 is q + 47(1-q). On the other hand, developing parcels (2) and (3) in period 1 forecloses the option of conserving these parcels in period 2 so the present value of this alternative is

11 47. It is optimal to conserve all parcels in period 1 if q + 47(1-q) , which holds for q ¼. The important take-away messages from this simple example are as follows. First, as in PLPN, the spatial dependencies in the ecosystem service benefits function mean that solving for the optimal landscape pattern requires information about the benefits of development and conservation across the whole landscape. The problem cannot, in general, be considered independently parcel by parcel. For example, in this simple example the optimal decision of what to do on parcel (3) depends upon the decision of what to do on parcel (2). Second, while the static optimal conservation problem of PLPN only requires knowledge of current benefits of development and conservation, solving the stochastic dynamic programming problem for optimal conservation under climate change requires information regarding current and future benefits of development and conservation. Finally, optimal choices require information about both the benefits of conservation and development. Without both of these pieces of information one cannot compare the net benefits of conservation. We now turn to the description of the dynamic subsidy auction mechanism that allows the regulator to gain information about the benefits of development and then to implement the optimal solution even with spatial dependency, asymmetric information and changing climate. 4. The Dynamic Subsidy Auction Mechanism In this section we describe an auction mechanism in which each landowner i submits a bid for conserving their land, the regulator chooses which bids to accept and what payments are given to landowners whose bids are accepted. We define a mechanism that will generate an optimal solution, one that maximizes social benefits. 11

12 In the auction, in period 1 each landowner i submits a bid with two parts, bb iiii, for t=1, 2. We assume that there is no collusion among landowners in the bidding process. Upon receiving the bids, the regulator chooses which bids to accept. If the bid is not accepted, the landowner develops and earns dd iiii for t=1, 2. If the bid for parcel i is accepted, the parcel is conserved in period 1 and the regulator gives the landowner a payment. Upon learning the climate state in period 2, the regulator either allows the landowner to develop or to continue to conserve. If development is allowed the landowner will develop and receive d i2. With continued conservation in period 2, the landowner receives an additional payment. We set payments to landowners using marginal social benefits of conservation. We define the marginal social benefits of conserving parcel i in period t net of the parcel s development value using the following steps: Step 1: Define the period t social benefits when parcel i is conserved as: WW iiii SS XX iiii SS = BB tt SS XX iiii SS + xx jj ii iiiiii dd jjjj (4) where XX SS iiii is the optimal landscape and xx iiiiii is the optimal choice for parcel j when choice is constrained to have parcel i conserved for all j i. Step 2: Define the period t social benefits when parcel i is developed net of the private development benefits of parcel i as: WW SS ~iiii XX SS ~iiii = BB SS tt XX SS ~iiii + jj ii xx ~iiiiii dd jjtt (5) where XX SS ~iiii is the optimal landscape and xx ~iiiiii is the optimal choice for parcel j when choice is constrained to have parcel i developed for all j i. 12

13 Step 3: The period t marginal social benefits of conserving parcel i is the difference: WW SS iiii = WW SS iiii XX SS iiii WW SS ~iiii XX SS ~iiii. (6) The regulator requires a rule for deciding which parcels to enroll in conservation, and a payment to enrolled parcels. Because achieving an optimal solution requires the regulator to know development value, the auction mechanism is designed to induce each landowner to truthfully reveal their development value in each period. We assume the regulator knows the current benefit function BB 1 (XX tt ), the future benefit function under alternative climate scenarios S, BB SS 2 (XX 2 ), and the probability density function of S. Therefore, the regulator can solve the stochastic dynamic programming problem to optimize conservation in the current period if they know development values. The dynamic optimality condition for enrolling parcel i in conservation in the period 1 is WW ii1 (XX ii1 ) + δδδδ SS mmmmmm WW SS ii2 (XX ii2 ) xxii1 =0, WW SS ~ii2 ) xxii1 =0 + dd ii2 WW ~ii1 XX ~ii,tt + dd ii1 + δδ(ee SS WW SS ~ii2 ) xxii1 =1 + dd ii2 ) (7) where EE SS is the expectation operator over the set of all possible climate scenarios S. Since the expectation operator is outside of the max operator in equation (7), the regulator can flexibly alter the conservation decision in period 2 in response to future climate information (Arrow and Fisher 1974, Albers 1996). The first term on the left side of equation (7), WW ii1 (XX ii1 ), is the period 1 social benefits when parcel i is conserved, while the second term on the left side of equation (7), δδδδ SS mmmmmm WW SS ii2 (XX ii2 ) xxii1 =0, WW SS ~ii2 ) xxii1 =0 + dd ii2, is the expected optimal social benefits in period 2 given that parcel i was conserved in period 1. If parcel i is optimally conserved under climate scenario S, the future landscape social benefits are WW ii2 SS (XX ii2 ); if parcel i is optimally 13

14 developed under climate scenario S, the future landscape social benefits are WW SS ~ii2 ) + dd ii2. The right side of equation (7) is the period one social benefits when parcel i is developed, WW ~ii1 XX ~ii,tt + dd ii1, plus the expected social benefits in period 2 when parcel i is developed in period 1, δδ(ee SS WW SS ~ii2 ) xxii1 =1 + dd ii2 ). Define SS as the set of climate states where i is optimally conserved in period 2, and define S' as the set of climate states where i is optimally developed in period 2. Further, define marginal benefits in period 2 as WW ii2 SS (XX ii2 ) = WW SS ii2 (XX ii2 ) xxii1 =0 WW SS ~ii2 WW SS ~ii2 SS ) = WW ~ii2 ) xxii1 =1 if i is optimally conserved in period 2, and define SS ) xxii1 =0 WW ~ii2 Therefore, equation (7) can be re-arranged, ) xxii1 =1 if i is not optimally conserved in period 2. WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) dd ii2 + δδ(1 qq)ee SS WW SS ~ii2 ) dd ii1 (8) where q is the probability that it is optimal to conserve parcel i in period 2 given that it was conserved in period 1. The middle term, δδδδδδ SS WW ii2 SS (XX ii2 ) dd ii2, represents the discounted option value of being able to conserve parcel i in period 2 should climate conditions warrant it (Arrow and Fisher 1974). This term is zero if parcel i is never optimally conserved in period 2. The third term, δδ(1 qq)ee SS WW SS ~ii2 ) represents the expected potential change in discounted period 2 social benefits arising from potentially different optimal conservation choices in period 1 when parcel i is conserved versus developed, leading to potentially different conservation choices on other parcels in period 2. Equation (8) can be slightly rearranged to be more convenient for the discussion that follows: WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) dd ii1 + δδδδδδ ii2 (9) 14

15 With this groundwork in place we now define the auction mechanism. In the auction mechanism the regulator accepts the bid from landowner i if and only if WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) bb ii1 + δδδδδδ ii2 (10) where the calculation of the optimal landscapes (ΔW) is done assuming that b jt = d jt for all j for t = 1, 2. If the bid from landowner i is accepted, the landowner receives WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ) in period 1. With enrollment in the conservation program, the landowner also agrees to allow the regulator to decide conservation status of the parcel in period 2. The regulator observes climate scenario S for period 2 and then decides whether to continue conservation on the parcel in period 2 or allow development. If parcel i is optimally conserved in period 2 (SS SS ), the regulator then pays the landowner WW ii2 SS (XX ii2 ); if the landowner is not optimally conserved in period 2 (SS SS ), the landowner is paid zero but is allowed to develop and earns dd ii2. It is assumed that the auction mechanism is common knowledge and that the regulator must commit to not changing the mechanism upon enrollment. Showing that the auction mechanism will achieve an optimal solution involves two parts. First, it must be the case that landowners truthfully reveal development value. Second, given this information the regulator optimally chooses which parcels to conserve and which to allow to develop. In the following propositions we show that the auction mechanism satisfies both points. Proposition 1. Under the subsidy auction mechanism described above, each landowner has a dominant strategy to bid bb iiii = dd iiii for t=1, 2. Proof. We show that bidding truthfully bb iiii = dd iiii is a dominant strategy by showing that truthful bidding leads to payoffs that are equal to or greater than over bidding (bb iiii > dd iiii ) or under- 15

16 bidding (bb iiii = dd iiii ), with strict inequality in payoffs for some potential outcomes. We begin by considering the bid for the second period, bb ii2. We first show that, conditional on being conserved in period 1, it is always optimal to truthfully bid period 2, bb ii2 = dd ii2. Part 1: bb ii2 = dd ii2. Suppose the landowner bids bb ii2 = dd ii2. If bb ii2 WW SS ii2 (XX ii2 ) the landowner s bid will be accepted and the landowner will receive a payment of WW SS ii2 (XX ii2 ). If bb ii2 > WW SS ii2 (XX ii2 ), the landowner s bid will be rejected and the landowner will be allowed to develop and receive dd ii2. We prove that bidding bb ii2 = dd ii2 is a dominant strategy by showing that this strategy generates equal or greater payoffs than overbidding (bb ii2 > dd ii2 ) or underbidding (bb ii2 < dd ii2 ) over the range of possible values of WW SS ii2 (XX ii2 ). First, suppose the landowner over-bids: bb ii2 > dd ii2. There is potentially some set of SS climate states S O for which bb ii2 > WW OO ii2 (XX ii2 ) dd ii2. The regulator would allow parcel i to be SS developed since bb ii2 > WW OO ii2 (XX SS ii2 ). However, since WW OO ii2 (XX ii2 ) > dd ii2, the landowner would be SS better off bidding truthfully, having the parcel be conserved with payment WW OO ii2 (XX ii2 ). For other values of WW SS ii2 (XX ii2 ), overbidding will yield the same outcome as truthful bidding. When dd ii2 > WW SS ii2 (XX ii2 ), overbidding is harmless since the bid will be rejected both under truthful bidding and overbidding. When WW SS ii2 (XX ii2 ) bb ii2 > dd ii2 the bid will be accepted regardless of overbidding so that payoffs are equal for overbidding and for truthful bidding. Therefore overbidding is dominated by truthful bidding: bb ii2 = dd ii2. Now suppose that the landowner under-bids: bb ii2 < dd ii2. There is potentially some set of SS climate states S U for which bb ii2 < WW UU ii2 (XX ii2 ) < dd ii2. The regulator would conserve parcel i since SS bb ii2 < WW UU ii2 (XX SS ii2 ). However, given that WW UU ii2 (XX ii2 ) < dd ii2 the landowner would be better off with truthful bidding and developing the parcel. For other values of WW SS ii2 (XX ii2 ), underbidding 16

17 will yield the same outcome as truthful bidding. When dd ii2 < WW SS ii2 (XX ii2 ), underbidding is harmless since the bid will be accepted both under truthful bidding and underbidding. When WW SS ii2 (XX ii2 ) bb ii2 < dd ii2 the bid will be accepted regardless of underbidding so that payoffs are equal for underbidding and for truthful bidding. Therefore underbidding is dominated by truthful bidding: bb ii2 = dd ii2. Part 2: bb ii1 = dd ii1. Here, we show that the dominant strategy for landowner i is to truthfully reveal the period 1 development value: bb ii1 = dd ii1. Part 1 of the proof established that the landowner has a dominant strategy to truthfully bid their second period development value, bb ii2 = dd ii2. Given that bb ii2 = dd ii2, if bb ii1 WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ) then equation (9) will be satisfied and the landowner s bid will be accepted. The landowner will receive a payment of WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ) in period 1 with continuation payoffs of either conservation or development as described above in period 2. If bb ii2 > WW ii2 SS (XX ii2 ), the landowner s bid will be rejected and the landowner will be allowed to develop and receive dd ii2. Case 1: Parcel i is optimally conserved in period 1: WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) dd ii1 + δδδδδδ ii2 (equation 9) First, suppose that the landowner overbids so that bb ii1 > dd ii1. If the bid is accepted, the landowner receives the same expected return as when the landowner bids truthfully (bb ii1 = dd ii1 ). If, however, the bid is rejected, the landowner develops and receives the development value in both periods, dd ii1 + δδdd ii2. With conservation in period 1, the landowner receives period 1 payment of WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ), and in period 2 the landowner receives the maximum of WW ii2 SS (XX ii2 ) or dd ii2. Given that equation (9) is satisfied, the landowner will do at least 17

18 as well, and for some realizations of S strictly better, with conservation than development in period 1. Overbidding in period 1 cannot improve the landowner s payoff and in some instances lowers the landowner s payoff. Now suppose that the landowner underbids such that bb ii1 < dd ii1. The bid will always be accepted, and the landowner receives the same expected return from conservation as when the landowner bids truthfully (bb ii1 = dd ii1 ). Underbidding in period 1 therefore does not improve the landowner s payoff. Case 2: Parcel i is optimally developed in period 1 WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) < dd ii1 + δδδδδδ ii2 Suppose the landowner overbids such that bb ii1 > dd ii1. In this case, the bid is always rejected, the landowner develops and receives the value of development, dd ii1 + δδδδ ii2, which is the same expected return as with truthful bidding. Overbidding in period 1 does not improve the landowner s payoff. Now suppose the landowner underbids such that bb ii1 < dd ii1. If the bid is not accepted, the landowner will be allowed to develop and receives the same payoff as bidding truthfully. If, however, the bid is accepted, the parcel will be conserved in period 1. The landowner receives period 1 payment of WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ), and in period 2 the landowner receives the maximum of WW ii2 SS (XX ii2 ) or dd ii2. Given that equation (9) is not satisfied, the landowner will do better with development than conservation in period 1. Underbidding in period 1 cannot improve the landowner s payoff and in some instances lowers the landowner s payoff. 18

19 Combining cases (1) and (2), we have shown that both overbidding and underbidding are dominated by the truthful bidding strategy bb ii1 = dd ii1 and bb ii2 = dd ii2. QED. The intuition for proposition 1 is as follows. First, consider the intuition for why truthful bidding in period 2 is dominant. Figure 2 depicts the potential losses from overbidding and from underbidding. By not bidding truthfully, the landowner alters the future climate scenarios in which the regulator accepts the bid such that they deviate from having bids accepted for the set of SS SS. However, since truthful bidding under the auction mechanism ensures the landowner always maximizes their payoffs for any given climate state seen with the bold line in figure 2 then any deviations from truthful bidding will alter their payoffs such that the landowner is worse off than truthful bidding of the period 2 development value. Similarly in period 1, the landowner can change whether the bid is accepted by changing the bid, but not the payment if the bid is accepted. By not bidding truthfully, the landowner will cause a deviation from the acceptance set that maximizes the landowner s expected payoffs. Hence, bidding truthfully is a dominant strategy. Using the result that landowners will bid truthfully, we now prove the main result of the paper that the auction mechanism will generate an optimal dynamic landscape that maximizes the sum of the values of ecosystem services plus private goods. Proposition 2. The subsidy auction mechanism in which i) the regulator accepts the bid from landowner i if and only if WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) bb ii1 + δδδδδδ ii2 ; 19

20 ii) if the bid is accepted, the regulator requires conservation in period 1 pays landowner i an amount equal to WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ) ; iii) if the bid is accepted, conditional on realized climate scenario S, the regulator requires conservation in period 2 if and only if WW ii2 SS (XX ii2 ) bb ii2 and pays landowner i an amount equal to WW ii2 SS (XX ii2 ); iv) if the bid is accepted, conditional on realized climate scenario S, the regulator allows development of parcel i in period 2 and makes no payment in period 2 to the landowner if WW ii2 SS (XX ii2 ) < bb ii2 ; v) the regulator rejects the bid and allows development in period 1 if WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) < bb ii1 + δδδδδδ ii2 ; generates an optimal dynamic landscape that maximizes the sum of ecosystem service value plus private goods value. Proof. First, proposition 1 established that landowners bid truthfully (bb ii1 = dd ii1 and bb ii2 = dd ii2 ) so that the regulator knows all development values in period 1 and 2. Therefore, the regulator can solve for the set of parcels to conserve in period 1 that maximizes expected social benefits. In the auction, parcel i is conserved in period 1 if and only if WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) bb ii1 + δδδδδδ ii2. But since landowners are bidding truthfully this expression is equivalent to WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) + δδ(1 qq)ee SS WW SS ~ii2 ) dd ii1 + δδδδδδ ii2 20

21 which is equation (9) that characterizes what must be true in an optimal solution. Therefore, the auction mechanism correctly solves the social benefits optimization problem in period 1. Further, in period 2, under the auction mechanism the regulator will continue to conserve parcels if and only if WW SS ii2 (XX ii2 ) bb ii2 = dd ii2, which again is the optimal rule for conservation in period 2. Therefore, the auction mechanism achieves the optimal solution. QED The auction mechanism generates the dynamically optimal landscape because the rule governing acceptance into the conservation program is based on the dynamic optimality condition for conservation. 5. The Dynamic Auction Tax Mechanism Significant shares of many landscapes are owned by governments rather than private individuals. For example, it has been estimated that 86% of the world s forests are owned by governments (Siry et al. 2009). 1 Rangelands including grasslands also tend to have significant government ownership, as about half of U.S. rangelands are government owned (federal, state, local) 2, and all of China s grasslands comprising 40% of the country s land area (Kang et al. 2007) are government owned. Further, many governments auction the development or use rights of some of their publicly-owned forest and grasslands to the highest bidder, e.g. U.S. Forest Service timber auctions, U.S. Bureau of Land Management grazing auctions, etc. Auctions are used to allocate development or use of public lands because governments generally do not know private returns to developing land. While most contemporary auctions are designed to maximize the government s rents from developing public lands, we show how a simple modification of the subsidy auction mechanism discussed in the 1 Sweden and the United States have a relatively low amount of public forestland ownership at 20 and 42% respectively, while other countries like China and Russia have 100% of forest land government owned. 2 See the U.S. Forest Service, (accessed 1/18/2016). 21

22 prior section can be made into a tax auction mechanism that can be used to implement the dynamically optimal provision of ecosystem services under climate change. As before, the landowner submits a bid bb iiii for the right to develop parcel i in t=1,2. The previously defined marginal benefits of conserving parcel i today ( WW ii1 ) and in the future under climate scenario S ( WW SS ii2 ) are now interpreted as environmental damages from developing parcel i. The bid to allow development is accepted and development occurs in period 1, which then allows developed use in both periods 1 and 2, if and only if WW ii1 (XX ii1 ) + δδδδ SS mmmmmm WW SS ii2 (XX ii2 ) xxii1 =0, WW SS ~ii2 ) xxii1 =0 + bb ii2 < WW ~ii1 XX ~ii,tt + bb ii1 + δδ(ee SS WW SS ~ii2 ) xxii1 =1 + bb ii2 ) (11) which can be re-written as: WW ii1 + δδδδδδ SS WW SS ii2 (XX ii2 ) bb ii2 + δδ(1 qq)ee SS WW SS ~ii2 ) < bb ii1 (12) Development should be approved in period 1 if the value of development in period 1 is greater than the loss in the value of ecosystem services in period 1 plus the loss in option value plus the loss (or gain) in the expected value of ecosystem services in period 2 due to development of parcel i in period 1. Since development is irreversible, a landowner who is granted development rights in period 1 will pay a tax in period 1 of WW ii1 + δδ(1 qq)ee SS WW SS ~ii2 ) In the second period once the climate state has been realized, the landowner must pay a tax of WW ii2 SS (XX ii2 ) if the parcel would have been optimally conserved were it not irreversibly developed in the first period (S* occurs). The tax requires the regulator to calculate WW ii2 SS (XX ii2 ) in period 1 22

23 for all S, and then commit to taxing the landowner the appropriate amount depending on the realization of S. From the perspective of period 1, the landowner who develops expects to pay a tax in period 2 of δδδδδδ SS WW ii2 SS (XX ii2 ) Note that the tax includes both the change in the value of period 1 ecosystem services and the discounted expected period 2 benefits had the parcel been conserved in period 1. If the bid is not accepted in period 1, then the landowner s development request is reconsidered in period 2 once the climate scenario S has been realized. The landowner is allowed to develop in period 2 if WW SS ii2 (XX ii2 ) < bb ii2, and is required to pay a tax of WW SS ii2 (XX ii2 ). If WW SS ii2 (XX ii2 ) > bb ii2, the landowner is not allowed to develop in period 2 and the parcel remains conserved. The dynamic auction tax mechanism generates the same incentives for the landowner to set their bid equal to their development value dd iiii because their tax payment in each period is independent of their bid. This dynamic tax mechanism also generates the same dynamicallyoptimal land-use outcome as the subsidy in that development occurs in t=1 if equation (11) holds. Similar to Coase (1960), the main difference between the auction tax and the auction subsidy mechanism is who pays whom: the landowners pay the government under the tax, while the government pays the landowners under the subsidy. The optimal land-use pattern remains with either mechanism. 6. Simple Example Revisited We revisit the simple example from section 3 to illustrate the auction mechanism. Table 1 shows the calculation of each component necessary to form the optimal payments for each 23

24 parcel of land. In this example, the term WW SS ~ii2 ) is zero for all three parcels since the period t=2 social benefits when parcel i is developed net of the private development benefits of parcel i are the same whether parcel i is initially conserved in t=1 or not: WW SS ~ii2 ) xxii1 =0 = WW SS ~ii2 ) xxii1 =1 for all i. Now consider the incentives offered to parcel (2). Their t=1 payment is WW ii1 =23, which is less than their t=1 development value dd ii1 =25. However, by conserving in t=1, they preserve the period t=2 option to be paid marginal benefits of conservation WW ii2 SS h (XX ii2 )=33 if climate state SS h occurs, or to develop and earn dd ii2 =25 if climate state SS ll occurs. If they develop in t=1, they would earn dd ii2 =25 with certainty. Suppose the probability of the high climate state q=0.5; then if parcel (2) conserves they earn a t=1 net loss of -2 but gain the option to earn an expected t=2 net gain of 4 (=0.5*(33-25)). So, conserving in t=1 is optimal for the risk-neutral landowner of parcel (2), and it is socially optimal as shown in section 3. Further, the components of their payment ( WW ii1, WW ii2 SS h (XX ii2 ), qq) are exogenous to their bid, and they cannot raise their payments by under or over-bidding as in the original Vickrey auction. This example illustrates that the regulator must communicate and commit to describing the expected gains to each landowner from conserving. 7. Discussion There are two dynamic problems when the static auction mechanism developed in PLPN is sufficient and the dynamic auction mechanism developed in this manuscript is not needed. First, if all future costs and benefits are known with certainty, then the PLPN mechanism can accommodate a dynamic problem whereby all costs and benefits are simply expressed as present values of the stream of future costs and benefits. Second, if development is entirely reversible, then the conservation problem is simply revisited every period with the PLPN mechanism. For 24

25 example, development of grassland into pasture could be (at least imperfectly) reversed, as could development of native rangeland into grazing habitat for cattle. In contrast, the static PLPN auction mechanism is insufficient when i) development is irreversible, and ii) the benefits of the future landscape are uncertain. Thus, the dynamic auction mechanism developed in this paper is most applicable to the very prominent problem of managing irreversible development in the face of climate change induced shifts in ecosystem service production functions. This paper uses a two-period framework to develop insights into the dynamic auction mechanism. However, the mechanism can be thought of in a more general dynamic framework where uncertainty over the spatial dependency exists at one future time t=t*, where all periods before t* are lumped together as t=1 and all periods after t* are lumped as t=2. With this view, all costs and benefits in period t=1 include the present discounted value of the stream of costs and benefits from the current period up to t*, while all costs and benefits in period t=2 represent the present value of an infinite stream of costs and benefits beginning at t*. For example, perhaps we have knowledge of costs and benefits for the next 50 years, but potential climate change induces an uncertainty as to what form the spatial ecosystem service production will take after 50 years. The auction mechanism requires landowners to bid current and future development values. This is akin to landowners simply bidding the sale price of land, since the sale price capitalizes the discounted stream of annualized rents that the land is expected to produce. Properly decomposing a capitalized price into a stream of rents requires information on the discount rate and the expected time path of rents. In the simplest case of constant annual rents, the annual rent simply equals the discount rate multiplied by the capitalized price. In the more complex case where the price of undeveloped land capitalizes future development rents that 25

26 would occur at some future date t f, there has been econometric work on how to decompose current prices into rents from undeveloped land and expected future rents (e.g. Plantinga et al. 2002). The mechanism in this paper is developed with the assumption that future development values are known with certainty and not a function of the future climate state. This assumption is useful for clarifying the key elements of a mechanism to implement optimal dynamic conservation, though it ignores the very real possibility that future development values are in fact a function of the uncertain climate state. Our mechanism can accommodate uncertain future development values if the uncertainty is known by the government and not private information e.,g. if dd ii2 = dd ıı2 + εε SS ii2 where εε SS ii2 is a random variable in t=1 with mean zero, and whose realization in t=2 is observed by the government once the climate state S is revealed. However, if εε ii2 SS is private information not known by the government, then another truth-revealing auction mechanism would need to be developed for t=2 which is beyond the scope of this paper. Analysis of the land conservation problem under uncertainty and irreversibility dates back to Arrow and Fisher s (1974) seminal article introducing option value into the conservation problem. Subsequent studies have clarified and refined the concept of option value (Haneman 1989; Dixit and Pindyck 1994; Albers 1996; Traeger 2014) that remains a fundamental part of the contemporary natural resource economist s toolkit. In their recent review paper, Mezey and Conrad (2010) argue that an important new application of the option value concept in resource economics would be to the problem of conservation planning under climate change. Thus, when conservation planning is described by an ecosystem service production function subject to spatial dependencies, the option value for any particular parcel will depend on the future development values of neighboring parcels. Therefore, when development values (conservation costs) are 26

27 private information, the option value concept is only operationalized under a truth-revealing mechanism such as developed in this manuscript. Finally, we note that our dynamic paper (along with that of the static PLPN paper) is distinguished from prior work on conservation auctions in one simple but fundamental way we examine optimal conservation as the problem of conserving parcels such that the social benefits from a landscape are maximized. An extensive prior literature exists that uses auctions to maximize environmental benefits less the government s costs of conservation (e.g. Latacz- Lohman and van der Hamsvoort 1997; Parkhurst and Shogren 2007; Arnold et al. 2013; Banerjee et al. 2014). However, government costs are not equal to social costs since this literature doesn t use truth-revealing auction mechanisms. Thus, our approach of optimal dynamic conservation aims to fit conservation planning more squarely into the realm of classical environmental economics devising policy mechanisms to internalize externalities and achieve social optimality. 27