metadata: authors: 0: author: Cai,Yongyang
1: author: Steinbuks,Jevgenijs
2: author: Judd,Kenneth L.
3: author: Jaegermeyr,Jonas
4: author: Hertel,Thomas W.
count: World
docna: 0: docna: Modeling Uncertainty in Large Natural
            Resource Allocation Problems
docty: Policy Research Working Paper
subtopic: Science of Climate Change,Climate Change and Environment,Climate Change and Health,Rural and Renewable Energy,Renewable Energy,Global Environment,Crops and Crop Management Systems,Climate Change and Agriculture,Energy Demand,Energy and Environment,Energy and Mining
teratopic: Science and Technology Development,Environment,Health, Nutrition and Population,Rural Development,Agriculture,Energy
abstracts: cdata!: The productivity of the world's
            natural resources is critically dependent on a variety of
            highly uncertain factors, which obscure individual investors
            and governments that seek to make long-term, sometimes
            irreversible investments in their exploration and
            utilization. These dynamic considerations are poorly
            represented in disaggregated resource models, as
            incorporating uncertainty into large-dimensional problems
            presents a challenging computational task. This study
            introduces a novel numerical method to solve large-scale
            dynamic stochastic natural resource allocation problems that
            cannot be addressed by conventional methods. The method is
            illustrated with an application focusing on the allocation
            of global land resource use under stochastic crop yields due
            to adverse climate impacts and limits on further
            technological progress. For the same model parameters, the
            range of land conversion is considerably smaller for the
            dynamic stochastic model as compared to deterministic
            scenario analysis. The scenario analysis can thus
            significantly overstate the magnitude of expected land
            conversion under uncertain crop yields.
repnme: repnme: Modeling Uncertainty in Large Natural
            Resource Allocation Problems
display_title: Modeling Uncertainty in Large
            Natural Resource Allocation Problems
listing_relative_url: /research/2020/02/31801092/modeling-uncertainty-large-natural-resource-allocation-problems
topicv3: Energy and Extractives m1326595 507,Biofuels Energy m1326600 532,Crop Production m1326627 174,Climate Change m1326829 567,Environment, Natural Resources and Blue
            Economy m1327391 566
publishtoextweb_dt: 2020-02-20T00:00:00Z
historic_topic: Science and Technology Development,Environment,Health, Nutrition and Population,Rural Development,Agriculture,Energy
origu: DEC-Sustainability & Infrastruct (DECSI)
no_of_pages: 64
author_keys: 0: author_key: 666646
1: author_key: 451732
model_extractions: Policy Research Working Paper 9159

# Modeling Uncertainty in Large Natural Resource Allocation Problems

_Yongyang Cai Jevgenijs Steinbuks Kenneth L. Judd Jonas Jaegermeyr Thomas W. Hertel_ Development Economics Development Research Group February 2020
Policy Research Working Paper 9159

# **Abstract**

The productivity of the world’s natural resources is critically dependent on a variety of highly uncertain factors, which obscure individual investors and governments that seek to make long-term, sometimes irreversible investments in their exploration and utilization. These dynamic considerations are poorly represented in disaggregated resource models, as incorporating uncertainty into large-dimensional problems presents a challenging computational task. This study introduces a novel numerical method to solve large-scale dynamic stochastic natural resource allocation problems that cannot be addressed by conventional methods. The method is illustrated with an application focusing on the allocation of global land resource use under stochastic crop yields due to adverse climate impacts and limits on further technological progress. For the same model parameters, the range of land conversion is considerably smaller for the dynamic stochastic model as compared to deterministic scenario analysis. The scenario analysis can thus significantly overstate the magnitude of expected land conversion under uncertain crop yields.

This paper is a product of the Development Research Group, Development Economics. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at jsteinbuks@worldbank.org.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# Modeling Uncertainty in Large Natural Resource Allocation Problems<sup>_⇤_</sup>

Yongyang Cai Jevgenijs Steinbuks Kenneth L. Judd Jonas Jaegermeyr Thomas W. Hertel JEL: C61, Q15, Q23, Q26, Q40, Q54 Keywords: Dynamic Stochastic Models, Extended Nonlinear Certainty Equivalent Approximation Method, Crop Yields, Land Use, Natural Resources, Uncertainty

> _⇤_ Cai: Department of Agricultural, Environmental and Development Economics, The Ohio State University, cai.619@osu.edu. Steinbuks: Development Research Group, The World Bank, jsteinbuks@worldbank.org. Judd: Hoover Institution, Stanford University, kennethjudd@mac.com. Jaegermeyr: Department of Computer Science, University of Chicago, NASA Goddard Institute for Space Studies, and Climate Impacts and Vulnerabilities, Potsdam Institute for Climate Impact Research, jaegermeyr@uchicago.edu. Hertel: Center for Global Trade Analysis, Purdue University, hertel@purdue.edu. Cai, Judd, and Hertel appreciate the financial support from United States Department of Agriculture NIFAAFRI grant 2015-67023-22905; Cai, Steinbuks, and Hertel appreciate the financial support from the National Science Foundation (SES-0951576 and SES-1463644) under the auspices of the RDCEP project at the University of Chicago. Cai would also like to thank Becker Friedman Institute at the University of Chicago and Hoover Institution at Stanford University for their support. This research is part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the State of Illinois. Blue Waters is a joint e↵ort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. Earlier versions of this paper include “The E↵ect of Climate and Technological Uncertainty in Crop Yields on the Optimal Path of Global Land Use” and “Optimal Path for Global Land Use under Climate Change Uncertainty”.
# **1 Introduction**

Understanding the allocation of the world’s natural resources over the course of the next century is an important research problem for agricultural and environmental economists. Analyzing natural resources use in the long run involves a complex interplay of di↵erent factors. These factors include, among others, continuing population increases, shifting diets among the poorest populations in the world, increasing the production of renewable energy, including biofuels, and growing demand for ecosystem services, including forest carbon sequestration (Foley et al., 2011). The problem is further confounded by faster than expected climate change, which is altering the biophysical environment of agriculture and forestry. Moreover, highly uncertain future productivities and valuations of ecosystem services, coupled with medium- to long-term irreversibilities in the extraction of nonrenewable or partially renewable resources, such as natural forests,<sup>1</sup> give rise to a challenging problem of decision-making under uncertainty.

While there is a large body of economic literature analyzing the problem of natural resource extraction and utilization under uncertainty theoretically or using stylized computational models (see e.g., Miranda and Fackler (2004), Tsur and Zemel (2014) and references therein), quantifying the e↵ects of uncertainty on natural resource use in a more realistic setting remains a challenging problem. This is because natural resource allocation problems, like environmental policy problems in general, involve highly nonlinear structure and damage functions, important irreversibilities, and long time horizons (Pindyck, 2007). Computational integrated models of the economy and environment are the standard workhorse mechanisms for modeling the long term allocation of the world’s natural resources, including particularly difficult land use problems.<sup>2</sup> These models have the important advantage of detailed spatial and sectoral (particularly, energy and agricultural sector) coverage, which allows them to capture a broad range of responses to changes in demand and supply factors a↵ecting utilization of natural resources. However, given the high computational complexity of these models, they are typically either static or based on myopic expectations, whereby decisions about production, consumption and resource extraction and conversion are made only on the basis of information in the period of the decision (Babiker et al., 2009). These models, therefore, have limited ability to address important intertemporal questions such as, for example, a dynamic trade-o↵between conservation, carbon sequestration, and renewable o↵sets for fossil fuels. Among the few forward-looking, dynamic economy and environment models, none explicitly incorporates uncertainty into the determination of the optimal path of natural resource use.<sup>3</sup> This is because introducing uncertainty into these models is confined by an array of computational obstacles that are very difficult (e.g., high dimensionality and kinks caused by occasionally binding constraints), if not impossible, to address using

> 1The biophysical and ecological literature suggests that restoration of forest structure and plant species takes at least 30–40 years and usually many more decades (Chazdon, 2008), costs several to $10,000 per hectare (Nessh ̈over et al., 2009), and is only partially successful in achieving reference conditions (Benayas et al., 2009).

> 2For a detailed overview of these models and their applications to natural resources and land use problems, see, e.g., F ̈ussel (2009), Schmitz et al. (2014), and Nikas et al. (2019).

> 3Several recent studies, most notably, Cai and Lontzek (2019) have successfully integrated uncertainty about economic and climate outcomes in a stochastic integrated assessment climate-economy framework. For a review of this related literature, see Cai (2019).

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standard computational methods in economics such as projection methods and value function iteration (see, e.g., Judd 1998; Miranda and Fackler 2004; Cai and Judd 2014; Cai 2019). To the extent that uncertainty in these models is considered, this is only through parametric or probabilistic sensitivity analysis or the use of alternative scenarios. Therefore, the high-dimensional resource use models have not e↵ectively dealt with optimal extraction and conversion decisions along the uncertain path of key drivers a↵ecting resource allocation in the face of costly reversal of conversion decisions.

In this study, we seek to address this important limitation of the economy-environment modeling of natural resource use. In doing so, we build on Cai et al. (2017), who have introduced a nonlinear certainty equivalent approximation method (NLCEQ) for solving large-scale infinite horizon stationary dynamic stochastic problems and demonstrated how this method could be used to achieve the accurate solution to a stylized stationary dynamic stochastic land use problem. While the original NLCEQ method can successfully solve many complicated problems in other fields of economics, particularly, macroeconomics, it has very limited applicability for solving environmental and resource economics problems. This is because many stochastic problems of utilization of natural resources feature nonstationary stochastic trends, such as, e.g., climate or technological trajectories, and some never converge to a stationary state. This paper introduces a novel algorithm, called Extended NLCEQ (ENLCEQ), that advances the original NLCEQ work to solve nonstationary dynamic stochastic problems and apply it to solve more complex dynamic stochastic multi-sectoral resource use problems with exogenous trends. Similar to the original NLCEQ method, the ENLCEQ method approximates the true solution to the underlying dynamic stochastic problem with globally valid, nonlinear certainty-equivalent decision rules. These rules are then used to generate simulation paths for nonstationary infinite or finite horizon resource use problems. In this paper, we show that the ENLCEQ approximation is highly accurate and achieves stable numerical solutions.

We illustrate the ENLCEQ method to solve for the dynamic optimal global land use allocation, which is a highly complicated resource use problem that features multiple crosssectoral and dynamic trade-o↵s. Specifically, we apply the method to a global land use model nicknamed FABLE (Forest, Agriculture, and Biofuels in a Land use model with Environmental services) in the face of uncertainty.<sup>4</sup> FABLE is a dynamic, forward-looking global multi-sectoral partial equilibrium model designed to analyze the evolution of global land use over the coming century. Prior applications of that model (Steinbuks and Hertel, 2013; Hertel et al., 2013, 2016; Steinbuks and Hertel, 2016) analyze competition for scarce global land resource in light of growing demand for food, energy, forestry and environmental services, and evaluate key drivers and policies a↵ecting global land use allocation. All these applications, however, assume perfect foresight, and treat uncertainty in a parametric fashion, thus ignoring the impact of future uncertainties on the optimal allocation of global land use.

By way of illustration, we choose to focus on uncertainty emanating from crop productivity over the next century. Along with energy prices, regulatory policies, and technological

> 4To evaluate the method’s accuracy, we also apply it to a simpler optimal growth model shown in the online appendix.

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change in food, timber and biofuels industries, this is one of four core uncertainties a↵ecting competition for global land use (Steinbuks and Hertel, 2013). To quantify the uncertainty in agricultural yields, we construct stochastic crop productivity index that captures two key uncertainty sources: technological progress and global climate change (Lobell et al., 2009; Licker et al., 2010; Foley et al., 2011).<sup>5</sup> Following Rosenzweig et al. (2014), we use projections from climate and crop simulation models, as well as the survey of recent agro-economic and biophysical studies to calibrate the index.

We simulate the results of the perfect foresight model under di↵erent realizations of the crop productivity index, focusing our attention on the current century. We then compare and contrast them with the results of the dynamic stochastic model, where the uncertainty in crop yields is brought to the model’s optimization stage. When the uncertainty in crop productivity is incorporated into the model, we see an additional redistribution of land resources aimed at o↵setting the impact of potentially lower yields. Owing to intertemporal substitution, some of that redistribution takes place even in the absence of the actual changes in the states of climate or technology a↵ecting crop yields. Moreover, the range of these alternative optimal paths of cropland is considerably smaller than the magnitude of possible land conversion resulting from the scenario analysis based on deterministic model simulations. This result indicates that the scenario analysis may significantly overstate the magnitude of expected agricultural land conversion under uncertain crop yields.

Besides the methodological innovation, our study also contributes to the growing environmental economics literature that analyzes the intertemporal allocation of land and other natural resources under uncertainty and irreversibility constraints. Most of that literature focuses on a particular type of resource or sector, where intertemporal issues are significant and cannot be ignored. One example of this literature is forestry management in the context of uncertain fire risks and climate mitigation policies (Sohngen and Mendelsohn, 2003, 2007; Daigneault et al., 2010). Another example is natural land conservation decisions under irreversible biodiversity losses (Conrad, 1997, 2000; Bulte et al., 2002; Leroux et al., 2009). While these models are undoubtedly helpful for understanding the broad implications of uncertainty on the intertemporal allocation of land resources, they fail to account for the e↵ect of uncertainty in supply and demand drivers on the optimal allocation of land resources in the long run. Our study is perhaps most closely related to the recent work of Lanz et al. (2017) who develop a two-sector stochastic Schumpeterian growth model with the endogenous allocation of global land use. They find, like our paper does, that optimal allocation of global land use requires more cropland conversion when the uncertainty in agricultural productivity is present. Lanz et al. (2017), however, focused on endogenous population dynamics, labor allocation, and technological progress, whereas our paper is concerned about the endogenous allocation of multiple types of land use and corresponding land-based goods and services. Our paper also advances on methodological grounds by introducing a novel algorithm that overcomes the computational difficulties of solving multidimensional stochastic land use models, which made Lanz et al. (2017) significantly

> 5Climate change will likely a↵ect the productivity of other land resources, such as forestland. Several recent modeling studies (see, e.g., Tian et al. (2016) and references therein) have suggested that climate change is likely to result in higher forest growth and greater timber yields, as well as in more forest dieback, with the net e↵ects varying over time and space. Incorporating these e↵ects is beyond the scope of this study and is left for future research.

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simplify their model by assuming that their binary shocks occur only in three time periods.

# **2 Extended NLCEQ**

Following the standard notation in the literature, let **St** be a vector of state variables (e.g., natural resource stock), and **a** _t_ be a vector of decision variables (e.g., resource extraction, transformation, and final consumption) at each time _t_ . The transition law of the state vector **S** is where _✏t_ is a serially uncorrelated random vector process,<sup>6</sup> and **G** _t_ is a vector of functions: its _i_ -th element, _Gt,i_ , returns the _i_ -th state variable at _t_ + 1: _St_ +1 _,i_ . For simplicity, we assume the mean of _✏t_ is 0.<sup>7</sup> We solve the following social planner’s problem: where _Ut_ is a utility function, _δ 2_ (0 _,_ 1) is the discount factor, E is the expectation operator, _T_<sup>_⇤_</sup> is the horizon ( _T_<sup>_⇤_</sup> = _1_ if it is an infinite-horizon problem), _VT ⇤_ ( **S** _T ⇤_ ) is a given terminal value function depending on the terminal state **S** _T ⇤_ (it is zero everywhere for an infinitehorizon problem), and _Dt_ ( **S** _t_ ) is a feasible set of actions **a** _t_ at time _t_ . And we assume that the initial state **S** 0 is given, as it can usually be observed or estimated.

In most economic problems of resource use under uncertainty, the social planner’s problem cannot be solved analytically, although certain inferences about potential e↵ects of uncertainty can be made from more stylized models.<sup>8</sup> Numerical dynamic programming with value function iteration (see, e.g., Cai and Judd 2014; Cai 2019) is a typical method to solve these dynamic stochastic problems. However, numerical dynamic programming faces challenging problems such as high dimensionality of state space, shape-preservation of value functions (Cai and Judd, 2013), and occasionally binding constraints. These challenges are common in modeling natural resource use and are hard to address even with the most advanced methods, such as parallel dynamic programming (Cai et al., 2015). Below we introduce a new algorithm, called the extended nonlinear certainty equivalent approximation

> 6In dynamic models with serially correlated random variables, they should be exogenous state variables, and we can use an uncorrelated vector _✏t_ in their transition laws.

> 7For notational simplicity we keep the same mathematical representation of a transition function even if some of its elements are redundant. For example, if _Gt,i_ is deterministic, we still denote it as _St_ +1 _,i_ = _Gt,i_ ( **S** _t,_ **a** _t, ✏t_ ) even though _St_ +1 _,i_ = _G_<sup>e</sup> _t,i_ ( **S** _t,_ **a** _t_ ) + 0 _· ✏t_ . Similarly, if there are some unused elements of _✏t_ or some redundant arguments in a function _Gt,j_ , we can multiply them by zero in _Gt,j_ and thus still use _St_ +1 _,j_ = _Gt,j_ ( **S** _t,_ **a** _t, ✏t_ ) .

> 8Pindyck (1984) shows that stochastic fluctuations add a risk premium to the rate of return required to keep a unit of renewable resource stock in situ. Conrad (2000) demonstrates that the presence of uncertainty over benefits and opportunity costs of extracting natural forest land attaches option value for wilderness preservation, and may require a higher return on development to induce deforestation. Daigneault et al. (2010) argue that the presence of uncertainty over forest fire requires more frequent thinning and shorter rotations of managed forests.

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method (ENLCEQ), that extends the original NLCEQ method of Cai et al. (2017) designed primarily for solving dynamic stochastic stationary problems in macroeconomics. Later in this study, we apply this method to solve a complicated dynamic stochastic land use problem, that features high dimensionality, occasionally binding constraints, and time-varying exogenous trends, with acceptable accuracy. It is important to note that although the focus of this study is on stochastic natural resource allocation problems, the ENLCEQ method is a general method that can be applied to approximately and numerically solve dynamic stochastic problems in other areas of the economics discipline.

The original NLCEQ method computes globally valid, nonlinear certainty-equivalent decision rules, and then uses them to generate simulation paths for stationary infinite horizon problems. For non-stationary problems, this involves computing such rules at each period _t_ along time-variant exogenous paths a↵ecting decisions. However, computing all these rules can be very time-consuming and unnecessary if our primary goal is to obtain simulation paths and their distributions until a time of interest, _T_ , even a long one (in environmental and climate change economics, for example, we are often interested in solutions for the coming century, and set the time of interest to 100 years and the problem horizon of more than 300 years to avoid a large impact of terminal conditions). Instead of solving for optimal decisions for all possible states at each time, we can approximately solve for optimal decisions for those simulated states along simulated paths. Thus, we introduce the following ENLCEQ algorithm.

**Algorithm 1** ENLCEQ for Dynamic Stochastic Problems with Time-Variant Exogenous <u>Paths</u>

- **Initialization.** _Given the initial state_ **S** 0 _and a time of interest T , choose a finite T_<sup>b</sup> _≫ T and an approximate terminal value function V_<sup>e</sup> b _T_<sup>(</sup><sup>**S**b</sup> _T_<sup>)</sup><sup>_foraninfinite-horizonproblem,_</sup> _or let T_<sup>b</sup> = _T_<sup>_⇤_</sup> _≥ T and V_<sup>e</sup> b _T_<sup>_⌘VT ⇤forafinitehorizonproblemwithT ⇤<1asthe_</sup> _terminal time. Simulate a sequence of ✏t to get m paths, denoted ✏_<sup>_i_</sup> _t_<sup>_forpathi,from_</sup> _t_ = 0 _to T_<sup>b</sup> _. For each i_ = 1 _, ..., m, repeat the following steps to get m simulated paths of states and decisions to obtain their distributions._

- **Step 1.** _Let_ **S**<sup>_i_</sup> 0<sup>=</sup><sup>**S**0</sup><sup>_.Fors_= 0</sup><sup>_,_1</sup><sup>_,_2</sup><sup>_, ..., T,iteratethroughsteps2and3._</sup>

**Step 2.** _Solve the following deterministic model starting from time s:_

**Step 3.** _Set_ **S**<sup>_i_</sup> _s_ +1<sup>=</sup><sup>**G**</sup><sup>_t_(</sup><sup>**S**</sup><sup>_i_</sup> _s_<sup>_,_</sup><sup>**a**</sup><sup>_i_</sup> _s_<sup>_, ✏i_</sup> _s_<sup>)</sup><sup>_,where_</sup><sup>**a**</sup><sup>_i_</sup> _s_<sup>_isthe optimaldecisionat timesofthe problem_</sup> _(2)._

Algorithm 1 obtains simulated pathways of optimal decisions and states. Its step 2 applies the original idea of NLCEQ method: for a given state at time _s_ , **S**<sup>_i_</sup> _s_<sup>,wereplaceall</sup> future stochastic variables by their corresponding expectations conditional on the current 6
state **S**<sup>_i_</sup> _s_<sup>,9and convert the dynamic stochastic problem (1) into a deterministic finite-horizon</sup> dynamic problem (2).

For an infinite-horizon problem, step 2 changes it to be a finite-horizon problem with a large horizon _T_<sup>b</sup> . This will have little impact on the solutions within the period of interest: [0 _, T_ ], because (i) most of infinite-horizon dynamic economic models assume that the system asymptotically evolves to its stationary state; (ii) the discount factor _δ <_ 1 makes the terms _δ_<sup>_t−T_</sup> _Ut_ ( **S** _t,_ **a** _t_ ) to have small magnitude for _t ≥ T_<sup>b</sup> with a large _T_<sup>b</sup> and a feasible and reasonable sequence of ( **S** _t,_ **a** _t_ ) so that the terms after _T_<sup>b</sup> has little changes on the objective function; (iii) we choose _T_<sup>b</sup> to be large enough such that any larger _T_<sup>b</sup> will almost have no change in the solutions in the periods of interest [0 _, T_ ]; and (iv) we may choose a good approximate terminal value function such as _V_<sup>e</sup> b _T_<sup>(</sup><sup>**S**b</sup> _T_<sup>)=</sup><sup>_U_b</sup> _T_ ⇣ **S** b _T_<sup>_,_e</sup><sup>**a**</sup><sup>_⇤_</sup> _T_ b<sup>(</sup><sup>**S**b</sup> _T_<sup>)</sup> ⌘ _/_ (1 _− δ_ ) to approximate the true value function at _T_<sup>b</sup> : _V_ b _T_<sup>(</sup><sup>**S**b</sup> _T_<sup>)=maxP</sup><sup>_1_</sup> _t_ = _T_<sup>b</sup><sup>_δt−_</sup> _T_<sup>b</sup> _Ut_ ( **S** _t,_ **a** _t_ ) subject to e the transition law of **S** _t_ and the feasibility constraint for **a** _t_ , where **a**<sup>_⇤_</sup> _T_ b<sup>(</sup><sup>**S**b</sup> _T_<sup>)isagoodguess</sup> of the true optimal policy function at _T_<sup>b</sup> .

In step 2 of Algorithm 1, we drop the uncertainty in the transition law of **S** _t_ of the original problem (1) by replacing _✏t_ by its zero mean, so that the expectation in the objective function of (1) is cancelled. We call this a certainty equivalent approximation.<sup>10</sup> We implement the optimal control method to solve (2) numerically, that is, we view (2) as a large-scale nonlinear constrained optimization problem with _{_ **a**<sup>_i_</sup> _t_<sup>:</sup><sup>_t≥s}_and</sup> _{_ **S**<sup>_i_</sup> _t_<sup>:</sup><sup>_t≥s}_asitsdecisionandstatevariables.Theproblemcanbedirectlysolvedwith</sup> an appropriate nonlinear optimization solver such as CONOPT (Drud, 1994).<sup>11</sup> Observe that we just need to keep the solution at time _s_ , **a**<sup>_i_</sup> _s_<sup>,foruseinthenextstep.Instep3</sup> of Algorithm 1 we use the optimal decision **a**<sup>_i_</sup> _s_<sup>togeneratethenext-periodstate,</sup><sup>**S**</sup><sup>_i_</sup> _s_ +1<sup>=</sup> **G** _t_ ( **S**<sup>_i_</sup> _s_<sup>_,_</sup><sup>**a**</sup><sup>_i_</sup> _s_<sup>_, ✏i_</sup> _s_<sup>),givenrealizationofshocks,</sup><sup>_✏i_</sup> _s_<sup>.Oncewereachthestate</sup><sup>**S**</sup><sup>_i_</sup> _s_ +1<sup>attime</sup><sup>_s_+ 1,</sup> we come back to implement step 2 and then step 3. In other words, Algorithm 1 uses an adaptive management way: decisions are made for the current period in the face of the future uncertain shocks; once the next-period shock is observed, decisions for the next period are made with re-optimization given the observed shock and new state variables at the next period. Observe that the serial correlation of random variables has been captured in their associated transition laws. Repeating this process iteratively through _T_ times,

> 9As _✏t_ is a serially uncorrelated stochastic process, we can replace _✏t_ by its zero mean in the functions of **F** _t_ and **G** _t_ in (2) if all transition laws are continuous. For problems with a discrete Markov chain in transition laws, we can use the same technique as described in Cai et al. (2017) for NLCEQ with a discrete stochastic state to obtain the corresponding deterministic model (1). That is, given realization of the Markov chain at time _s_ , we can compute expectations of the Markov chain at all times after _s_ conditional on the value at time _s_ and then replace the stochastic process by the path of the conditional expectations in step 2 of Algorithm 1.

> 10If the utility _Ut_ is a common power function with a relative risk aversion parameter _γ_ , then the role of the risk aversion disappears in the certainty equivalent approximation model (2). As _γ_ is also the inverse of the intertemporal elasticity of substitution (IES) for time-separable power utilities, the IES, 1 _/γ_ , still a↵ects the solution to the deterministic model (2). For this reason, the NLCEQ and the ENLCEQ methods cannot work for dynamic portfolio problems, where the risk aversion is important for risky portfolio choices, dynamic stochastic problems with Epstein–Zin preferences (e.g., Cai and Lontzek (2019)) in which the risk aversion and the IES are separated, or static problems, where _γ_ has only the risk aversion role. For the stochastic land use problem in the next section and the optimal growth model in the appendix, we use time-separable utilities, so the ENLCEQ is an appropriate method. As we cannot _a priori_ determine the implications of certainty equivalence assumption on the solution accuracy, we should always check the errors of solutions of ENLCEQ. As we show below, for examples analyzed in this study, the solution errors are minimal.

> 11See Cai (2019) for discussion on the optimal control method.

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we compute a representative simulated pathway of optimal decisions, _{_ **a**<sup>_i_</sup> _s_<sup>_}T_</sup> _s_ =0<sup>,andstates,</sup> _{_ **S**<sup>_i_</sup> _s_<sup>_}T_</sup> _s_ =0<sup>,whichcorrespondstotherealizedpathofshocks,</sup><sup>_{✏i_</sup> _s_<sup>_}T_</sup> _s_ =0<sup>.Repeatingover</sup><sup>_i_,we</sup> compute _m_ simulated paths of optimal states and decisions, ( **S** _t,_ **a** _t_ ), from time 0 to _T_ , and then obtain their distributions. This simulation process can be naturally parallelized.<sup>12</sup> For an illustration of the method’s accuracy, in the appendix, we solve a simple optimal growth problem with stochastic discrete total factor productivity (TFP),<sup>13</sup> represented as a Markov chain. TFP and capital are state variables (where TFP is discrete, and capital is continuous), and consumption is a continuous decision variable. Using ENLCEQ, we generate 1,000 simulated solution paths of capital, productivity, and optimal consumption over the first 20 periods. That is, at each realized state of productivity and capital, **S**<sup>_i_</sup> _s_<sup>,</sup> we have its corresponding optimal consumption, **a**<sup>_i_</sup> _s_<sup>,for</sup><sup>_i_=1</sup><sup>_, ...,_1000and</sup><sup>_s_=0</sup><sup>_, ...,_19.</sup> We also solve the same growth problem using value function iteration (VFI) with a highdegree Chebyshev polynomial approximation on the value function and obtain the optimal consumption policy (approximated by a high-degree Chebyshev polynomial for each discrete value of productivity). We compare the ENLCEQ solution of consumption, **a**<sup>_i_</sup> _s_<sup>,andthe</sup> values of the VFI optimal consumption policy at the corresponding ENLCEQ state, **S**<sup>_i_</sup> _s_<sup>,for</sup> each _i_ = 1 _, ...,_ 1000 and _s_ = 0 _, ...,_ 19.<sup>14</sup> We find that the _L_<sup>1</sup> relative error (i.e., the average of absolute relative errors across _i_ = 1 _, ...,_ 1000 and _s_ = 0 _, ...,_ 19) of the ENLCEQ solution is 3 _._ 7 _⇥_ 10<sup>_−_3</sup> , and the _L_<sup>_1_</sup> relative error (i.e., the maximum of absolute relative errors) is 5 _._ 5 _⇥_ 10<sup>_−_3</sup> .<sup>15</sup> This shows that ENLCEQ can solve a dynamic stochastic problem with 2 to 3 digit accuracy, which is similar to the approximation errors of Cai et al. (2017). We also check the normalized Euler errors for our dynamic stochastic land use problem below and find that the _L_<sup>1</sup> error of solutions for the first 100 years (the periods of interest) among 1,000 simulated paths is only 8 _._ 6 _⇥_ 10<sup>_−_4</sup> , and the corresponding _L_<sup>_1_</sup> error is only 0 _._ 02.<sup>16</sup> This is within range of acceptable accuracy for the most dynamic stochastic natural resource problems.

# Compared to the original NLCEQ method, the ENLCEQ method can solve nonstation-

> 12The numerical solution approach may resemble a well-known Monte Carlo procedure, but there are important di↵erences. Unlike the Monte Carlo procedure, for a given initial-time state, ENLCEQ only needs to solve for one case of (2) with _s_ = 0 to obtain an initial-time solution, whereas the Monte Carlo method needs to solve thousands of same-size cases, whereby each case corresponds to one simulated path of shocks. The ENLCEQ method solves (2) period by period to obtain solutions for the next periods. That is, the decision-maker learns realized shocks in the previous periods, forms certainty equivalent approximation over future shocks conditional on the current-period state, and then finds the solution at the current period. The Monte Carlo method does not have this learning and adaptive property. Both ENLCEQ and Monte Carlo methods ignore the risk aversion and are thus unable to solve dynamic portfolio problems or dynamic stochastic problems with Epstein–Zin preferences. See Cai (2019) for more discussion on the Monte Carlo and certainty-equivalence methods.

> 13We choose this model because it is a standard benchmark for testing novel computational methods, see, e.g., Den Haan et al. (2011).

> 14Note that we may also use Chebyshev polynomials to approximate the ENLCEQ solution of consumption with the data �� **S**<sup>_i_</sup> _s_<sup>_,_</sup><sup>**a**</sup><sup>_i_</sup> _s_ � : _i_ = 1 _, ...,_ 1000; _s_ = 0 _, ...,_ 19 , and then compare these Chebyshev polynomials with the VFI optimal consumption policy. But this will introduce policy function approximation errors to ENLCEQ. Moreover, decision-makers are more interested in estimating errors on policies at the realized states instead of the whole state space, particularly for non-stationary problems with time-varying policy functions. Furthermore, for problems with high-dimensional state spaces, 1,000 simulated paths are often not enough to obtain a good approximation to its policy function. Therefore, we compare the ENLCEQ solution, **a**<sup>_i_</sup> _s_<sup>,directlywiththevaluesoftheVFIoptimalconsumptionpolicyatthecorrespondingENLCEQ</sup> state, **S**<sup>_i_</sup> _s_<sup>.</sup>

> 15We view the VFI optimal consumption policy to be the “true” solution as there is no analytical solution for the optimal growth problem.

> 16For more details of the error checking, please refer to the appendix.

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ary problems which the NLCEQ method cannot unless it is applied at every period _t_ . Even for stationary problems, ENLCEQ does not require explicit approximation of the optimal decision rules so it may be more suitable for implementation of high-dimensional problems that require sparse grid approximation and are difficult to code. Moreover, ENLCEQ is more efficient than the original NLCEQ method, if the number of simulations, _m_ , is less than the number of approximation nodes used in the original NLCEQ method. For example, for the dynamic stochastic model of global land use in this study, it has 15 continuous state variables and 2 discrete state variables (totally 5 _⇥_ 3 = 15 discrete states). If we use only 5 nodes per continuous state variable to get degree-4 complete Chebyshev polynomial approximation, NLCEQ needs to solve 5<sup>15</sup> _⇥_ 15 _⇡_ 4 _._ 6 _⇥_ 10<sup>11</sup> optimization problems at every period _t_ . Even if we use only 3 nodes per continuous state variable to get a quadratic polynomial approximation, NLCEQ needs to solve 3<sup>15</sup> _⇥_ 15 _⇡_ 2 _._ 2 _⇥_ 10<sup>8</sup> optimization problems at every period _t_ . The ENLCEQ method only needs to solve _m_ optimization problems at every period _t_ . Typically we can choose _m_ = 1 _,_ 000, so it is about 22,000 times faster than NLCEQ with quadratic polynomial approximation, or 4 _._ 6 _⇥_ 10<sup>8</sup> times faster than NLCEQ with degree-4 polynomial approximation. Furthermore, because the ENLCEQ method does not use an explicit approximation to continuous value / policy functions, there are no functional approximation errors, and solutions from ENLCEQ may be more accurate than those from the original functional approximation-based NLCEQ method.

# **3 Stochastic FABLE model**

This section presents a modeling framework for analyzing nonlinear dynamic stochastic models of natural resource use with multiple sectors, in which preferences, production technology, resource endowments, and other exogenous state variables evolve stochastically over time according to a Markov process with time-varying transition probabilities.<sup>17</sup> Specifically, we develop a stochastic version of a global land use model nicknamed FABLE (Forest, Agriculture, and Biofuels in a Land use model with Environmental services), a dynamic multi-sectoral model for the world’s land resources over the next century (Steinbuks and Hertel, 2012, 2016). This model brings together recent strands of agronomic, economic, and biophysical literature into a single, intertemporally consistent, analytical framework, at the global scale. FABLE is a discrete dynamic partial equilibrium model, where the population, labor, physical and human capital, and other variable inputs are assumed to be exogenous. Total factor productivity and technological progress in non-land intensive sectors are also predetermined. The model focuses on the optimal allocation of scarce land across competing uses across time and solves for the dynamic paths of alternative land uses, which together maximize global economic welfare.

The FABLE model accommodates a complex dynamic interplay between di↵erent types of global land use, whereby the societal objective function places value on processed crops and livestock, energy services, timber products, ecosystem services, and other non-land goods and services (Figure 1). There are three accessible primary resources in this partial

> 17 The constructed model belongs to the class of stochastic growth models with multiple sectors studied in Brock and Majumdar (1978), Majumdar and Radner (1983), and Stokey et al. (1989) among others.

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<!-- Start of picture text --> Technological Progress,<br>Climate Change<br><!-- End of picture text --> <!-- Start of picture text --> Utility<br><!-- End of picture text --> <!-- Start of picture text --> Technological Progress, Utility<br>Climate Change<br>Final Ecosystem Processed Processed Processed Energy Other<br>Goods Services Timber Crops Livestock Services G&S<br>Livestock 1 st Gen. 2 nd Gen.<br>Raw Biofuels Biofuels<br>Intermediate Timber<br>products Food Crops<br>Biofuels’ Crops Petroleum<br> Animal Feed Fertilizers Products<br>Managed Crop Pasture<br>Forests Land Land<br>Primary<br>Inputs<br>Protected Unmanaged Fossil Other Primary<br>Forests Forests Fuels Inputs<br><!-- End of picture text --> <!-- Start of picture text --> Final<br>Goods<br><!-- End of picture text --> <!-- Start of picture text --> Intermediate<br>products<br><!-- End of picture text --> <!-- Start of picture text --> Primary<br>Inputs<br><!-- End of picture text --> <!-- Start of picture text --> Fossil<br>Fuels<br><!-- End of picture text --> <!-- Start of picture text --> Protected<br>Forests<br><!-- End of picture text --> <!-- Start of picture text --> Unmanaged<br>Forests<br><!-- End of picture text --> <!-- Start of picture text --> Other Primary<br>Inputs<br><!-- End of picture text --> Figure 1: Structure of the FABLE Model Note. State variables, **S** _t_ are shown as oval shapes. Decision variables, **a** _t_ , are shown as rectangular shapes. Utility function, _Ut_ , is shown as octagonal shape. Transition laws, **G** _t_ , are shown as dotted arrows. Stochastic model terms incorporating random processes, _✏t_ , are shown as dashed shapes or arrows.

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equilibrium model of the global economy: land, liquid fossil fuels, and other primary inputs, e.g., labor and capital (see the bottom part of Figure 1). The supply of land is fixed and faces competing uses that are determined endogenously by the model. They include unmanaged forest lands - which are in an undisturbed state (e.g., parts of the Amazon tropical rainforest ecosystem), agricultural (or crop) land, pasture land, and commercially managed forest land.<sup>18</sup> As trees of di↵erent age have di↵erent timber yields and di↵erent propensities to sequester carbon, the model keeps track of various tree vintages in managed forests,<sup>19</sup> which introduces additional numerical complexity for solving the model. The flow of liquid fossil fuels evolves endogenously along an optimal extraction path, allowing for exogenously specified new discoveries of fossil fuel reserves. Other primary inputs include variable inputs, such as labor, capital (both physical and human), and intermediate materials. The endowment of other primary inputs is exogenous and evolves along a pre-specified global economic growth path.

There are six intermediate inputs used in the production of land-based goods and services in FABLE: petroleum products, fertilizers, crops, liquid biofuels,<sup>20</sup> live animals, and raw timber (see the middle part of Figure 1). Fossil fuels are refined and converted to either petroleum products, that are further combusted, or to fertilizers, that are used to boost yields in the agricultural sector. Cropland and fertilizers are combined to grow crops, that can be further converted into processed food and biofuels, or used as an animal feed. Specifically, we assume that agricultural land _L_<sup>_A,c_</sup> _t_ and fertilizers _x_<sup>_n,c_</sup> _t_ are imperfect substitutes in production of food crops, _x_<sup>_c_</sup> _t_<sup>,withspecificproductiontechnologygivenbythefollowing</sup> constant elasticity of substitution (CES) function: where _✓t_<sup>_c_is stochastic crop technology index, and</sup><sup>_↵n_and</sup><sup>_⇢n_are, respectively, the input share</sup> and substitution parameters. Equation (3) captures three key responses within the model to changes in crop technology index: (i) demand response (change in consumption of food crops), (ii) adaptation on the extensive margin (substitution of agricultural land for other land resources), and (iii) adaptation on the intensive margin (substitution of agricultural land for fertilizers).

The biofuels substitute imperfectly for liquid fossil fuels in final energy demand. The food crops used as animal feed and pasture land are combined to produce raw livestock. Harvesting managed forests yield raw timber, that is further used in timber processing.

The land-based consumption goods and services take the form of processed crops, livestock, and timber, and are, respectively, outcomes of food crops, raw livestock and timber

> 18We ignore other land use types, such as savannah, grasslands, and shrublands, which are largely unmanaged and often of limited productivity. This makes them difficult to incorporate into an economic model of land use. Consequently, they are typically left out of most contemporary analyses of global land use change (Hertel et al., 2009). We also ignore residential, retail, and industrial uses of land in this partial equilibrium model of agriculture and forestry.

> 19We do not keep track of vintages for natural lands and assume they are primarily old grown forest.

> 20In FABLE, bioenergy does not include the potential use of biomass in power generation. This limitation is acknowledged in Steinbuks and Hertel (2016, p. 566): “A more serious limitation to this study is our omission of the potential demand for biomass in power generation. Under some scenarios, authors have shown this to be an important source of feedstock demand by mid-century (Rose et al. 2012). However, absent a full representation of the electric power sector, our framework is ill-suited to addressing this issue.”

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processing. The production of energy services combines non-land energy inputs (i.e., liquid fossil fuels) with the biofuels, and the resulting mix is further combusted. Finally, all land types have the potential to contribute to other ecosystem services, a public good to society, which includes recreation, biodiversity, and other environmental goods and services.<sup>21</sup> To close the demand system, we also include other non-land goods and services (e.g., manufacturing goods and retail, construction, financial, and information services), which involve ’consumption’ of other primary inputs not spent on the production of land-based goods and services. As the model focuses on the representative agent’s behavior, the final consumption products are all expressed in per-capita terms.

To preserve space, a complete description of model equations, variables, and parameter values is presented in the online appendix.

# **4 Modeling Crop Yield Uncertainty**

This section characterizes uncertainty in future agricultural yields over the coming century, which is one of the core uncertainties shown to a↵ect land use in the long run (Steinbuks and Hertel, 2013). Crop yields are subject to two types of uncertainties: those related to the development and dissemination of new technologies, and those related to changes in the climatic conditions under which the crops are grown. The former type of uncertainty has until recently dominated the pattern of the evolution of the global crop yields, whereas the latter is becoming an increasingly important factor (Lobell and Field, 2007; IPCC, 2014a). While it is plausible to hypothesize that accelerating climate impacts may, in turn, induce further technological advances in an e↵ort to facilitate adaptation to climate change, this hypothesis is not supported by limited empirical evidence.<sup>22</sup> Therefore, in this paper, these two sources of uncertainty are treated separately, although they are both characterized by the use of combined climate and crop simulation models run over a global grid.

We characterize future uncertainty in yields by constructing a stochastic crop productivity index, _✓t_<sup>_c_,whichcapturestheevolutionoffuturecropyieldsunderdi↵erentrealizations</sup> of uncertainty in crop productivity based on the most recent projections in the agronomic and environmental science studies. An important characteristic of staple grains yields is that they tend to grow linearly, adding a constant amount of gain (e.g., ton/ha) each year (Grassini et al., 2013). This suggests that the proportional growth rate should fall gradually over time. However, crop physiology dictates certain biophysical limits to the rate at which sunlight and soil nutrients can be converted to the grain. And there is some recent agronomic evidence (Cassman et al., 2010; Grassini et al., 2013) showing that yields appear to be reaching a plateau in some of the world’s most important cereal-producing countries. Cassman (1999) suggests that average national yields can be expected to plateau when they

> 21It is well established in the environmental science literature that managed lands have positive environmental externalities. For example, managed forests provide timber, but also help to retain soils and moisture, as well as creating microclimates. Crop and pastures provide food, but also facilitate pollination, wild animal feed, and biological control of prey species and reduction of herbivory by top predators (Kumar, 2010). As FABLE aims to find the socially optimal path of global land use, it does account, albeit in a stylized way, for these complex contributions of the world’s land resources to human welfare.

> 22For example, in a recent study of climate change adaptation in the United States, arguably one of the most technologically developed countries, Burke and Emerick (2016) conclude that “longer-run adaptations appear to have mitigated less than half—and more likely none—of the large negative short-run impacts of extreme heat on productivity.”

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reach 70–80% of the genetic yield potential ceiling. Based on these observations from the agronomic literature, we specify the following logistic function determining the evolution of the crop productivity index over time: where _✓_ 0<sup>_c_is the value of the crop productivity index in period 0, which we calibrate to match</sup> observed weighted yields in key staple crops (corn, rice, soybeans, and wheat), _✓T_<sup>_c_is the crop</sup> yield potential in the end of the current century, that is, “the yield an adapted crop cultivar can achieve when crop management alleviates all abiotic and biotic stresses through optimal crop and soil management” (Evans and Fischer, 1999), and _c_ is the logistic convergence rate to achieving potential crop yields.

Though the initial value of the crop productivity index is known with certainty, potential crop yields are highly uncertain. We assume that potential crop yields are a↵ected by a two-dimensional stochastic process of climate and technological shocks, _J_ 1 _,t_ , and _J_ 2 _,t_ , respectively. For the technological shock, _J_ 2 _,t_ , we assume that there are three states of technology: “bad” (indexed by _J_ 2 _,t_ = 1), “medium” (indexed by _J_ 2 _,t_ = 2), and “good” (indexed by _J_ 2 _,t_ = 3). In the optimistic (i.e., “good”) state of advances in crop technology, we assume that yields continue to grow linearly throughout the coming century, eliminating the yield gap by 2100. In the “medium” state of technology, rather than closing the yield gap by 2100, average yields in 2100 are just three-quarters of yield potential at that point in time. In the “bad” state of technology, there is no technological progress, and the crop yields stay the same as at the beginning of the coming century.

For the climate shock, _J_ 1 _,t_ , we assume it is a Markov chain with five possible states at each time _t_ . To construct these states, we use the results of Rosenzweig et al. (2014), who conducted a globally consistent, protocol-based, multi-model climate change assessment for major crops with the explicit characterization of uncertainty.<sup>23</sup> Based on this assessment, we construct five states that correspond to quintiles of the distribution of di↵erent outcomes of four global crop simulation models and five global climate models, with and without CO2 fertilization e↵ects for potential crop yields by 2100. Under two optimistic states of the world, we observe a 2 and 15 percent increases in potential crop yields relative to model baseline, respectively, whereby significant CO2 fertilization e↵ects o↵set the negative e↵ects of climate change. For the next two states, we see a 15 and 19 percent declines in potential crop yields relative to model baseline whereby CO2 fertilization e↵ects are assumed to be either small or non-existent, and the negative e↵ects of climate change tend to prevail. Finally, under the most pessimistic state of the world, drastic adverse e↵ects of climate change combined with the absence of any CO2 fertilization e↵ects result in a 36 percent decline in potential crop yields relative to model baseline.

Further details of constructing climate and technological states can be found in the

> 23 Given the partial equilibrium nature of FABLE, we cannot directly capture all sources of GHG emissions and, therefore, endogenize their e↵ect on global land use. However, as global land use emissions account for less than a quarter of global GHG emissions (IPCC, 2014b), climate-induced changes in land use will be relatively small to have a major e↵ect on global temperatures.

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Appendix.

The path of technological change in crop yields evolves by reversible transitions across these states. The stochastic path of the crop productivity index is then given by where _AT_ ( _i, j_ ) represents the crop productivity index at the terminal time _T_ at the state _J_ 1 _,t_ = _i_ and _J_ 2 _,t_ = _j_ , for _i_ = 1 _,_ 2 _, ...,_ 5 and _j_ = 1 _,_ 2 _,_ 3.<sup>24</sup> Thus, _At_ is a Markov chain, which takes one of 15 possible time-varying values at each time period. This can be seen as a discretization of a mean-reverting process with continuous values and time trend, but a finer Markov chain with more values can only marginally change our solution. As _At_ is completely dependent on _J_ 1 _,t_ and _J_ 2 _,t_ , it is not a state variable, whereas _J_ 1 _,t_ and _J_ 2 _,t_ are both state variables. Having characterized the realizations of crop productivity under alternative states of the agricultural technology and climate change impacts, we still need to calibrate transition probabilities for the climate and technology shocks to construct the stochastic crop productivity index. As regards climate shock, the environmental and climate science literature acknowledges some degree of persistence but does not provide much guidance on the transition dynamics between alternative climate states a↵ecting crop yields. In the absence of reliable estimates, for constructing the transition probability matrix of _J_ 1 _,t_ we assume simple transition dynamics, where each state has a 50 percent probability of retaining itself next period and a 25 percent probability of moving upwards or downwards to an adjacent state. As regards the technology shock, since we do not have the historical data for evolution of agricultural technology, we assume that technological advances in agriculture follow a similar trend to advances in the rest of the economy, and use the probability transition matrix of _J_ 2 _,t_ estimated by Tsionas and Kumbhakar (2004) for a comprehensive panel of 59 countries over the period of 1965–1990. These estimates correspond to a 20 percent probability of the “bad” technological state, 56 percent of the “medium” state, and 24 percent of the “good” state. The transition probability matrices of _J_ 1 _,t_ and _J_ 2 _,t_ are shown in the Appendix.

Figure 2 shows the deterministic-baseline path (the solid line) used in the perfect foresight model and the range of the stochastic crop productivity index based on 1,000 simulation paths over the entire 21st century, with additional summary statistics presented in Appendix. The simulations start at the “medium” states of climate and technology in the initial year. The deterministic-baseline path is calculated by taking expectations of the stochastic crop productivity index conditional on the initial “medium” states (equation 5). It also takes the same values as the median path (the “o” line) of simulations, whereby the climate and technological states are kept at “medium”, while the average line (the “+” line) deviates a bit after the year 2070. At every time _t_ , there are 1,000 realized values of _At_ among which there are only 15 di↵erent values. The 10% and 90% quantile lines (the dashed and dash-dotted lines) represent the 10% and 90% quantiles of these 1,000 simulated values of

> 24It is straghtforward to demonstrate that lim _t!_ 0  _AT_ ( _AJT_ 1 ( _,tJ,J_ 1 _<u>,</u>_ 2 _t,t,J_ )+2 _<u>,tA</u>_ <u>)</u> _A_ 0(0 _ee_<sup>_<u>ctct</u>_</sup> _−_ 1) � = _A_ 0 and lim _t!1_  _AT_ ( _AJT_ 1 ( _,tJ,J_ 1 _<u>,</u>_ 2 _t,t,J_ )+2 _<u>,tA</u>_ <u>)</u> _A_ 0(0 _ee_<sup>_<u>ctct</u>_</sup> _−_ 1) � = _AT_ .

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<!-- Start of picture text --> Crop Productivity Index<br>28<br>Range of all sample paths of stochastic model<br>Deterministic-Baseline<br>26 average<br>10% quantile<br>50% quantile<br>24 90% quantileone sample path<br>22<br>20<br>18<br>16<br>14<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year<br><!-- End of picture text --> Figure 2: Crop Productivity Index _At_ at time _t_ , so they are not realized sample paths, but Figure 2 also displays one realized sample path of _At_ which is the dotted line.

# **5 Model Results**

This section describes the results of the impact of crop yield uncertainty on the optimal path of global land use based on the dynamic stochastic model simulations. While it is unlikely that the world’s land will be optimally allocated in the coming century, knowledge of this path can provide important insights into how uncertainty and irreversibility shape the desired path for global land use decisions. We solve the model over the 400 years with 5-year time steps and present the results for the first 100 years to minimize the e↵ect of terminal period conditions on our analysis.<sup>25</sup> We first present the results of the perfect foresight model, wherein the optimal land allocation decisions are made based on the values of the crop productivity index in the absence of climate and technology shocks. This deterministic analysis is a useful reference point for further discussion when the uncertainty in food crop yields is introduced. We then present the results of the dynamic stochastic model, where the impact of the intrinsic climate and technology uncertainty is brought into the model optimization stage. Specifically, we generate 1,000 sample paths of optimal global land use under di↵erent realizations of the stochastic crop productivity index. The results are presented as the di↵erence between the stochastic path and deterministic reference solution. **5.1 Optimal Path of Global Land Use under Crop Yield Uncertainty** Figure 3 depicts the optimal allocation of global land use over the next century. The left-hand side of Figure 3 shows the deterministic paths of di↵erent types of land considered

> 25The model converges to its stationary state around 2150. The di↵erences in land use allocations between 2100 and 2150 are small and therefore not reported.

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in this study, i.e., when the food crop yields are perfectly anticipated. Specifically, it shows three scenarios, where the value of the crop technology index corresponds to (i) expected values of the stochastic crop productivity index (deterministic-baseline scenario), (ii) the most pessimistic climate and bad technology states (deterministic-pessimistic scenario), and (iii) the most optimistic climate and good technology states (deterministic-optimistic scenario). The right-hand side of Figure 3 shows the di↵erence range between the 1,000 simulation paths based on di↵erent ex-ante realizations of the stochastic crop productivity index and the deterministic-baseline path. The 10%, 50% and 90% quantile lines represent 10%, 50% and 90% quantiles of 1,000 simulated values respectively at each time, and the average line (the “+” line) represent the average of 1,000 simulated values respectively at each time. The right-hand side of Figure 3 also shows two extreme cases of optimal land-use paths conditional on period _t_ realizations of crop productivity index states _At_ ( _J_ 1 _,t, J_ 2 _,t_ ). The realized crop productivity index always takes the highest possible value in a _stochasticoptimistic_ case (the line of squares), and the lowest possible value in a _stochastic-pessimistic_ case (the line of marks). As future realizations of the stochastic crop productivity index are uncertain, these extreme stochastic solutions are not the same as the corresponding deterministic solutions, where the values of future crop productivity index are known with certainty. In the stochastic-optimistic case, for example, potentially lower realizations of future crop productivity index result in larger current-period agricultural land allocation as compared to the deterministic-optimistic solution. For other model variables, due to resource limits (e.g., the total land area is unchanged over time) and other constraints, the impact of uncertainty is theoretically difficult to assess.

We start with the left-hand side of panels (a)-(e) of Figure 3 that shows the optimal land use paths under the perfect foresight. Beginning with the description of the baseline scenario we see that, in the first half of the coming century, the area dedicated to food crops increases by 350 million hectares or 22 percent compared to 2004, reaching its maximum of 1.88 billion hectares around mid-century (panel a). Continuing population growth, intensification of livestock production, and increasing demand for food, stemming from economic growth are the key drivers for this cropland expansion. In the second half of the coming century, slower population growth, and technology improvements in crop yields and food processing result in a smaller demand for cropland. By 2100 cropland area declines significantly relative to its peak value, falling to 1.45 billion hectares, which is, 6 percent smaller than in 2004. In the first half of the coming century land allocation for the second-generation biofuels is close to zero (panel b). Consistent with the recent analysis of 2G biofuels’ deployment potential (National Research Council, 2011), absent aggressive GHG regulations and biofuels’ policies, this technology is suboptimal because of low extraction and refining costs of fossil fuels and high production and deployment costs of the second-generation biofuels. In the second half of the coming century, the second-generation technology becomes viable, as fossil fuels become scarce and costs of producing the second-generation biofuels decline. This results in greater land requirements for the second-generation biofuels crops. By the end of the coming century, agricultural land dedicated to the second-generation biofuels crops adds 500 million hectares. Consistent with recent trends, global pasture area declines throughout the entire century (panel c), reflecting increased substitution of pasture land for 16
a) <!-- Start of picture text --> Agricultural Land Area, Food Crops (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>100<br>Range of all sample paths<br>1.9 average<br>80 10% quantile<br>50% quantile<br>90% quantile<br>60 Stochastic-optimistic<br>1.8 Stochastic-pessimistic<br>40<br>1.7 20<br>0<br>1.6<br>-20<br>-40<br>1.5<br>-60<br>Deterministic-baseline<br>1.4 Deterministic-optimisticDeterministic-pessimistic -80<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion hectares million hectares<br><!-- End of picture text -->

# b)

<!-- Start of picture text --> Agricultural Land Area, 2G Biofuels Crops (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>0.55<br>Deterministic-baseline 80 Range of all sample paths<br>0.5 Deterministic-optimistic average<br>Deterministic-pessimistic 10% quantile<br>60 50% quantile<br>0.45 90% quantile<br>0.4 40 Stochastic-optimisticStochastic-pessimistic<br>0.35 20<br>0.3 0<br>0.25 -20<br>0.2 -40<br>0.15<br>-60<br>0.1<br>-80<br>0.05<br>-100<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>c)<br>Pasture Land Area (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>2.7 Range of all sample paths<br>average<br>2.65 6 10% quantile50% quantile<br>90% quantile<br>2.6 Stochastic-optimistic<br>4 Stochastic-pessimistic<br>2.55<br>2.5<br>2<br>2.45<br>2.4 0<br>2.35<br>2.3 -2<br>2.25<br>Deterministic-baseline<br>2.2 Deterministic-optimistic -4<br>Deterministic-pessimistic<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion hectares million hectares<br>billion hectares million hectares<br><!-- End of picture text --> 17
# d)

<!-- Start of picture text --> Total Managed Forest Land Area (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>1.9 10<br>1.85 5<br>1.8 0<br>1.75<br>-5<br>Range of all sample paths<br>1.7 -10 average10% quantile<br>50% quantile<br>Deterministic-baseline 90% quantile<br>1.65 Deterministic-optimistic Stochastic-optimistic<br>Deterministic-pessimistic -15 Stochastic-pessimistic<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>e)<br>Natural Land Area (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>2.45 10 Range of all sample paths<br>average<br>2.4 8 10% quantile50% quantile<br>90% quantile<br>2.35 6 Stochastic-optimisticStochastic-pessimistic<br>2.3<br>4<br>2.25<br>2<br>2.2<br>2.15 0<br>2.1 -2<br>2.05<br>-4<br>2 Deterministic-baseline<br>Deterministic-optimistic -6<br>1.95 Deterministic-pessimistic<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion hectares million hectares<br>billion hectares million hectares<br><!-- End of picture text -->

# f)

<!-- Start of picture text --> Protected Land Area (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>0.5 5<br>Deterministic-baseline<br>Deterministic-optimistic<br>0.45 Deterministic-pessimistic<br>0.4 0<br>0.35<br>-5<br>0.3<br>Range of all sample paths<br>average<br>10% quantile<br>0.25 50% quantile<br>90% quantile<br>-10 Stochastic-optimistic<br>Stochastic-pessimistic<br>2020 2040 2060 2080 2100 2020 2040 2060 2080 2100<br>Year Year<br>billion hectares million hectares<br><!-- End of picture text --> Figure 3: Optimal Global Land Use Paths 18
animal feed in livestock production (Taheripour et al., 2013). Managed forest area increases throughout the entire century reaching 1.95 billion Ha by 2100 (panel d). In contrast, in response to greater requirements for agricultural land, unmanaged forest area declines by about 500 million Ha over the course of the 21st century (panel e). The decline in unmanaged forest land is less environmentally damaging in the second half of the coming century, as deforestation is limited, with most of the unmanaged forests being converted to managed or protected forest land. Finally, protected forest area more than doubles by the end of the coming century, in light of strong growth in the demand for ecosystem services (panel f). The other two scenarios exhibit broadly similar dynamics. As expected, compared to the baseline scenario, the most pessimistic scenario foresees a greater expansion of agricultural land for food crops and reduction in other types of land (except for protected lands) in response to expected lower realizations of the crop technology index. The situation is reversed for the optimistic scenario. There is a significant variation in the range of anticipated expansion of the agricultural area for food crops between optimistic and pessimistic scenarios, which amounts to 200 million hectares or 14 percent of total cropland in 2100. Variation in expected land use change for other types of land accounts for an approximately equal proportion of the variation in agricultural land for food crops, whereas protected forest areas change very little across di↵erent climate scenarios.

Uncertainty in the crop productivity index results in additional redistribution of land resources so to o↵set the impact of potentially lower yields. As social preferences exhibit intertemporal substitution in this stochastic application of the FABLE model, some of that redistribution takes place even in the absence of the actual changes in the states of climate or technology.<sup>26</sup> Compared to the deterministic scenario, the median (i.e., the 50 percent quantile) path of global land use that correspond to the “medium” state of climate ( _J_ 1 _,t_ = 3) and the “medium” technological state ( _J_ 2 _,t_ = 2) foresees a smaller use of agricultural land for food crops (panel a), and greater use of agricultural land for 2G, biofuels’ crops (panel b), pasture land (panel c), managed forest land (panel d). There is also a decline in the protected land area (panel e) and an increase in unmanaged natural land (panel d). Unlike the deterministic scenario, almost all of the variation in global land resources in stochastic simulations happens across agricultural land for food crops and 2G biofuels crops (panels a and b). For all other land resources, the di↵erences between stochastic and deterministic paths are small and economically insignificant. This is because land conversion costs of agricultural land for other types of land become larger in the presence of uncertainty. These other types of land have higher adjustment costs of conversion associated with additional time cost of regrowing lumber and livestock, and irreversibilities in accessing protected land areas. Land rotation between food crops and 2G biofuels crops is less costly in the FABLE model. This result is consistent with earlier studies that find that closer integration with the energy sector o↵ers greater potential for food-energy substitution, and thus also a greater resilience against adverse climate conditions a↵ecting food crop yields (Di↵enbaugh et al., 2012; Verma et al., 2014).

While the direction of the e↵ect of the uncertainty in the crop productivity on land

> 26 This result is similar to the theoretical findings and numerical simulations of Lanz et al. (2017).

19
conversion can be inferred from the economic theory of environmental and natural resource management under uncertainty (see, e.g., Tsur and Zemel (2014) and references therein), the extent to which this uncertainty propagates into land conversion depends critically on chosen model structure and parameters. For example, Alexander et al. (2017, p.1) find that even in the absence of intrinsic uncertainty “systematic di↵erences in land cover areas are associated with the characteristics modeling approach are at least as great as the di↵erences attributed to scenario variations”. Depending on the assumptions on the substitution of land for other resources, the size of technological progress, and the responsiveness of demand for land-based goods and services to changes in the crop productivity, this magnitude can be substantially di↵erent for other land use models. However, for the same model parameters, we can see that the range of land conversion is considerably smaller for the dynamic stochastic model as compared to the deterministic scenario analysis. As we see from Figure 3, panel (a), the di↵erence between the most extreme paths of the stochastic crop productivity index is about 170 million hectares at 2100 or about 12 percent of the total agricultural area dedicated to food crops. About half of that variation can be attributed to the most extreme (i.e., falling beyond 10<sup>th</sup> and above 90<sup>th</sup> percent quantiles) realizations of crop productivity. This is because the stochastic model assumes that climate and technological states a↵ecting crop yields are reversible (that is, if the current state is “bad” (or “good”), it could be “good” (or “bad”) in future). In comparison with the deterministic model under the pessimistic (or optimistic) scenario, the social optimum in the stochastic model requires smaller (or greater) conversion of other types of land to cropland. Thus, agricultural land area in the deterministic pessimistic (or optimistic) scenario is larger (or smaller) than the largest (or the smallest) path in the stochastic simulations. For example, in 2100, under the deterministic-pessimistic scenario, the cropland deviation from the deterministic-baseline scenario is about 100 million hectares, which is 25 percent larger than the largest deviation under the stochastic simulations, and about twice as large than the deviation above the 90% quantile of the stochastic crop technology index. This result demonstrates that scenario analysis can significantly overstate the magnitude of expected agricultural land conversion under uncertain crop yields.

# **5.2 Optimal Path of Land-Based Goods and Services under Crop Yield Uncertainty**

The left-hand side panels of Figure 4 report the optimal paths of land-based goods and services under the deterministic model scenarios. Beginning with the baseline scenario, the first panel of Figure 4 shows the production path of food crops, which increases steadily in the first half of the coming century. Compared to 2004, production of food crops (including livestock and biofuels feedstock) nearly doubles, reaching its maximum of about 11 billion tons around 2050. As with cropland expansion, rapid population growth and rising incomes are the key drivers for growing consumption on the demand side. On the supply side, the increase in the production of food crops is further boosted by growing crop yields. At the end of the coming century, production of food crops moderates, as consumers satiate their food requirements and the technology of food marketing and processing improves. By 2100 crop production for the livestock feed has leveled o↵and even begins to decline. There is also 20
a) <!-- Start of picture text --> Food Crops (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>Range of all sample paths<br>average<br>10% quantile<br>12 50% quantile<br>2 90% quantile<br>Stochastic-optimistic<br>Stochastic-pessimistic<br>11<br>1<br>10<br>0<br>9<br>-1<br>8<br>-2<br>Deterministic-baseline<br>7 Deterministic-optimistic<br>Deterministic-pessimistic<br>-3<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>b)<br>Biofuels (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>20<br>Deterministic-baseline Range of all sample paths<br>35 Deterministic-optimistic average<br>Deterministic-pessimistic 10% quantile<br>15 50% quantile<br>90% quantile<br>30 Stochastic-optimistic<br>Stochastic-pessimistic<br>10<br>25<br>5<br>20<br>0<br>15<br>-5<br>10<br>-10<br>5<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>c)<br>2G Biofuels (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>1600<br>Deterministic-baseline Range of all sample paths<br>Deterministic-optimistic 200 average<br>1400 Deterministic-pessimistic 10% quantile<br>150 50% quantile<br>90% quantile<br>1200 100 Stochastic-optimistic<br>Stochastic-pessimistic<br>50<br>1000<br>0<br>800<br>-50<br>600 -100<br>-150<br>400<br>-200<br>200 -250<br>-300<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion tons billion tons<br>million tonnes of oil equivalent million tonnes of oil equivalent<br>million tonnes of oil equivalent million tonnes of oil equivalent<br><!-- End of picture text --> 21
# d)

<!-- Start of picture text --> Livestock (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>Range of all sample paths<br>1.5 average<br>10% quantile<br>200 50% quantile<br>90% quantile<br>1.4 Stochastic-optimistic<br>Stochastic-pessimistic<br>100<br>1.3<br>0<br>1.2<br>1.1 -100<br>1 -200<br>Deterministic-baseline<br>0.9 Deterministic-optimisticDeterministic-pessimistic -300<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion tons million tons<br><!-- End of picture text -->

# e)

<!-- Start of picture text --> Timber (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>3.2<br>40<br>3<br>30<br>2.8 20<br>2.6 10<br>0<br>2.4<br>-10<br>2.2<br>-20<br>2 Range of all sample paths<br>-30 average<br>10% quantile<br>1.8 Deterministic-baseline -40 50% quantile90% quantile<br>Deterministic-optimistic Stochastic-optimistic<br>1.6 Deterministic-pessimistic -50 Stochastic-pessimistic<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>f)<br>Output of Ecosystem Services (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>3 Deterministic-baseline 20<br>Deterministic-optimistic<br>2.9 Deterministic-pessimistic 15<br>10<br>2.8<br>5<br>2.7 0<br>-5<br>2.6<br>-10<br>2.5 -15<br>-20 Range of all sample paths<br>2.4 average10% quantile<br>-25 50% quantile<br>90% quantile<br>2.3 -30 Stochastic-optimistic<br>Stochastic-pessimistic<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>billion tons million tons<br>billion USD<br>million of 2004 US dollars<br><!-- End of picture text --> Figure 4: Optimal Paths of Land-Based Goods and Services 22
a significant variation in the range of production of food crops between the most optimistic and pessimistic scenarios, which amounts to the sizable amount of 5.3 billion tons.

The results of the dynamic stochastic model simulations also show that uncertainty in the crop productivity index has a profound e↵ect on the optimal production path of food crops. Between the most extreme paths of the stochastic crop productivity index, production of food crops varies by about 5.4 billion tons compared to the corresponding deterministic path of the stochastic crop productivity index. This is a sizable change, which suggests a significant variation in levels of consumption in 2100 along di↵erent paths of stochastic crop productivity index. In the FABLE model, much of the variation in the optimal path of food crops come on the demand side, with the crop productivity decline resulting primarily in the reduced consumption of processed crops and livestock. As shown above, the uncertainty-induced supply response is relatively small along the extensive margin in the dynamic stochastic model (i.e., land conversion). In the online appendix (Figure C.1, panel a) we show that the supply response on the intensive margin is smaller, with the ratio of fertilizers to cropland increasing by less than 6 kg/Ha (or eight percent) under extreme realizations of climate and technology uncertainties. About a half of that di↵erence, however, corresponds to the most extreme (i.e., falling beyond 10<sup>th</sup> and above 90<sup>th</sup> percent quantiles) realizations of crop productivity. This result indicates that extreme uncertainty in crop productivity could have a significant impact on food consumption over the coming century.

Production of both first- and second- generation biofuels in the deterministic model grows as oil becomes scarcer along the baseline path and agricultural yields increase (panels b and c). Along that optimal path, characterized by the absence of climate and renewables policies and the abundance of cheap fossil fuels in the first part of the coming century, firstgeneration biofuels never become a large source of energy consumption. In 2100 production of first- generation biofuels is a mere 20 million tonnes of oil equivalent (Mtoe) in the baseline scenario, and 40 Mtoe in the optimistic scenario. These numbers are considerably higher compared to 2004 but are still small in relative terms (less than one percent of total liquid fuel consumption: see technical appendix, Figure C.1, panel b). In contrast, the production of second-generation biofuels takes o↵sharply and expands rapidly after 2040 as they become cost-competitive relative to increasingly costly fossil fuels. In 2100 production of second-generation biofuels reaches 1.55 billion tonnes of oil equivalent (Btoe) in the baseline scenario. Uncertainty in food crop yields has important implications for the production of the first-generation biofuels that are directly a↵ected by both climate and technology states of food crop yields. The di↵erence between the best and worst states of the crop productivity index is about 31 million tons of oil equivalent, which exceeds their expected baseline production in 2100. Although climate and technology states of food crop yields do not directly a↵ect yields of the second- generation biofuels crops, production of second-generation biofuels is nonetheless a↵ected through indirect substitution e↵ects of food for energy in FABLE demand system. There is a sizable variation in the production of second-generation biofuels between extreme paths of the stochastic crop productivity index, which accounts for 450 Mtoe, or about 30 percent of their total production in 2100.

Production of livestock in the deterministic model increases throughout the coming century (panel d) reflecting shifting diets and the growing demand for processed meat as popula23
tion income increases (Foley et al., 2011). By the end of the coming century, the production of livestock in the baseline scenario increases by about 1.5 times compared to 2004, reaching 1.28 billion tons. Given the important contribution of the livestock feed in the production of livestock, we can see its production is smaller in the pessimistic scenario and larger in the optimistic scenario. The di↵erence in livestock production between the optimistic and pessimistic scenarios accounts for about 550 million tons. This range is similar to the dynamic stochastic model. As the significance of animal feed in livestock production grows over time, the e↵ect of uncertain crop yields becomes more pronounced. Similar to the result for food crops, the most extreme paths of crop productivity account for about a third of all variation in livestock production.

Production of timber in the deterministic model also expands with the growing demand for timber products and further improvements in forest yields (panel e). By 2100, production of merchantable timber crops reaches 3.2 billion tons in the deterministic baseline scenario, which is twice as large as in 2004. The consumption of ecosystem services declines in the near decades, as unmanaged natural forest lands are converted to croplands (panel f). It then increases throughout the remaining part of the coming century as the demand for ecosystem services increases, and more natural forest lands become institutionally protected. By 2100 consumption of ecosystem services is 36 percent larger than in 2004. Crop productivity has a very small e↵ect on consumption paths of merchantable timber and ecosystem services in either deterministic or stochastic models. This result is not very surprising as the crop productivity does not directly a↵ect the production of either timber or ecosystem services, whereas indirect land use change e↵ects are relatively small in this stochastic application of the FABLE model.

# **6 Conclusions**

This paper demonstrates how the uncertainties associated with nonstationary biophysical processes and technological change can be incorporated into an economic analysis of the optimal allocation of natural resources in the long run. In doing so, it introduces a novel computational method, ENLCEQ, for solving nonstationary dynamic high-dimensional stochastic problems and applies it to FABLE, a recently developed multi-sectoral dynamic model of global land use.

For illustrative purposes, the study focuses on uncertainty in future crop yields, one of the core uncertainties a↵ecting the evolution of global land use in the long run. Combining scenarios from global climate models and high-resolution output from spatial crop simulation models for four major crops, it comes up with a plausible range of realizations of climate shocks and their e↵ect on future crop yields. These estimates are supplemented with an extensive survey of recent agro-economic and biophysical studies assessing the potential for closing yield gaps as well as attaining further advances in potential yields through plant breeding.

The paper’s key insight is to illustrate the magnitude of optimal land conversion decisions in the context of di↵erent realizations of the stochastic crop productivity. Consistent with the economic theory of natural resource management under uncertainty, the agricultural productivity shocks, due either to adverse climate impacts or unexpected limits on 24
further technological progress, result in additional conversion of scarceland resource to o↵set the impact of potentially lower yields. Owing to intertemporal substitution, some of that conversion takes place even in the absence of actual realization of the climate shocks or technology outcomes. This expansion is accompanied by the changes in the consumption of processed food, livestock, and biofuels - the land-based products most a↵ected by changes in crop productivity.

This study is primarily a methodological contribution, and the chosen model illustration (FABLE) seeks to balance computational complexity and economic tractability. It thus ignores many features standard in more advanced computational land and other resource use models. The future research should focus on integrating economic decisions under uncertainty into large dynamic natural resource models that feature spatial disaggregation at the regional or zonal level, a more extensive representation of the energy sector, and di↵erent types of resources and their production derivatives. Another promising research direction would be to incorporate a more detailed representation of uncertain states backed by an econometric analysis that recovers underlying distributions of uncertain natural resource drivers over time.

# **References**

- Alexander, P., R. Prestele, P. H. Verburg, A. Arneth, C. Baranzelli, F. Batista e Silva, C. Brown, A. Butler, K. Calvin, N. Dendoncker, et al. (2017). Assessing Uncertainties in Land Cover Projections. _Global Change Biology 23_ (2), 767–781.

- Babiker, M., A. Gurgel, S. Paltsev, and J. Reilly (2009). Forward-looking versus Recursivedynamic Modeling in Climate Policy Analysis: A Comparison. _Economic Modelling 26_ (6), 1341–1354.

- Benayas, J., A. Newton, A. Diaz, and J. Bullock (2009). Enhancement of Biodiversity and Ecosystem Services by Ecological Restoration: a Meta-analysis. _Science 325_ (5944), 1121–1124.

- Brock, W. A. and M. Majumdar (1978). Global Asymptotic Stability Results for Multisector Models of Optimal Growth under Uncertainty when Future Utilities are Discounted. _Journal of Economic Theory 18_ (2), 225–243.

- Bulte, E., D. van Soest, G. Van Kooten, and R. Schipper (2002). Forest Conservation in Costa Rica when Nonuse Benefits are Uncertain but Rising. _American Journal of Agricultural Economics 84_ (1), 150–160.

- Burke, M. B. and K. Emerick (2016). Adaptation to Climate Change: Evidence from US Agriculture. _American Economic Journal: Economic Policy 8_ (3), 106–140.

- Cai, Y. (2019). Computational methods in environmental and resource economics. _Annual Review of Resource Economics_ (11), 59–82.

- Cai, Y. and K. Judd (2013). Shape-preserving Dynamic Programming. _Mathematical Methods of Operations Research 77_ (3), 407–421.

25
- Cai, Y. and K. Judd (2014). Advances in Numerical Dynamic Programming and New Applications. In K. L. Judd and K. Schmedders (Eds.), _Handbook of Computational Economics_ . Elsevier.

- Cai, Y., K. Judd, and J. Steinbuks (2017). A Nonlinear Certainty Equivalent Approximation Method for Dynamic Stochastic Problems. _Quantitative Economics 8_ (1), 117–147.

- Cai, Y., K. Judd, G. Thain, and S. Wright (2015). Solving Dynamic Programming Problems on a Computational Grid. _Computational Economics 45_ (2), 261–284.

- Cai, Y. and T. Lontzek (2019). The social cost of carbon with economic and climate risks. _Journal of Political Economy 127_ (6), 2684–2734.

- Cassman, K., P. Grassini, and J. van Wart (2010). Crop Yield Potential, Yield Trends, and Global Food Security in a Changing Climate. In D. Hillel and C. Rosenzweig (Eds.), _Handbook of Climate Change and Agroecosystems_ , pp. 37–51. Imperial College Press.

- Cassman, K. G. (1999). Ecological Intensification of Cereal Production Systems: Yield Potential, Soil Quality, and Precision Agriculture. _Proceedings of the National Academy of Sciences 96_ (11), 5952–5959.

- Chazdon, R. (2008). Beyond Deforestation: Restoring Forests and Ecosystem Services on Degraded Lands. _Science 320_ (5882), 1458–1460.

- Conrad, J. (1997). On the Option Value of Old-growth Forest. _Ecological Economics 22_ (2), 97–102.

- Conrad, J. (2000). Wilderness: Options to Preserve, Extract, or Develop. _Resource and Energy Economics 22_ (3), 205–219.

- Daigneault, A. J., M. J. Miranda, and B. Sohngen (2010). Optimal Forest Management with Carbon Sequestration Credits and Endogenous Fire Risk. _Land Economics 86_ (1), 155–172.

- Den Haan, W. J., K. L. Judd, and M. Juillard (2011). Computational Suite of Models with Heterogeneous Agents II: Multi-country Real Business Cycle Models. _Journal of Economic Dynamics and Control 35_ (2), 175–177.

- Di↵enbaugh, N. S., T. W. Hertel, M. Scherer, and M. Verma (2012). Response of Corn Markets to Climate Volatility under Alternative Energy Futures. _Nature climate change 2_ (7), 514.

- Drud, A. S. (1994). CONOPT?A Large-Scale GRG Code. _ORSA Journal on Computing 6_ (2), 207–216.

- Evans, L. and R. Fischer (1999). Yield Potential: its Definition, Measurement, and Significance. _Crop Science 39_ (6), 1544–1551.

- Foley, J. A., N. Ramankutty, K. A. Brauman, E. S. Cassidy, J. S. Gerber, M. Johnston, N. D. Mueller, C. O’Connell, D. K. Ray, P. C. West, et al. (2011). Solutions for a Cultivated Planet. _Nature 478_ (7369), 337–342.

26
- F ̈ussel, H.-M. (2009). An Updated Assessment of the Risks from Climate Change Based on Research Published since the IPCC Fourth Assessment Report. _Climatic change 97_ (3-4), 469.

- Grassini, P., K. M. Eskridge, and K. G. Cassman (2013). Distinguishing between Yield Advances and Yield Plateaus in Historical Crop Production Trends. _Nature Communications 4_ .

- Hertel, T., S. Rose, and R. Tol (2009). _Economic Analysis of Land Use in Global Climate Change Policy_ . Routledge.

- Hertel, T., J. Steinbuks, and W. Tyner (2016). What is the Social Value of Second Generation Biofuels? _Applied Economic Perspectives and Policy 38_ (4), 599–617.

- Hertel, T. W., J. Steinbuks, and U. Baldos (2013). Competition for Land in the Global Bioeconomy. _Agricultural Economics 44_ (s1), 129–138.

- IPCC (2014a). _Climate Change 2014: Impacts, Adaptation and Vulnerability: Contribution of Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change_ . Cambridge University Press.

- IPCC (2014b). _Climate Change 2014: Impacts, Adaptation and Vulnerability: Contribution of Working Group III to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change_ . Cambridge University Press.

- Judd, K. L. (1998). _Numerical Methods in Economics_ . The MIT press.

- Kumar, P. (2010). _The Economics of Ecosystems and Biodiversity: Ecological and Economic Foundations_ . Routledge.

- Lanz, B., S. Dietz, and T. Swanson (2017). Global Economic Growth and Agricultural Land Conversion under Uncertain Productivity Improvements in Agriculture. _American Journal of Agricultural Economics 100_ (2), 545–569.

- Leroux, A., V. Martin, and T. Goeschl (2009). Optimal Conservation, Extinction Debt, and the Augmented Quasi-option Value. _Journal of Environmental Economics and Management 58_ (1), 43–57.

- Licker, R., M. Johnston, J. A. Foley, C. Barford, C. J. Kucharik, C. Monfreda, and N. Ramankutty (2010). Mind the Gap: How do Climate and Agricultural Management Explain the ’Yield Gap’ of Croplands around the World? _Global Ecology and Biogeography 19_ (6), 769–782.

- Lobell, D. B., K. G. Cassman, and C. B. Field (2009). Crop Yield Gaps: their Importance, Magnitudes, and Causes. _Annual Review of Environment and Resources 34_ (1), 179.

- Lobell, D. B. and C. B. Field (2007). Global Scale Climate - Crop Yield Relationships and the Impacts of Recent Warming. _Environmental Research Letters 2_ (1), 014002.

- Majumdar, M. and R. Radner (1983). Stationary Optimal Policies with Discounting in a Stochastic Activity Analysis Model. _Econometrica 51_ (6), 1821–1837.

27
- Miranda, M. J. and P. L. Fackler (2004). _Applied Computational Economics and Finance_ . MIT press.

- National Research Council (2011). _Renewable Fuel Standard: Potential Economic and Environmental E↵ects of U.S. Biofuel Policy_ . National Academies Press.

- Nessh ̈over, C., J. Aronson, J. Blignaut, D. Lehr, A. Vakrou, and H. Wittmer (2009). Investing in Ecological Infrastructure. In P. ten Brink (Ed.), _The Economics of Ecosystems and Biodiversity: National and International Policy Making. An Output of TEEB_ , pp. 401–448. Earthscan.

- Nikas, A., H. Doukas, and A. Papandreou (2019). A Detailed Overview and Consistent Classification of Climate-Economy Models. In _Understanding Risks and Uncertainties in Energy and Climate Policy_ , pp. 1–54. Springer.

- Pindyck, R. S. (1984). Uncertainty in the Theory of Renewable Resource Markets. _The Review of Economic Studies 51_ (2), 289–303.

- Pindyck, R. S. (2007). Uncertainty in Environmental Economics. _Review of Environmental Economics and Policy 1_ (1), 45–65.

- Rosenzweig, C., J. Elliott, D. Deryng, A. Ruane, A. Arneth, K. Bootef, C. Folberth, M. Glotter, N. Khabarov, C. M ̈uller, K. Neumann, F. Piontek, T. Pugh, E. Schmid, E. Stehfest, and J. Jones (2014). Assessing Agricultural Risks of Climate Change in the 21st Century in a Global Gridded Crop Model Intercomparison. _Proceedings of the National Academy of Sciences 111_ (9), 3268–3273.

- Schmitz, C., H. van Meijl, P. Kyle, G. C. Nelson, S. Fujimori, A. Gurgel, P. Havlik, E. Heyhoe, D. M. d’Croz, A. Popp, et al. (2014). Land-Use Change Trajectories Up to 2050: Insights from a Global Agro-Economic Model Comparison. _Agricultural Economics 45_ (1), 69–84.

- Sohngen, B. and R. Mendelsohn (2003). An Optimal Control Model of Forest Carbon Sequestration. _American Journal of Agricultural Economics 85_ (2), 448–457.

- Sohngen, B. and R. Mendelsohn (2007). A Sensitivity Analysis of Forest Carbon Sequestration. In M. Schlesinger (Ed.), _Human-Induced Climate Change: An Interdisciplinary Assessment_ , pp. 227–237. Cambridge University Press.

- Steinbuks, J. and T. Hertel (2012). Forest, Agriculture, and Biofuels in a Land use model with Environmental services (FABLE). GTAP Working Paper 71, Center for Global Trade Analysis, Department of Agricultural Economics, Purdue University.

- Steinbuks, J. and T. Hertel (2016). Confronting the Food-Energy-Environment Trilemma: Global Land Use in the Long Run. _Environmental and Resource Economics 63_ (3), 545– 570.

- Steinbuks, J. and T. W. Hertel (2013). Energy Prices Will Play an Important Role in Determining Global Land Use in the Twenty First Century. _Environmental Research Letters 8_ (1), 014014.

28
- Stokey, N. L., R. E. Lucas, and E. C. Prescott (1989). _Recursive Methods in Economic Dynamics_ . Harvard University Press.

- Taheripour, F., C. Hurt, and W. E. Tyner (2013). Livestock Industry in Transition: Economic, Demographic, and Biofuel Drivers. _Animal Frontiers 3_ (2), 38–46.

- Tian, X., B. Sohngen, J. B. Kim, S. Ohrel, and J. Cole (2016). Global Climate Change Impacts on Forests and Markets. _Environmental Research Letters 11_ (3), 035011.

- Tsionas, E. G. and S. C. Kumbhakar (2004). Markov Switching Stochastic Frontier Model. _The Econometrics Journal 7_ (2), 398–425.

- Tsur, Y. and A. Zemel (2014). Dynamic and Stochastic Analysis of Environmental and Natural Resources. In M. Fischer and P. Nijkamp (Eds.), _Handbook of Regional Science_ , pp. 929–949. Springer-Verlag Berlin Heidelberg.

- Verma, M., T. Hertel, and N. Di↵enbaugh (2014). Market-Oriented Ethanol and Corn-Trade Policies can Reduce Climate-Induced US Corn Price Volatility. _Environmental Research Letters 9_ (6), 064028.

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# **Technical Appendix**

## **A.1 FABLE Model Description**

This section and the following section describe key elements of FABLE model, as well as its equations, variables, and model parameters. For a full description of the model, including details on model baseline calibration and extensive sensitivity analysis, please refer to Hertel _et al._ (2016) and Steinbuks & Hertel (2016) and technical appendices therein.

### **A.1.1 Primary Resources**

Primary resources comprise of land, liquid fossil fuels, and other primary inputs, e.g., labor and capital. The supply of land is fixed and faces competing uses that are determined endogenously by the model. The flow of liquid fossil fuels evolves endogenously along their optimal extraction path, accounting for exogenous discoveries in new fossil fuel reserves. The endowment of other primary inputs is exogenous and evolves along the prespecified global economy growth path.

#### **A.1.1.1 Land**

The total land endowment in the model, _L_<sup>_total_</sup> _,_ is fixed. Each period of time _t_ there are four profiles of land in the economy. They include unmanaged forest land, _L_<sup>_N_</sup> _,_ agricultural land, _L_<sup>_A_</sup> _,_ pasture land, _L_<sup>_P_</sup> _,_ and commercially managed forest land, _L_<sup>_C_</sup> . The agricultural land area can be allocated for the cultivation of food crops (denoted _L_<sup>_A,c_</sup> ), and second-generation biofuels feedstocks (denoted _L_<sup>_A,b_2</sup> ). We assume that the natural forest land consists of two types. Institutionally protected land, _L_<sup>_R_</sup> _,_ includes natural parks, biodiversity reserves and other types of protected forests. This land is used to produce ecosystem services for society, and cannot be converted to commercial land. Unmanaged natural land, _L_<sup>_N_</sup> _,_ can be accessed and either converted to managed land or to protected natural land. Once the natural land is converted to managed land, its potential to yield ecosystem services is diminished. This potential can be partially restored for managed forests with significant land rehabilitation costs incurred. The use of managed land can be shifted between cropland, forestland, and pasture land (see Figure 1 in the main manuscript for a graphical representation of these transitions). We denote land transition flows from land type _i_ 1
to land type _j_ as ∆<sup>_i,j_</sup> (a negative value means a transition from land type _j_ to land type _i_ ). Equations describing allocation of land across time and di↵erent uses are as follows : Equations (A.1) and (A.2) define, respectively, the composition of total land and agricultural land in the economy. Equations (A.3)-(A.5) describe the transitions for unmanaged land, agricultural land, and pasture land.<sup>1</sup> Equation (A.6) shows the growth path of protected natural land.

Accessing the natural lands comes at a cost associated with building roads and other infrastructure (Golub _et al._ , 2009). In addition, converting natural land to reserved land entails additional costs associated with passing legislation to create new natural parks. We denote the natural land access, rehabilitation, and protection costs as _C_<sup>_N,A,R_</sup> _, C_<sup>_C,N_</sup> _,_ and _C_<sup>_N,R_</sup> _,_ respectively. There are also costs of switching between the cropland and the pasture land, denoted as _C_<sup>_A,P_</sup> . We assume that all these costs are continuous, monotonically increasing, and strictly convex functions of converted land. There are no additional costs of natural land conversion to commercial land, as the revenues from deforestation o↵set these costs.

Managed forests are characterized by _v_ max vintages of tree species with vintage ages _v_ = 1 _, ..., v_ max. At the end of period _t_ each hectare of managed forest

> 1Equations (A.2) and (A.4) do not account for the transition from forestry to pasture land. Throughout the past century tropical forests, particularly in the Latin America region, have been extensively converted to the pasture land (Barbier _et al._ , 1994). However, in the FABLE model, conversion of forest land to pasture is never optimal as cropland has higher productivity for cattle breeding at the same conversion (stumpage) cost.

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land, _L_<sup>_C_</sup> _v,t_<sup>_,_hasanaveragedensityoftreevintageage</sup><sup>_v,_withtheinitialalloca-</sup> tion given and denoted by _L_<sup>_C_</sup> _v,_ 0<sup>.Theforestrotationagesandmanagementare</sup> endogenously determined. Each period the managed forest land can be either planted, harvested, or left to mature. The newly planted trees occupy ∆<sup>_C,C_</sup> hectares of land, and reach the average age of the first tree vintage next period. The harvested area of tree vintage age _v_ occupies ∆<sup>_C,H_</sup> _v_ hectares of forest land. The di↵erence between the harvested area of all tree vintage ages and the newly planted area is used for cropland, i.e., The following equations describe land use of managed forests: Equation (A.7) describes the composition of managed forest area across vintages. Equation (A.8) illustrates the harvesting dynamics of forest areas with the ages _v_ max _−_ 1 and _v_ max. Equation (A.10) shows the transition from the planted area to new forest vintage area.

The average harvesting and planting costs per hectare of new forest planted, _c_<sup>_o,H_</sup> , and _c_<sup>_o,C_</sup> _,_ are invariant to scale and are the same across all vintages. Harvesting managed forests and conversion of harvested forest land to agricultural land is subject to additional near term adjustment costs, _c_<sup>_H_</sup> _._ The specific functional forms of land conversion costs are shown in section C.1, equations (C.34)(C.39).

Thus, we have defined the vector of land state variables: and its associated transition laws.

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# **A.1.1.2 Fossil Fuels**

The initial stock of liquid fossil fuels, _X_<sup>_F_</sup> _,_ is exogenous, and each period of time _t_ adds a new amount of fossil fuels, ∆<sup>_F,D_</sup> _,_ which reflects exogenous technological progress in fossil fuel exploration.<sup>2</sup> The economy extracts fossil fuels, which have two competing uses in our partial equilibrium model of landuse. A part of extracted fossil fuels, ∆<sup>_F,n_</sup> _t_ , is converted to fertilizers that are further used in the agricultural sector. The remaining amount of fossil fuels, ∆<sup>_F,E_</sup> _t ,_ is combusted to satisfy the demand for energy services. The following equation describes supply of fossil fuels: The cost of fossil fuels, _c_<sup>_F_</sup> _,_ reflects the expenditures on fossil fuels’ extraction, refining, transportation and distribution, as well the costs associated with emissions control (e.g., Pigovian taxes) in the non-land-based economy. We assume that the cost of fossil fuels is a nonlinear quadratic function with accelerating costs as the stock of fossil fuels depletes (Nordhaus & Boyer, 2000): where the parameter _⇠_ 1<sup>_F_capturesthecurvatureoftheliquidfossilfuelcost</sup> function.

# **A.1.1.3 Other Primary Resources**

The initial endowment of all other primary resources in the non-land-based economy, such as labor, physical and human capital, and materials inputs, _X_<sup>_O_</sup> , is exogenous in this model. We assume that the growth rate of all other primary resources is a weighted average of the population growth, which reflects demographic changes, and the physical capital growth, __<sup>_o,X_</sup> . The following equation describes the supply of other primary inputs: where ⇧ _t_ is the economy’s population, and _↵_<sup>_o,l_</sup> is the share of population growth to the growth rate of all other primary resources. Other primary inputs can be

> 2This technological progress comprises of both discoveries on new exploitable oil and gas fields, as well as development of new technologies for extraction of non-conventional fossil fuels.

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used for the production of land-based goods and services or converted to final goods and services in the non-land economy. Thus, state variables for resources other than land are as: As _X_<sup>_O_</sup> is exogenous and deterministic, it is a degenerated state variable and not counted as a state variable for model solution purposes.<sup>3</sup>

# **A.1.2 Intermediate Inputs**

We analyze six intermediate inputs used in the production of land-based goods and services: petroleum products, fertilizers, crops, biofuels, and raw timber. Fossil fuels are refined and converted to either petroleum products, _x_<sup>_p_</sup> , that are further combusted, or to fertilizers, _x_<sup>_n_</sup> , that are used to boost yields in the agricultural sector. Agricultural land and fertilizers are combined to grow food crops, _x_<sup>_c_</sup> or 2G biofuels crops, _x_<sup>_c,b_2</sup> . Food crops can be further converted into processed food and 1G biofuels, _x_<sup>_b_1</sup> _,_ or used as an animal feed, _x_<sup>_c,l_</sup> . 2G biofuels crops can only be converted into 2G biofuels, _x_<sup>_b_2</sup> . 1G biofuels substitute imperfectly for liquid fossil fuels in final energy demand, whereas 2G biofuels and liquid fossil fuels are the perfect substitutes The food crops used as animal feed and pasture land are combined to produce raw livestock, _x_<sup>_l_</sup> . Harvesting managed forests yield raw timber, _x_<sup>_w_</sup> , that is further used in timber processing. The production functions for intermediate inputs can be illustrated by the following equations where ∆<sup>_F,_</sup> _t_<sup>_{E,n}_</sup> represents that either ∆<sup>_F,E_</sup> _t_ or ∆<sup>_F,n_</sup> _t_ is an argument of _g_<sup>_j_</sup> , similarly for _L_<sup>_{_</sup> _t_<sup>_A,P }_</sup> and _x_<sup>_c,{l,b}_</sup> . The specific functional forms of _g_<sup>_j_</sup> ( _·_ ) are shown in section C.1, equations (C.14)-(C.22).

# **A.1.3 Final Goods and Services**

We consider five per capita land-based services that are consumed in the final

> 3 It would be, however, counted as a state variable for the model solution purposes if we assume that the endowment is a stochastic process.

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demand: services from processed crops, _y_<sup>_f_</sup> , livestock, _y_<sup>_l_</sup> , energy, _y_<sup>_e_</sup> , timber, _y_<sup>_w_</sup> , and ecosystem services, _y_<sup>_r_</sup> . Processed crops, livestock, and timber are respectively products of food crops, raw livestock, and timber processing. The production of energy services combines liquid fossil fuels with the biofuels, and the resulting mix is further combusted. The ecosystem services are the public good to society, which captures recreation, biodiversity, and other environmental goods and services. To close the demand system, we also include other goods and services, _y_<sup>_o_</sup> , which comprise of consumption of other primary inputs not spent on the production of land-based goods and services.

We have all state variables for the deterministic model: and the vector of decision variables is where **x** _t ⌘_ [ _x_<sup>_p_</sup> _t_<sup>_, xn_</sup> _t_<sup>_, xc_</sup> _t_<sup>_, xb_</sup> _t_<sup>_, xl_</sup> _t_<sup>_, xw_</sup> _t_<sup>_, xc,l_</sup> _t_<sup>_, xc,b_</sup> _t_<sup>]and</sup><sup>**y**</sup><sup>_t_=</sup> _yt_<sup>_f, y_</sup> _t_<sup>_l, y_</sup> _t_<sup>_e, y_</sup> _t_<sup>_w, y_</sup> _t_<sup>_r, y_</sup> _t_<sup>_o_</sup> . ⇣ ⌘ The production functions for final per capita land-based goods and services can be illustrated by the following equation: where some arguments in _F_<sup>_i_</sup> ( _·_ ) could be redundant. It follows from equation (A.15) that production of final goods and services involves the combination of land resources and intermediate inputs. The specific functional forms of _F_<sup>_i_</sup> ( _·_ ) are shown in section C.1, equations (C.23)-(C.29), which are functions of **L** and _{x_<sup>_j_</sup> _}_ . All these equations constitute a part of the feasibility constraint **a** _t 2 Dt_ ( **S** _t_ ).

The production of intermediate inputs or final land-based goods and services _i_ incurs costs, _c_<sup>_o,i_</sup> , that are subtracted from available other primary resources. The remaining amount of other primary resources is converted into other goods and services, which are subsequently consumed in final demand. As the focus of this model is on the utilization of land-based resources, we introduce the other goods and services, _y_<sup>_o_</sup> , in a very simplified manner. We introduce no additional cost of producing other goods and services, assuming that it is reflected in the size of the endowment of other primary inputs. The specific functional form for _y_<sup>_o_</sup> is shown in section C.1, equation (C.28).

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# **A.1.4 Preferences**

The economy’s per-capita utility, _u_ , is derived from the per capita consumption of processed crops, livestock, timber, energy and ecosystem services, and other goods and services. Following the macro economic literature, we assume constant relative risk aversion utility, where is the per capita consumption bundle of goods and services, _C_ ( **y** ) is a nonlinear aggregator over **y** , and _γ_ is the coefficient of relative risk aversion, which captures the economy’s attitude to uncertain events. We choose a nonhomothetic AIDADS preference (Rimmer & Powell, 1996) to compute _C_ ( **y** ) implicitly: where _↵_ , _β_ , and _<u>yq</u>_ are positive parameters with<sup>P</sup> _q_<sup>_↵q_=P</sup> _q_<sup>_βq_=1.These</sup> preferences place greater value on eco-system services, and smaller value on additional consumption of food, energy and timber products as society becomes wealthier. When _γ_ = 1, our utility function is equivalent to the AIDADS utility.

# **A.1.5 Welfare**

We denote the transition laws of land, (A.3)-(A.6) and (A.8)-(A.10), as and the transition laws for other resources, (A.11)-(A.13), as Combining (A.18) and (A.19), we have for the deterministic model in the notations of Section 2.

The objective of the planner is to maximize the total expected welfare, which is the cumulative expected utility of the population’s consumption of final goods and services, **y** , discounted at the constant rate _δ >_ 0 _._ The planner allocates 7
managed agricultural, pasture and forest lands for crop, livestock, and timber production, the scarce fossil fuels and protected natural forests to solve the following problem: subject to the transition laws (A.20) and the feasibility constraints **a** _t 2 Dt_ ( **S** _t_ )which include (A.15), (C.23)-(C.29), (C.28), (A.17) and nonnegativity constraints for the variables. Here is the utility function in the notations of Section 2.

# **B.1 Quantifying the Uncertainty in Crop Yields**

## **B.1.1 Uncertainty in Agricultural Technology**

Advances in crop technology are very difficult to predict due to four interconnected factors (Fischer _et al._ , 2011). First, there is significant uncertainty about the potential for exploiting large and economically significant yield gaps (i.e., the di↵erences between observed and potential crop yields) in developing countries, especially those in Sub-Saharan Africa. A second and closely related point is that it is unclear how fast available yield-enhancing technologies can be adopted at a global scale.<sup>4</sup> Third, there is a significant variation in developing countries’ institutions and policies that make markets work better and provide a conducive environment for agricultural technology adoption.<sup>5</sup> Finally, while plant breeders continue to make steady gains in further advancing crop yields, progress depends on the level of funding provided for agricultural research. This has proven to be somewhat volatile, with per capita funding falling in the decades leading up to the recent food crisis (Alston & Pardey, 2014). The food price rises since 2007 have stimulated new investments. However, whether this interest will be sustained remains to be seen. Overall, progress from conventional breeding is becoming more difficult. Transgenic (genetic modification) technologies have a proven record of more than a decade of safe and environmentally sound use, and thus o↵er huge potential to address critical biotic and abiotic stresses in

> 4These technologies include conservation farming approaches based on no-tillage, the genetic modification technology revolution, and information and communication technologies for more efficient and precise management of modern inputs.

> 5These best practices include the adoption of better risk management, market development, rural finance, farmers’ organizations, and the provision of advisory services to farmers.

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the developing world. However, expected yield gains, costs of further developing these technologies, and the political acceptance of genetically modified foods are all highly uncertain.

To quantify the extent to which the advances in crop technology can further boost agricultural yields over the next century, we first need to assess the magnitude of existing yield gaps at the global scale. In a comprehensive study, Lobell _et al._ (2009) report a significant variation in the ratios of actual to potential yields for major food crops across the world, ranging from 0.16 for tropical lowland maize in Sub-Saharan Africa to 0.95 for wheat in Haryana, India. For the purposes of this study, we employ the results of Licker _et al._ (2010), who conduct comprehensive yield gap analysis using global crop dataset of harvested areas and yields for 175 crops on a 0.5<sup>_◦_</sup> geographic grid of the planet for the year 2000. Using these estimates, we calculate the global yield gap as the grid-level output-weighted yield gap of the four most important food crops (wheat, maize, soybeans, and rice). The resulting estimate suggests that average yields are 53% of potential yields, which is close to the median estimates by Lobell _et al._ (2009). As a further robustness check we employ the Decision Support System for Agrotechnology Transfer (DSSAT) crop simulation model (Jones _et al._ , 2003), run globally on a 0.5 degree grid in the parallel System for Integrating Impacts Models and Sectors (pSIMS; Elliott _et al._ 2014b) to simulate yields of the same four major food crops under best agricultural management conditions and compare simulated yields to their observed yields. The resulting yield gap estimates were not substantially di↵erent.

In the optimistic (i.e., “good”) state of advances in crop technology, we assume that yields continue to grow linearly throughout the coming century, eliminating the yield gap by 2100. This high yield scenario rests on the assumption of continued strong growth in investment in agricultural research and development, widespread acceptance of genetically modified crops, continuing institutional reforms in developing countries, and public and private investments in the dissemination of new technologies. The erosion of any one of these component assumptions will likely result in a slowing of crop technology improvements. And there are some grounds for pessimism. In a comprehensive statistical analysis of historical crop production trends, Grassini _et al._ (2013) note that “despite the increase in investment in agricultural R&D and education [...] the relative rate of yield gain for the major food crops has 9
decreased over time together with evidence of upper yield plateaus in some of the most productive domains. For example, investment in R&D in agriculture in China has increased threefold from 1981 to 2000. However, rates of increase in crop yields in China have remained constant in wheat, decreased by 64% in maize as a relative rate and are negligible in rice. Likewise, despite a 58% increase in investment in agricultural R&D in the United States from 1981 to 2000 (sum of public and private sectors), the rate of maize yield gain has remained strongly linear.” To capture the possibility of much slower technological improvement in the coming century, we specify two more pessimistic scenarios. In the “medium” state of technology, rather than closing the yield gap by 2100, average yields in 2100 are just three-quarters of yield potential at that point in time. In the “bad” state of technology, there is no technological progress, and the crop yields stay the same as at the beginning of the coming century. This is the path on which we begin the simulation in 2004. As previously noted, we then specify probabilities with which the crop technology index evolves across the di↵erent states of technology.

# **B.1.2 Uncertainty in Climate Change Impacts**

In addition to crop technology uncertainty, there is great uncertainty about the physical environment in which this technology will be deployed. In particular, long-run changes in both temperature and precipitation are likely to have an important impact on the productivity of land in agriculture (IPCC, 2014), and therefore, the global pattern of land use. Quantification of the impact of climate change on agricultural yields requires coming to grips with three interconnected factors (Alexandratos, 2011). First, there is significant uncertainty in future GHG concentrations along the long-run growth path of the global economy. Second, the General Circulation Models (GCMs) developed by climate scientists to translate these uncertain GHG concentrations into climate outcomes disagree about the spatially disaggregated deviations of temperature and precipitation from baseline levels. Finally, there is significant uncertainty in the biophysical models used to determine how changes in temperature and precipitation will a↵ect plant growth and the productivity of agriculture in di↵erent 10
agro-ecological conditions. The impact of climate change on food crop yields depends critically on their phenological development, which, in turn, depends on the accumulation of heat units, typically measured as growing degree days (GDDs). More rapid accumulation of GDDs as a result of the climate change speeds up phenological development, thereby shortening key growth stages, such as the grain filling stage, hence reducing potential yields (Long, 1991). However, rising concentrations of CO2 in the atmosphere result in an increase in potential yields due to improved water use efficiency, often dubbed the “CO2 fertilization e↵ect” (Long _et al._ , 2006). Sorting out the relative importance of these e↵ects and achieving greater confidence in evaluations of climate impacts on agricultural yields remains an important research question in the agronomic literature (Cassman _et al._ , 2010; Rosenzweig _et al._ , 2014).

To quantify the uncertainty in climate impacts on agricultural yields we follow the approach of Rosenzweig _et al._ (2014), who have recently conducted a globally consistent, protocol-based, multi-model climate change assessment for major crops with the explicit characterization of uncertainty. To quantify the uncertainty of impacts of temperature increases due to climate change on potential crop yields we obtain results of four crop simulation models: GEPIC (Liu _et al._ , 2007), LPJmL (Bondeau _et al._ , 2007), pDSSAT (Jones _et al._ , 2003), and PEGASUS (Deryng _et al._ , 2011).<sup>6</sup> All models are run globally on a 0.5<sup>_◦_</sup> grid over the period between 1971 and 2099 and weighted by the agricultural output of four major food crops (maize, soybeans, wheat, and rice). To ensure simulation results comparability with the structural parameters of FABLE model all models are run under Representative Concentration Pathways 6.0W _/_ m<sup>2</sup> (RCP6) GHG forcing scenario (Moss _et al._ , 2008). We also consider alternative assumptions on CO2 fertilization e↵ects. To quantify uncertainty in temperature increases due to climate change we employ outputs for five global climate models (GCM): GFDL-ESM2M (Dunne _et al._ , 2013), HadGEM2-ES (Collins _et al._ , 2008), IPSL-CM5A-LR (Dufresne _et al._ , 2012), MIROC-ESMCHEM (Watanabe _et al._ , 2011), and NorESM1-M (Bentsen _et al._ , 2012). For each of the simulations, we fit a linear trend in order to parsimoniously characterize the evolution of crop yields in the face of climate change over the

> 6Our results are based on four crop simulation models though Rosenzweig _et al._ (2014) consider seven crop simulation models. The remaining three models have fewer crops and/or temporal frames for model baseline and are thus omitted. Rosenzweig _et al._ (2014) find that five models, including GEPIC, LPJmL, and pDSSAT models considered in this analysis, yield broadly similar predictions. One model (LPJ-GUESS) not covered here has much higher variation in predicted crop yields under di↵erent climate scenarios. Our results may, therefore, understate the range of uncertainty of climate change impacts on potential crop yields.

11
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yields (20-30 percent). The GEPIC model predicts that on average crop yields will be little changed, showing a small increase in crop yields for some climate models and a small decline for other models. The predictions of LPJmL and pDSSAT models are reversed when CO2 fertilization e↵ects are removed, showing a decline of about 10-15 percent in potential yields. The PEGASUS model predicts an even larger decline in potential yields (30-35 percent), whereas the predictions of GEPIC model show a moderate decline of about 5-10 percent in potential yields.

Given a large variation in model predictions, we construct 5 states for potential crop yields under uncertain climate change. These states correspond to quintiles of the distribution of di↵erent model outcomes for potential crop yields by 2100. Under two optimistic states of the world, we observe a 2 and 15 percent increases in potential crop yields relative to model baseline whereby significant CO2 fertilization e↵ects o↵set the negative e↵ects of climate change. For the next two states, we see a 15 and 19 percent declines in potential crop yields relative to model baseline whereby CO2 fertilization e↵ects are either small or nonexistent, and the negative e↵ects of climate change tend to prevail. Finally, under most pessimistic states of the world, drastic adverse e↵ects of climate change combined with the absence of any CO2 fertilization e↵ects result in a 36 percent decline in potential crop yields relative to model baseline.

13
# **C.0.3 Transition probabilities**

The five possible values of the climate state _J_ 1 _,t_ are _J_ 1 _,_ 1 = 0 _._ 64, _J_ 1 _,_ 2 = 0 _._ 85, _J_ 1 _,_ 3 = 0 _._ 89, _J_ 1 _,_ 4 = 1 _._ 02, and _J_ 1 _,_ 5 = 1 _._ 15, and its probability transition matrix is where _P_ 1 _,i,j_ represents the probability from the _j_ -th value of _J_ 1 _,t_ to the _i_ -th value, for 1 _ i, j _ 5. The three possible values of the technological state _J_ 2 _,t_ are _J_ 2 _,_ 1 = 1 _._ 45, _J_ 2 _,_ 2 = 1 _._ 675, and _J_ 2 _,_ 3 = 1 _._ 9, and its probability transition matrix is where _P_ 2 _,i,j_ represents the probability from the _j_ -th value of _J_ 2 _,t_ to the _i_ -th value for 1 _ i, j _ 3. We assume that _J_ 2 _,t_ is independent of _J_ 1 _,t_ .

# **C.0.4 Model**

After we add the risks, the state vector becomes where **J** _t_ = ( _J_ 1 _,t, J_ 2 _,t_ ). And **J** is a Markov chain so it can be represented as **J** _t_ +1 = **G**<sup>_J_</sup> _t_<sup>(</sup><sup>**J**</sup><sup>_t, ✏t_)where</sup><sup>_✏t_isavectorofshockswithzeromeans.Theproblem</sup> is subject to 14
and **a** _t 2 Dt_ ( **S** _t_ ) representing the feasibility constraints, that is, inequality constraints and the equations other than the above transition laws. The above transition laws are just a special case of in the notations of Section 2, so we can implement our ENLCEQ method to solve the dynamic stochastic programming problem. Since our time of interest is _T_ = 100 years, in ENLCEQ we choose _T_<sup>b</sup> = 400 so that a larger _T_<sup>b</sup> will have little change on our solution at the first 100 years.

In the step 2 of Algorithm 1 for solving the solution at time _s_ , we replace _✏t_ by its zero mean to have **S** _t_ +1 = **G** _t_ ( **S** _t,_ **a** _t,_ 0), that is, **J** _t_ +1 = **G**<sup>_J_</sup> _t_<sup>(</sup><sup>**J**</sup><sup>_t,_0).But</sup> this **J** _t_ +1 = **G**<sup>_J_</sup> _t_<sup>(</sup><sup>**J**</sup><sup>_t,_0)isonlyforsimplicityinnotations.Infact,since</sup><sup>**J**isa</sup> Markov chain, we replace **J** _t_ by its mean conditional on the realized value of **J** _s_ (i.e., its certainty equivalent approximation): for all _t ≥ s_ , where _J_ 1 = ( _J_ 1 _,_ 1 _, ..., J_ 1 _,_ 5), _J_ 2 = ( _J_ 2 _,_ 1 _, J_ 2 _,_ 2 _, J_ 2 _,_ 3), _⇡_ 1 _,t,s_ and _⇡_ 2 _,t,s_ are two column vectors representing probability distributions of _J_ 1 _,t_ and _J_ 2 _,t_ conditional on the realized values of _J_ 1 _,s_ and _J_ 2 _,s_ respectively. If the realized values of _J_ 1 _,s_ and _J_ 2 _,s_ are _J_ 1 _,i_ and _J_ 2 _,j_ respectively, then we have _⇡_ 1 _,t,s_ = _P_ 1<sup>_t−s_</sup> _⇡_ 1 _,s,s_ and _⇡_ 2 _,t,s_ = _P_ 2<sup>_t−s_</sup> _⇡_ 2 _,s,s_ , where _⇡_ 1 _,s,s_ is a length-5 column vector with 1 at the _i_ th element and 0 everywhere else, and _⇡_ 2 _,s,s_ is a length-3 column vector with 1 at the _j_ th element and 0 everywhere else.

# **C.1 Model Equations, Variables and Parameters**

## **C.1.1 Equations**

_Land Use_ 15
_Fossil Fuels_ _Other Primary Resources_ 16
(( ) C ~~C~~ ()( <u>JE</u> ) ( ) ( oy)
_Technology (deterministic)_ _Technology (stochastic)_ _Costs_ 18
_Preferences_ _Population_ _Welfare_ with _U_ ( **S** _t,_ **a** _t_ ) = _u_ ( **y** _t_ )⇧ _t_ , **S** := ( **L** _,_ **X** _,_ **J** ), and 19
# **C.1.2 Tables**

Table C.1: Model Exogenous Variables

|Parameter|Description|Units|
|---|---|---|
|_Exogenous _|_Variables_||
|∆<sup>_F,D_</sup><br>_t_|Flow of Newly Discovered Fossil Fuels|trillion toe|
|_X_<sup>_O_</sup><br>_t_|Other Primary Goods|trillion USD|
|_At_<br>|Crop Technology Index||
|_✓_<sup>_c,b_2</sup><br>_t_<br>|2G biofuels Crop Technology Index||
|_✓_<sup>_b_2</sup><sup>_,K_</sup><br>_t_|2G Biofuels Fixed Factor Decay Index||
|_✓_<sup>_w_</sup><br>_v,t_<br>|Logging Productivity Index||
|_✓_<sup>_f_</sup><br>_t_|Food Processing Productivity Index||
|_✓_<sup>_e_</sup><br>_t_<br>|Energy Efciency Index||
|_✓_<sup>_l_</sup><br>_t_<br><sup>_w_</sup>|Livestock Processing Productivity Index||
|_✓_<sup>_y_</sup><br>_t_|Wood Processing Productivity Index||
|_✓_<sup>_o_</sup><br>_t_<br>|Total Factor Productivity Index||
|_C_<sup>_F_</sup><br>_t_<br>|Fossil Fuel Extraction Cost|share of _X_<sup>_O_</sup><br>_t_<br>|
|_C_<sup>_N_</sup><br>_t_<br>|Natural Land Access Cost|share of _X_<sup>_O_</sup><br>_t_<br>|
|_C_<sup>_R_</sup><br>_t_<br>|Natural Land Protection Cost|share of _X_<sup>_O_</sup><br>_t_<br>|
|_C_<sup>_H_</sup><br>_t_|Managed Forest Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|
|_C_<sup>_P_</sup><br>_t_<br>|Pasture Land Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|
|_C_<sup>_C,N_</sup><br>_t_|Natural Land Restoration Cost|share of _X_<sup>_O_</sup><br>_t_|
|⇧_t_|Population|billion people|

20
Table C.2: Model Endogenous Variables

|Parameter|Description|Units|
|---|---|---|
|_L_<sup>_A_</sup><br>_t_<br>|Agricultural Land Area|GHa|
|_L_<sup>_A,c_</sup><br>_t_<br>|Agricultural Land Area, food crops|GHa|
|_L_<sup>_A,b_2</sup><br>_t_|Agricultural Land Area, 2G biofuels crops|GHa|
|_L_<sup>_P_</sup><br>_t_|Pasture Land Area|GHa|
|_L_<sup>_C_</sup><br>_t_|Commercial Forest Land Area|GHa|
|_L_<sup>_N_</sup><br>_t_|Unmanaged Natural Land Area|GHa|
|_L_<sup>_R_</sup><br>_t_<br>|Protected Natural Land Area|GHa|
|∆<sup>_N,A_</sup><br>_t_|Flow of Deforested Natural Land|GHa|
|∆<sup>_N,R_</sup><br>_t_<br>|Flow of Protected Natural Land|GHa|
|∆<sup>_C,N_</sup><br>_t_<br>|Flow of Restored Natural Land|GHa|
|∆<sup>_C,A_</sup><br>_t_<br>|Managed Forest Land Converted to Agriculture|GHa|
|∆<sup>_C,C_</sup><br>_t_<br>|Replanted Forest Land Area|GHa|
|∆<sup>_C,H_</sup><br>_v,t_<br>|Harvested Forest Land Area of Vintage _v_|GHa|
|∆<sup>_A,P_</sup><br>_t_|Agricultural Land Converted to Pasture|GHa|
|_X_<sup>_F_</sup><br>_t_<br>|Stock of Fossil Fuels|Ttoe|
|∆<sup>_F,E_</sup><br>_t_<br>|Flow of Fossil Fuels Converted to Petroleum|Ttoe|
|∆<sup>_F,n_</sup><br>_t_|Flow of Fossil Fuels Converted to Fertilizers|Ttoe|
|_x_<sup>_p_</sup><br>_t_|Petroleum Products|Gtoe|
|_x_<sup>_n_</sup><br>_t_|Fertilizers|Gton|
|_x_<sup>_c_</sup><br>_t_<br>|Food Crops|Gton|
|_x_<sup>_c,b_2</sup><br>_t_<br>|2G Biofuels Crops|Gton|
|_x_<sup>_b_1</sup><br>_t_|1G Biofuels|Gtoe|
|_x_<sup>_b_2</sup><br>_t_<br>|2G Biofuels|Gtoe|
|_x_<sup>_l_</sup><br>_t_|Livestock|Gtoe|
|_x_<sup>_w_</sup><br>_t_|Raw Timber|Gton|
|_Y _<sup>_f_</sup><br>_t_|Services from Processed Food|billion USD|
|_Y _<sup>_e_</sup><br>_t_|Energy Services|billion USD|
|_Y _<sup>_l_</sup><br>_t_|Services from Processed Livestock|billion USD|
|_Y _<sup>_w_</sup><br>_t_|Services from Processed Timber|billion USD|
|_Y _<sup>_r_</sup><br>_t_|Eco-system Services|billion USD|
|_Y _<sup>_o_</sup><br>_t_|Other Goods and Services|trillion USD|

21
Table C.3: Baseline Parameters

|Parameter|Description|Units|Value|
|---|---|---|---|
|_Population_||||
|⇧0|Population in 2004|billion people|6.39|
|⇧_T_|Population in time T|billion people|10.1|
|__<sup>_⇡_</sup>|Population Convergence Rate||0.042|
|_Land Use_||||
|_L_|Total Land Area|billion Ha|8.56|
|_L_<sup>_A_</sup><br>0<br>|Area of Agricultural Land in 2004|billion Ha|1.53|
|_L_<sup>_P_</sup><br>0|Area of Pasture Land in 2004|billion Ha|2.73|
|_L_<sup>_C_</sup><br>0|Area of Commercial Forest Land in 2004|billion Ha|1.62|
|_L_<sup>_N_</sup><br>0<br>|Area of Unmanaged Natural Land in 2004|billion Ha|2.47|
|_L_<sup>_R_</sup><br>0|Area of Protected Natural Land in 2004|billion Ha|0.207|
|_⇠_<sup>_n_</sup><br>0|Access Cost Function Parameter||0.6|
|_⇠_<sup>_n_</sup><br>1|Access Cost Function Parameter||105|
|_⇠_<sup>_R_</sup><br>0|Protection Cost Function Parameter||4.5|
|_⇠_<sup>_R_</sup><br>1<br>|Protection Cost Function Parameter||400|
|_⇠_<sup>_P_</sup><br>1|Pasture Conversion Cost Function Parameter||170|
|_⇠_<sup>_H_</sup><br>0|Forest Conversion Cost Function Parameter||80|
|_⇠_<sup>_H_</sup><br>1<br>|Forest Conversion Cost Function Parameter||0.004|
|_⇠_<sup>_C,N_</sup><br>0<br>|Natural Land Restoration Cost Parameter||0.8|
|_⇠_<sup>_C,N_</sup><br>1|Natural Land Restoration Cost Parameter||400|
|_Fossil Fuels_||||
|_X_<sup>_F_</sup><br>0|Endowment of Fossil fuels in 2004|trillion toe|0.343|
|∆<sup>_F,D_</sup>|Flow of Newly Discovered Fossil Fuels|trillion toe|0.008|
|_⇠_<sup>_F_</sup><br>1|Fuel Extraction Cost Function Parameter||2000|
|_Other Prima_|_ry Goods_|||
|_X_<sup>_O_</sup><br>0|Endowment of Other Primary Goods in 2004|USD _⇥_10<sup>13</sup>|3.16|
|__<sup>_o,X_</sup>|Growth Rate of Physical Capital||0.0035|
|_↵_<sup>_o,l_</sup>|Share of demographic factors in growth of _X_<sup>_O_</sup><br>_t_||0.39|
|_Intermediate_|_Products_|||
|_✓_<sup>_p_</sup>|Petroleum Conversion Factor|per toe of ∆<sup>_F,E_</sup><br>_t_|0.5|
|_c_<sup>_o,p_</sup>|Petroleum Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|0.0157|

22
Table C.3: Baseline Parameters (continued)

|Parameter|Description|Units|Value|
|---|---|---|---|
|_✓_<sup>_n_</sup>|Fertilizer Conversion Factor|Tton / Ttoe|1.071|
|_c_<sup>_o,n_</sup><br>|Fertilizer Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|0.0021|
|_✓_<sup>_b_1</sup>|1G Biofuels Conversion Rate|toe/ton|0.283|
|_✓_<sup>_b_2</sup>|2G Biofuels Conversion Rate|toe/ton|0.467|
|_K_<br>|2G Biofuels Fixed Factor Index||0.005|
|_c_<sup>_o,b_1</sup>|1G Biofuels Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|0.00025|
|_c_<sup>_o,b_2</sup>|2G Biofuels Conversion Cost|share of _X_<sup>_O_</sup><br>_t_|0.00033|
|_a_<sup>_n_</sup>|Share of Agricultural Land in CES function||0.55|
|_⇢n_|CES Parameter for Agricultural Land and<br>Fertilizers||0.123|
|_A_0|Crop Technology Index in 2004||13.89|
|__<sup>_c_</sup>|Logistic Growth Rate of Crop Technology Index||0.025|
|_c_<sup>_o,c_</sup>|Food Crop Production Cost|share of _X_<sup>_O_</sup><br>_t_|0.016|
|_✓_<sup>_c,b_2</sup><br>0|2G Biofuels Crop Technology Index in 2004||14.89|
|_b_2|2G Biofuels Fixed Factor Decay Rate||0.05|
|_↵_<sup>_b_2</sup>|Fixed Factor Cost Share in 2G Biofuels<br>Production||0.6|
|_⇢b_2|CES Parameter for Fixed Factor and<br>Agricultural<br>Land||-1.5|
|_c_<sup>_o,c_</sup>|2G Biofuels Crops Production Cost|share of _X_<sup>_O_</sup><br>_t_|0.022|
|_✓_<sup>_P_</sup>|Livestock Technology Index in 2004||0.69|
|_a_<sup>_l_</sup>|Share of Pasture Land in CES function||0.35|
|_⇢l_|CES Parameter for Pasture Land and Feed||-0.33|
|_c_<sup>_o,l_</sup>|Livestock Production Cost|share of _X_<sup>_O_</sup><br>_t_|0.0055|
|_a_|Merchantable Timber Yield Parameter 1||5.62|
|_b_|Merchantable Timber Yield Parameter 2||76.5|
|_~~v~~_|Minimum Age for Merchantable Timber|Years|11|
|__<sup>_w_</sup><br>_v_|Timber Yield Gains of Vintage _v_|Share of Yield 0|0.011|
|_c_<sup>_p_</sup>|Forest Planting Cost|share of _X_<sup>_O_</sup><br>_t_<br>|0.0001|
|_c_<sup>_o,w_</sup>|Forest Harvesting Cost|share of _X_<sup>_O_</sup><br>_t_|0.0021|
|_Final Goods _|_and Services_|||
|_✓_<sup>_f_</sup><br>0|Food Processing Technology Index in 2004||1.5|
|__<sup>_f_</sup>|Food Processing Technology Index Growth<br>Rate||0.0225|
|_c_<sup>_o,f_</sup>|Food Processing Cost|share of _X_<sup>_O_</sup><br>_t_|0.015|
|_✓_<sup>_l_</sup><br>0|Livestock Processing Technology Index in 2004||1.7|
|__<sup>_l_</sup>|Livestock Processing Technology Growth Rate||0.0025|
|_c_<sup>_o,yl_</sup>|Livestock Processing Cost|share of _X_<sup>_O_</sup><br>_t_|0.0068|

23
Table C.3: Baseline Parameters (continued)

|Parameter|Description|Units|Value|
|---|---|---|---|
|_✓_<sup>_e_</sup><br>0|Energy Technology Index in 2004||1.195|
|__<sup>_e_</sup>|Energy Technology Index Growth Rate||0.0225|
|_⇢e_|CES Parameter for Petroleum and Biofuels||0.5|
|_↵_<sup>_e_</sup>|Share of Biofuels in CES Function||0.09|
|_✓_<sup>_yw_</sup><br>0|Timber Processing Technology Index in 2004||1.52|
|__<sup>_yw_</sup>|Timber Processing Technology Growth Rate||0.0225|
|_c_<sup>_o,yw_</sup>|Timber Processing Cost|share of _X_<sup>_O_</sup><br>_t_|0.0224|
|_✓_<sup>_r_</sup><br>|Ecosystem Services Technology Index||0.71|
|_↵_<sup>_A,r_</sup><br>|Share of Agricultural Land in CES Function||0.02|
|_↵_<sup>_P,r_</sup><br>|Share of Pasture Land in CES Function||0.14|
|_↵_<sup>_C,r_</sup>|Share of Managed Forest Lands in CES<br>Function||0.26|
|_⇢r_|CES Parameter for Ecosystem Services||0.123|
|_✓_<sup>_R_</sup>|E↵ectiveness Index of Protected Lands||10|
|_c_<sup>_o,r_</sup>|Cost of Recreation Services||0.0296|
|_✓_<sup>_o_</sup><br>0|Total factor Productivity Index in 2004||1.854|
|__<sup>_o_</sup>|Total Factor Index Growth Rate||0.0225|
|_Preferences _|_and Welfare_|||
|_↵f_|AIDADS Marginal Budget Share at Subsistence||0.189|
||Income for Services from Processed Food|||
|_↵l_|AIDADS Marginal Budget Share at Subsistence<br>Income for Services from Processed Livestock||0.035|
|_↵e_|AIDADS Marginal Budget Share at Subsistence<br>Income for Energy Services||0.112|
|_↵w_|AIDADS Marginal Budget Share at Subsistence<br>Income for Services from Processed Timber||0.036|
|_↵r_|AIDADS Marginal Budget Share at Subsistence<br>Income for Ecosystem Services||0.049|
|_↵o_|AIDADS Marginal Budget Share at Subsistence||0.579|
||Income for Other Goods and Services|||

24
Table C.3: Baseline Parameters (continued)

|Parameter|Description|Units|Value|
|---|---|---|---|
|_βf_|AIDADS Marginal Budget Share at High<br>Income for Services from Processed Food||0.028|
|_βl_|AIDADS Marginal Budget Share at High<br>Income for Services from Processed Livestock||0.011|
|_βe_|AIDADS Marginal Budget Share at High<br>Income for Energy Services||0.049|
|_βw_|AIDADS Marginal Budget Share at High<br>Income for Services from Processed Timber||0.032|
|_βr_|AIDADS Marginal Budget Share at High<br>Income for Ecosystem Services||0.104|
|_βo_|AIDADS Marginal Budget Share at High<br>Income for Other Goods and Services||0.776|
|_γ_<sup>_f_</sup>|AIDADS Subsistence Parameter for Processed<br>Food||0.45|
|_γ_<sup>_l_</sup>|AIDADS Subsistence Parameter for Processed<br>Livestock||0.003|
|_γ_<sup>_e_</sup>|AIDADS Subsistence Parameter for Energy<br>Services||0.026|
|_γ_<sup>_w_</sup>|AIDADS Subsistence Parameter for Processed<br>Timber Products||0.027|
|_γ_<sup>_r_</sup>|AIDADS Subsistence Parameter for Ecosystem<br>Services||0.028|
|_γ_<sup>_o_</sup>|AIDADS Subsistence Parameter For Other<br>Goods and Services||0.346|
|_γ_<br>_δ_|Risk Aversion Parameter<br>Social Discount Rate||2<br>0.95|

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# **D.1 Supplementary Figures**

<!-- Start of picture text --> a)<br><!-- End of picture text --> <!-- Start of picture text --> 140 Fertilizers (Deterministic-Baseline) 100 Difference between Stochastic and Deterministic-Baseline Solutions<br>Range of all sample paths<br>average<br>10% quantile<br>6 50% quantile<br>90% quantile<br>Stochastic-optimistic<br>Stochastic-pessimistic<br>4<br>120 80<br>2<br>0<br>100 60<br>-2<br>Fertilizers (million tons)<br>Ratio of Fertilizers to -4<br>Food Crop Land Area (kg per Ha)<br>80 40<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>million tons kg per Ha<br>Ratio of Fertilizers to Food Crop Land Area (kg per Ha)<br><!-- End of picture text --> b) <!-- Start of picture text --> Biofuels (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>Deterministic-baseline 20 Range of all sample paths<br>35 Deterministic-optimisticDeterministic-pessimistic average10% quantile<br>15 50% quantile<br>90% quantile<br>30 Stochastic-optimistic<br>10 Stochastic-pessimistic<br>25<br>5<br>20<br>0<br>15<br>-5<br>10<br>-10<br>5<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>c)<br>2G Biofuels (Deterministic) Difference between Stochastic and Deterministic-Baseline Solutions<br>1600<br>Deterministic-baseline Range of all sample paths<br>Deterministic-optimistic 200 average<br>1400 Deterministic-pessimistic 10% quantile<br>150 50% quantile<br>90% quantile<br>1200 100 Stochastic-optimistic<br>Stochastic-pessimistic<br>50<br>1000<br>0<br>800<br>-50<br>600 -100<br>-150<br>400<br>-200<br>200 -250<br>-300<br>2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100<br>Year Year<br>26<br>million tonnes of oil equivalent million tonnes of oil equivalent<br>million tonnes of oil equivalent million tonnes of oil equivalent<br><!-- End of picture text --> Figure D.1: Consumption of Fertilizers and Biofuels
# **D.1 Error Checking**

For a simulated path _i_ , at the current time _s_ , we use the step 2 of Algorithm 1 to find the optimal decision **a**<sup>_i_</sup> _s_<sup>atitscurrentstate</sup><sup>**S**</sup><sup>_i_</sup> _s_<sup>.Itsnextstateinthe</sup> original stochastic model (1) is **G** _s_ ( **S**<sup>_i_</sup> _s_<sup>_,_</sup><sup>**a**</sup><sup>_i_</sup> _s_<sup>_, ✏s_)whichisrandomanddependent</sup> on _✏s 2_ ⇥, where ⇥represents the set of all possible vectoral values of _✏s_ . The associated Euler equation for the stochastic model is where _r_ **S** _s_ +1 is the gradient operator over **S** _s_ +1, _δ_<sup>_t_</sup> _λt_ is the multiplier (a column vector) of the constraint **S** _t_ +1 = **G** _t_ ( **S** _t,_ **a** _t, ✏t_ ) and _δ_<sup>_t_</sup> _μt_ is the multiplier (a column vector) of the feasibility constraints **a** _t 2 Dt_ ( **S** _t_ ) which are represented by **H** _t_ ( **S** _t,_ **a** _t_ ) _≥_ 0 here.<sup>8</sup> In the literature, the multipliers are often substituted by marginal utilities or some other expressions derived from Karush–Kuhn–Tucker conditions, but we keep them in the equation as such substitutions are often complicated (e.g., in our examples) and we can directly get the multipliers from numerical optimization solvers. The multipliers and **a** _s_ +1 can be approximated by the solutions in the deterministic model (2) at their corresponding states. For example, we compute **S**<sup>_i,j_</sup> _s_ +1<sup>=</sup><sup>**G**</sup><sup>_s_(</sup><sup>**S**</sup><sup>_i_</sup> _s_<sup>_,_</sup><sup>**a**</sup><sup>_i_</sup> _s_<sup>_, ✏j_</sup> _s_<sup>)foreverypossible</sup><sup>_✏j_</sup> _s_<sup>_2_⇥and</sup> then use the step 2 of Algorithm 1 to find the optimal decision **a**<sup>_i,j_</sup> _s_ +1<sup>atitsasso-</sup> ciate state **S**<sup>_i,j_</sup> _s_ +1<sup>attime</sup><sup>_s_+ 1andtheircorrespondingmultipliers,andthenwe</sup> can compute the expectation in equation (D.1) as the conditional probability distribution of _✏s_ is given. When _✏s_ is a continuous random variable vector, we can use its quadrature nodes _✏_<sup>_j_</sup> _s_<sup>andthenimplementquadraturerulestoesti-</sup> mate the expectation in equation (D.1). Thus, for the simulated path _i_ , we can compute the normalized Euler error at time _s_ :

> 8An equality constraint _f_ ( _x_ ) = 0 can be written as a combination of _f_ ( _x_ ) _≥_ 0 and _−f_ ( _x_ ) _≥_ 0.

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where **1** is a vector of ones,<sup>_λ_</sup> _λ_<sup>_s_</sup><sup><u>+</u></sup> _s_<sup>1</sup> and<sup>_<u>μ</u>_</sup> _λ_<sup>_s_</sup><sup><u>+</u></sup> _s_<sup>1</sup> are elementwise divisions, and _k·k_ is a norm operator. Note that the normalized Euler error is unit free.

For our stochastic FABLE model, we find that the approximate _L_<sup>1</sup> error of the first 100 years solutions among 1,000 simulated paths, defined as is only 8 _._ 6 _⇥_ 10<sup>_−_4</sup> , and the corresponding approximate _L_<sup>_1_</sup> error, defined as is only 0 _._ 02. Thus, we see that ENLCEQ solves our stochastic model within an acceptable accuracy.

# **D.2 An Illustrative Example**

Below we use a simple optimal growth model to illustrate ENLCEQ. We assume the total factor of productivity, _At_ , is a Markov chain. It has three possible values: _A_ 1 = 0 _._ 9, _A_ 2 = 1 _._ 0, and _A_ 3 = 1 _._ 1. Its transition probability matrix is We use _At_ +1 = _G_<sup>_A_</sup> ( _At, ✏t_ ) to represent the transition law of _At_ , where _✏t_ is a random variable with zero mean. We solve the following optimal growth problem: where _ct_ is consumption at time _t_ , _β_ = 0 _._ 96 is the discount factor, _kt_ is capital, _δ_ = 0 _._ 1 is the depreciation rate, _↵_ = 0 _._ 3, and _u_ ( _c_ ) = _−c_<sup>_−_1</sup> is the utility function. Assume that we are interested in the solutions in the first 20 periods (i.e., _T_ = 20). Here in the notations of Section 2, **S** := ( _k, A_ ) is the vector of 28
state variables, **a** := _c_ is the decision variable, and the transition laws are which can be written as **S** _t_ +1 = **G** ( **S** _t,_ **a** _t, ✏t_ ).

In ENLCEQ, we choose _T_<sup>b</sup> = 200 which has been large enough for the solutions in the periods of our interest (i.e., the first 20 periods). If we are interested in solutions at more periods, then we can also increase _T_<sup>b</sup> . In the step 2 of Algorithm 1 for solving problems at time _s_ , we let **S** _t_ +1 = **G** ( **S** _t,_ **a** _t,_ 0), that is, _At_ +1 = _G_<sup>_A_</sup> ( _At,_ 0), which in fact means that we replace _At_ by its mean conditional on the realized values of _As_ : _A⇡t,s_ , for all _t ≥ s_ , where _A_ = ( _A_ 1 _, A_ 2 _, A_ 3) and _⇡t,s_ is a column vector representing the probability distribution conditional on the realized values of _As_ . If the realized values of _As_ is _Ai_ , we have _⇡t,s_ = _P_<sup>_t−s_</sup> _⇡s,s_ , where _⇡s,s_ is a length-3 column vector with 1 at the _i_ th element and 0 everywhere else.

In the GAMS code attached in the end of the appendix, we compute 1,000 simulated paths of the first 20 periods which are the periods of interest. It took 15 minutes on a laptop, much slower than NLCEQ, as ENLCEQ solves 20,000 optimization methods while NLCEQ just needs to solve a few (e.g., 33 optimization problems to obtain a degree-10 Chebyshev polynomial of _k_ for each of three values of _A_ to approximate the optimal policy functions) for this simple illustrative case. But as discussed in Section 2, ENLCEQ can solve many problems that NLCEQ cannot or is less efficient or less accurate.

Each simulation path _i_ from ENLCEQ contains a pair of ( _kt,i_<sup>ENLCEQ</sup> _, A_<sup>ENLCEQ</sup> _t,i , c_<sup>ENLCEQ</sup> _t,i_ ) at each time _t_ , where _A_<sup>ENLCEQ</sup> _t,i_ is simulated exogenously. We also apply value function iteration to solve this simple problem and then get the optimal policy function for consumption: _C_<sup>_V F I_</sup> ( _k, A_ ), which is approximated by a degree-20 Chebyshev polynomial of _k_ on [0 _._ 5 _,_ 5] for each possible value of _A_ . We then compare _c_<sup>ENLCEQ</sup> _t,i_ and _C_<sup>_V F I_</sup> ( _kt,i_<sup>ENLCEQ</sup> _, A_<sup>ENLCEQ</sup> _t,i_ ) for all 1 _ i _ 1000 and _t _ 20. We find that the approximate _L_<sup>1</sup> relative error is 3 _._ 7 _⇥_ 10<sup>_−_3</sup> and the approximate _L_<sup>_1_</sup> relative error is 5 _._ 5 _⇥_ 10<sup>_−_3</sup> . This tells us that ENLCEQ can achieve 2 to 3 digit accuracy, which is consistent with the accuracy of NLCEQ in Cai _et al._ (2017).

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