Accounting for spatial economic interactions at local and meso scales in integrated assessment model (IAM) frameworks: challenges and recent progress

The scientific and policy needs to assess and manage climate change impacts have spawned new coupled, multi-scale integrated assessment model (IAM) frameworks that link global climate and economic processes with high-resolution data and models of human-environmental systems at local and meso scales (Fisher-Vanden and Weyant 2020 Annu. Rev. Resour. Econ. 12 471–87). A central challenge is in accounting for the fundamental interdependence of people, firms, and economic activities across space at multiple scales. This requires modeling approaches that can incorporate the relevant spatial details at each scale while also ensure consistency with spatially varying feedbacks and interactions across scales—a condition economists refer to as spatial equilibrium. In this paper, we provide an overview of how economists think about and model spatial interactions, particularly those at the local level. We describe challenges and recent progress in accounting for greater spatial heterogeneity at individual (field, agent) scales and incorporating heterogeneous spatial interactions and dynamics into consistent IAM frameworks. We conclude that the most notable progress is in advancing global IAMs with spatial heterogeneity and dynamics embedded in spatial equilibrium frameworks and that less progress has been made in incorporating features of spatial equilibrium into highly detailed multi-scale IAMs.


Introduction
The scientific and policy needs to better assess climate change impacts have spawned new coupled modeling frameworks that link global climate and economic processes with high-resolution data and models of human-natural systems at local and meso scales. These newer, multi-scale integrated assessment model (IAM) frameworks are motivated by the pressing need for models that can assess vulnerabilities and inform climate adaptation policies at local and sub-national regional scales (Fisher-Vanden and Weyant 2020). A key challenge of multi-scale IAMs is the coupling of detailed modeling components, i.e. that are specified with high-resolution temporal and spatial data and more realism of fine-scale processes and their interactions, with modeling components defined at more aggregate spatial and temporal scales (Kling et al 2017, Fisher-Vanden and Weyant 2020, Piontek et al 2021. Doing so requires knowledge of how the underlying socioeconomic and biophysical processes evolve and interact across disparate scales, sufficient data to specify these relationships, and substantial computational resources to iteratively solve for a consistent set of outcomes (Meyfroidt et al 2018).
The goal of this paper is to review key challenges and progress in representing detailed spatial economic processes within IAM frameworks at local and meso scales. In this context, the local scale can be thought of as those economic and environmental phenomena that manifest at scales of 1 km or less and involve the decisions and interactions of individual agents, e.g. landowners, with surrounding natural systems (Johnson et al 2023). The meso scale is the mediating layer of processes, systems, and institutions, including markets, that connect local scale phenomena to larger, global scale systems and vice versa (Johnson et al 2023). Spatial economic interactions, i.e. the geographic movements and interactions of people, goods, and information that link economic activities across locations, are fundamental components of human systems that exist at local and meso scales. Accounting for these spatially interdependent flows and the price feedbacks that mediate them is critical for consistently modeling and projecting the environmental impacts of economic activities and visa-versa.
First generation IAMs (e.g. Nordhaus 1993) ignored spatial interactions altogether and focused on the global economy as a single region. These seminal models consist of a dynamic model of the global economy specified with aggregate socioeconomic data and a simplified representation of the biophysical processes. Integration of global trade models into IAMs has succeeded in accounting for greater detail in sector-specific trade flows and the spatial adjustments of economic activities across multi-country regions in response to climate change (Meyfroidt et al 2013, Liu et al 2015, Tol 2018, Hertel et al 2019, Cruz and Rossi-Hansberg 2021. Subsequent models have accounted for more detailed processes, e.g. interregional migration (Benveniste et al 2020, Burzyński et al 2022, production reallocation and sectoral specialization (Conte et al 2021, Cruz and, and ecosystem service production (Baldos et al 2020, Johnson et al 2021, Golub et al 2022.
By incorporating greater spatial and sectoral details, these models have increased the relevance of global IAMs for country-specific policies. However, greater resolution comes at a cost. First, these models are largely static in terms of agents' behaviors. They do not account for how future outcomes, e.g. future changes in emissions or uncertainty over future climate change, affect current decision making, even though these dynamic factors have a substantial influence on economic outcomes and optimal policies (Cai and Lontzek 2019). Second, they omit fine-scale, heterogeneous patterns of land use, built infrastructure, amenities, natural lands, and people that determine regional growth, consumption, and environmental impacts (Verburg and Overmars 2009, Bateman et al 2013, Ahlfeldt et al 2015, Liu et al 2017, Monte et al 2018, Duranton and Puga 2020, Suh et al 2020. Accounting for these temporal and spatial complexities in multi-scale IAMs is conceptually and computationally challenging. It entails grappling with key problems such as accounting for heterogeneity in local agents and landscapes, understanding how individual agents interact with complex economic and ecological systems, and incorporating dynamical representations of temporal processes such as land use change, carbon sequestration, and capital investment decisions. It is ultimately not feasible, nor desirable, to build a super model to account for all these considerations. Instead, a range of models with different strengths have emerged and progress is being made on multiple fronts in addressing these challenges.
The remainder of the paper is structured as follows. In the next section we provide an overview of spatial economic interactions with a focus on land use and the location of people and economic activities, which are key spatial interactions that arise in IAMs. Section 3 discusses the challenges and recent progress to modeling fine-scaled spatial economic interactions at local and meso scales and integrating them into IAM frameworks. Section 4 concludes.

Accounting for space in economics
Spatial economics concerns the allocation of scarce resources over space resulting from the decisions of firms, residents, workers, and governments and the corresponding locations of economic activity (Proost and Thisse 2019). As such, spatial economics provides consistent frameworks for understanding the fundamental interdependence among economic activities at different locations that arises through connected flows of goods, people, factors of production, and information. The spatially varying distribution and interactions of people, production, resources, and institutions implies that land and locations are differentiated in important ways. Economists have traditionally conceptualized these differences as 'first nature' or 'second nature' advantages (Krugman 1993, Duranton andPuga 2014). First nature advantages include exogenous, i.e. fixed, features such as resource endowments, soil quality, or water access that make some locations more advantageous for production, travel, and trade. Second nature advantages also confer an economic advantage, but are endogenous-meaning they are not fixed and instead and evolve over time via the interactions of people, firms, and institutions both within and across locations (Ahlfeldt and Pietrostefani 2019). For example, knowledge spillovers are a second nature advantage that emerge from the concentration of population in a city of place, which changes over time due to spatial interactions (Duranton and Puga 2014).
These sources of economic advantage are also mediated, reinforced, or offset by natural advantages, which leads to a fundamental interdependence between spatial interactions and natural systems. The distribution of many natural advantages and disadvantages is often not fixed and co-evolves with human activities. For example, the water quality of coastal or inland lakes, which determines the location of people and commercial activities, is influenced by human activities within the watershed, including upstream agricultural practices that depend on economic and behavioral factors .
Spatial interactions occur at local, meso scales, and global scales, ranging from conversations between neighbors to work commutes within a metro region to global supply chain and trade flows among nations. Spatial interactions at local and meso scales exhibit a high degree of spatial heterogeneity, implying that the localized effects of a common global shock will depend on the specific configuration of land use, firm and household location, transportation and trade networks, or other spatially differentiated resources (Desmet et  Spatial equilibrium is a fundamental concept in economics that ensures consistency in economic outcomes from one location to the next, given that these outcomes are interdependent across space (Proost and Thiesse 2019). Heterogeneity across space generates push and pull forces that influence location and land use choices. Agents respond to these spatial forces by choosing the location that most improves their utility or profitability. Examples include a firm that relocates its production to a new location in response to a transportation improvement or a household that migrates for better labor and educational opportunities. These choices are subject to constraints, including a location's affordability and potential market frictions, such as transport or information costs. Spatial equilibrium arises when there is no further incentive for an individual household or firm to alter their location, land use, production, or consumption decision given the decisions of all other economic agents. Spatial equilibrium is a standard condition of economic models and arises due to prices that adjust to the locations of households and firms such that no individual household or firm can make themselves better off by relocating. For example, Albouy (2016) illustrates that differences in natural amenities generate large differences in local productivity, quality of life, and land rents across U.S. cities. Spatial equilibrium arises because of the greater demand for high-amenity places, which bids up land rents that ultimately offset the desirability of these locations.
The spatial equilibrium framework allows researchers to reflect the fundamental spatial interdependencies among people, firms, and economic activities without having to explicitly represent each of the many individual-level interactions. Instead, these interactions are assumed to be mediated by the price mechanism in which prices and aggregate flows of people, firms, and economic goods adjust to each other. Rather than iteratively modeling these adjustments, most spatial economic models simplify this complexity by assuming that all price and quantity adjustments are made fully and simultaneously. This approach provides a tractable way to ensure the consistency of location and land use decisions, prices, and second-nature feedbacks across space. However, the greater the spatial heterogeneity and model complexity, the more difficult it can be to define the conditions for spatial equilibrium.

Challenges and progress in representing spatial heterogeneity and interactions in IAMs
Representing spatial socioeconomic and biophysical processes and their interactions within IAM frameworks in ways that consistently account for prices and other endogenous feedbacks presents many conceptual and computational challenges (Kling et al 2017, Hertel et al 2019, Fisher-Vanden and Weyant 2020. These challenges necessitate modeling tradeoffs across temporal and spatial dimensions. At one end of the spectrum are single-scale, dynamic IAMs that incorporate space in a highly aggregated way, e.g. as multi-country regions. These models simplify space to focus on other aspects, including on the dynamic decision making by agents, i.e. forward-looking investment, migration, or resource use decisions that consider the future evolution of capital, resource, or other stocks over time. By simplifying the spatial dimensions, these models can be solved for a set of equilibrium conditions that fully characterize the two-way interactions between the economy and biophysical systems, including the spatial equilibrium that describes the interdependence across aggregate regions. At the other end of the spectrum are multiscale IAMs with multiple models of highly detailed socioeconomic and biophysical processes, including high-resolution spatial models that specify individual agent decision making and may reflect substantial heterogeneity in several dimensions, e.g. behavioral, environmental, or sectoral. These models are too detailed to be analytically or computationally solved for a set of equilibrium conditions or to incorporate dynamic decision making. Instead they require an iterative approach to account for feedbacks (Robinson et al 2018). With many sources of spatial and agent heterogeneity, iterating to a set of outcomes consistent across submodules may not be possible.
In between are a continuum of modeling frameworks that balance model complexity across spatial and temporal dimensions. In so doing, they also balance theoretical frameworks that assign structure to the model, such as spatial equilibrium, with datadriven approaches that are more flexible and able to take advantage of the increasing amounts of highresolution data. In the remainder of this section we further examine the challenges and recent progress across a continuum of theoretical and datadriven approaches with a focus on: (a) accounting for heterogeneity at individual (field, agent) scales and (b) incorporating heterogeneous spatial interactions and dynamics into IAM frameworks.

Heterogeneity at individual (field, agent) scales
IAMs that represent processes at local scales must contend with many sources of spatial heterogeneity, which quickly exhausts the computational budget of the model. It is therefore essential to recognize when and where local heterogeneities matter for evaluating meso or global outcomes and to identify the key spatial interactions within and across these scales.
Theoretical descriptions of heterogeneous spatial processes are an essential guide for modeling processes at a local and meso scale. For example, models of household sorting provide a systematic understanding of how residents choose their location over space in response to changes in amenities and resources, e.g. temperature or water availability (Fan et al 2018, Sinha et al 2018. By focusing on the changes that are most influential in agents' location choices, this framework provides a means to determine which variables should be included. Another theory-driven example is found in the application of basic land use theory which posits that landowners allocate land the different uses to maximize their profits. Given prices, it is then possible to determine the optimal land use shares across space. This approach is used by Bateman et al (2013) to generate hypothetical agricultural landscapes in the U.K. and evaluate ecosystem services across future scenarios.
Advances in remote sensing and other data collection technologies have spurred tremendous gains in the ability to model spatially detailed processes across large, even global, extents. Substantial progress has been made in accounting for fine-scaled human adaptations to climate change by incorporating substantial spatial heterogeneity into theory-based models. In many cases, access to richer, finer-scale data can augment theory-driven approaches by using these data to estimate the so-called 'structural' parameters governing theoretically derived relationships. For example, newly assembled, highly detailed datasets on mortality, productivity, agricultural yields have been used in combination with data on weather and temperature shocks to estimate the structural parameters of models that quantify the human consequences of extreme weather events (Carleton and Hsiang 2016, Hsiang 2016, Hsiang et al 2017, Carleton et al 2022. Similarly, the proliferation of satellite imagery has allowed for modeling population and economic activity at a fine spatial scale using night light and other image features (Donaldson and Storeygard 2016, Ch et al 2021, Dingel et al 2021. These data coupled with theoretical frameworks, e.g. that allow relationships among night-time lights, urban density, and worker attributes to be specified, has advanced fine-scale modeling of urbanization in locations around the globe where traditional sources of data are unavailable (Harari 2020, Dingel et al 2021.
Alternatively, the proliferation in data availability and computational power can give rise to data-driven approaches that are not constrained by theoretical frameworks. An example is in modeling highresolution income and population changes in the U.S. using deep learning approaches applied to daytime satellite imagery to achieve predictive power well beyond previous models (Khachiyan et al 2022). Similar data have been combined to shed light on the spatially heterogeneous determinants of land use transitions and to specify data-driven models of urbanization, agricultural intensification, and deforestation (Verburg and Overmars 2009, Nelson et  Heterogeneous spatial interactions may be incorporated in a variety of ways into these models (Alexander et al 2017, Prestele et al 2017, Plantinga 2021. In principle this requires some type of theoretical framework to guide the hypothesized mechanisms by which land parcels (or cells) are interdependent over space, e.g. as described by a set of spatial equilibrium conditions, and how changes at local scales interact with changes at the meso and global scales. In the absence of a theoretical framework, it is more challenging to ensure that local-scale processes are consistent with meso-scale spatial dependencies and market forces (Prestele et al 2016, Verburg et al 2019. Efforts to augment data-rich, machine learning (ML) approaches with theory-based models provide a potential means to incorporate higher-level spatial equilibrium relationships (Karpatne et al 2017). For example, innovations in the emerging field of knowledge-guided ML (KGML) have proven useful in combining the predictive power of ML with theoretical knowledge of the system (as reviewed by Song et al this issue), but as far as we know the development of KGML models for the purposes that we describe here has not yet to be attempted.
In other cases, local-scale land use decisions may be influenced by meso-or global-scale processes, but not generate feedbacks to these higher scales. In these cases, there is not the same two-way feedback across scales that necessitates spatial equilibrium conditions to ensure consistency. Accounting for a multitude of agent and spatial heterogeneity is then possible, but limited by model tractability and data availability. Direct information on decision making agents from surveys or detailed sampling might be used to generate decision heuristics or types of agents, which can be used to account for heterogeneity while reducing model complexity (Verburg et al 2019). An example of this approach can be found in Liu et al (2020), Kast et al (2021), andBeetstra et al (2022), in which choice experiment results are used to identify classes of farmers with regards to management practice behavior using latent class analysis to define farmer types.
Agent-based models (ABMs) offer an alternative to theory-driven and data-driven approach to accounting for multiple sources of heterogeneity and cross-scale interactions (An 2012). They explicitly model individual interactions of heterogeneous agents based on specified decision-making rules of heterogeneous agents. Recursive simulation methods are then used to reveal the bottom-up emergence of aggregate outcomes from heterogeneous individual agent choices. Because these models omit any topdown constraints, such as a set of spatial equilibrium conditions, they can incorporate considerable detail and complexity. For example, Murray-Rust et al (2014) presents a land use modeling framework designed to represent a diversity of human behavior and land management decisions; differences in land use intensity and productivity, as a consequence of biophysical factors as well as agent's decisions; multifunctional land uses and the trade-offs between the provision of different bundles of goods and services; a wide range of ecosystem goods and services, including those that are not explicitly defined in monetary terms such as biodiversity; and institutional agents and the various mechanisms by which they influence land use change. They use an adhoc (i.e. a theoretic) approach to valuing the investment of land and other inputs into producing ecosystem services at a spatially-explicit, grid cell scale based on the aggregate residual demand for these services.
A key challenge with this approach is the availability of individual-level data that is needed to specify 'types' of agents and the rules that govern their individual production and consumption decisions. Without sufficient data and in the absence of theory to guide this specification, the decision-making rules are often based on heuristics and not readily testable. Another challenge is ensuring that individual agent interactions at local scales are consistent with higher-scale processes without imposing topdown constraints. For these reasons, ABMs are most appropriate in modeling social-ecological systems in which heterogeneous agents engage in direct interactions with each other that are not mediated by a price mechanism, e.g. the governance and management of small-scale fisheries to examine the conditions under which collective action and cooperation among owners may emerge (Lindkvist et al 2020).

Incorporating heterogeneous spatial interactions and dynamics into IAM frameworks
The challenges of incorporating greater spatial heterogeneity into an equilibrium framework are compounded by the amount of model coupling that must occur in multi-scale IAMs, in which multiple submodules of economic and biophysical processes pass information and outputs among each other (Fisher-Vanden and Weyant 2020). Feedbacks filter through space, time, and across multiple scales, creating challenges for causal identification and more practically for coupling models of interrelated systems. To make models tractable, assumptions about the spatial and temporal scales over which causal relationships and feedbacks operate are necessary. Within local land markets, for example, field-level allocations of land use may be assumed to be fully substitutable across space. This assumption implies that equilibrium allocations are constrained by the total amount, and not the fine-scale pattern, of land uses within a local market area.
Linking local attributes of land parcels to global conditions and understanding how local or mesolevel characteristics may mediate global impacts is critical for understanding observed phenomena that cannot be explained only by spatial heterogeneity at local scales. For example, in evaluating how agricultural productivity evolves in response to climate change, poorer quality land brought into production might be more vulnerable to weather shocks, but also may have market access advantages, e.g. due to trade linkages of the region at meso or global scales. In this case, understanding the mediating effects of market accessibility is critical to explaining land conversion at local scales and why, despite declining marginal productivity, there is more pressure on these lands to meet increasing demand (Suh et al 2020).
Recent progress in the development of fine-scaled quantitative spatial economic models has enabled a significant advance in incorporating more detailed spatial processes into IAMs. These models, developed by trade and urban economists over the past decade, formalize the representation of economic interactions across geographic space at a high degree of spatial resolution. They integrate the insights of spatial equilibrium from urban economics with the key assumptions of trade models, including initial location endowments, market structure, and frictions that arise from the costly movement of goods, people, and ideas across space (Redding and Rossi-Hanberg 2017). The spatial scale of analysis determines the types of economic interactions that are relevant. Locations are characterized by differences in amenities, wages, housing prices, and productivities and individuals and firms are heterogeneous in their preferences and technologies respectively (Ahlfeldt et al 2015, Conte et al 2021. By allowing for linkages across space through migration and trade, and commuting in the case of models that focus on a single region, these models account for the endogenous characterization of land, housing prices, and wages and spatial patterns of population location, density, firm location, amenities, and economic activities. Once calibrated, they allow for counterfactual policy exercises that account for these complex spatial interactions (Fretz et al 2017, Allen andArkolakis 2019).
A dynamic version of these models, building on a global dynamic economy model developed by Desmet et al (2018), has been integrated with reduced-form representations of climate dynamics to study the spatial impacts of global climate change on trade and migration and implications for social welfare. Dynamic spatial models introduce time by modeling migration or innovation as investment decisions in which workers choosing destinations face a tradeoff between the present value of expected future utility flows and a one-off relocation cost (Ahlfeldt et al 2020, Cruz and. The approach also accounts for the impact of economic activities on carbon emissions and temperature and therefore provide a consistent means of representing the twoway coupling between the global economy and climate system. For example, Cruz and Rossi-Hansberg (2021) link temperature changes to local productivity and amenities via climate damage functions and use their calibrated model to simulate the global economy to 2200 to compare welfare losses over space due to climate change impacts on productivity and amenability of different global locations. This approach has also been applied to study sectoral specialization (Conte et al 2021), the consequences of higher temperatures on economic activity (Krusell and Smith Jr 2022), the efficacy of carbon taxes (Cruz and Rossi-Hansberg 2022), coastal flooding impacts , and migration responses to climate change Rossi-Hansberg 2015, Burzyński et al 2022).
These models are specified with high resolution spatial data on population, economic output, and temperature, and therefore can quantify global changes in the spatial pattern of population and economic activities at highly disaggregate scales, e.g. one-degree grid size. They are made tractable by assuming population dynamics can be simplified to migration choices that depend only on current variables, and not past or future ones, and that the system of equations that defines a spatial equilibrium in each period can be reduced to a system of equations for population and wages in each location. Given these assumptions, all other variables, including firm investments and other dynamic conditions, can be computed.
The integration of quantitative spatial models that are grounded in spatial equilibrium with a reducedform representation of climate dynamics represents an important and exciting advance in global IAMs. These models can incorporate spatial heterogeneity and interactions at a high spatial resolution with key dynamics, including of population, economic activity, and carbon emissions. However, they do not account for the many other sources of spatial heterogeneity in biophysical and socioeconomic processes that are relevant at local scales nor do they capture multi-scale interactions. While these models are useful for projecting changes in economic activities and climate impacts at a high resolution, they are more limited in their ability to inform adaptation and mitigation policies at local and meso scales.

Discussion and conclusions
As we have highlighted here, there has been significant progress in addressing substantial gaps in modeling frameworks that combine high resolution spatial heterogeneity with structural spatial equilibrium models that capture spatial population, economic, and other flows across space. Recent advances in quantitative spatial models integrated with stylized global IAMs of climate change and the global economy are a major step toward incorporating high spatial resolution into these dynamic, structural IAMs. However, these models omit many of the realistic features of human and biophysical features that are relevant at local and meso scales. On the other hand, multiscale IAMs can represent detailed socioeconomic and biophysical processes, including heterogeneous agent decision making and interactions with heterogeneous environmental processes, but relatively less progress has been made in accounting for spatial equilibrium to ensure consistency across scales in detailed multiscale IAMs. There is clearly a continuing need for further progress in both directions.
Given the inevitable modeling trade-offs, it is neither possible nor desirable to have a single model that accounts for all spatial and dynamic considerations. The most suitable approach depends on the research questions and objectives. Nonetheless, it is clear that research questions focusing on humanenvironmental interactions at local and meso scales must account for critical spatial economic interactions at and across these scales. This will require further developing multi-scale IAMs to devise appropriate methods for aggregating individual-level activities and impacts in ways that preserve the essential forces of spatial equilibrium, while also continuing to advance models that account for the high-resolution spatial heterogeneity that is needed to assess environmental impacts and feedbacks to individual-level decisions.

Data availability statement
No new data were created or analyzed in this study.