The risk and consequences of multiple breadbasket failures: an integrated copula and multilayer agent-based modeling approach

Okuyama (2007) highlights three aspects of climate shocks that models need to incorporate: time, space and nonlinear feedback effects. In the last few decades, several waves of modeling efforts have taken place that led to considerable methodological developments in the field of natural disaster analysis in that regard. This includes the early input–output (I–O) models which conducted extensive research on lifeline losses and disruption of transportation networks but focused exclusively on high-income countries like the USA and Japan (Dacy and Kunreuther 1969; Wilson 1982; Rose and Miernyk 1989; Rose et al. 1997). These models were later extended toward more developing countries and emerging economies (Okuyama 2004, 2011). Better availability of data also allowed I–O models to include regional impacts (Okuyama and Santos 2014; Cavallo et al. 2014). However, these models have been criticized for their strong assumptions of linear and fixed relationships across parameters and purely supply-side-driven adjustments.

Therefore, in addition to I–O models, computable general equilibrium (CGE) and dynamic stochastic general equilibrium (DSGE) models were developed and are now the most commonly used framework for analyzing post-shock outcomes (Cole 1995, 1998, 2004). Both CGE and DSGE models focus on long-run equilibrium using rational optimizing agents. Models in this field include national-level (Ueda et al. 2001; Rose and Guha 2004; Rose and Liao 2005) and regional- level studies (Tsuchiya et al. 2007; Hallegatte and Dumas 2009). Both CGE and DSGE models assume perfectly functioning markets, perfect foresight and smooth transitions to a new equilibrium after shocks. While they can now handle regional analysis both due to availability of better computational power and advances in numerical solutions (Ishiwata and Yokomatsu 2018), they still suffer from certain drawbacks. For example, strong assumptions of fixed and exogenously defined parameters remain unchanged during and after the shock and the exclusive focus on long-run equilibrium with little or no discussion on the transition processes. Additionally, the mathematical structure of these models makes it extremely challenging to include a large number of sectors and regions, or other nonlinear behavioral aspects, for example endogenous parameter changes (Rose 2004; Okuyama 2007).

Some recent models have taken non-mainstream approaches. For example, Hallegatte and Ghil (2008) and Hallegatte et al. (2007) introduce business cycles in a mixed demand and supply-driven framework. Albala-Bertrand 1993 shows that while impacts might be negligible at the macro level, they can completely disrupt regional economies. Albala-Bertrand 1993 also highlights the role of labor in estimating overall losses, an aspect that is missing in mainstream models which are mostly driven by capital stock accumulation through investments (Skoufias 2003; Hallegatte and Przyluski 2010).

While the above methods are relevant from a long-term policy planning perspective where economic trends are assumed to stabilize and behave “normally,” short-run outcomes, which are relevant for immediate policy response under an uncertain environment need alternative modeling tools (Skoufias 2003; Naqvi and Rehm 2014b). The above literature points out to several gaps, especially two worth noting. First, little has been done to include probabilistic climate risks in macroeconomic models with only some recent literature touching upon this topic (Hochrainer-Stigler et al. 2011; Ishiwata and Yokomatsu 2018; Poledna et al. 2018). Second, the evolution of short-run direct and indirect losses is not clearly understood with competing and conflicting views in the empirical literature (Kahn 2005; Loayza et al. 2012; Horwich 2000; Strobl 2012; Skidmore and Toya 2002; Hallegatte et al. 2007; Crespo et al. 2008; Hallegatte and Dumas 2009).

We discuss in the following subsections on how copulas and agent-based models can be used to model the short-run outcomes following climate shocks to breadbasket regions.

While the above-mentioned models focus on economic relationships and the impact of climate shocks, copulas are especially useful to address issues of tail dependency usually not looked at in such models. For example, a copula can explicitly address the probability of a large loss in one region given another region faces a large loss (Sklar 1959; Joe 1997; Nelsen 2007). Initially developed for the financial sector (Aas 2004; Dißmann et al. 2013), copulas are now increasingly applied to the field of natural hazard and disaster risk modeling. For example, the phenomena of increasing tail dependency are noticeable in hydrological processes such as water discharge levels across basins. In the work of Jongman and colleagues (Jongman et al. 2014), it was shown that large-scale atmospheric processes can result in strongly correlated extreme discharges across river basins leading to extreme flooding across large regions over Europe. Also in the case of drought events, the risk of simultaneous large-scale events that cause drought-induced crop losses in several regions at once are found to exhibit tail dependency (Prudhomme and Genevier 2011; Ratnam et al. 2016; Vicente-Serrano and López-Moreno 2006; Gaupp et al. 2019).

Copulas are able to model nonlinear codependencies of risks as in the case of joint climate extremes. They can also be used to model the dependence structure between multiple variables such as climate indicators and crop yields. In agricultural sciences, copulas are typically used to depict the joint effects of temperature and precipitation on crop yields (Cong and Brady 2012). A common class of multivariate copulas used in this field is the so-called Vine copulas (Aas et al. 2009; Bedford and Cooke 2002; Czado et al. 2013) which consist of a cascade of bivariate conditional and unconditional pair-copulas. For our case study, we use the most advanced approach currently available, namely regular vines, or RVines, which can model complex dependencies even in large dimensions (Dißmann et al. 2013). As we will see further below, copulas are a flexible statistical tool to model the dependencies between risks. Due to Sklar’s Theorem (Sklar 1959), the underlying univariate marginal distributions, for example of individual risks, can be modeled separately from the dependence structure (Aas et al. 2009). In this way, droughts in different regions, and with varying degrees of duration, severity and distributions, can still be modeled jointly using copulas (Shiau 2006). For our case, we use logistically de-trended wheat yield data in the Indian breadbasket states to demonstrate the importance of incorporating tail dependencies in drought risk estimations. Subsequent cascading impacts are modeled through an agent-based multilayer network approach as discussed in the next section.

While inter-dependent risk can be modeled through copula approaches, economic modeling approaches, as discussed above, do not capture the intrinsic dynamics of shocks within systems, and therefore, agent-based modeling approaches are coming nowadays to the forefront (Elsner et al. 2014; Naqvi and Rehm 2014a). Agent-based models, or ABMs, are a bottom-up methodology that allows agents to make decisions based on past and current outcomes as well as on the behavior of others (Bowles 2006). ABMs have the ability to handle a large set of different agent sets allowing for heterogeneity within each agent set. Within such a framework, the interaction of agents determines meso- and macro-outcomes, which can further feedback into the micro-decision-making of individual agents resulting in nonlinear path-dependent outcomes (Arthur 2006; Elsner et al. 2014; Tesfatsion 2006). Thus, ABMs are well equipped to handle large parameter spaces, nonlinear thresholds, boundary conditions and out-of-equilibrium states thus going beyond state-of-the-art modeling tools typically used in disaster analysis (Farmer and Foley 2009; Axtell 2005).

However, it is not enough, as currently done, to look at the economic system as a single-layer entity with heterogeneous agents. We suggest applying multilayer network approaches instead (Kivelä et al. 2014; Battiston et al. 2016). Multilayer networks are represented by unique but interconnected layers. Assuming regions can be represented as nodes in a network structure, these nodes can interact with each other in different ways. For example, goods flow can be represented by trade networks and population flows by migration networks. The nodes interact with and across layers based on behavioral rules usually derived from the economic theory or empirical literature based on the region under study (Alfarano and Milaković 2009; Naqvi and Rehm 2014a; Naqvi 2017). This network structure allows us to track changes resulting from climate and economic shocks that might occur in one part of the network, but can cascade throughout the system through behavioral interactions. The specific network structure is typically determined by the number of locations or nodes, the number of layers, endowment of stocks and the strength of connections across the layers.

The multilayer approach was first applied to financial networks after the 2008 global crisis to understand how shocks cascade and can cause “systemic risk” (Bardoscia et al. 2015; Battiston et al. 2012; Poledna et al. 2015). Unlike financial networks, where all transactions are in monetary units across different types of financial institutions or assets (Hurd and Gleeson 2011; Stolbova et al. 2018), food production involves networks comprising of different monetary and non-monetary units. For example, land is used to produce a physical output of crops, which is later converted into monetary units through market transactions. Similarly, agriculture labor, which is typically a large proportion of the total population in developing countries, engages in farm work in exchange for usually low wages. A climate shock like flood or drought to such a system implies that food output is lost through less land being available. A more catastrophic shock like earthquakes can even cause high casualties reducing labor power in a region. Thus, losses in different layers cannot be treated in the same way as losses within financial networks. Additionally, unlike financial networks which are governed with relatively well-defined homogeneous rules and regulations, the agriculture–food web and its corresponding agrarian and farming population need to be set up with layer-specific behavioral rules. In order to do so, the dynamics within each dimension need to be modeled through the behavioral interaction of different agent sets belonging to different layers across a network of spatially explicit locations.

In the next section, we describe how the three methods of copulas, multilayer networks and agent-based models can be combined together to tackle the challenges addressed above, that is, how to model simultaneous risks and cascading impacts across regions.

Copula approaches gained their importance due to Sklar’s Theorem (Sklar 1959). It states that every multivariate distribution F with margins \(F_1, \dots , F_d\) can be written as:with C as a d-dimensional copula, a multivariate distribution on the unit hypercube \([0,1]^d\) with uniform marginal distributions. F is continuous with strictly increasing continuous marginals \(F_1, \dots , F_d\) such that:where c is the copula density, f, the multi-dimensional probability density function of F, and \(f_k\), the marginal probability density function of \(F_k\), \(k=1,\dots ,d\). d in our example refers to the number of breadbasket states in India. The advantage of this approach is that the copula, and thereby the dependence structure, can be chosen independently from the marginal distributions. There are different copula families capturing different dependence structures (Jaworski et al. 2010) but generally speaking, any multivariate distribution can be used as the basis for a copula (Aas 2007). To structure our multivariate copula model, we used regular vines (RVines) (Aas et al. 2009; Bedford and Cooke 2002; Kurowicka and Cooke 2006; Dißmann et al. 2013), a flexible graphical model that consists of a cascade (or tree structure) of conditional and unconditional bivariate copulas that decompose the multivariate probability density. A d-dimensional RVine model consists of \(d-1\) trees. Each tree consists of nodes and edges joining the nodes. The first tree identifies \(d-1\) pairs of variables with a directly modeled distribution. The second tree identifies \(d-2\) pairs with a distribution conditional on a single variable, modeled by a pair-copula. Continuing in that way, the last tree consists of a single pair-copula with a distribution conditional on all remaining variables. For a graphical representation of a RVine tree, see (Dißmann et al. 2013) or (Brechmann and Schepsmeier 2013). The RVine tree structure in this study is selected using the maximum spanning tree approach with Kendall’s \(\tau \) as edge weights (Dißmann et al. 2013):with \({\hat{\tau }}_{m,n}\) as pairwise empirical Kendall’s \(\tau \) (that is, a rank correlation coefficient, see Embrechts et al. 1997). As will be discussed in the results section, this approach was adopted for India on the state level, which showed large tail dependencies for some regions and therefore is an excellent example for showing the advantages of a copula approach when it comes to determine systemic risks.

Going to the ABM modeling component, we first want to note that every ABM needs to be contextualized, that is, it must be set up for a specific purpose of analysis (Gilbert 2008; Klabunde and Willekens 2016; Naqvi 2017), which in our case are drought-related production shocks in India. A shock in the food system through a strong reduction in crop yields will cascade through the affected population as well as through the unaffected one. For example, in the case of India, a state’s economy can be assumed to produce output across different sectors that can be divided into agricultural or other goods (physical goods and services). Each state holds a stock of workers which are employed across the different sectors that interact with each other to form a simple circular flow economy. Cross-state interactions result in the exchange of goods (including food) and services. Underlying this production and trade interactions are labor markets, which determine how much labor is employed across different sectors. Like trade which is driven by price and profit signals, income differentials result in migration across locations. This, in turn, determines population and income distributions. Migration decisions are also influenced by social signals, for example social- or community-based networks. The household and the production layer combined represents a multilayer network structure as shown in Fig. 1.

Each dot in Fig. 1 represents a location divided across two layers. A production layer employs workers to produce output. This output is in demand not only in one location but in other locations as well. The second layer represents the demand side, where households contribute via two channels: by providing labor for production in exchange for income and demand for consumption purchased through income and savings. Labor can also move around based on market signals or when faced with high-distress scenarios as in the case of shocks.

To give an understanding of the underlying principle of dependent crop losses, we start with an example of two dependent Indian states, namely Uttar Pradesh and Rajasthan, both part of the Indian wheat breadbasket. Figure 2 shows the univariate marginal distributions of logistically de-trended (Gaupp et al. 2017) wheat yields in Uttar Pradesh and Rajasthan. Both de-trended yields follow a normal distribution tested with the Shapiro–Wilk test.

In the middle, their scatter plot is shown as well as the contour plot of the Frank copula which was selected as the bestfitting copula type. It joins the marginals of de-trended wheat yields in Uttar Pradesh and Rajasthan and has the following form:with \(\theta \) as the copula parameter determining the strength of tail dependence. To stress the importance of dependence, we take a look at the 1974 food crisis in India. A combination of adverse weather conditions and a rise in oil prices with consequent shortages of petroleum-based fertilizer led to unexpectedly low yields in India (Weinraub 1974a, b; Joerin and Joerin 2013). Based on our data, the likelihood of yields as low as the 1974 yields occurring simultaneously in Uttar Pradesh and Rajasthan equals \(C_\theta [F_\mathrm{{UP}}(-0.29), F_\mathrm{{R}}(-0.16)] = 0.027\). If we would not take into account the dependence structure between the two states, the likelihood of both states experiencing a 1974-type event would be \(F_\mathrm{{UP}}(-0.29), F_\mathrm{{R}}(-0.16)= 0.007\). To estimate the joint probability of all eight breadbasket states, we use an RVine tree structure as described in the methods section. Figure 3 shows the first tree of the de-trended wheat yields in the Indian breadbasket. Edges are labeled with the copula families connecting the nodes via bivariate copulas.

The full tree structures which include all conditional copulas are shown in Fig. 7 in “Appendix B”. In 2003, following the 2002 monsoon failure, India experienced a major wheat yield shock (with shock defined as deviation from the long-term logistic trend). Based on our analysis in eight breadbasket states (see Table 2), the wheat yield shock in 2003 in the Indian breadbasket was a 1-in-17 years event. The importance of not only considering the yield distributions in the different states in an agricultural risk analysis but also the dependence structure between them becomes clear when we compare the return period of the 2003 yield shock with the return period if we would ignore the dependencies. If we assume independence between the states, the likelihood of a 2003 event would decrease to a 1-in-370 years event, a 20-fold increase in the magnitude. Therefore, by ignoring linkages between the production in different states, the risks of crop failure are underestimated dramatically. In addition, further spillover effects of those spatially inter-dependent yield shocks have to be modeled to fully grasp the systemic risks of our current food system.

The above framework is applied to India to estimate the impact of a breadbasket failure on internal displacement. Figure 4b shows the states of India of which eight (shown as green dots) are major food crop producing states.This represents 40–45% of all employment in India, most of which is at extremely low end of the income distribution. The 2003 crop failure resulted in a rise in food prices. Since India is not a net importer of food items, the loss in crops resulted in an internal market adjustment to the climate shock.

We conduct four experiments to show the usefulness of our approach, using a combination of no-displacement and displacement scenarios and independent and copula-based probability distributions. The displacement scenario allows populations to move from one state to another if they see better real-income opportunities. Due to a lack of state-to-state migration or displacement data, which can allow us to calculate migration propensities, all states are given equal weights in terms of destination choices, where the only negative factor is average distance where farther away locations are considered less attractive than nearby ones. In a more real-world scenario, propensity to move would also be determined by community and language ties, work opportunities and other social and cultural factors (Black et al. 2013; Klabunde and Willekens 2016; Cattaneo and Peri 2016). Thus, our displacement scenarios represent a highly stylized baseline outcome, which, in reality, is likely to be much worse. The 36 states are set up in the simulation model using data from Table 1 for calibration. The table provides information on low-income populations, total output produced in each sector, and the share of agriculture in this output. Low-income workers are divided into these sectors accordingly. The simulations run till states achieve stable trends in prices and population distributions (Fig. 4b).

The breadbasket states, shown in green in Fig. 4, are subjected to a production shock to simulate the drought event. As described above, the shocks can either be modeled as independent or follow an inter-dependent copula structure. The probabilities of changes in yields using both independent probability distributions and a copula-based probability distribution for a 100 year production shock event are estimated and provided in Table 1. Figure 5 shows the temporal trends for the four scenarios for four different indicators: real income, average consumption, marginal propensity to consume and the VRank index. Figure 5a shows that the real income, or the price of labor relative to the price of food, falls across all scenarios. The independent, no-displacement scenario shows the smallest change, while the copula with displacement scenario shows the highest impact. This can be explained by the fact that wheat yields are strongly positively correlated between the states which means that a shock in one state leads to a shock in surrounding states with a high probability. A copula-based shock shows on average a larger decline in relative price changes. Since income falls, and food becomes more expensive, due to an overall decline in output, the average consumption levels, shown in Fig. 5b, also fall.

As consumption levels decline, households allocate more of their income toward food in order to stay above the minimum consumption threshold. Figure 5a, b indicates that including and excluding dependencies between states have a much higher impact than with or without displacement. Figure 5c shows the development of this indicator, which rises steeply if one assumes tail dependence with displacement. This scenario also takes the longest to stabilize. Figure 5d shows the VRank index, which on average increases for all scenarios with the copula scenarios showing the highest increase. This last graph highlights that underestimating both the joint probability risk and the multilayer displacement effects can result in underestimating the full extent of the shocks. Figure 6 shows the results as spatial choropleth maps at the end of the six- month simulation period, when the indicators stabilize. The graphs show changes relative to the baseline no-shock scenario. Displacement scenarios, shown on the right column in Fig. 6, exhibit a relatively higher vulnerability as the shock spreads from the affected states to the non-affected states through migration. In the non-displacement scenario, the copula- based distribution of the shock shows a higher increase in vulnerability.

The results provide important insights in terms of dynamics as well as policy-wise. For example, while an aggregated macroeconomic performance measure such as the GDP exhibits similar patterns across two displacement scenarios they can completely differ in the distributional effects. Furthermore, such different outcomes also imply that completely different policy responses are needed to tackle such risk. As recently suggested by Pflug and Pichler (2018), the difference in distributional impacts across the dependent and independent case can be used for determining the share of systemic risk (due to the dependency between states) to overall risk. If we conducted a complete probabilistic assessment of all possible agriculture-related climate risk scenarios, which in principle is possible as both the copula and marginal distributions are available, it would be possible to apply classic risk management strategies as well as to analyze humanitarian aid responses. However, this would be extremely difficult to operationalize in short time span due to the high computational requirements (for example, thousand of risk distributions need to be generated and analyzed via the ABM model). Alternatively, so-called risk-layer approaches which focus on risk reduction for more frequent risks and risk financing for infrequent ones (Mechler et al. 2014) can provide guidance on best strategies forward. As we were looking at independence and 100-year event cases, from a risk perspective, these events would still belong to the “relatively frequent” set. Therefore, risk reduction measures such as crop storage or subsidies during time of crisis could be, in principle, applied and studied with our approach. However, there are limitations due to data scarcity, especially as displacement patterns on such large scales are difficult to be empirically tested and validated. Nevertheless, current efforts are underway to get a more complete picture (Kc et al. 2018).

While these results are derived from a highly stylized model, where adjustments in markets are perfect, a fully calibrated model would include more frictions where the results would show much worse outcomes and distributions. Thus, our model results can be taken as an upper bound for disaster impacts in a best-case scenario.

Model calibration and application to countries are always a challenge for agent-based models. A model on a subregional level, for example provinces or states, has relatively abundant data, allowing for calibration since such regions typically represent standard administrative units (like provinces or districts) at which data are collected. Thus, a subregional level allows the model to be flexible enough to be adopted to different countries. The model can be scaled further down, for example to the village or even household level, but data limitations and the model scale would make the results both highly intractable and potentially subject to extensive sensitivity testing (Naqvi 2017).