#  Simultaneous supply and demand constraints in input-output networks: The
case of Covid-19 in Germany, Italy, and Spain

Anton Pichler1,2,3 and J. Doyne Farmer1,2,4
1 Institute for New Economic Thinking at the Oxford Martin School, University
of Oxford, UK
2 Mathematical Institute, University of Oxford, UK
3 Complexity Science Hub Vienna, Austria
4 Santa Fe Institute, US

###### Abstract

Natural and anthropogenic disasters frequently affect both the supply and
demand side of an economy. A striking recent example is the Covid-19 pandemic
which has created severe disruptions to economic output in most countries.
These direct shocks to supply and demand will propagate downstream and
upstream through production networks. Given the exogenous shocks, we derive a
lower bound on total shock propagation. We find that even in this best case
scenario network effects substantially amplify the initial shocks. To obtain
more realistic model predictions, we study the propagation of shocks bottom-up
by imposing different rationing rules on industries if they are not able to
satisfy incoming demand. Our results show that economic impacts depend
strongly on the emergence of input bottlenecks, making the rationing
assumption a key variable in predicting adverse economic impacts. We further
establish that the magnitude of initial shocks and network density heavily
influence model predictions.
Keywords: Covid-19; production networks; input-output models; rationing;
linear programming; economic shocks; shock propagation; economic impact
JEL codes: C61; C67; D57; E23

_Acknowledgments:_ We thank F. Lafond for helpful feedback. This work was
supported by Baillie Gifford, Partners for a New Economy, the UK’s Economic
and Social Research Council (ESRC) via the Rebuilding Macroeconomics Network
(Grant Ref: ES/R00787X/1), the Oxford Martin Programme on the Post-Carbon
Transition, and the Institute for New Economic Thinking at the Oxford Martin
School. This research is based upon work supported in part by the Office of
the Director of National Intelligence (ODNI), Intelligence Advanced Research
Projects Activity (IARPA), via contract no. 2019-1902010003. The views and
conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies, either
expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for governmental
purposes notwithstanding any copyright annotation therein.
_Corresponding author:_ Anton Pichler<EMAIL_ADDRESS>
##  1 Introduction

An immanent feature of many natural and anthropogenic disasters is that they
affect both the supply and the demand side of the economy. In this paper we
study the Covid-19 pandemic as an exemplary case of simultaneous supply and
demand shocks. Supply shocks from the pandemic arise from different sources.
While deaths and sickness of employees can limit productive capacity, these
effects are minor compared to nation-wide lockdown measures imposed by
governments to curb the spreading of the virus. During lockdown workers
employed in non-essential industries who cannot work remotely are unable to
perform their jobs (del Rio-Chanona et al., 2020; Dingel and Neiman, 2020;
Koren and Pető, 2020).

The pandemic also affects demand in heterogeneous ways (Congressional Budget
Office, 2006; del Rio-Chanona et al., 2020; Chetty et al., 2020; Carvalho et
al., 2020; Chen et al., 2020). While we expect demand shocks to be
comparatively small for some industries (e.g. manufacturing), other industries
are strongly affected by demand shocks. An illustrative example is the
transportation industry which is considered as essential in many countries and
thus would not experience adverse supply side effects. But the transportation
industry faces large demand shocks, since consumers reduce demand for air
travel and public transport to avoid infectious exposure.

Since economic agents are embedded in production networks, we expect that the
overall economic impact is larger than the initial shocks to supply and demand
suggest. Demand shocks will reduce the sales of firms and, by backward
propagation, also diminish the sales of their suppliers. Supply shocks, on the
other hand, will spread downstream and upstream. Downstream effects
materialize if the limited productive capacity of suppliers creates input
bottlenecks for customers. Due to lower productive capacities, firms will also
require less inputs for production, thus adversely affecting upstream
suppliers of these inputs.

Input-output (IO) models are frequently used to model higher-order economic
impacts arising from such exogenous shocks. Most of these models account for
either supply or demand shocks but do not incorporate them concurrently
(Galbusera and Giannopoulos, 2018). In this paper we revisit existing IO
modeling techniques and introduce a novel dynamic approach, to account for
both types of shocks. Our analysis is based on industry-level data but could
easily be extended to firm-level analysis that has become the focus of recent
studies (Diem et al., 2021; Borsos and Mero, 2020; Mandel and Veetil, 2020b;
Inoue and Todo, 2019). In particular, our results relate directly to the
effects of data aggregation on impact estimates.

Several macroeconomic models that account for the production network structure
have been suggested to study the economic impacts of the Covid-19 pandemic.
Inoue and Todo (2020) use an agent-based model calibrated to more than a
million Japanese firms to model nation-wide economic effects of a lockdown in
Tokyo. Using the World Input-Output Database (WIOD), Mandel and Veetil (2020a)
study the effects of national lockdowns on global GDP in a non-equilibrium
framework. Fadinger and Schymik (2020); Baqaee and Farhi (2020); Bonadio et
al. (2020) and Barrot et al. (2020) use general equilibrium models to quantify
economic impacts of social distancing. All these studies focus on the supply-
side shocks of the pandemic and without further considering changes in final
consumption. Exceptions are Pichler et al. (2020) and Guan et al. (2020) who
incorporate both supply and demand shocks in their production network models.

In contrast to these studies, we consider simpler modeling techniques derived
from traditional IO analysis which we extend to account for simultaneous
supply and demand shocks. We follow this approach to better isolate underlying
mechanisms of shock propagation in production networks that can be hard to
disentangle in more sophisticated macroeconomic models. When applying pandemic
shocks to data from Germany, Italy and Spain, we find that existing IO
modeling techniques yield infeasible solutions of economic impacts. To
understand the nature of this problem we introduce a simple linear programming
method that allows us the determine feasible market allocations, representing
a lower bound of minimal shock propagation. To find more realistic impact
estimates, we further introduce a new bottom-up approach based on rationing
outputs. This setup allows us to dynamically model the propagation of shocks
in production networks and uncover several theoretical implications, such as
the effect of alternative behavioral assumptions on economic impacts.

We intentionally keep our modeling approach simple. As mentioned above, even
more complicated models currently applied to assess the economic effects of
the pandemic do not incorporate supply and demand shocks simultaneously. The
focus of this paper is on the theoretical implications of simultaneous supply
and demand constraints for IO-based impact assessment methods. For more
realistic empirical impact assessment of natural disasters, we stress that
further aspects that we do not consider here could play an important role,
e.g. inventory dynamics (Bak et al., 1993; Pichler et al., 2021), price and
substitution effects, adaptive behavior and governmental intervention (see
Oosterhaven (2017) for a recent discussion).

Our paper contributes to the field of IO models in disaster impact assessment,
for which a recent review can be found in Galbusera and Giannopoulos (2018).
For recent demand-side and supply-side focused contributions see Klimek et al.
(2019) and Avelino and Dall’erba (2019), respectively. Similar to our
derivation of a lower bound of shock propagation, Koks and Thissen (2016) and
Oosterhaven and Bouwmeester (2016) rely on optimization techniques to account
for supply constraints in demand-driven models. Constrained optimization
techniques are frequently used in extended IO models (for example Duchin
(2005) and Duchin and Levine (2012)). The modeling of alternative rationing
algorithms is related to the studies of Steenge and Bočkarjova (2007); Li et
al. (2013) and Koks et al. (2015) which allow for imbalanced IO tables
immediately after the disaster strikes and evaluate different recovery paths.

Our findings show that shock amplification is much larger in the bottom-up
approaches than what best case scenarios would suggest. Micro-level
coordination failures can severely exacerbate adverse economic shocks.
Rationing assumptions play a key role in impact estimates and alternative
behavioral rules can lead to very different aggregate outcomes. These effects
strongly interact with the overall shock magnitude as well as the level of
connectedness in the production network. While we find that economic impacts
increase with higher levels of network density in general, the extent to which
this is true strongly depends on the underlying rationing mechanisms. Our
results also imply that estimates of economic impact can be highly sensitive
with respect to data quality and aggregation.

This paper is organized as follows. We first discuss first-order pandemic
shocks to supply and demand, as well as the datasets used (Section 2). In
Section 3 we introduce the basic IO framework and discuss how existing
approaches have difficulties in incorporating simultaneous supply and demand
constraints. Section 4 discusses the main results of this paper. We propose an
optimization method in Section 4.1 and introduce alternative rationing
algorithms in Section 4.2 to model shock propagation in supply and demand
constrained production networks. We show empirical results and discuss further
theoretical implications in Section 4.3 before concluding in Section 5.

##  2 Pandemic shocks to supply and demand

Our analysis is based on the most recent year (2014) of the World-Input-
Output-Database (WIOD) (Timmer et al., 2015).111 All code and data to
reproduce this paper are made accessible online:
https://www.doi.org/10.5281/zenodo.4326815.  We use estimates of supply and
demand shocks from the literature to determine maximum final consumption and
production values for 54 industries in three major European economies, i.e.
Germany, Italy and Spain.222 We removed industries _T_ (Household activities)
and _U_ (Extraterritorial activities) from our analysis since they are not
connected to other industries in the data and thus don’t play any role in the
propagation of shocks in the production network.  We focus on these economies
because existing research provides us with lists of essential industries and
thus allows us to make reasonable estimates of supply shocks. For this reason,
we only allow for domestic supply shocks in our analysis, despite
acknowledging the importance of international supply chain disruptions.

We follow the approach of del Rio-Chanona et al. (2020) to compute supply
shocks for every industry during a lockdown. A supply shock is caused by the
removal of labor in non-essential industries due to social distancing
measures. In contrast, an industry which is defined as essential will not be
affected. Even if employed in non-essential industries, workers who can
accomplish their work from home will do so and are assumed to keep pre-
lockdown productivity levels. The Remote Labor Index $\text{RLI}_{i}$
indicates the share of an industry’s labor force that can work from home. The
supply shock to industry $i$ is then computed as
$\epsilon_{i}^{S}=(1-\text{RLI}_{i})(1-\text{e}_{i})$, where $\text{e}_{i}$ is
the share of an industry defined as essential. Thus, the supply shock of an
industry during lockdown is the share of labor in non-essential industries
that cannot work from home. We assume that the supply shock to an industry
determines the maximum production of an industry, i.e.

$x_{i}^{\text{max}}=(1-\epsilon_{i}^{S})x_{i,0},$ (1)

where $x_{i,0}$ denotes the production in industry $i$ before lockdown.
Following this approach, every industry faces binding supply side constraints,
limiting its production to $x_{i}\in[0,x_{i}^{\text{max}}]$. The Remote Labor
Index is taken from del Rio-Chanona et al. (2020) and industry-specific
essential ratings for Germany, Italy and Spain are obtained from Fana et al.
(2020).

To determine first-order demand shocks to every industry, $\epsilon_{i}^{D}$,
we use estimates of a prospective Congressional Budget Office (CBO) study
aiming at quantifying the demand-side impact of a pandemic (Congressional
Budget Office, 2006). Demand and supply shock data have been mapped to WIOD
industry categories by Pichler et al. (2021) and are presented in detail in
Appendix A. As for supply shocks, we assume that demand shocks determine
maximum final consumption values according to

$f_{i}^{\text{max}}=(1-\epsilon_{i}^{D})f_{i,0},$ (2)

where $f_{i,0}$ represents pre-lockdown final consumption. Since consumption
cannot be negative, we have $f_{i}\in[0,f_{i}^{\text{max}}]$.333 In principle,
$f_{i}$ could be negative if there is extremely large inventory depletion.
This is not the case in our data. In national accounts inventory adjustment
merely represents a variable to rebalance row and column sums of IO tables. We
therefore do not consider the possibility of negative final demand.  Note that
WIOD distinguishes different final consumption categories.444 Given the multi-
regional nature of WIOD, imports from and exports to other industries are
accounted for in the intermediate consumption matrix. Since we treat
international trade as exogenous, we aggregate all intermediate exports to a
single final consumption category, as is usually the case with national IO
tables. Similarly, we treat intermediate imports as an exogenous input that
does not inhibit production, or equivalently, that is always available if
demanded.  We apply the CBO estimates to final consumption of households and
non-profit organizations and assume a 10% shock to investments and exports as
in Pichler et al. (2021). Due to our methodological focus we do not further
distinguish between different final consumption categories but for simplicity
only consider total final consumption values for every industry. It is
important to point out that this is a major simplification and we expect the
various types of final consumption to be highly relevant in empirical
assessments. Table 1 shows country aggregates of essentialness scores, Remote
Labor Index, supply and demand shocks.

| $x$ | $f$ | $\epsilon^{S}$ | $\epsilon^{D}$ | RLI | $e$
---|---|---|---|---|---|---
Germany | 7,066.74 | 4,447.11 | 0.31 | 0.09 | 0.42 | 0.49
Italy | 4,075.40 | 2,343.12 | 0.27 | 0.11 | 0.41 | 0.55
Spain | 2,567.91 | 1,552.88 | 0.33 | 0.13 | 0.40 | 0.44
Table 1:  Country aggregates of gross output, final consumption and supply and
demand shock inputs. Columns $x=\sum_{i}x_{i,0}$ and $f=\sum_{i}f_{i,0}$ are
country gross output and final consumption in billion USD based on 2014
values. $\epsilon^{S}=1-\sum_{i}x_{i}^{\text{max}}/x$ and
$\epsilon^{D}=1-\sum_{i}f_{i}^{\text{max}}/f$ are total supply and demand
shocks, respectively. $\text{RLI}=\sum_{i}\text{RLI}_{i}x_{i,0}/x$ and
$e=\sum_{i}e_{i}x_{i,0}/x$ denote the country-wide Remote Labor Index and
essentialness score on which supply shocks are based. In all three countries
total direct supply shocks are larger than demand shock.

##  3 IO framework

We first introduce basic national accounting identities that build the basis
for most IO models. Let us consider an economy consisting of $n$ industries,
each producing a unique, industry-specific good. Inter-industry purchases and
sales are encoded in the intermediate consumption matrix $\mathbf{Z}$ where an
element $z_{ij}$ denotes the total monetary value of goods produced by
industry $i$ that are consumed by industry $j$. For the $n$-dimensional
vectors of gross output, total final consumption and value added, we write
$\bm{x}$, $\bm{f}$ and $\bm{v}$, respectively. In this economy the following
identities hold

$\displaystyle\bm{x}=\mathbf{Z}\mathbf{i}+\bm{f}=\mathbf{Z}^{\top}\mathbf{i}+\bm{v},$
(3)

where $\mathbf{i}$ is a $n$-dimensional column vector of ones.

A core assumption in a Leontief-inspired modeling framework is that industries
produce based on fixed production recipes, allowing us to rewrite the first
identity of Eq. (3) as

$\bm{x}=\mathbf{A}\bm{x}+\bm{f}=\mathbf{L}\bm{f},$ (4)

where $\mathbf{A}$ is the technical coefficient matrix with elements
$a_{ij}\equiv z_{ij}/x_{j}$ (the production recipe) and $\mathbf{L}$ the
Leontief inverse $\mathbf{L}\equiv(\mathbf{I}-\mathbf{A})^{-1}$. A
conventional assumption is that gross output is determined endogenously,
whereas final consumption is taken to be as exogenous. It then follows that
value added is obtained as a residual variable.

Eq. (4) is at the core of many IO models that are frequently used for disaster
impact analysis. This specification defines a demand-driven model. However, as
we have already stressed, many economic shocks actually act on the supply side
of the economy. For example, natural catastrophes such as earthquakes or
floods destroy physical capital, putting upper limits on an industry’s
production in the disaster aftermath. Eq. (4), however, neglects supply
capacity constraints and implies an infinite elasticity of supply with respect
to demand. This is particularly problematic given the frequent short-term
focus of IO studies.

We should point out that there are also supply-driven IO models building upon
Ghosh’s model (Ghosh, 1958) that assumes exogenous primary factors and derives
final consumption values endogenously. Ghosh’s model does not comply with a
Leontief production function and has been heavily criticized amongst other
aspects for the assumption of perfect demand elasticities and perfect
substitution of inputs (Oosterhaven, 1988; Gruver, 1989; De Mesnard, 2009).
The IO inoperability model (Haimes and Jiang, 2001; Santos and Haimes, 2004)
is another notable supply-side focused model. Dietzenbacher and Miller (2015)
show, however, that it closely corresponds to conventional IO modeling
approaches.

Neither the demand- nor the supply-driven specification of IO models are able
to incorporate supply and demand constraints at the same time. A potential
remedy is the mixed endogenous/exogenous model (MEEM), which has been applied
in several studies including Steinback (2004); Kerschner and Hubacek (2009)
and Arto et al. (2015). The MEEM acknowledges that not only final demand is
constrained but that for some industries supply constraints are more severe
and therefore binding. A difficulty in empirical analysis is to define which
industries are supply and which are demand constrained. In some cases there
might be “natural” distinctions. For example, Arto et al. (2015) study global
supply chain disruption effects of the 2011 Tōhoku earthquake by solving the
MEEM where only the Japanese transport equipment industry is supply
constrained, whereas for all other industries gross output is endogenous. In
case of simultaneous (severe) supply and demand shocks the line of distinction
will be more blurred.

Note that the MEEM incorporates both supply and demand shocks, but not
_simultaneously_ for a single industry. Instead, an industry is either supply
or demand constrained. As a consequence, solutions to the MEEM might not
comply with exogenous industry-level constraints to supply and demand, i.e.
they can be economically _infeasible_. As is shown in detail in Appendix B,
this is exactly what happens if we apply the MEEM to the pandemic shocks
discussed in the previous section. For all three countries, the MEEM yields
final consumption values that are either negative or larger than the exogenous
constraints.

##  4 Propagation of simultaneous supply and demand shocks

The industry-specific effects to supply and demand during the Covid-19
pandemic and the difficulty of incorporating them in the existing models
motivated us to explore possible extensions of the IO modeling framework.
Rather than using more complicated models that can deal with simultaneous
supply and demand shocks, such as computable general equilibrium (CGE) models
or those discussed in the introduction, we intentionally keep the modeling
approach simple. We remain close to the Leontief framework, the “work horse”
of many more advanced IO-based models. This allows us to better isolate the
basic mechanisms of shock propagation that are easily conflated with other
effects in more sophisticated models.

To determine lower bounds of shock propagation with respect to output and
consumption, we study the idealized case of a social planner who allocates
goods to maximize either total gross output or total final consumption within
the given economic constraints. This will give us a best case scenario of
negative economic impacts, i.e. the minimal decrease in total output and final
consumption necessary to arrive within the set of feasible solutions given the
exogenous constraints to supply and demand. Of course, alternative objectives
could be optimized as well (e.g. employment).

While deriving a lower bound is valuable, it is unlikely to be realistic. We
therefore consider a second approach by comparing alternative rationing
algorithms. If an industry is supply constrained, it will not be able to
satisfy all of its demand and thus needs to make a decision which customer to
serve and to what extent. We implement several decision rules and investigate
how this choice influences the estimated economic impact. In contrast to the
optimization method or the MEEM, which simply compute an equilibrium, this
approach explicitly computes the transient dynamics that lead to a new
equilibrium.

###  4.1 Feasible market allocations

Given exogenous constraints to supply and demand, what is the feasible market
allocation that maximizes final consumption and/or total output? The solution
needs to lie within exogenous bounds on supply and demand and also needs to
satisfy the assumption of Leontief production, Eq. (4). We seek market
allocations $\\{\bm{x}^{*},\bm{f}^{*}\\}$ that (a) respect given production
recipes $\bm{x}^{*}=\mathbf{L}\bm{f}^{*}$ and (b) satisfy basic output and
demand constraints $\bm{x}^{*}\in[\bm{0},\bm{x}^{\text{max}}]$ and
$\bm{f}^{*}\in[\bm{0},\bm{f}^{\text{max}}]$.

We follow a mathematical optimization procedure to map out the solution space
of feasible market allocations. As a first case, we determine the market
allocation that maximizes gross output under the assumptions specified. Large
levels of gross output indicate high economic activity, which in turn entail
high levels of primary factors such as labor compensation. As a second case we
look at market allocations that maximize final consumption given current
production capacities. Due to the linearity of the Leontief framework, the
problems boil down to linear programming exercises.

##### Maximizing gross output:

$\displaystyle\underset{\bm{f}\in[\bm{0},\bm{f}^{\text{max}}]}{\max}\qquad\mathbf{i}^{\top}(\mathbf{I}-\mathbf{A})^{-1}\bm{f},$
(5) $\displaystyle\text{subject
to}\qquad(\mathbf{I}-\mathbf{A})^{-1}\bm{f}\in[\bm{0},\bm{x}^{\text{max}}].$

##### Maximizing final consumption:

$\displaystyle\underset{\bm{x}\in[\bm{0},\bm{x}^{\text{max}}]}{\max}\qquad\mathbf{i}^{\top}(\mathbf{I}-\mathbf{A})\bm{x},$
(6) $\displaystyle\text{subject
to}\qquad(\mathbf{I}-\mathbf{A})\bm{x}\in[\bm{0},\bm{f}^{\text{max}}].$

To maximize gross output of the economy, $\sum_{i}x_{i}^{*}$, requires us to
the find the vector of final consumption $\bm{f}^{*}$. The constraint
$\mathbf{L}\bm{f}\in[\bm{0},\bm{x}^{\text{max}}]$ ensures that industry output
levels lie within the respective production capacities. The problem is similar
when maximizing final consumption where a vector of output levels $x^{*}$ is
chosen to maximize final consumption, $\sum_{i}f_{i}^{*}$. The auxiliary
constraint enforces that final consumption levels do not exceed given demand.
The optimization problem always admits a solution since the trivial allocation
of a full collapse $\\{\bm{x}^{*},\bm{f}^{*}\\}=\\{\bm{0},\bm{0}\\}$ always
exists, although we expect positive values for realistic input data.

Note that the market allocations are “optimal” in a mathematical sense. That
is, they represent maximum values of output and consumption given the
exogenous constraints to supply and demand. Or stated differently, they
determine the minimum level of shock propagation measured in aggregate final
consumption and gross output. Thus, this optimization method is not intended
to give realistic or necessarily desirable economic outcomes, but rather
probes the system boundaries by mapping out the solution space. Any feasible
solution, under the above-specified assumptions, has to lie within this space.

###  4.2 Input bottlenecks and rationing variations

As our second method we implement different rationing schemes for output
constrained industries. In contrast to the optimization methods, this
represents a bottom-up approach for finding feasible market allocations.
Industries place orders to their suppliers based on incoming demand. Since
suppliers can be output constrained, they might not be able to satisfy demand
fully. A supplier therefore needs to make a decision about how much of each
customer’s demand it serves. Intermediate consumers transform inputs to
outputs based on fixed production recipes. Thus, if a customer receives less
inputs than she asked for, she faces an input bottleneck further constraining
her production. As a consequence, the customer reduces her demand for other
inputs as they are not further needed under limited productive capacities. We
iterate this procedure forward until the algorithm converges.

We run this algorithm with four alternative rationing rules: (a) strict
proportional rationing, (b) proportional rationing to intermediate demand but
priority of intermediate over final demand, (c) priority rationing serving
largest customers first and (d) random rationing, where customers are served
based on a random order. We then compare the results obtained from the four
competing behavioral rules.

These rationing approaches are frequently applied in the literature, although
there are differences in the exact specifications. In general it is hard to
calibrate how firms distribute output in case of supply constraints to
empirical data and so the rationing choice is often ad-hoc. Here, we apply all
four rules within a consistent dynamic framework to better understand how
alternative behavioral assumptions affect impact estimates. Strict
proportional rationing is frequently assumed in the literature (Henriet et
al., 2012; Hallegatte, 2014; Guan et al., 2020; Mandel and Veetil, 2020a;
Pichler et al., 2020) and also mixed proportional/priority rationing has been
considered (Battiston et al., 2007; Hallegatte, 2008; Li et al., 2013; Diem et
al., 2021). The agent-based framework of Inoue and Todo (2019) and Inoue and
Todo (2020) assumes a variation of priority rationing while the random
matching of suppliers and customers in the agent-based model proposed by
Poledna et al. (2018) and Poledna et al. (2019) most closely resembles a
random rationing approach.

##### (a) Strict proportional rationing.

If industries are unable to satisfy total incoming demand completely, they
distribute output proportional to their customers’ demand, where no
distinction is made between intermediate and final customers. More
specifically, if an industry’s output, $x_{i}$, is smaller than incoming
demand, $d_{i}$, it will supply $z_{ij}=o_{ij}\frac{x_{i}}{d_{i}}$ to customer
$j$, where $o_{ij}$ denotes the demand from customer $j$ to industry $i$. We
implement the rationing algorithm in the following way: First, industries
determine their total demand as if there were no supply-side constraints, i.e.
$\bm{d}=\mathbf{L}\bm{f}^{\text{max}}$. Industries then evaluate if they are
able to satisfy demand given their constrained production capacities. If an
industry $i$ can satisfy demand only partially, it will create a bottleneck of
size $r_{i}=\frac{x^{\text{max}}_{i}}{d_{i}}$ to other industries due to
proportional rationing. Since industries produce according to fixed Leontief
input recipes, their largest input bottleneck,
$s_{i}=\underset{j:a_{ji}>0}{\min}\\{r_{j},1\\}$ will be the binding
constraint in production. Thus, in case of input bottlenecks where $s_{i}<1$,
production of $i$ reduces to
$x_{i}=\underset{j:a_{ji}>0}{\min}\left\\{x_{i}^{\text{max}},\frac{s_{i}a_{ji}d_{i}}{a_{ji}}\right\\}=\min\\{x_{i}^{\text{max}},s_{i}d_{i}\\}<d_{i}$.
This in turn reduces the amount of goods delivered to the final consumer
$f_{i}=\min\\{x_{i}-\sum_{j}a_{ij}x_{j},0\\}$. The new final demand vector
$\bm{f}$ now implies a new, lower level of aggregate demand,
$\bm{d}=\mathbf{L}\bm{f}$, and we again let industries evaluate whether they
can satisfy total demand within given production constraints. We iterate this
procedure forward until all demand is met and no input bottleneck further
constrains production. We can write the proportional rationing algorithm more
compactly as follows:

###### Algorithm 1

Proportional rationing; industries are not prioritized over the final
consumer. Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as
given, implying an initial aggregated demand vector
$\bm{d}[1]=\mathbf{L}\bm{f}[0]$. By looping over the index $t=\\{1,2,...\\}$,
the following system is iterated forward:

$\displaystyle r_{i}[t]$ $\displaystyle=\frac{x_{i}^{\text{max}}}{d_{i}[t]},$
(7) $\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{j}[t],1\\},$ (8)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (9)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (10)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (11)

The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.

Eq. (7) indicates whether an industry is output constrained, where $r_{i}$ is
the share of demand that can be met given existing productive capacities. If
$r_{i}\geq 1$, industry $i$ is able to meet demand completely, whereas demand
can only be partially satisfied if $r_{i}<1$. If any supplier of industry $i$
(which is the set $\\{j:a_{ji}>0\\}$) is sufficiently output constrained,
industry $i$ faces an input bottleneck according to Eq. (8). Due to perfect
complementarity of inputs prescribed by the Leontief production function,
industry $i$ can only produce a fraction of total demand as indicated in Eq.
(9), reducing its delivery to final consumers as specified in Eq. (10). The
new total demand to an industry $i$ is then again derived through the weighted
sum of final demand values where the weights are obtained from the Leontief
inverse, Eq. (11).

##### (b) Mixed proportional/priority rationing.

It has been argued that firm-firm relationships are stronger than firm-
household ties, so that intermediate demand should be prioritized over final
demand (Hallegatte, 2008; Inoue and Todo, 2019). While this assumption might
make sense for households, this is not necessarily the case for other final
demand categories. Final demand categories in national IO tables can include
demand by governments and non-profit organizations, exports and investments
and it is debatable whether intermediate demand should be prioritized over
these categories.

To quantify the effect of this assumption on the amplification of initial
shocks, we implement a mixed proportional/priority rationing algorithm. Here,
industries ration intermediate demand proportional analogously to the
proportional rationing algorithm (a), but prioritize intermediate demand over
final demand. We outline this algorithm below.

###### Algorithm 2

Proportional rationing among industries; industries are prioritized over final
consumers. Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as
given, implying an initial aggregated demand vector
$\bm{d}[1]=\mathbf{L}\bm{f}[0]$. By looping over the index $t=\\{1,2,...\\}$,
the following system is iterated forward:

$\displaystyle r_{i}[t]$
$\displaystyle=\frac{x_{i}^{\text{max}}}{\sum_{j}a_{ij}d_{j}[t]},$ (12)
$\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{j}[t],1\\},$ (13)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (14)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (15)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (16)

The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.

The algorithm is similar to the proportional rationing algorithm (a) but
differs mainly in one aspect. Only intermediate demand affects the extent of
an industry’s output constraints, Eq. (12). As a consequence, final demand
does not play a role in the creation of input bottlenecks, Eq. (13).

#####  (c) Priority rationing (“largest first”).

Since it is not obvious that industries should pass on their output
proportionally in case they are not able to meet demand fully, we next
consider a type of priority rationing. Industries rank their customers based
on demand magnitude and serve larger customers before smaller customers. In
the proportional rationing setting an output constrained supplier affects all
customers in the same way. Under a priority rationing scheme, however, supply
shocks propagate downstream heterogeneously. For example, if a supplier cannot
meet demand by only a small margin, most customers will not be affected by the
priority rationing scheme. Only the smallest customers will face input
bottlenecks, whereas every customer would experience the same small shock in
the proportional rationing setup.

Intuitively, a priority rationing rule could make sense, as firms might have
an interest in serving more important (large) customers fully, or at least as
well as possible, before focusing on less important customers. It also seems
closer to practice that firms process orders one-by-one instead of working
through all orders simultaneously and leaving them incomplete to the same
degree. While a priority rationing scheme could be plausible on the basis of
firms or single transactions, it might be less so for more aggregate industry-
level data. A link between two industries in IO tables corresponds to many
firm/establishment level transactions and so it is not clear if large inter-
industry links are due to a few big orders or many small orders. This becomes
particularly evident when considering final consumers. Several industries face
large demand from private consumers which effectively is the sum of many small
orders (e.g. restaurants, grocery, theaters). Since we only consider an
aggregate of final demand representing several distinct categories such as
private consumers, government and investments, we exclude final demand from
the priority rationing scheme. Thus, in the same manner as in the mixed
prop./prior. rationing algorithm (b), we assume that intermediate demand is
always prioritized over final demand. By adopting this convention, we can
formulate the priority rationing algorithm as follows.

###### Algorithm 3

Largest first rationing; industries are prioritized over the final consumer.
Take an initial demand vector $\bm{f}[0]=\bm{f}^{\text{max}}$ as given,
implying an initial aggregate demand vector $\bm{d}[1]=\mathbf{L}\bm{f}[0]$.
Every industry $i$ ranks each customers $j$ based on initial demand size:
$h_{ij}=\\{k_{(1)},k_{(2)},...,k_{(j)}\,:\,a_{ik_{(1)}}d_{k_{(1)}}[1]\geq
a_{ik_{(2)}}d_{k_{(2)}}[1]\geq...\geq a_{ik_{(j)}}d_{k_{(j)}}[1]\\}$. By
looping over the index $t=\\{1,2,...\\}$, the following system is iterated
forward:

$\displaystyle r_{ij}[t]$ $\displaystyle=\frac{x_{i}^{\text{max}}}{\sum_{n\in
h_{ij}}a_{in_{(j)}}d_{n_{(j)}}[t]},$ (17) $\displaystyle s_{i}[t]$
$\displaystyle=\underset{j:a_{ji}>0}{\min}\\{r_{ji}[t],1\\},$ (18)
$\displaystyle x_{i}[t]$
$\displaystyle=\min\\{x_{i}^{\text{max}},s_{i}[t]d_{i}[t]\\},$ (19)
$\displaystyle f_{i}[t]$
$\displaystyle=\max\left\\{x_{i}[t]-\sum_{j}a_{ij}x_{j}[t],0\right\\},$ (20)
$\displaystyle d_{i}[t+1]$ $\displaystyle=\sum_{j}l_{ij}f_{j}[t].$ (21)

The algorithm converges to a new feasible economic allocation if
$d_{i}[t+1]=d_{i}[t]$ for all $i$. In this case output and final consumption
levels are given as $x_{i}=d_{i}[t+1]=x_{i}[t+1]$ and $f_{i}=f_{i}[t+1]$,
respectively.

##### (d) Random rationing.

As our final case, we again consider priority rationing. Rather than using a
fixed ordering scheme based on demand magnitude, we use random priority.
Industries rank their customers randomly and serve customers based on their
position in the ranking. While largest-first rationing makes intuitive sense
in some cases, it is unlikely to be a good approximation for all real-world
settings. In practice, it is likely that other factors such as timing of
orders matter. In that case industries could adopt a first-come-first-serve
principle to process orders. Since IO data rarely comes with granular time
information, we adopt a random ordering of incoming orders that mimics a
first-come-first-serve principle under a uniform prior of which orders are
coming in first. The algorithm is presented below.

###### Algorithm 4

Random rationing; industries are prioritized over the final consumer. The
algorithm is identical to Algorithm 3, except that the ranking of customer $j$
by industries $i$, $h_{ij}=\\{k_{(1)},k_{(2)},...,k_{(j)}\\}$, is randomly
drawn.

While these algorithms are not guaranteed to converge to a steady-state
equilibrium, we observe convergence in the vast majority of our simulation
experiments. If the algorithm converges, the economic allocations obtained are
automatically feasible. This means that no industry has negative output or
produces more than its productive capacities allow, final consumption is non-
negative and below given exogenous maximum consumption levels, and there are
no input bottlenecks left that further constrain production of downstream
industries. The Leontief equation $\bm{x}=\mathbf{L}\bm{f}$ holds, which
implies that all sales add up to total output.

###  4.3 Results

We initialize these four rationing algorithms with the supply and demand
shocks and IO data of Germany, Italy and Spain. We then compare the steady-
state equilibrium economic outcomes predicted by the rationing algorithms with
the optimization results and the direct shock computations (Appendix A, Tables
2 and 3).

Fig. 1 summarizes the main result visually, where a diamond indicates the
aggregate final consumption and gross output levels predicted by the different
approaches. Note that a data point in the top-right corner indicates high
gross output and final consumption values, i.e. limited negative impacts,
whereas diamonds in the bottom-left corner represent extremely adversely
impacted economies. As already reported in Table 1, for all three countries
supply shocks are substantially larger than final consumption shocks, pushing
the _Direct shock_ diamond substantially below the 45 degree line. Note that
the direct shock market allocations are not feasible, as they ignore higher-
order effects, such that a direct supply shock to an industry reduces the
inputs of downstream industries and thus reduces their output too.

The best case feasible market allocations are given by the two optimization
methods (maximize output vs. consumption), which yield exactly the same
predictions on the aggregate and the industry level. This is true even though
the two optimization methods are not equivalent.555 When perturbing the
economic systems we can find cases where the two optimization methods do not
yield exactly the same results. Practically, we find that aggregate
predictions are always similar for the two methods although industry-level
results can sometimes differ significantly.  The two optimization methods have
the highest output, but they still amplify the initial supply shocks to reduce
the output from about $69\%$ to $63\%$ in Germany and from $67\%$ to $53\%$ in
Spain.

All the other feasible market allocations, obtained from the rationing
algorithms, lie substantially below the best case scenarios. The wide range of
predicted outcomes is the most striking aspect of Fig. 1. Both the random
rationing scheme and the priority rationing scheme essentially collapse the
entire economy. Proportional rationing substantially collapses the economy
(with an output below $20\%$ of normal for all countries) and the mixed scheme
reduces output for Germany and Spain by more than 70% and about $60\%$ for
Italy. Interestingly, all feasible market allocations lie close to the
identity line, suggesting similar impacts to output and consumption.

Figure 1:  Comparison of different shock propagation mechanisms. The y-axis
denotes aggregate gross output levels normalized by pre-shock levels as
predicted by the rationing and optimization methods. The x-axis shows the same
for aggregate final consumption levels. The _Random rationing_ diamond
represents the average taken from 100 samples and the error bars indicate the
interquartile range.

On a sectoral level, we find the overall orderings of economic impacts are
highly correlated across methods and countries, despite a few pronounced
differences. An interesting example is Forestry (A02) which represents the
only industry that experiences less adverse impacts under proportional
rationing compared to mixed rationing. Due to being non-essential and having a
low Remote Labor Index (Appendix A, Table 3), forestry is the industry with
the largest supply shocks. Thus, it virtually always faces stronger supply
constraints than demand constraints. In case of no prioritization of
industries, Forestry receives less demand and thus creates smaller input
bottlenecks for downstream industries. This, in turn, leads to smaller input
bottlenecks for Forestry as its downstream industries can be found upstream as
well.

There is also substantial variation in how close industries are to their
theoretical maximum value as determined by the optimization method. When
considering the proportional rationing algorithm in Italy, for example, the
output levels of the best faring industries (such as Education (P85) or
Telecommunications (J61)) reach around 30% of their theoretical maximum. On
the other hand, the hardest hit industry, Accommodation-Food (I), is only at a
5% level of what is possible. Thus, Accommodation-Food is not only hit
extremely hard by direct shocks, but also experiences large higher-order
effects through the rationing dynamics.

Fig. 1 makes it clear that the behavioral assumptions imposed on suppliers
matter enormously for economic impact predictions. In fact, results vary more
across alternative assumptions than across countries.

In the absence of further modeling refinements (such as inventories, adaptive
behavior, substitution effects), shock amplification is always pronounced if
basic national accounting identities are required to hold. But the actual
extent of shock amplification depends strongly on how input bottlenecks are
created and passed on downstream in the supply chain.

####  4.3.1 Shock magnitude effects

We next investigate the sensitivity of economic impact predictions with
respect to shock magnitude and network connectedness. While we have seen in
Section 4.3 that different assumptions on rationing behavior can lead to
entirely different estimates of impact, it is not clear whether these results
are specific to the three datasets considered. We therefore conduct a series
of simulation experiments to gain a better understanding of the generality of
the results.

First, we investigate how the estimates of the various methods depend on the
magnitude of shocks. To do this, we rescale supply and demand shocks and we
apply the optimization methods and the rationing algorithms to the new shock
data. We then redo this analysis for various shock scales. To better
differentiate between the qualitative effects of demand and supply shock
propagation, we allow for different scaling factors for demand and supply
constraints, i.e.

$\displaystyle x_{i}^{\text{max}}$
$\displaystyle=(1-\alpha^{S}\epsilon_{i}^{S})x_{i,0},$ (22) $\displaystyle
f_{i}^{\text{max}}$ $\displaystyle=(1-\alpha^{D}\epsilon_{i}^{D})f_{i,0},$
(23)

where $\alpha^{S},\alpha^{D}\in[0,1]$.

Fig. 2(a) shows aggregate output levels for all cases when only demand shocks
are scaled between zero and one and when there are no further supply
constraints being present ($\alpha^{S}=0,\alpha^{D}\in[0,1]$).666 Since
results for aggregate final consumption and aggregate gross output are very
similar, we only present figures of the latter in this section. We show
results for scaling supply and demand shocks concurrently
($\alpha^{S}=\alpha^{D}\in[0,1]$) in Appendix C.  It becomes evident that
predictions made by the rationing algorithms and the optimization methods are
identical and scale linearly with the demand shock magnitude. Thus, the
rationing algorithms do not differ with respect to upstream shock propagation
and always arrive at the optimal solution in absence of further supply side
constrictions.

Figure 2:  Economic impact as a function of shock magnitude. (a) Aggregate
gross output levels as a function of scaling only demand shocks between zero
and one ($\alpha^{S}=0,\alpha^{D}\in[0,1]$). All methods yield the same
economic impact estimates which scale linearly with $\alpha^{S}$. The black
horizontal line indicates that there is no adverse economic impact from supply
side constraints. (b) Aggregate gross output levels as a function of scaling
only supply shocks between zero and one ($\alpha^{D}=0,\alpha^{S}\in[0,1]$).
The _Random rationing_ line represents the average taken from 100 samples and
the shades indicate the interquartile range. We do not distinguish further
between the two maximization methods as results are almost identical.

The picture becomes very different when rescaling only supply shocks and
turning off demand shocks ($\alpha^{D}=0,\alpha^{S}\in[0,1]$) as demonstrated
in Fig. 2(b). Here, the direct impact reduces gross output linearly as the
shock magnitude is increased and puts an upper bound to the solutions of the
methods considered here. When there are no shocks, $\alpha^{S}=0$, all methods
recover the empirical IO data as expected. For small shocks we observe that
estimated impacts are very similar across different methods, but the
proportional rationing algorithm consistently returns the smallest output
values. The results of the other algorithms (mixed, priority, random) lie
between those proportional rationing and optimization predictions, and are
identical for small shocks up to about $\alpha^{S}=0.5$. As $\alpha^{S}$ is
increased further they deviate dramatically. Under the priority algorithm even
a small increase in $\alpha^{S}$ causes the economy to collapse. The random
algorithm also causes the economy to collapse, though more slowly, and the
mixed algorithm is substantially better.

Fig. 2 thus makes clear that the ranking of the impact assessment methods as
seen in Fig. 1 is not generic, but strongly depends on the shock magnitude. If
there are only small supply shocks present, a priority rationing rule is
better than proportional rationing. In this case most of the demand of
downstream industries can be met and small shocks only propagate to few
customers. In a proportional rationing setup, on the other hand, shocks are
passed on to every customer. Despite small shock magnitudes, the wider breadth
of shock propagation leads to comparatively larger aggregate impacts.

In contrast, priority rationing exacerbates shock propagation compared to
proportional rationing in case of large supply shocks. If industries are
severely output constrained and ration based on a priority rule, the demand of
several customers’ might not be satisfied at all. The Leontief production
function, in turn, implies a complete shutdown of these downstream industries
due to the input bottlenecks created. More input bottlenecks will be created
in further rounds of shock propagation, potentially causing massive collapses.
If supply shocks are that large, shock amplification is milder if passed on
proportionally to customers. While here every customer experiences some shocks
from a constrained supplier, this effect is outweighed by the fact that
proportional rationing avoids the creation of even larger idiosyncratic input
bottlenecks.

####  4.3.2 Network density

Intuitively, the extent of shock propagation not only depends on the size of
initial shocks but also on how industries are connected with each other. If
the production network is dense, i.e. most of the potential links are present,
idiosyncratic shocks will spread out very quickly to many other industries in
the network. In contrast, if the network is sparse, shock propagation might be
more local, at least initially, and takes more steps to spread out in the
network.

We therefore conduct an experiment where we again apply the different shock
propagation mechanisms to the data, but control for the IO network density. We
do this in the following way. First, we randomly eliminate a given number of
links in the intermediate consumption matrix $\mathbf{Z}$.777 We also
experimented with eliminating smallest links first instead of randomly
selection. Results are similar to the ones presented here and shown in the
Appendix D.  We only consider deleting links instead of adding links since the
aggregate IO networks we are using are highly dense
($\sum_{ij}\mathds{1}_{\\{z_{ij}>0\\}}/n^{2}>99\%$). Note that setting a link
$z_{ij}>0$ equal to zero without changing final consumption values reduces
total output of supplier $i$, since the accounting identity
$x_{i}=\sum_{j}z_{ij}+f_{i}$ has to hold. The output of customer $j$ will not
be affected if we assume that the reduction in intermediate consumption is
absorbed by a respective increase of $j$’s value added (which does not affect
the simulations). After deleting nodes we thus rebalance the economic system
by adjusting gross output of the relevant industries such that the basic
national accounting identity is satisfied. Next, we recompute the technical
coefficient matrix $\mathbf{A}$, the Leontief matrix $\mathbf{L}$ and the
vector of productive constraints $\bm{x}^{\text{max}}$ to initialize the
optimization and rationing simulations. We repeat this procedure 50 times for
a given level of density and explore the whole range of possible density
values.

We visualize aggregate output levels predicted by the various impact
assessment methods as a function of network density in Fig. 3. The plot again
confirms that the observed ranking of methods in Fig. 1 is not generic but is
strongly affected by the network density. It becomes clear that shock
amplification is comparatively small if the network is very sparse. On the
other hand, if the network is very dense, as the empirical data would suggest,
economic impacts are substantial for all cases. In particular, the gap between
the optimal solution and the bottom-up rationing approaches widens with higher
network density. Interestingly, the minimal amplification of direct shocks
also increases with higher density values, although the size of this effect is
rather small.

Figure 3:  Economic impact as a function of network density. The network
density is changed by randomly eliminating links in the IO network. Solid
lines are predicted output values obtained from averaging over 50 samples and
shades indicate the interquartile range. Since random rationing is stochastic
by itself, we apply this algorithm 50 times to every network sample and take
averages and quantiles over the full (pooled) density-specific sample. The
actual data corresponds to the very right of the x-axis, as indicated by the
diamonds.

If industries have only few suppliers and customers, proportional rationing
yields by far the most pessimistic predictions of aggregate impact. If there
are many connections among industries, on the contrary, proportional rationing
mitigates shock propagation dynamics compared to priority and random
rationing. Shock amplification is always mildest if industries use a mixed
proportional/priority rationing rule, although even in this case economic
impacts increase substantially with higher density.

These results indicate that estimates of economic impact are highly sensitive
with respect to the network structure which, in turn, often depends on the
level of data aggregation. More aggregated data necessarily implies higher
levels of density. For example, industry-level links are the result of pooling
many firm links. Firm-level production networks, on the other hand, are an
aggregation of many individual contractual relationships and payments. Imagine
applying the rationing mechanisms to a firm-level production network in two
ways: first to the actual firm-level data and second to an industry aggregate
of the same data. Our results suggest that predicted economic impacts could be
substantially larger in the second case, despite using the same underlying
data.

Our findings also point out that economic impact predictions can be sensitive
with respect to data quality. Many disaggregate firm-level production network
datasets are substantially biased, as they frequently include only specific
supplier-customer relationships which are subject to specific reporting rules.
Thus, we would expect real world production networks to be substantially
denser than what the data suggests. Our analysis indicates that such biases
could have important consequences for impact assessment.

##  5 Discussion

We have shown that existing IO models have difficulties in dealing with
simultaneous supply and demand shocks. However, both types of shocks play an
important role in situations such as natural disasters and during a pandemic.
We have introduced a simple optimization method that allows us to find best
case market allocations that are consistent with the exogenous shocks. To
obtain more realistic, bottom-up impact estimates, we studied alternative
rationing dynamics which differ in how suppliers serve customers in case of
output constraints. Using IO data for Germany, Italy and Spain, we found that
these bottom-up approaches lead to substantial amplifications of initial
shocks, which are much higher than optimal solutions would suggest. We further
established that different rationing assumptions can lead to dramatically
different economic impact predictions. Moreover, these predictions are highly
sensitive with respect to the magnitude of first-order shocks and the
production network structure.

It is clear that adequate macroeconomic predictions of pandemic impacts
require more sophisticated modeling techniques than the ones studied here. Yet
the downside of more complicated models is that the underlying mechanisms of
predictions can be difficult to isolate. We therefore studied relatively
simple economic models to better carve out key mechanisms of shock propagation
in twofold constrained production networks. The choice for simplicity in this
study also entails limitations which are important to bear in mind. We did not
depart from the assumption of industry-specific Leontief production functions
throughout our analysis, although the choice of production function has been
shown to be a key variable for impact assessment (Pichler et al., 2020). While
input substitutions might be limited in the short-run, fixed production
recipes are nevertheless a strong assumption, in particular for the aggregate
data considered here.

It is also important to stress that we did not take any adaptive behavior of
economic agents into account and did not allow for the possibility of
inventory buildup and depletion. By imposing feasible market solutions, we
essentially forced the economy to converge into a new equilibrium. It is not
clear, however, that a perturbed complex system such as national economies
would quickly approach a new equilibrium state instead of following transient
paths for an extended period. In general, we refrained from making the time
dimension explicit, although we acknowledge its relevance for shock
propagation dynamics.

Despite the caveats mentioned, our analysis makes clear that level of
aggregation and data quality play an important role in aggregate predictions
of shock amplification. Detailed and high-quality production network data are
rare, necessitating researchers to study biased or aggregate data. Inevitably,
this will affect the connectedness of economic agents and thus influence shock
propagation mechanisms. For example, studies based on large-scale firm level
databases report network density values well below 0.01% (Kumar et al., 2021;
Borsos and Stancsics, 2020; Peydró et al., 2020; Demir et al., 2018; Huneeus,
2018; Spray, 2017). We should also mention that estimates of direct shocks to
supply and demand are subject to large uncertainties. This could have
important consequences for model predictions, due to the sensitivity of shock
propagation mechanisms with respect to first-order shock magnitudes.

We find that the number of links between industries strongly influences how
different behavior rules amplify direct shocks. The level of connectedness is
a very simple aggregate network measure, and we would expect that further
aspects, such as community structure, degree/strength heterogeneity or (dis-)
assortative mixing, play an important role too. Understanding how alternative
rationing assumptions interact with structural properties of complex networks
would require more detailed production network data, e.g. at the firm level,
and could be an interesting avenue for future research.

We conclude by stressing that our analysis is not constrained to pandemic
shocks. While the Covid-19 pandemic is a main motivation of our study,
simultaneous supply and demand constraints are ubiquitous features of any
economy, in particular in the short run. Supply shocks are a prominent
characteristic of many natural hazards (floods, earthquakes, hurricanes) and
tools of mostly demand-driven IO analysis are frequently applied for impact
assessment. Our results indicate that adequately modeling shock propagation in
production networks will require a better integration of both types of
economic constraints.

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## Appendix

##  Appendix A Details on first-order shocks to supply and demand

|  |  $f_{i}$ (%) |  $\epsilon_{i}^{D}$ (%)
---|---|---|---
ISIC | Industry | DEU | ESP | ITA | DEU | ESP | ITA
A01 | Agriculture | 0.5 | 1.4 | 1.0 | 10.0 | 9.9 | 10.0
A02 | Forestry | 0.1 | 0.0 | 0.1 | 10.0 | 8.6 | 3.8
A03 | Fishing | 0.0 | 0.1 | 0.0 | 10.0 | 10.0 | 10.0
B | Mining | 0.3 | 0.5 | 0.4 | 10.0 | 10.0 | 10.0
C10-12 | Manuf. Food-Beverages | 4.1 | 5.0 | 4.0 | 10.0 | 10.0 | 10.0
C13-15 | Manuf. Textiles | 0.6 | 1.4 | 2.9 | 10.0 | 10.0 | 10.0
C16 | Manuf. Wood | 0.4 | 0.1 | 0.3 | 10.0 | 10.0 | 9.9
C17 | Manuf. Paper | 0.6 | 0.4 | 0.5 | 10.0 | 10.0 | 10.0
C18 | Media print | 0.1 | 0.1 | 0.1 | 10.0 | 9.9 | 10.0
C19 | Manuf. Coke-Petroleum | 1.7 | 2.6 | 1.4 | 10.0 | 10.0 | 10.0
C20 | Manuf. Chemical | 3.4 | 2.1 | 1.5 | 9.9 | 9.9 | 10.0
C21 | Manuf. Pharmaceutical | 1.1 | 1.0 | 1.2 | 8.1 | 9.3 | 9.8
C22 | Manuf. Rubber-Plastics | 1.4 | 0.7 | 1.0 | 10.0 | 9.9 | 10.0
C23 | Manuf. Minerals | 0.6 | 0.5 | 0.7 | 10.0 | 10.0 | 10.0
C24 | Manuf. Metals-basic | 1.5 | 1.3 | 1.4 | 10.0 | 10.0 | 10.0
C25 | Manuf. Metals-fabricated | 1.8 | 1.0 | 1.6 | 10.0 | 10.0 | 10.0
C26 | Manuf. Electronic | 2.0 | 0.4 | 0.9 | 9.9 | 9.9 | 10.0
C27 | Manuf. Electric | 2.3 | 0.8 | 1.4 | 10.0 | 10.0 | 10.0
C28 | Manuf. Machinery | 5.8 | 1.4 | 4.7 | 10.0 | 10.0 | 10.0
C29 | Manuf. Vehicles | 8.2 | 3.7 | 2.1 | 10.0 | 10.0 | 10.0
C30 | Manuf. Transport-other | 1.0 | 1.1 | 1.0 | 9.9 | 9.9 | 10.0
C31-32 | Manuf. Furniture | 1.4 | 0.6 | 1.4 | 9.9 | 9.9 | 10.0
C33 | Repair-Installation | 0.5 | 0.3 | 0.6 | 10.0 | 10.0 | 10.0
D35 | Electricity-Gas | 1.6 | 1.6 | 1.1 | 2.3 | 0.8 | 1.2
E36 | Water | 0.2 | 0.4 | 0.3 | 1.5 | 0.9 | 0.6
E37-39 | Sewage | 0.7 | 0.7 | 0.6 | 3.6 | 2.5 | 1.7
F | Construction | 5.5 | 7.6 | 7.5 | 10.0 | 9.9 | 9.9
G45 | Vehicle trade | 0.9 | 1.7 | 1.4 | 10.0 | 10.0 | 10.0
G46 | Wholesale | 3.4 | 4.1 | 4.6 | 9.8 | 9.5 | 9.9
G47 | Retail | 3.4 | 5.5 | 5.7 | 9.5 | 9.5 | 9.8
H49 | Land transport | 0.8 | 1.7 | 1.8 | 56.9 | 38.8 | 55.2
H50 | Water transport | 0.7 | 0.2 | 0.5 | 11.4 | 27.5 | 47.3
H51 | Air transport | 0.6 | 0.6 | 0.4 | 50.3 | 24.7 | 47.3
H52 | Warehousing | 0.3 | 0.8 | 1.1 | 22.3 | 19.7 | 33.5
H53 | Postal | 0.1 | 0.0 | 0.1 | 3.4 | 2.2 | 3.4
I | Accommodation-Food | 2.4 | 8.6 | 4.7 | 73.1 | 75.5 | 79.1
J58 | Publishing | 0.4 | 0.3 | 0.3 | 3.8 | 5.4 | 4.0
J59-60 | Video-Sound-Broadcasting | 0.6 | 0.5 | 0.3 | 4.3 | 3.8 | 3.5
J61 | Telecommunications | 0.9 | 1.3 | 1.2 | 1.1 | 1.6 | 3.0
J62-63 | IT | 1.5 | 1.6 | 1.1 | 8.8 | 9.5 | 8.5
K64 | Finance | 1.9 | 0.8 | 0.9 | 2.9 | 3.8 | 1.6
K65 | Insurance | 1.4 | 1.1 | 1.0 | 1.3 | 0.8 | 1.1
K66 | Auxil. Finance-Insurance | 0.0 | 0.2 | 0.1 | 2.0 | 1.9 | 4.9
L68 | Real estate | 7.2 | 7.8 | 9.8 | 0.2 | 0.1 | 0.5
M69-70 | Legal | 0.7 | 0.6 | 0.4 | 9.3 | 7.9 | 5.2
M71 | Architecture-Engineering | 1.0 | 1.0 | 0.2 | 9.4 | 9.3 | 8.4
M72 | R&D | 0.9 | 0.5 | 0.6 | 8.3 | 7.9 | 9.8
M73 | Advertising | 0.1 | 0.1 | 0.1 | 10.0 | 9.1 | 9.2
M74-75 | Other Science | 0.3 | 0.1 | 0.3 | 3.5 | 3.5 | 5.0
N | Private Administration | 1.2 | 1.2 | 1.1 | 4.2 | 4.2 | 4.9
O84 | Public Administration | 6.1 | 6.3 | 7.1 | 0.2 | 0.7 | 0.0
P85 | Education | 4.1 | 5.1 | 3.9 | 0.7 | 1.0 | 0.0
Q | Health | 8.3 | 7.2 | 7.5 | 0.1 | 0.1 | 0.1
R_S | Other Service | 3.2 | 3.3 | 3.2 | 4.3 | 4.3 | 4.7
T | Household activities | 0.2 | 0.8 | 1.1 | 0.0 | -0.0 | -0.0
Table 2:  Industry-specific demand shock details. $f_{i}$ denotes final consumption per industry as fraction of aggregate final consumption. $\epsilon_{i}^{D}$ is the total demand shock per industry.  |  |  $x_{i}$ (%) |  $\epsilon_{i}^{S}$ (%) |  $e_{i}$ (%) | $\text{RLI}_{i}$
---|---|---|---|---|---
ISIC | Industry | DEU | ESP | ITA | DEU | ESP | ITA | DEU | ESP | ITA | (%)
A01 | Agriculture | 0.9 | 2.2 | 1.7 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 13.6
A02 | Forestry | 0.1 | 0.0 | 0.0 | 85.0 | 85.0 | 85.0 | 0.0 | 0.0 | 0.0 | 15.0
A03 | Fishing | 0.0 | 0.1 | 0.1 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.7
B | Mining | 0.2 | 0.3 | 0.3 | 48.3 | 48.3 | 34.5 | 30.0 | 30.0 | 50.0 | 31.0
C10-12 | Manuf. Food-Beverages | 3.5 | 6.9 | 4.1 | 0.0 | 26.0 | 26.0 | 100.0 | 66.7 | 66.7 | 22.1
C13-15 | Manuf. Textiles | 0.4 | 1.0 | 2.6 | 68.5 | 68.5 | 59.4 | 0.0 | 0.0 | 13.3 | 31.5
C16 | Manuf. Wood | 0.5 | 0.3 | 0.5 | 73.1 | 73.1 | 60.6 | 0.0 | 0.0 | 17.0 | 26.9
C17 | Manuf. Paper | 0.7 | 0.6 | 0.7 | 34.3 | 0.0 | 48.7 | 50.0 | 100.0 | 29.0 | 31.5
C18 | Media print | 0.4 | 0.4 | 0.4 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 39.0
C19 | Manuf. Coke-Petroleum | 1.5 | 2.4 | 1.7 | 0.0 | 6.4 | 0.0 | 100.0 | 90.0 | 100.0 | 36.0
C20 | Manuf. Chemical | 2.6 | 2.5 | 1.6 | 19.0 | 52.5 | 8.2 | 70.0 | 17.0 | 87.0 | 36.7
C21 | Manuf. Pharmaceutical | 0.9 | 0.7 | 0.8 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 40.3
C22 | Manuf. Rubber-Plastics | 1.4 | 0.9 | 1.3 | 35.3 | 70.5 | 47.2 | 50.0 | 0.0 | 33.0 | 29.5
C23 | Manuf. Minerals | 0.8 | 0.8 | 1.0 | 63.9 | 63.9 | 61.4 | 0.0 | 0.0 | 4.0 | 36.1
C24 | Manuf. Metals-basic | 1.9 | 2.1 | 1.8 | 72.6 | 72.6 | 72.6 | 0.0 | 0.0 | 0.0 | 27.4
C25 | Manuf. Metals-fabricated | 2.4 | 1.4 | 2.6 | 66.3 | 66.3 | 66.3 | 0.0 | 0.0 | 0.0 | 33.7
C26 | Manuf. Electronic | 1.4 | 0.4 | 0.7 | 43.1 | 43.1 | 38.8 | 0.0 | 0.0 | 10.0 | 56.9
C27 | Manuf. Electric | 1.9 | 0.8 | 1.1 | 63.1 | 63.1 | 50.5 | 0.0 | 0.0 | 20.0 | 36.9
C28 | Manuf. Machinery | 4.5 | 1.2 | 3.6 | 61.8 | 61.8 | 47.0 | 0.0 | 0.0 | 24.0 | 38.2
C29 | Manuf. Vehicles | 6.3 | 2.5 | 1.5 | 69.7 | 69.7 | 69.7 | 0.0 | 0.0 | 0.0 | 30.3
C30 | Manuf. Transport-other | 0.8 | 0.8 | 0.7 | 59.7 | 59.7 | 59.7 | 0.0 | 0.0 | 0.0 | 40.3
C31-32 | Manuf. Furniture | 0.9 | 0.6 | 1.2 | 65.2 | 59.7 | 58.0 | 0.0 | 8.5 | 11.0 | 34.8
C33 | Repair-Installation | 0.7 | 0.5 | 0.6 | 60.6 | 60.6 | 34.0 | 0.0 | 0.0 | 44.0 | 39.4
D35 | Electricity-Gas | 2.4 | 4.7 | 2.7 | 0.0 | 5.8 | 0.0 | 100.0 | 90.0 | 100.0 | 41.6
E36 | Water | 0.2 | 0.5 | 0.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 33.5
E37-39 | Sewage | 0.9 | 0.8 | 1.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 29.8
F | Construction | 5.2 | 6.4 | 6.7 | 71.6 | 71.6 | 49.6 | 0.0 | 0.0 | 30.7 | 28.4
G45 | Vehicle trade | 1.1 | 1.4 | 1.1 | 18.0 | 27.3 | 13.7 | 67.0 | 50.0 | 75.0 | 45.4
G46 | Wholesale | 3.9 | 4.9 | 5.3 | 0.0 | 30.0 | 33.5 | 100.0 | 40.0 | 33.0 | 50.1
G47 | Retail | 3.0 | 3.9 | 3.9 | 24.7 | 25.7 | 25.2 | 51.0 | 49.0 | 50.0 | 49.7
H49 | Land transport | 1.8 | 2.5 | 3.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 31.4
H50 | Water transport | 0.5 | 0.2 | 0.4 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.3
H51 | Air transport | 0.5 | 0.5 | 0.4 | 0.0 | 24.2 | 0.0 | 100.0 | 66.0 | 100.0 | 28.8
H52 | Warehousing | 2.3 | 2.1 | 2.1 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 29.6
H53 | Postal | 0.6 | 0.2 | 0.2 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 35.6
I | Accommodation-Food | 1.6 | 5.8 | 3.3 | 64.6 | 64.6 | 56.6 | 0.0 | 0.0 | 12.5 | 35.4
J58 | Publishing | 0.6 | 0.4 | 0.3 | 0.0 | 7.6 | 0.0 | 100.0 | 75.0 | 100.0 | 69.8
J59-60 | Video-Sound-Broadcasting | 0.6 | 0.6 | 0.5 | 0.0 | 11.0 | 0.0 | 100.0 | 75.0 | 100.0 | 56.1
J61 | Telecommunications | 1.2 | 1.8 | 1.3 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 55.1
J62-63 | IT | 2.1 | 1.4 | 1.6 | 7.2 | 14.4 | 0.0 | 75.0 | 50.0 | 100.0 | 71.1
K64 | Finance | 2.7 | 2.1 | 2.9 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.4
K65 | Insurance | 1.4 | 1.0 | 0.8 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.3
K66 | Auxil. Finance-Insurance | 0.6 | 0.4 | 1.0 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 71.7
L68 | Real estate | 7.2 | 6.7 | 7.5 | 51.3 | 51.3 | 51.3 | 0.0 | 0.0 | 0.0 | 48.7
M69-70 | Legal | 2.5 | 1.5 | 2.3 | 36.3 | 18.1 | 0.0 | 0.0 | 50.0 | 100.0 | 63.7
M71 | Architecture-Engineering | 1.2 | 1.1 | 1.0 | 45.9 | 45.9 | 0.0 | 0.0 | 0.0 | 100.0 | 54.1
M72 | R&D | 0.6 | 0.3 | 0.4 | 41.1 | 41.1 | 0.0 | 0.0 | 0.0 | 100.0 | 58.9
M73 | Advertising | 0.4 | 0.5 | 0.5 | 39.7 | 39.7 | 39.7 | 0.0 | 0.0 | 0.0 | 60.3
M74-75 | Other Science | 0.4 | 0.3 | 0.7 | 19.6 | 19.6 | 0.0 | 50.0 | 50.0 | 100.0 | 60.8
N | Private Administration | 3.9 | 2.6 | 3.0 | 42.7 | 54.3 | 41.6 | 33.3 | 15.2 | 35.0 | 36.0
O84 | Public Administration | 4.6 | 4.4 | 4.2 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 44.6
P85 | Education | 2.9 | 3.4 | 2.4 | 0.0 | 46.0 | 0.0 | 100.0 | 0.0 | 100.0 | 54.0
Q | Health | 5.4 | 4.8 | 4.9 | 0.0 | 0.0 | 0.0 | 100.0 | 100.0 | 100.0 | 36.0
R_S | Other Service | 2.8 | 2.7 | 2.7 | 61.2 | 56.0 | 49.2 | 0.0 | 8.6 | 19.6 | 38.8
T | Household activities | 0.1 | 0.5 | 0.6 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 50.0 | 100.0
Table 3:  Industry-specific supply shock details. $x_{i}$ denotes gross output
per industry as share of aggregate output. $\epsilon_{i}^{S}$ is the total
supply shock per industry. $e_{i}$ is the extent to which an industry is
considered as essential following government policies. $\text{RLI}_{i}$ is the
Remote Labor Index.

##  Appendix B Details on mixed endogenous/exogenous modeling

A core motivation to study shock propagation based on alternative, bottom-up
approaches was our observation that existing methods such as the mixed
endogenous/exogenous model (MEEM) yields infeasible solution when applied to
pandemic shocks. In the MEEM the $n$ industries are divided into two groups.
The first group is the set of $n^{s}$ supply constrained industries and the
second group the $n^{d}$ demand-constrained industries (we have
$n^{s}+n^{d}=n$). We can use this setup to partition Eq. (4) into

$\begin{pmatrix}\bm{x}^{s}\\\
\bm{x}^{d}\end{pmatrix}=\begin{pmatrix}\mathbf{A}^{ss}&\mathbf{A}^{sd}\\\
\mathbf{A}^{ds}&\mathbf{A}^{dd}\end{pmatrix}\begin{pmatrix}\bm{x}^{s}\\\
\bm{x}^{d}\end{pmatrix}+\begin{pmatrix}\bm{f}^{s}\\\ \bm{f}^{d}\end{pmatrix},$
(24)

where superscripts $s$ and $d$ denote supply and demand constrained
industries, respectively. The matrix block $\mathbf{A}^{sd}$ indicates the
input recipes of demand constrained customers with respect to output
constrained suppliers and analogously for the other blocks. In this framework
vectors $\bm{f}^{s}$ and $\bm{x}^{d}$ are endogenous and $\bm{f}^{d}$ and
$\bm{x}^{s}$ exogenous. Rearranging Eq. (24) such that all endogenous
variables appear on the left-hand side and all exogenous variables on the
right-hand side, yields

$\begin{pmatrix}\bm{f}^{s}\\\
\bm{x}^{d}\end{pmatrix}=\begin{pmatrix}\mathbf{I}&\mathbf{A}^{sd}\\\
\bm{0}&\mathbf{I}-\mathbf{A}^{dd}\end{pmatrix}^{-1}\begin{pmatrix}\mathbf{I}-\mathbf{A}^{ss}&\bm{0}\\\
\mathbf{A}^{ds}&\mathbf{I}\end{pmatrix}\begin{pmatrix}\bm{x}^{s}\\\
\bm{f}^{d}\end{pmatrix}$ (25)

(for details see Miller and Blair (2009), 621-633).

To apply the MEEM in the pandemic context, we categorize industries into a
demand and a supply constrained group based on which shock is larger. If the
supply shock of an industry exceeds its demand shock in absolute terms, we
treat this industry as supply constrained and vice versa. The shock magnitudes
for supply and demand for each industry are given as

$\displaystyle x_{i}^{\text{SS}}$
$\displaystyle=x_{i,0}-x_{i}^{\text{max}}=-\epsilon_{i}^{S}x_{i,0},$ (26)
$\displaystyle f_{i}^{\text{DS}}$
$\displaystyle=c_{i,0}-f_{i}^{\text{max}}=-\epsilon_{i}^{D}f_{i,0},$ (27)

where $d_{i}^{\text{SS}}$ denotes the total _supply shock_ and
$f_{i}^{\text{DS}}$ the _demand shock_. If
$x_{i}^{\text{SS}}>f_{i}^{\text{DS}}$, we consider this industry as supply
constrained and we will use its gross output values on the right hand side of
Eq. (25). Otherwise we treat it as demand constrained.

Following this approach, we apply the MEEM to the IO data of Germany, Italy
and Spain by calibrating it to the estimated pandemic supply and demand
shocks. As shown in Fig. 4, the MEEM does not yield a feasible solution for
any of the three countries. Violations of feasibility conditions are most
frequent for Spain, which faces the largest shocks to supply and demand. The
model does not compute any negative final consumption values for Germany, but
still allocates final consumption values to industries which are larger than
$f_{i}^{\text{max}}$. Note that the results obtained by the MEEM are out of
the solution space delimited by the exogenous constraints on output and
consumption. Thus, these results are difficult to interpret in this context
and we thus refrain from comparing them more systematically with the results
shown in the main text.

Figure 4:  Final consumption values from mixed exogenous/endogenous IO
modeling with simultaneous supply and demand shocks. Black circles indicate
initial pre-shock values, red circles demand-constrained industries and blue
circles supply-constrained industries. Note that demand-constrained
consumption values are exogenously determined by the first-order shocks, while
in other cases the change in output depends on the higher order effects on the
whole economy. Red lines indicate the overall change in production for each
sector. Numbers to the right of both panels indicate the value change in
percentages and are colored red if the result is infeasible.

The MEEM can yield infeasible economic allocations and it depends on the
context whether negative final consumption values are meaningful or not
(Miller and Blair, 2009, p. 628). To see why the MEEM can give negative final
consumption values, let us consider the output-constrained part of Eq. (25),
which can be rewritten as

$\displaystyle\bm{f}^{s}=[\mathbf{I}-\mathbf{A}^{ss}]\bm{x}^{s}-\mathbf{A}^{sd}[\mathbf{I}-\mathbf{A}^{dd}]^{-1}(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d}).$
(28)

Note that the vector $(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d})$ is always non-
negative by definition, as is every element of the matrix
$\mathbf{A}^{sd}[\mathbf{I}-\mathbf{A}^{dd}]^{-1}$. (This is clear from
invoking the Hawkins-Simon conditions). Thus, for $\bm{f}^{s}\geq\bm{0}$,
$[\mathbf{I}-\mathbf{A}^{ss}]\bm{x}^{s}$ must be non-negative and larger than
the part to the right of the minus in Eq. (28). However, this term can be
negative for supply shocks that are sufficiently heterogeneous. For example,
consider the case of an economy with only two supply-constrained industries
without self-loops, $i$ and $j$. Any supply shocks which lead to
$x_{i}^{\text{max}}<a_{ij}^{ss}x_{j}^{\text{max}}$ yield negative final
consumption values for industry $i$. This demonstrates that the MEEM framework
is unlikely to yield plausible solutions in the current context.

The MEEM can also lead to final consumption values that lie above any given
pre-specified upper limit of consumption $f_{i}^{\text{max}}$. Let us again
consider an example of two industries, $i$ and $j$, where $i$ is supply
constrained and $j$ is demand constrained. If industry $i$ only supplies
industry $j$ and final consumers and industry $j$ only supplies to final
consumers, it can be verified that $f_{i}^{s}>f_{i}^{\text{max}}$ if
$x_{i}^{\text{SS}}-f_{i}^{\text{DS}}<a_{ij}^{sd}f_{j}^{\text{DS}}$. Thus, in
case industry $i$ is only slightly supply constrained (supply shocks are only
slightly larger than demand shocks) and industry $j$ faces comparatively
serious demand constraints, the MEEM would compute larger final consumption
values for industry $i$ than are possible.

Note that gross output values of demand constrained industries always lie
within the feasible range $x_{i}^{d}\in[0,x_{i}^{\text{max}}]$ if only adverse
supply shocks to the economy are allowed ($\epsilon_{i}^{S}\geq 0$). By noting
that
$\bm{x}^{d}=[\mathbf{I}-\mathbf{A}^{dd}]^{-1}(\mathbf{A}^{ds}\bm{x}^{s}+\bm{f}^{d})$,
it can easily be verified that $x_{i}^{d}\geq 0$. The condition that
$\epsilon_{i}^{S}x_{i}^{d}<\epsilon_{i}^{D}f_{i}^{d}$ ensures that output of
demand constrained industries cannot exceed the maximum output value
$x_{i}^{\text{max}}$.

The supply and demand shocks do not need to be large for the MEEM to generate
infeasible solutions. To show this we scale down the size of the supply and
demand shocks by varying a parameter $\alpha\in[0,1]$ to obtain new maximum
output and consumption values, according to

$\displaystyle x_{i}^{\text{max}}$
$\displaystyle=(1-\alpha\epsilon_{i}^{S})x_{i,0},$ (29) $\displaystyle
f_{i}^{\text{max}}$ $\displaystyle=(1-\alpha\epsilon_{i}^{D})f_{i,0}.$ (30)

Since we scale demand and supply shocks by the same proportion this does not
change which industries are supply or demand constrained.

Fig. 5 shows the MEEM results for varying directs shocks. If $\alpha=0$, there
is no direct shock, resulting in a feasible market allocation since in this
case the MEEM simply recovers the pre-shock economy. This is indicated by the
green colors at the very left of all three panels. But even for very small
$\alpha>0$ we obtain infeasible solutions for all three countries as shown by
the gray colors. If we increase shock sizes further, it becomes more likely
that the model computes negative final consumptions values as can be seen from
the transition of red colors into gray when following the x-axis from left to
right.

Figure 5:  Infeasible results for final consumption for the MEEM model as a
function of shock size. The parameter $\alpha$ that scales the supply and
demand shocks varies along the x-axis. The color codes indicate the ratio
$f/f^{\text{max}}$ where $f$ is the MEEM result of final consumption. Note
that this ratio is always equal to one (green color) for industries which are
demand constrained. Infeasible values, $f\notin[0,f^{\text{max}}]$, are
indicated in gray.

##  Appendix C Details on shock magnitude effects

In the main text we have shown the effect of scaling either supply or demand
shocks on aggregate output. We also explored shock amplification effects when
scaling both, supply and demand shocks, simultaneously. Fig. 6 shows the
effect of different shock magnitudes on aggregate output and final consumption
values. Results are fairly similar for aggregate values of output and
consumption and are also in qualitative agreement with the results presented
in Fig. 2.

Figure 6:  Economic impact as a function of shock magnitude. Aggregate gross
output and final consumption levels as a function of scaling demand and supply
shocks equally between zero and one ($\alpha=\alpha^{S}=\alpha^{D}\in[0,1]$).

##  Appendix D Details on network density

In Section 4.3.2 we removed links randomly and repeated this procedure
multiple times for any desired network density value. While random edge
removal is one possible approach, other procedures could be followed too. A
natural alternative to random edge deletion is to first delete small links.
While the high aggregation of the data results in almost complete graphs, link
sizes are highly heterogeneous, implying the existence of many very small
links (McNerney et al., 2013; Cerina et al., 2015). It could be argued that
many of these links are rather an artifact of data aggregation instead of
encoding a fixed production recipe. We therefore repeat the procedure of
Section 4.3.2 but eliminate smaller before larger links to achieve a given
level of network density. Doing this results in Fig. 7 which indicates
qualitatively similar results as Fig. 3 of the main text.

Figure 7:  Economic impact as a function of network density. The figure is the
same as Fig. 3 in the main text, except that network density is changed by
eliminating links based on their size instead of random deletion.

The elimination of existing IO links changes key properties of the underlying
economic system. As discussed in Section 4.3.2, removing intermediate
consumption values will reduce aggregate output. Figs. 8(a) and 8(b) visualize
the relationship between output and network density following the random link
and “smallest-first” removal approach, respectively. Similarly, the ratio
intermediate consumption over aggregate output will be reduced as a
consequence. A density value equal to zero means that there is no intermediate
consumption left and firms only use primary factors as inputs in production.
Fig. 8 also shows that the average output multiplier decreases when making the
network sparser, although not necessarily monotonously. Overall, the economic
indicators change fairly linearly with respect to network density if links are
randomly removed, whereas these relationships are highly nonlinear if smaller
edges are eliminated before larger ones.

Figure 8:  Key economic measures as a function of network density. (a) The
network density is changed by eliminating links randomly (corresponding to
Fig. 3). (b) The network density is changed by eliminating smaller before
larger links (corresponding to Fig. 7). _Multiplier_ refers to the
(unweighted) average multiplier of the economy, $\sum_{ij}l_{ij}/n$, after
rebalancing as percentage of the initial economy. _Intermediate_ denotes the
share of intermediate consumption in total output,
$\sum_{ij}z_{ij}/\sum_{i}x_{i}$ after rebalancing the economy. _Output_ is
total output after rebalancing divided by initial total output.