# Relief and Stimulus in a Cross-sector Multi-product
Scarce Resource Supply Chain Network

Xiaowei Hu  Xiaowei Hu  Xiaowei Hu Corresponding author<EMAIL_ADDRESS>Peng
Li<EMAIL_ADDRESS>
(https://doi.org/10.1016/j.tre.2022.102932
Received 25 May 2022; Revised 31 August 2022; Accepted 3 October 2022
Transportation Research Part E: Logistics and Transportation Review (2022),
Volume 168, pp 102932.)

Abstract

In the era of a growing population, systemic changes to the world, and the
rising risk of crises, humanity has been facing an unprecedented challenge of
resource scarcity. Confronting and addressing the issues concerning the scarce
resource’s conservation, competition, and stimulation by grappling its
characteristics and adopting viable policy instruments calls the decision-
maker’s attention with a paramount priority. In this paper, we develop the
first general decentralized cross-sector supply chain network model that
captures the unique features of scarce resources under a unifying fiscal
policy scheme. We formulate the problem as a network equilibrium model with
finite-dimensional variational inequality theories. We then characterize the
network equilibrium with a set of classic theoretical properties, as well as
with a set of properties that are novel to the network games application
literature, namely, the lowest eigenvalue of the game Jacobian. Lastly, we
provide a series of illustrative examples, including a medical glove supply
network, to showcase how our model can be used to investigate the efficacy of
the imposed policies in relieving supply chain distress and stimulating
welfare. Our managerial insights inform and expand the political dialogues on
fiscal policy design, public resource legislation, social welfare
redistribution, and supply chain practice toward sustainability.

Keywords: networks; resource scarcity; game theory; supply chain management;
fiscal policy

This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.

##  1 Introduction

Humanity depends on the supply of an array of essential resources. These
resources, such as agricultural crops, fisheries, wildlife, forests,
petroleum, metals, minerals, even air, soil, and water, are not only critical
to the flourishing of humanities but also considered assets for businesses
operations, product innovation, and government affairs (Rosenberg, 1973;
Wagner and Light, 2002; Krautkraemer, 2005). The impact of resource scarcity
has also become a popular theme that impels some of the futuristic opinions
and critiques on potential societal collapse and humanity trajectory (see
also, Friedman (2009a, b), Gilding (2012), inter alia). Therefore, meeting the
resource demands is a substantive economic, commercial, and societal matter
for the decision-makers.

However, modern history has never been immune to resource scarcity; in fact,
we are experiencing shortages with greater magnitude and frequency. In the
1970s, the western world faced oil crises, which had a lasting impact on the
United States and extended the oil shortage of 1979 to the farm crisis in the
1980s, resulting in low crop prices, foreclosed farmlands, and thereby the low
farm income (Corbett, 2013; FDIC, 1997). Since the 2000s, many regions around
the world, including North and South Africa, South and Central Asia, and some
parts of the U.S., have been under enormous water stress. For instance,
California of the U.S. declared to have experienced the driest year in 2013,
and again in 2021 (Worland, 2014; Jones, 2021). Since 2019, induced by the
COVID-19 global pandemic, the world has experienced an overwhelming shortage
of commodities, ranging from raw materials such as lumber, precious metals,
and labor, to the end-products such as medical supplies, semiconductor chips,
grocery items, and household cleaning products. (Gasparro et al., 2020; Chou,
2020; Valinsky, 2021). As is summarized by an Economist (2021) article: we are
in a shortage economy.

Some of the key causes of the resource scarcity, as is pointed out by
Population Reference Bureau, include supply, demand, and structure (UN, 2012).
Today, with the world’s population projected to reach 8.5 billion in 2030 (UN,
2019), the demand for the planet’s scarce resources continues to grow at a
staggering rate. Meanwhile, the industrialization in the past two centuries
has ended the “ever-cheaper” commodity resources era and set the fierce
competition with rising prices for the “ever-scarcer” resources as the new
norm (Krautkraemer, 1998). Furthermore, traumatic crises, such as the
aforementioned COVID-19 pandemic, also induce the scarcity of specific
supplies.

As a means to relieve resource shortages, stimulate growth, and influence
economic outcomes, especially during critical times, supply-side fiscal
policies are commonly used by governments as strategic instruments. The fiscal
policy describes changes to government spending and revenue behavior (Keynes,
1936). In the energy crisis of the 1970s, the U.S. Senate advanced supply-side
fiscal incentives to mitigate the ramification of the energy problems (Long,
1973). Those policies included investment credits for domestic exploratory
drilling, research and development on the commercial exploitation of
alternative energy sources. In the global financial crisis of 2007-2008, the
U.S. Congress passed the $787 billion American Recovery and Reinvestment Act
in 2009, after the $125 billion provided by the Economic Stimulus Act of 2008
(Davig and Leeper, 2011). In the COVID-19 pandemic of 2020, worldwide
governments adopted economic packages including fiscal, monetary, and
financial policy measures to mitigate the negative effects of the public
health crisis on the economy and to sustain public welfare (Gourinchas, 2020).
To date, the estimated global stimulus effort has totaled $ 10.4 trillion
(Economist, 2021) with $5.2 trillion by the U.S. (Romer and Romer, 2021).
Monitoring the global fiscal policy in the wake of the COVID-19 pandemic, IMF
(2021) sees the worldwide fiscal stimulus in many advanced economies (e.g.,
the European Union and the U.S.) starting to make it more productive,
equitable, and sustainable.

From the industrial standpoint of a scarce resource value chain, the supply-
side fiscal policies, such as production incentives, tax credits, or
expansionary funds, serve as a cost reduction and revenue increase for firms
in short term and entrepreneurial processes in long term (Arestis and Sawyer,
2003; Braunerhjelm, 2022). In particular, the provision of fiscal policies to
a resource sector in scarcity, directly or indirectly, can have an extreme
preserve effect on the management of the targeted sector and the broader
economy (Young, 2015). For example, in the face of groundwater depletion in
Indian states like Punjab and Gujarat, farmers are given free access to
electricity to pump groundwater, in compensating for the massive
underinvestment in water-saving technologies. Such a benefit results in a more
profound impact on the resilience of the supply chain as well. Today, the
emerging resource scarcity, especially from the latest pandemic, has served as
a wake-up call by exposing the vulnerability of the supply chains. Along with
an overwhelming pleading for fiscal stimulation in popular press (Blinder,
2021; Winegarden, 2021; Smith, 2021), it bears merit to investigate how the
supply-side fiscal policies could relieve the shortage and stimulate the
social outcome.

In existential resource scarcity where the demand is not limited to one single
resource type, governments must identify their optimal grand strategies. But
often, policies and regulations alone can inadvertently create sub-optimal
signals to economic or environmental concerns. As a response to the need for
expanding and informing political dialogues, growing acknowledgment of the
cross-sector consideration such as the well-known water-energy-food (WEF)
nexus concept (Hoff, 2011) (see also, Bazilian et al. (2011) and the reference
therein) has emerged in the past decade. Similarly, the cross-sector construct
considers key issues in each commodity sector’s security through a holistic
lens in order to achieve long-term sustainability (cf. Biggs et al. (2015)).
For instance, the discussions on integrated water resources management have
now emphasized issues through analyses of inter-sectoral competition for
surface freshwater resources, integration of water management at farm, system,
and basin scales, or balancing tradeoffs with electricity generation and fuel
supply (Kurian, 2017; Zhang and Vesselinov, 2016). To date, the cross-sectoral
approach has tended toward technical assessments to enable knowledge and
information sharing, support intersectoral cooperation, and enhance
productivity and synergies (Biggs et al., 2015; Olawuyi, 2020).

In this paper, we construct a general supply chain network equilibrium (SCNE)
model to better understand the scarce resources vis-à-vis conservation,
competition, transportation, and policy intervention, by consolidating a
resource’s innate scarcity and the supply-demand caused shortage (more in
§2.1). Our supply chain network model incorporates the interdependency of the
scarce resources by allowing for cross-resource sector links. With the
inclusion of multiple resource products that are differentiated by markets,
such a network emulates the scarce resource’s heterogeneity feature in a
similar approach by Zhang (2006). The interdependency and heterogeneity are
two features inspired by Pfeffer and Salancik (2003), and Hunt (2000).
Furthermore, in the model, we install a unifying supply-side fiscal policy
scheme in relation to the scarce resource ownership and production. We assess
the efficacy of utilizing incentives and taxes in addressing the supply chain
shortage from an industrial organization’s viewpoint. Also, we incorporate a
non-cooperative behavior of decision-makers, considering the unique ownership
of the resources and the nature of quantity competition under the context of
scarcity. We believe such behavior is best represented by a decentralized
network, a construct known for its generality, efficiency, and ability to
synthesize local information in supply chains (see also, Lee and Billington
(1993), inter alia). Finally, in illustration, we provide a series of
numerical examples, including one that concerns medical personal protective
equipment (PPE) supplies, to illustrate the utility of our model.

The remainder of this paper is organized as follows. Section 2 is a literature
review and the contributions of our work. Section 3 presents the scarce
resource supply chain network model with fiscal policies and its mathematical
formulation. In Section 4, we show the solution concept of the model. Section
5 establishes the theoretical properties of the equilibrium in the context of
both classic and novel network game studies. Section 6 proposes an algorithmic
procedure to find the network equilibrium. Section 7 demonstrates the utility
of the model with small-scale illustrative examples. Section 8 provides the
decision-makers and practitioners with real-world managerial insights. Lastly,
Section 9 is a highlight of our study and foresight to its extension.

##  2 Literature review

###  2.1 Scarce resources

The broader studies on resource scarcity lie primarily in the fields of
economics and management, each of which gives a self-containing viewpoint
toward the characteristics of scarce resources. On one hand, the equilibrium
and monopoly theories of Adam Smith, John Stuart Mill, and John Nash, as well
as the rent and location theories of Alfred Marshall and Alfred Weber, tie the
scarcity to the location and quality of resources needed for industrial and
agricultural purposes (Weber and Friedrich, 1929). On the other hand, among
the rising organizational theories, the Resource Dependence Theory (Pfeffer
and Salancik, 2003) claim the importance of resource by which the owners,
producers, and suppliers are connected and interdependent (cf. Caniëls and
Gelderman (2007)), whereas the Resource Advantage Theory (Hunt, 2000) posits
resource’s features, including their demand heterogeneity and roles, in a firm
setting. Overall, from an economic perspective, scarce resources can be
characterized as of oligopolistic supply-side power, heterogeneous intra-
sector demand, and low price-elasticity of products. From a management
perspective, the possession of scarce resources is indicative of a firm’s
competitive advantage and negotiation power. The coalescence of these two
avenues of studies in resource scarcity has yet to occur in literature.

The scholarly interests in the interconnectedness of scarce resources started
to agglomerate since the release of the World Economic Forum report introduced
the WEF nexus as a novel concept (Hoff, 2011). Nearly all WEF-related
literature that we are aware of has sought integrated solutions or suggested
the interconnection of resources. For instance, Bai et al. (2016) investigated
the interplay between agricultural and fuel products; Zhang and Vesselinov
(2016) studied the interaction between water and energies; Bakker et al.
(2018) exemplified a regional economy with water and agriculture in focus; Mun
et al. (2021) provided an expansionary solution to the hydro networks for
energy, irrigation and flood control for developing countries. Almost all of
these works, at the same time, amalgamate with agricultural supply chain
management (SCM), in which resource scarcity was associated with a range of
societal and economic issues.

In the tumultuous times of critical resource shortages, especially in the wake
of the COVID-19 pandemic, a wealth of studies have emerged to demonstrate the
utility of quantitative supply chain models for the urgency of healthcare,
agricultural, logistical, and humanitarian challenges. For example, Kazaz et
al. (2016) developed a malaria medicine supply chain model to improve supply
and reduce price volatility via interventions. Arifoğlu and Tang (2022)
devised a decentralized system to combat the frequent supply and demand
mismatches of the vaccines. Both Chen and Tang (2015) and Liao et al. (2019)
used stylized models to examine the impact of information provision policies
on farmers in developing countries. Yu et al. (2020) constructed three-echelon
supply chain models to study donors’ optimal subsidy strategies for the
development of under-developed areas. Chintapalli and Tang (2021) modeled,
analyzed, and evaluated the impact of a credit-based scheme on the welfare of
small-scale, risk-averse farmers. Shen and Sun (2021) explained the impact of
the pandemic on supply chain resilience, identified the challenges that retail
supply chains had experienced in China, and showcased the practical response
of JD.com throughout the pandemic.

###  2.2 Fiscal policies

The research of fiscal policies such as economic incentives in behavioral
economics is also related to our work. Under such subdomain, the design of
fiscal policies often becomes a delicate and complex matter when presented
with a decentralized system of multiple decision-making agents (cf. Arrow and
Kruz (2013)). Generally, to assess the interplay of incentives and behaviors,
microeconomic approaches are often adopted (see also, Frey and Jegen (2001),
Huck et al. (2012), inter alia). With such approaches, incentives may be
linked to the network topology and its design (cf. Belhaj and Deroïan (2013),
Jackson and Zenou (2015)). For instance, Calvo-Armengol and Jackson (2004)
showed that education subsidies and other labor market regulation policies
display local increasing returns due to the network structure. Hu et al.
(2019) studied the price fluctuation in agricultural markets by discussing the
role a carefully designed preseason buyout contract plays in individual
farmer’s welfare. Srai et al. (2022) explored how the interplay between
competing and coexisting policy regimens can affect the supply dynamics
between producers, customers, and their intermediaries in a supply network.

As noted before, today’s fragmentation and globalization of supply chains give
impetus to new value-adding opportunities with the inclusion of supply-side
fiscal policies. A plethora of recent works is dedicated to designing supply
chain smart tax- and incentive-based schemes to promote new energies, foster
sustainability, and develop e-commerce, health system, or broader economies.
For example, Alizamir et al. (2019) analyzed the impact of two U.S. farmer’s
subsidy programs on consumers, farmers, and the government. Jiang et al.
(2021) studied the government’s penalty provision in a bioenergy supply chain.
Tao et al. (2020) evaluated the impact of various carbon policies on planning
and operation decisions in emerging markets. Xu and Choi (2021) investigated
the effect of the blockchain technology on the manufacturer’s operational
decisions under cap-and-trade regulation. Arifoğlu and Tang (2022) developed a
two-sided incentive program that proposes “vaccination incentives” to be given
to both the demand and supply side to combat the frequent supply and demand
mismatches. Levi et al. (2022) analyzed the effectiveness of various
government interventions in improving consumer welfare under an artificial
shortage in agricultural supply chains.

###  2.3 Supply chain network equilibrium models

Also akin to this study are the supply chain network models with multi-agent
decision-making, in which case, game theory is often used (e.g., Hu et al.
(2021), inter alia). Among the wealth of existing literature, the SCNE models,
inaugurated by Nagurney et al. (2002), permit one to represent the
interactions between decision-makers in the supply chain for general
commodities in terms of connections, flows, and prices. Since then, this sub-
domain has proliferated to a wide range of applications with timely interests.
In conjunction with resource scarcity and shortages, we note, for instance, Wu
et al. (2006) and Matsypura et al. (2007) modeled power transmissions. Masoumi
et al. (2012), Dutta and Nagurney (2019) each modeled the supply and demand of
a unique medical item; Besik and Nagurney (2017) and Wu et al. (2018) modeled
the logistics of fresh produce in the scope of their qualities and
preservation. In conjunction with supply-side fiscal policies, on the other
hand, we are aware that Yu et al. (2018), Wu et al. (2019), and Yang et al.
(2021, 2022), for example, captured the design of an environmental tax or
subsidy policy in a green supply chain. Nagurney et al. (2019a, b) studied how
global trade policies could impact the suppliers and markets.

To date, albeit the research in SCNE models has been well developed, new
studies continue to proliferate. In surveying the SCNE studies since 2020, we
now identify a few trends of its application. First, the Chinese e-commerce
industry has evolved rapidly with explosive consumer traffic in online
shopping, payment, marketing, and services, as the works of Zhang et al.
(2020), Guoyi et al. (2020), and Chen et al. (2020), all affiliated with
Chinese institutions, demonstrated the opportunities and utilities of such
model in this industry. Second, the studies in socially responsible practices,
led by firms’ environmentally conscientious strategies, have reflected the
zeitgeist of our renewed societal values. In particular, Chen et al. (2020),
Zhang et al. (2021), and Yang et al. (2021, 2022) all showcased that
corporate-level efforts like de-carbonization have embodied today’s global
supply chain practitioners’ response. Last but not least, the COVID-19
pandemic has evidently created a host of complex problems in the logistics of
medical supplies and labor, among which, for example, are the works by
Nagurney (2021a, b), inter alia.

###  2.4 Research gap and current contribution

While most of the literature in SCM focuses on either a category of general
products or one specific type, it remains largely sparse on the
interdependence of firms and the specific resource commodities they produce in
a cross-sector framework (cf. Bell et al. (2012)). Such a gap can be
exemplified by the papers referenced in §2.1, as they mainly emphasize one or
two specific products. To the best of our knowledge, we are not aware of a
study in SCM that explores the characteristics of resource scarcity and the
shortage conditions from both economic and management perspectives. A
systematic review by Matopoulos et al. (2015) has underscored a need for
further research on understanding the implications of resource scarcity for
supply chain relationships and their impact on supply chain configurations.

Moreover, the fiscal policy scheme we implement in our model is inclusive of
both incentives and taxes. We also set forth our scope of policy calibration
as the broader social outcome and firm’s collective profits, instead of
specific performance measures (e.g., prices and quantities), as what appeared
in the referenced papers in §2.2. The coalescence of these two aspects has
been limited in the theoretical literature on supply chain networks. Such a
topic would have provided invaluable managerial insights and should,
therefore, merit attention. The major contributions of this paper are
summarized as follows:

  1. 1.

The construction of the first decentralized 4-tier multi-product scarce
resource SCNE model with supply-side fiscal instruments. The model could
handle not only the supply chains of scarce resources but also those of
general resources that pertain to the aforementioned characteristics of
scarcity.

  2. 2.

The first work to examine a general supply-side fiscal policy that encompasses
both economic incentives and taxes in one model, with the social outcome as a
calibration.

  3. 3.

A rigorous variational inequality formulation of the modeled general Nash
equilibrium problem due to the consideration of both competition and resource
capacities; The establishment of a set of novel theoretical conditions to
characterize the network equilibria.

  4. 4.

Our examples and the resulted managerial insights instilled for governments
and supply chain practitioners to handle scarce resource management, fiscal
policy design, shortage relief, and sustainable development.

Finally, we use Table LABEL:table:compare_SR to compare and differentiate our
work with other closely related SCNE models mentioned in this section.

Table 1: A comparison of closely related SCNE articles  Article | Multi-product | Oligopolistic Competition | Capacitated | Policy Instrument
---|---|---|---|---
Single | Unifying
Masoumi et al. (2012) |  | $\bullet$ |  |  |
Besik and Nagurney (2017) |  | $\bullet$ |  |  |
Yu et al. (2018) |  | $\bullet$ |  | $\bullet$ |
Yang et al. (2021) |  | $\bullet$ |  | $\bullet$ |
Yang et al. (2022) |  | $\bullet$ |  | $\bullet$ |
Zhang (2006) | $\bullet$ |  |  |  |
Nagurney and Dutta (2021) | $\bullet$ |  |  |  |
Wu et al. (2006) |  |  |  | $\bullet$ |
Nagurney et al. (2019b) |  |  |  | $\bullet$ |
Wu et al. (2019) |  |  |  | $\bullet$ |
Hu and Qiang (2013) |  |  | $\bullet$ |  |
Nagurney et al. (2017) |  |  | $\bullet$ |  |
This Paper | $\bullet$ | $\bullet$ | $\bullet$ |  | $\bullet$

##  3 The cross-sector multi-product scarce resource supply chain network
model

In the network, there are $I$ types of scarce resources, denoted as
$i=1,...,I.$ We assign $i$’s alias $j$ in order to use it in different tiers.
Naturally, $j=1,...,I$. Each type of resource is affiliated with $N_{i}$
owners, $M_{j}$ producers, and $S_{j}$ suppliers. The unfinished or end-
product throughout the entire supply chain network is assumed to be
homogeneous by the types of resources. See Figure 1 for the network topology.
Later, when referring to a certain type of resource, we will omit the phrase
“type” or “type of”.

Figure 1: A cross-sector scarce resource supply chain network model

The first tier of nodes in the network represents the resource owner. For
instance, a soybean-producing farmer is an owner of a resource type "soybean
products". As the stakeholder of a scarce resource, an owner may supply such a
resource to any producer. A typical resource owner is denoted as $(i,n)$,
where $n=1,...,N_{i}$.

The second tier of nodes represents the resource producer. An example of a
resource producer can be a food manufacturer who uses the soybeans purchased
from the farmers as raw material and produces soybean-related foods. A
resource producer is able to obtain any type of resource from any owner, as
stated above, but is only able to sell and ship its products to the suppliers
who are specialized in the same type of resource. A typical resource producer
is denoted as $(j,m)$, where $m=1,...,M_{j}$.

The third tier of nodes represents the resource supplier. A resource supplier
can only obtain and supply one type of resource to all markets. A soybean
supplier, for instance, does not supply any other resources. A typical
resource supplier is denoted as $(j,s)$, where $s=1,...,S_{j}$.

The fourth tier of nodes represents the physical markets. A typical physical
market, consuming some resources from its suppliers, is denoted as $k$, where
$k=1,...,K$. Each supplier has $t$ transportation modes, where
$t=1,...,T_{j}$, to ship a resource product to any physical market. To emulate
the heterogeneity of the scarce resources characterized by Hunt (2000), we
assume that each market has its own perception of the same resource, but is
unable to distinguish the same resource from different suppliers or
transportation modes. We assume also all markets to be competitive and,
thereby, price-takers.

We consider the network to be decentralized, i.e., each firm of a given tier
competes with all others in a non-cooperative fashion to maximize its own
profit by determining its optimal production quantity and shipments, given the
simultaneous best response of all competitors. All of the notations regarding
the model are listed in Table LABEL:table:notations.

Table 2: Notations of the model Notation |  Definition
---|---
Indices: |
$i,j$ |  The index of resource. $j$ is an alias of $i$. $i,j\in\\{1,...,I\\}$.
$n$ |  The index of resource owner $i$. $n\in\\{1,...,N_{i}\\}$.
$m$ |  The index of resource producer $j$. $m\in\\{1,...,M_{j}\\}$.
$s$ |  The index of resource supplier $j$. $s\in\\{1,...,S_{j}\\}$.
$t$ |  The index of transportation mode of resource product $j$. $t\in\\{1,...,T_{j}\\}$
$k$ |  The index of a physical market. $k\in\\{1,...,K\\}$.
$g$ |  The index of an incentive payment bracket. $g\in\\{1,...,G\\}$.
Parameters: |
$U_{i}$ |  The total amount of resource $i$. $U\equiv\\{U_{1},U_{2},...,U_{I}\\}$ is a column vector.
$A^{i}_{g}$ |  The $g$’th incentive payment bracket of resource owners $(i,n)$
$B^{j}_{g}$ |  The $g$’th incentive payment bracket of resource producers $(j,m)$
$\psi^{in}_{jm}$ |  The non-negative conversion rate at resource producers $(j,m)$ from owner $(i,n)$
$w_{ts}$ |  The weight on market demand over supplier $s$ via transportation mode $t$
Variables: |
$x^{in}_{jm}$ |  The flow between resource owner $(i,n$) and producer $(j,m)$.
$x^{jm}_{s}$ |  The flow between resource producer $(j,m)$ and supplier $(j,s)$.
$x^{js}_{tk}$ |  The flow between resource supplier $(j,s)$ and market $k$ via transportation mode $t$.
$x^{in},x_{jm},x^{js}$ |  The column vectors that collect $x^{in}_{jm}$, $x^{jm}_{s}$, $x^{js}_{tk}$, respectively.
$d_{jk}$ |  Demand of resource $j$ at market $k$.
$p^{in}_{0jm}$ |  The transaction price between resource owner $(i,n)$ to producer $(j,m)$.
$p^{jm}_{1s}$ |  The transaction price between resource producer $(j,m)$ to supplier $(j,s)$.
$p^{js}_{2tk}$ |  The sales price by supplier $(j,s)$ for market $k$ via transportation mode $t$.
$p^{js}_{3tk}$ |  The transaction price at market $k$ from supplier $(j,s)$ via transportation mode $t$.
$\delta^{in}_{g}$ |  Resource owner $(i,n)$’s excess of output quantity to bracket $g$.
$\delta^{jm}_{g}$ |  Resource producer $(j,m)$’s excess of output quantity to bracket $g$.
$\lambda^{0}_{i}$ |  The Lagrange multipliers associated with constraint (9).
$\lambda^{1}_{jm}$ |  The Lagrange multipliers associated with constraint (18).
$\lambda^{2}_{js}$ |  The Lagrange multiplier associated with constraints (24).
$\mu^{0}_{ing}$ |  The Lagrange multipliers associated with constraint (10).
$\mu^{1}_{jmg}$ |  The Lagrange multipliers associated with constraint (19).
$\pi_{in},\pi_{jm},\pi_{js}$ |  The profit of resource owner $(i,n)$, producer $(j,m)$, and supplier $(j,s)$, respectively.
$CS_{jk}$ |  The consumer surplus of resource product $j$ at market $k$.
$SW$ |  Social welfare of the entire supply chain network.
Functions: |
$\alpha^{i}_{g}(.)$ |  The incentive payment to resource owners in bracket $g$.
$\beta^{j}_{g}(.)$ |  The incentive payment to resource producers in bracket $g$.
$f^{in}=f^{in}(x^{in})$ |  The operating cost incurred by resource owner $(i,n)$ in terms of outgoing shipments.
$c^{in}_{jm}=c^{in}_{jm}(x^{in}_{jm})$ |  The transaction cost incurred by resource owner $(i,n)$ with producer $(j,m)$.
$f^{jm}=f^{jm}(x_{jm})$ |  The operating cost incurred by producer $(j,m)$ in terms of incoming shipments.
$c^{jm}_{s}=c^{jm}_{s}(x^{jm}_{s})$ |  The transaction cost incurred by producer $(j,m)$ with supplier $(j,s)$.
$f^{js}=f^{js}(x^{js})$ |  The operating cost incurred by resource supplier $(j,s)$ in terms of outgoing shipments.
$c^{js}_{tk}=c^{js}_{tk}(x^{js}_{tk})$ |  The transaction cost incurred by supplier $(j,s)$ for market $k$ via transportation mode $t$.
$\hat{c}^{js}_{tk}=\hat{c}^{js}_{tk}(x^{js}_{tk})$ |  The transaction cost incurred by market $k$ with supplier $(j,s)$ via transportation mode $t$.
$p^{j}_{3k}=p^{j}_{3k}(d_{jk})$ |  The price-demand function of resource $j$ at the market $k$.

###  3.1 The fiscal policy scheme

For the ease of conceptualization, in our model, we apply a supply-side fiscal
incentive to resource owners and producers. We design a piece-wise scheme to
represent some of the most commonly used governmental incentive policies. The
incentive scheme devised here can easily be expanded to a general fiscal
policy to include taxation.

We now define the incentive scheme. Let $A^{i}_{g}$ denote the cutoff bracket
of the incentive on resource $i$, with $g$ enumerating between $1,...G$. Let
$\delta^{in}_{g}$ denote the resource owner $(i,n)$’s output quantity excess
over the bracket $A^{i}_{g}$. We assume the incentives given to each entity to
be a series of linear functions, only affected by its output quantity, as most
commonly used governmental economic policies are (cf. Sadka (1976); Huck et
al. (2012); Yu et al. (2018), inter alia). As such, we define the incentive
payment function as

$\alpha^{i}_{0}(.)+\sum^{G}_{g=1}\alpha^{i}_{g}(.),$ (1)

where $\alpha^{i}_{0}(.)$ is usually non-negative, both $\alpha^{i}_{0}(.)$
and $\alpha^{i}_{g}(.)$ are linear. For an arbitrary output quantity of $x$,
the payment scheme for the resource owners is piece-wise linear in shape and
can be expressed as:

If $0\leq x<A^{i}_{1}$, then the incentive payment is $\alpha^{i}_{0}(x)$;

If $A^{i}_{1}\leq x<A^{i}_{2}$, then $\delta^{in}_{1}=x-A^{i}_{1}$, and the
incentive payment is $\alpha^{i}_{0}(x)+\alpha^{i}_{1}(\delta^{in}_{1})$;

If $A^{i}_{2}\leq x<A^{i}_{3}$, then $\delta^{in}_{2}=x-A^{i}_{2}$, and the
incentive payment is
$\alpha^{i}_{0}(x)+\alpha^{i}_{1}(\delta^{in}_{1})+\alpha^{i}_{2}(\delta^{in}_{2})$;

If $A^{i}_{g}\leq x<A^{i}_{g+1}$, then $\delta^{in}_{g}=x-A^{i}_{g}$, and the
incentive payment is
$\alpha^{i}_{0}(x)+\alpha^{i}_{1}(\delta^{in}_{1})+\alpha^{i}_{2}(\delta^{in}_{2})+...+\alpha^{i}_{g}(\delta^{in}_{g})$;

If $x\geq A^{i}_{G}$, then the incentive payment is
$\alpha^{i}_{0}(x)+\alpha^{i}_{1}(\delta^{in}_{1})+\alpha^{i}_{2}(\delta^{in}_{2})+...+\alpha^{i}_{g}(\delta^{in}_{G})=\alpha^{i}_{0}(x)+\sum^{G}_{g=1}\alpha^{i}_{g}(\delta^{in}_{g})$.

Finally, a resource owner’s general incentive payment function in terms of
$\delta^{in}_{g}$ can be rewritten in a compact form as

$\alpha^{i}_{0}(x)+\sum^{G}_{g=1}\alpha^{i}_{g}(\delta^{in}_{g}),\quad
i=1,...,I,\quad n=1,...,N,$ (2)

where,

$\delta^{in}_{g}=max\\{x-A^{i}_{g},\ 0\\},\quad g=1,...,G.$ (3)

In particular, when $\alpha^{i}_{0}(.)$ is set to be a constant and
$\alpha^{i}_{g}(.)$ to be negative, expression (2) represents a regressive
incentive.111A regressive incentive is one whose marginal rate decreases (cf.
Sadka (1976)). An example of this incentive scheme is the United States
Economic Impact Payments of COVID-19.222See: https://www.irs.gov/coronavirus-
tax-relief-and-economic-impact-payments. Naturally, with the properly chosen
parameter values, expression (2)-(3) can also represent a general tax scheme.
We will use this scheme to express the fiscal policies for the rest of the
paper. Specifically, in §3.3, we will invariably define the incentives
administered to the resource producers.

###  3.2 The resource owner’s problem

A typical resource $i$ has a total capacity of $U_{i}$, shared by all $N_{i}$
owners of the same resource. A resource owner $(i,n)$ strategically determines
its shipment quantity $x^{in}_{jm}$ to a resource producer $(j,m)$. We use
$Q^{0}$, a column vector with $Q^{0}\in
R_{+}^{\sum_{i=1}^{I}\sum_{j=1}^{I}N_{i}M_{j}}$, to group all $x^{in}_{jm}$.
We assume an owner incurs an operating cost of

$f^{in}=f^{in}(x^{in}),\ \forall i,n,$ (4)

where $x^{in}\equiv(x^{in}_{11},...,x^{in}_{jm},...,x^{in}_{IM_{j}})$, and a
transaction cost of

$c^{in}_{jm}=c^{in}_{jm}(x^{in}_{jm}),\ \forall i,n,j,m.$ (5)

The incentive payment received by the resource owner $(i,n)$ is

$\alpha^{i}_{0}(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in}_{jm})+\sum^{G}_{g=1}\alpha^{i}_{g}(\delta^{in}_{g}),\
\forall i,n,$ (6)

where, the quantity excess

$\delta^{in}_{g}=max\Bigg{\\{}\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in}_{jm}-A^{i}_{g},\
0\Bigg{\\}},\quad\forall g=1,...,G.$ (7)

We group all $\delta^{in}_{g}$ into a column vector $\mathfrak{\Delta}^{0}$,
with $\mathfrak{\Delta}^{0}\in R_{+}^{\sum_{i=1}^{I}GN_{i}}$, and assume all
of the cost functions above, i.e., $f^{in},\ c^{in}_{jm}$, to be continuously
differentiable.

The price of products between the resource owner $(i,n)$ and the producer
$(j,m)$ is denoted by $p^{in}_{0jm}$. We further denote the equilibrium of
such a price as $p^{in*}_{0jm}$. Hence, the total revenue of each owner is
$\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}p^{in}_{0jm}x^{in}_{jm}$.

As a profit-maximizer, each typical owner faces the following problem.

Maximize: $\displaystyle\
\pi_{in}=\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}p^{in}_{0jm}x^{in}_{jm}-f^{in}(x^{in})-\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}c^{in}_{jm}(x^{in}_{jm})+\alpha^{i}_{0}(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in}_{jm})+\sum^{G}_{g=1}\alpha^{i}_{g}(\delta^{in}_{g})$
(8) Subject to: $\displaystyle\
\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}x^{in}_{jm}\leq U_{i},$
$\displaystyle(\lambda^{0}_{i})$ (9) $\displaystyle\
\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in}_{jm}-\delta^{in}_{g}\leq
A^{i}_{g},\quad\forall g,$ $\displaystyle(\mu^{0}_{ing})$ (10) $\displaystyle\
x^{in}_{jm}\geq 0,\ \forall j,m,\qquad\delta^{in}_{g}\geq 0,\ \forall g.$

The constraint (9) is the resource capacity. The constraint (10), along with
$\delta^{in}_{g}\geq 0$, is the equivalence of expression (7).
$\lambda^{0}_{i}$ and $\mu^{0}_{ing}$ are the Lagrange multipliers333For the
details of such reformulation technique, see Bertsekas and Tsitsiklis (1989).
associated with inequality constraints (9) and (10), respectively. We
subsequently group all $\lambda^{0}_{i}$ and $\mu^{0}_{ing}$ into a column
vector $\lambda^{0}$ and $\mu^{0}$, with $\lambda^{0}\in R_{+}^{I}$ and
$\mu^{0}\in R_{+}^{\sum_{i=1}^{I}GN_{i}}$, respectively.

In this decentralized supply chain, the resource owners compete in a non-
cooperative fashion, a behavior of classic Cournot-Nash (Cournot, 1838; Nash,
1950, 1951) stating that each owner determines the optimal output, given the
simultaneously optimal ones of all competitors. As such, the optimal behavior
of all resource owners can be expressed as a variational inequality
(VI)444Gabay and Moulin (1980) were among some of the earliest landmark works
to establish the equivalence between the optimization problem and variational
inequality problem. See also, Nagurney (1999).: determine
$(Q^{0*},\mathfrak{\Delta}^{0*},\lambda^{0*},\mu^{0*})\in\mathcal{K}^{1}$,
satisfying:

$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\Bigg{[}\frac{\partial
f^{in}(x^{in*})}{\partial x^{in}_{jm}}+\frac{\partial
c^{in}_{jm}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}-p^{in*}_{0jm}-\frac{\partial\alpha^{i}_{0}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}+\lambda^{0*}_{i}+\sum_{g=1}^{G}\mu^{0*}_{ing}\Bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\alpha^{i}_{g}(\delta^{in*}_{g})}{\partial\delta^{in}_{g}}-\mu^{0*}_{ing}\bigg{]}\times(\delta^{in}_{g}-\delta^{in*}_{g})+\sum_{i=1}^{I}\bigg{[}U_{i}-\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}x^{in*}_{jm}\bigg{]}$
$\displaystyle\times(\lambda^{0}_{i}-\lambda^{0*}_{i})$ (11)
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}A^{i}_{g}-\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in*}_{jm}+\delta^{in*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{0}_{ing}-\mu^{0*}_{ing})\geq 0,$
$\displaystyle\forall(Q^{0},\mathfrak{\Delta}^{0},\lambda^{0},\mu^{0})\in\mathcal{K}^{1},$

where,
$\mathcal{K}^{1}\equiv\\{(Q^{0},\mathfrak{\Delta}^{0},\lambda^{0},\mu^{0})|(Q^{0},\mathfrak{\Delta}^{0},\lambda^{0},\mu^{0})\in
R_{+}^{\sum_{i=1}^{I}(\sum_{j=1}^{I}N_{i}M_{j}+2GN_{i})+I}\\}.$

Due to constraint (9), the solution to the optimality condition (3.2) is a
variational equilibrium (Kulkarni and Shanbhag, 2012), a refinement of the
general Nash equilibrium (GNE)555GNE problem has been frequently used in
common-pool resource studies as a mathematical-economic tool. For more
information on general Nash equilibrium problems (GNEP), see Harker (1991),
Facchinei and Kanzow (2007).. In a variational equilibrium, all of the
Lagrange multipliers associated with the shared constraints are equal. By
formulating a GNE in VI, we can resort to a set of well-established
theoretical properties (more in Section 5) and algorithmic schemes than with a
formulation by quasi-inequality, a formalism for GNE problems (Facchinei and
Kanzow, 2007).

In addition, condition (3.2) proffers a readily interpretable mathematical
form from an economic perspective. According to the first summand, a resource
owner will ship a positive amount of the product to a resource producer if the
price that the producer is willing to pay for the product is exactly equal to,
the owner’s marginal cost less the marginal incentive benefit. If the owner’s
marginal cost, less the marginal incentive benefit, exceeds what the retailer
is willing to pay for the product, then there will not be any shipments from
the owner to the producer.

###  3.3 The resource producer’s problem

A resource producer $(j,m)$ strategically determines its shipment quantity
$x^{jm}_{s}$ to a resource supplier $(j,s)$. We group all $x^{jm}_{s}$ into a
column vector $Q^{1}$, with $Q^{1}\in R_{+}^{\sum_{j=1}^{I}M_{j}S_{j}}$. We
introduce a non-negative conversion rate of production $\psi^{in}_{jm}$. We
assume that the producer’s operating cost is associated with its total raw
material quantity receivables from the owners and that it can be written as

$f^{jm}=f^{jm}(x_{jm}),\quad\forall j,m,$ (12)

where, $x_{jm}\equiv(x_{jm}^{11},...,x_{jm}^{in},...,x_{jm}^{IN_{i}}).$

The producer’s transaction cost is associated with the flow between each pair
of producer and supplier

$c^{jm}_{s}=c^{jm}_{s}(x^{jm}_{s}),\quad\forall j,m,s.$ (13)

Similar to the incentive scheme for the resource owners, we let $B^{j}_{g}$
denote the cutoff bracket of the incentives, with $g$ enumerating between
$1,...G$. Let $\delta^{jm}_{g}$ denote the excess of output quantity over
bracket $B^{j}_{g}$. As such, we define the resource producer’s incentive
payment function as

$\beta^{j}_{0}(.)+\sum^{G}_{g=1}\beta^{j}_{g}(.),$ (14)

Specifically, the incentive payment received by the resource producer $(j,m)$
is a function of its total output quantity, i.e.,

$\beta^{j}_{0}(\sum^{S_{j}}_{s=1}x^{jm}_{s})+\sum^{G}_{g=1}\beta^{j}_{g}(\delta^{jm}_{g}),\
\forall j,m,$ (15)

where, the quantity excess

$\delta^{jm}_{g}=max\Bigg{\\{}\sum^{S_{j}}_{s=1}x^{jm}_{s}-B^{j}_{g},\
0\Bigg{\\}},\quad\forall g=1,...,G.$ (16)

We group all $\delta^{jm}_{g}$ into a column vector $\mathfrak{\Delta}^{1}$,
and assume $f^{jm}$, $c^{jm}_{s}$ to be continuously differentiable,
$\beta^{j}_{0}$, and $\beta^{j}_{g}$ to be linear.

The price of products between the resource producer $(j,m)$ and the supplier
$(j,s)$ is denoted by $p^{jm}_{1s}$. We further denote the equilibrium of such
a price as $p^{jm*}_{1s}$. Hence, the total revenue of a producer is
$\sum_{s=1}^{S_{j}}p^{jm}_{1s}x^{jm}_{s}$.

As a profit-maximizer, each typical producer faces the following problem.

Maximize:
$\displaystyle\pi_{jm}=\sum_{s=1}^{S_{j}}p^{jm*}_{1s}x^{jm}_{s}-\sum_{j=1}^{I}\sum_{n=1}^{N_{i}}p^{in*}_{0jm}x^{in}_{jm}-f^{jm}(x_{jm})-\sum_{s=1}^{S_{j}}c^{jm}_{s}(x^{jm}_{s})+\beta^{j}_{0}(\sum^{S_{j}}_{s=1}x^{jm}_{s})+\sum^{G}_{g=1}\beta^{j}_{g}(\delta^{jm}_{g})$
(17) Subject to:
$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}x^{in}_{jm}\cdot\psi^{in}_{jm}\geq\sum_{s=1}^{S_{j}}x^{jm}_{s},$
$\displaystyle(\lambda^{1}_{jm})$ (18)
$\displaystyle\sum^{S_{j}}_{s=1}x^{jm}_{s}-\delta^{jm}_{g}\leq
B^{j}_{g},\quad\forall g,$ $\displaystyle(\mu^{1}_{jmg})$ (19) $\displaystyle
x^{jm}_{s}\geq 0,\ \forall s,\qquad\delta^{jm}_{g}\geq 0,\ \forall g.$

The constraint (18) is the flow conservation in light of the inter-resource
conversion. The constraint (19), along with $\delta^{jm}_{g}\geq 0$, is the
equivalence of expression (16). $\lambda^{1}_{jm}$ and $\mu^{1}_{jmg}$ are the
Lagrange multipliers associated with inequality constraints (18) and (19),
respectively. We subsequently group all $\lambda^{1}_{jm}$ and $\mu^{1}_{jmg}$
into a column vector $\lambda^{1}$ and $\mu^{1}$, with $\lambda^{1}\in
R_{+}^{\sum_{j=1}^{I}M_{j}}$ and $\mu^{1}\in R_{+}^{\sum_{j=1}^{I}GM_{j}}$,
respectively. The optimal behavior of all resource producers can be expressed
as a VI: determine
$(Q^{0*},Q^{1*},\mathfrak{\Delta}^{1*},\lambda^{1*},\mu^{1*})\in\mathcal{K}^{2}$,
satisfying:

$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\Bigg{[}\frac{\partial
f^{jm}(x^{*}_{jm})}{\partial
x^{in}_{jm}}+p^{in*}_{0jm}-\psi^{in}_{jm}\lambda^{1*}_{jm}\Bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\Bigg{[}\frac{\partial
c^{jm}_{s}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}-p^{jm*}_{1s}-\frac{\partial\beta^{j}_{0}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}+\lambda^{1*}_{jm}+\sum_{g=1}^{G}\mu^{1*}_{jmg}\Bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$ (20)
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\beta^{j}_{g}(\delta^{jm*}_{g})}{\partial\delta^{jm}_{g}}-\mu^{1*}_{jmg}\bigg{]}\times(\delta^{jm}_{g}-\delta^{jm*}_{g})+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}B^{j}_{g}-\sum^{S_{j}}_{s=1}x^{jm*}_{s}+\delta^{jm*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{1}_{jmg}-\mu^{1*}_{jmg})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}x^{in*}_{jm}\cdot\psi^{in}_{jm}-\sum_{s=1}^{S_{j}}x^{jm*}_{s}\bigg{]}$
$\displaystyle\times(\lambda^{1}_{jm}-\lambda^{1*}_{jm})\geq 0,$
$\displaystyle\forall(Q^{0},Q^{1},\mathfrak{\Delta}^{1},\lambda^{1},\mu^{1})\in\mathcal{K}^{2},$

where,
$\mathcal{K}^{2}\equiv\\{(Q^{0},Q^{1},\mathfrak{\Delta}^{1},\lambda^{1},\mu^{1})|(Q^{0},Q^{1},\mathfrak{\Delta}^{1},\lambda^{1},\mu^{1})\in
R_{+}^{\sum_{j=1}^{I}(\sum_{i=1}^{I}N_{i}M_{j}+M_{j}S_{j}+2GM_{j}+M_{j})}\\}.$

###  3.4 The resource supplier’s problem

A typical resource supplier is denoted as $(j,s)$. Each supplier has $t$ modes
to transport the resources to Market $k$. The supplier’s strategic variable is
$x^{js}_{tk}$, its shipment quantity to market $k$ with transportation mode
$t$. We group all $x^{js}_{tk}$ into a column vector $Q^{2}$, with $Q^{2}\in
R_{+}^{\sum_{j=1}^{I}KS_{j}T_{j}}$. We assume the supplier incurs an operating
cost that is associated with the resource product flow from the producers,
i.e.,

$f^{js}=f^{js}(x^{js}),$ (21)

where, $x^{js}\equiv(x^{j1}_{s},...,x^{jm}_{s},...,x^{jM_{j}}_{s_{j}}),\ \
\forall j,s$. The supplier also incurs a transaction cost with each outgoing
shipment to the markets:

$c^{js}_{tk}=c^{js}_{tk}(x^{js}_{tk}).$ (22)

We assume $f^{js}$ and $c^{js}_{tk}$ to be continuously differentiable.

The price of goods between the resource supplier $(j,s)$ and the market $k$
via transportation mode $t$ is denoted by $p^{js}_{2tk}$. We further denote
the equilibrium of such a price as $p^{js*}_{2tk}$. Hence, the total revenue
of each supplier is $\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}p^{js}_{2tk}x^{js}_{tk}$.

As a profit-maximizer, each typical supplier faces the following problem.

Maximize:
$\displaystyle\quad\pi_{js}=\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}p^{js*}_{2tk}x^{js}_{tk}-\sum_{m=1}^{M_{j}}p^{jm*}_{1s}x^{jm}_{s}-f^{js}(x^{js})-\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}c^{js}_{tk}(x^{js}_{tk})$
(23) Subject to:
$\displaystyle\quad\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}x^{js}_{tk}\leq\sum_{m=1}^{M_{j}}x^{jm}_{s},$
$\displaystyle(\lambda^{2}_{js})$ (24) $\displaystyle\quad x^{js}_{tk}\geq
0,\quad\forall t,k.$

The constraint (24) is the flow conservation of the finished resource
commodities at each supplier, with $\lambda^{2}_{js}$ being the associated
Lagrange multipliers. We subsequently group all $\lambda^{2}_{js}$ into a
column vector $\lambda^{2}$ with $\lambda^{2}\in R_{+}^{\sum_{j=1}^{I}S_{j}}$.
The optimal behavior of all resource suppliers can be expressed as a VI:
determine $(Q^{1*},Q^{2*},\lambda^{2*})\in\mathcal{K}^{3}$, satisfying:

$\displaystyle\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}\Bigg{[}\frac{\partial
c^{js}_{tk}(x^{js*}_{tk})}{\partial
x^{js}_{tk}}-p^{js*}_{2tk}+\lambda^{2*}_{js}\Bigg{]}$
$\displaystyle\times(x^{js}_{tk}-x^{js*}_{tk})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\bigg{[}\frac{\partial
f^{js}(x^{js*})}{\partial x^{jm}_{s}}+p^{jm*}_{1s}-\lambda^{2*}_{js}\bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$ (25)
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\Bigg{[}\sum_{m=1}^{M_{j}}x^{jm*}_{s}-\sum_{k=1}^{K}\sum_{t=1}^{T_{j}}x^{js*}_{tk}\Bigg{]}$
$\displaystyle\times(\lambda^{2}_{js}-\lambda^{2*}_{js})\geq
0,\qquad\forall(Q^{1},Q^{2},\lambda^{2})\in\mathcal{K}^{3},$

where,
$\mathcal{K}^{3}\equiv\\{(Q^{1},Q^{2},\lambda^{2})|(Q^{1},Q^{2},\lambda^{2})\in
R_{+}^{\sum_{j=1}^{I}(KS_{j}T_{j}+M_{j}S_{j}+S_{j})}\\}.$

###  3.5 Demand market equilibrium conditions

Scarce resources, such as clean water, energy, foods, and industrial
materials, are consumed in demand markets with unique needs. Concerned with
its willingness to pay, a market determines the purchasing quantity from each
supplier via each available transportation mode.

Because the difference in the suppliers or the transportation modes can cause
different product preservation in the network, without loss of generality, for
an arbitrary market $k$ ($k=1,...,K$) receiving resource commodity $j$
($j=1,...,I$), we assign a set of ex-ante weights $w_{ts}$ across all flows
via transportation modes $t$ ($t=1,...,T_{j}$) and suppliers $s$
($s=1,...,S_{j}$). In other words, the demand for product $j$ at market $k$ is

$d_{jk}=\sum_{t=1}^{T_{j}}\sum_{s=1}^{S_{j}}w_{ts}\cdot
x^{js}_{tk},\quad\forall j,k.$ (26)

We group all $d_{jk}$ from (26) into a column vector $d$, i.e.,
$d\equiv(d_{11},...,d_{jk},...,d_{JK})$, where $d\in R^{IK}_{+}$. We further
denote the equilibrium of such demands as $d^{*}_{jk}$ and $d^{*}$,
respectively. Because each scarce-resource product in each market has a unique
demand function, as is in (26), it corresponds with a market price
$p^{j}_{3k}$ in the form of price-demand function

$p^{j}_{3k}=p^{j}_{3k}(d).$ (27)

Market $k$ incurs a transaction cost from supplier $(j,s)$ via transportation
mode $t$

$\hat{c}^{js}_{tk}=\hat{c}^{js}_{tk}(x^{js}_{tk}),\quad\forall j,s,k,t.$ (28)

We assume $\hat{c}^{js}_{tk}$ to be continuously differentiable. Finally, we
adopt the classic Spatial Price Equilibrium (SPE) model666The Spatial Price
Equilibrium model appeared in the early literature of Samuelson (1952),
Takayama and Judge (1964), inter alia, and was thereafter extended with VI
theories by Dafermos and Nagurney (1984). to represent the markets. The
corresponding equilibrium condition is given by

$p^{js*}_{2tk}+\hat{c}^{js*}_{tk}(x^{js*}_{tk})\left\\{\begin{array}[]{ll}=p^{j}_{3k}(d^{*})&\quad
if\quad x^{js*}_{tk}>0\\\ \geq p^{j}_{3k}(d^{*})&\quad if\quad
x^{js*}_{tk}=0\end{array}\quad\forall j,s,t,k.\right.$ (29)

The intuition of equation (29) is, that the consumption of resource at the
markets will remain positive if the supplier’ purchase price of such a
resource plus all associated cost is equal to the price consumers are willing
to pay; however, if the supplier’ purchase price plus cost turns out to be
higher than what consumers are willing to pay, then there will be no shipments
between the supplier and the market.

As such, the optimal behaviors of all demand markets as a spatial price
equilibrium can be expressed in a VI form: determine
$(Q^{2*},d^{*})\in\mathcal{K}^{4}$, satisfying:

$\displaystyle\sum_{j=1}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}[p^{js*}_{2tk}+\hat{c}^{js}_{tk}(x^{js*}_{tk})]\times(x^{js}_{tk}-x^{js*}_{tk})-\sum^{I}_{j=1}\sum^{K}_{k=1}p^{j}_{3k}(d^{*})\times(d_{jk}-d^{*}_{jk})\geq
0,\qquad\forall(Q^{2},d)\in\mathcal{K}^{4},$ (30)

where, $\mathcal{K}^{4}\equiv\\{(Q^{2},d)|Q^{2}\in
R_{+}^{\sum_{j=1}^{I}KS_{j}T_{j}},\ d\in R^{IK}_{+}\\}.$

###  3.6 Welfare estimates

To assess the social impact of the model elements in our supply chain network,
we provide an estimate of consumer surplus and social welfare. The consumer
surplus, termed to measure the aggregate of consumers’ benefits upon
purchasing a product (cf. Willig (1976)) of resource $j$ at demand market $k$
is given by:

$CS_{jk}=\int_{0}^{d^{*}_{jk}}p^{j}_{3k}(z)dz-p^{j*}_{3k}d^{*}_{jk},\quad\forall
j,k.$ (31)

We also denote social welfare of the network to be the aggregation of all firm
profits and consumer surplus, i.e.,

$SW=\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\pi_{in}+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\pi_{jm}+\sum_{j=1}^{I}\sum_{s=1}^{S_{j}}\pi_{js}+\sum_{j=1}^{I}\sum_{k=1}^{K}CS_{jk},\quad\forall
i,j,n,m,s,k.$ (32)

##  4 The network equilibrium

######  Definition 1 (Cross-sector multi-product scarce resource supply chain
network equilibrium with fiscal policy)

A product flow pattern
$(Q^{0*},Q^{1*},Q^{2*},\mathfrak{\Delta}^{0*},\mathfrak{\Delta}^{1*},d^{*},\lambda^{0*},\lambda^{1*},\lambda^{2*},\mu^{0*},\mu^{1*})\in\mathcal{K}$
is said to constitute a cross-sector multi-product scarce resource supply
chain network equilibrium with fiscal policy if it satisfies condition (3.2),
(3.3), (3.4), and (30).

######  Theorem 1 (Variational inequality formulation)

The equilibrium conditions governing the cross-sector multi-product scarce
resource supply chain network, according to Definition 1, are equivalent to
the solution to the following VIs:

Determine
$(Q^{0*},Q^{1*},Q^{2*},\mathfrak{\Delta}^{0*},\mathfrak{\Delta}^{1*},d,\lambda^{0*},\lambda^{1*},\lambda^{2*},\mu^{0*},\mu^{1*})\in\mathcal{K}$,
satisfying

$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\Bigg{[}\frac{\partial
f^{in}(x^{in*})}{\partial x^{in}_{jm}}+\frac{\partial
f^{jm}(x^{*}_{jm})}{\partial x^{in}_{jm}}+\frac{\partial
c^{in}_{jm}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}-\frac{\partial\alpha^{i}_{0}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}+\lambda^{0*}_{i}-\psi^{in}_{jm}\lambda^{1*}_{jm}+\sum_{g=1}^{G}\mu^{0*}_{ing}\Bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\Bigg{[}\dfrac{\partial
f^{js}(x^{js*})}{\partial x^{jm}_{s}}+\frac{\partial
c^{jm}_{s}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}-\frac{\partial\beta^{j}_{0}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}+\lambda^{1*}_{jm}-\lambda^{2*}_{js}+\sum_{g=1}^{G}\mu^{1*}_{jmg}\Bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}\Bigg{[}\frac{\partial
c^{js}_{tk}(x^{js*}_{tk})}{\partial
x^{js}_{tk}}+\hat{c}^{js}_{tk}(x^{js*}_{tk})+\lambda^{2*}_{js}\Bigg{]}$
$\displaystyle\times(x^{js}_{tk}-x^{js*}_{tk})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\alpha^{i}_{g}(\delta^{in*}_{g})}{\partial\delta^{in}_{g}}-\mu^{0*}_{ing}\bigg{]}\times(\delta^{in}_{g}-\delta^{in*}_{g})+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\beta^{j}_{g}(\delta^{jm*}_{g})}{\partial\delta^{jm}_{g}}-\mu^{1*}_{jmg}\bigg{]}$
$\displaystyle\times(\delta^{jm}_{g}-\delta^{jm*}_{g})$ (33)
$\displaystyle-\sum^{I}_{j=1}\sum^{K}_{k=1}p^{j}_{3k}(d^{*})\times(d_{jk}-d^{*}_{jk})+\sum_{i=1}^{I}\bigg{[}U_{i}-\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}x^{in*}_{jm}\bigg{]}$
$\displaystyle\times(\lambda^{0}_{i}-\lambda^{0*}_{i})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}x^{in*}_{jm}\cdot\psi^{in}_{jm}-\sum_{s=1}^{S_{j}}x^{jm*}_{s}\bigg{]}\times(\lambda^{1}_{jm}-\lambda^{1*}_{jm})+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\bigg{[}\sum_{m=1}^{M_{j}}x^{jm*}_{s}-\sum_{k=1}^{K}\sum_{t=1}^{T_{j}}x^{js*}_{tk}\bigg{]}$
$\displaystyle\times(\lambda^{2}_{js}-\lambda^{2*}_{js})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}A^{i}_{g}-\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in*}_{jm}+\delta^{in*}_{g}\bigg{]}\times(\mu^{0}_{ing}-\mu^{0*}_{ing})+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}B^{j}_{g}-\sum^{S_{j}}_{s=1}x^{jm*}_{s}+\delta^{jm*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{1}_{jmg}-\mu^{1*}_{jmg})\geq 0,$
$\displaystyle\forall(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\lambda^{2},\mu^{0},\mu^{1})\in\mathcal{K},$

where,
$\mathcal{K}\equiv\mathcal{K}^{1}\times\mathcal{K}^{2}\times\mathcal{K}^{3}\times\mathcal{K}^{4}=\big{\\{}(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\lambda^{2},\mu^{0},\mu^{1})|(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\\\
\lambda^{2},\mu^{0},\mu^{1})\in
R_{+}^{\sum_{i=1}^{I}(\sum_{j=1}^{I}2N_{i}M_{j}+2GN_{i})+\sum_{j=1}^{I}(2M_{j}S_{j}+2KS_{j}T_{j}+2GM_{j}+S_{j}+M_{j})+IK+I}\big{\\}}.$

Proof: See Appendix A.

It should be noted that variable $p^{in*}_{0jm}$, $p^{jm*}_{1s}$, and
$p^{js*}_{2tk}$ do not appear within the formulation of Theorem 1. They are
endogenous to the model and can be retrieved once the solution is obtained. To
retrieve $p^{js*}_{2tk}$, for all $j,s,t,k$, recall equilibrium condition
(29). Since $p^{j}_{3k}(d^{*})$ is readily available from (27), if
$x^{js*}_{tk}>0$ for some $j,s,t,k$, then $p^{js*}_{2tk}$ can be obtained by
the equality

$p^{js*}_{2tk}=p^{j*}_{3k}(d^{*})-\hat{c}^{js}_{tk}(x^{js*}_{tk}).$ (34)

Invariably, if $x^{jm*}_{s}>0$ for some $j,m,s$, then from the second summand
in (3.4), one may immediately obtain

$p^{jm*}_{1s}=\lambda^{2*}_{js}-\frac{\partial f^{js}(x^{js*})}{\partial
x^{jm}_{s}}.$ (35)

And, if $x^{in*}_{jm}>0$ for some $i,n,j,m$, then from the first summand of
(3.3),

$p^{in*}_{0jm}=\psi^{in}_{jm}\lambda^{1*}_{jm}-\frac{\partial
f^{jm}(x^{*}_{jm})}{\partial x^{in}_{jm}}.$ (36)

##  5 Theoretical properties

We provide a few classic theoretical properties of the solution to VI (1),
based on Gabay and Moulin (1980), Nagurney (1999), and Melo (2018), inter
alia. In particular, we derive existence and uniqueness results by
incorporating strategic measure of network games from the latest theoretical
advancement.

To facilitate the development in this section, we rewrite VI problem (1) in
standard form as follows: determine $X^{*}\in\mathcal{K}$ satisfying

$\langle F(X^{*}),\ X-X^{*}\rangle\geq 0,\quad\forall X^{*}\in\mathcal{K},$
(37)

where,
$X\equiv(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\lambda^{2},\mu^{0},\mu^{1})$,
with an indulgence of notation,
$F(X)\equiv(F_{Q^{0}},F_{Q^{1}},F_{Q^{2}},F_{\mathfrak{\Delta}^{0}},F_{\mathfrak{\Delta}^{1}},F_{d},F_{\lambda^{0}},\\\
F_{\lambda^{1}},F_{\lambda^{2}},F_{\mu^{0}},F_{\mu^{1}})$, in which each
component of $F$ is given by each respective summand expression in (1), and,
$\mathcal{K}\equiv\mathcal{K}^{1}\times\mathcal{K}^{2}\times\mathcal{K}^{3}\times\mathcal{K}^{4}.$
The notation $\langle\cdot,\cdot\rangle$ represents the inner product in a
Euclidean space. Both $F$ and $X$ are $\mathcal{N}$-dimensional column
vectors, where
$\mathcal{N}=\sum_{i=1}^{I}(\sum_{j=1}^{I}2N_{i}M_{j}+2GN_{i})+\sum_{j=1}^{I}(2M_{j}S_{j}+2KS_{j}T_{j}+2GM_{j}+S_{j}+M_{j})+IK+I$.

First, we provide the existence properties. While $F$ in (37) is continuous,
the feasible set $\mathcal{K}$ is not compact. This causes the existence
condition of (1) not readily available. But one can impose a weak condition to
guarantee the existence of a solution pattern, as in Nagurney et al. (2002).
Let

$\displaystyle\mathcal{K}_{b}\equiv\big{\\{}(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\lambda^{2},\mu^{0},\mu^{1})\
|\ 0\leq Q^{0}\leq b_{1},0\leq Q^{1}\leq b_{2},0\leq Q^{2}\leq
b_{3},0\leq\mathfrak{\Delta}^{0}\leq b_{4},$ $\displaystyle
0\leq\mathfrak{\Delta}^{1}\leq b_{5},0\leq d\leq b_{6},0\leq\lambda^{0}\leq
b_{7},0\leq\lambda^{1}\leq b_{8},0\leq\lambda^{2}\leq b_{9},0\leq\mu^{0}\leq
b_{10},0\leq\mu^{1}\leq b_{11}\big{\\}},$

where
$b=(b_{1},b_{2},b_{3},b_{4},b_{5},b_{6},b_{7},b_{8},b_{9},b_{10},b_{11})\geq
0$ and $Q^{0}\leq b_{1}$, $Q^{1}\leq b_{2}$, $Q^{2}\leq b_{3}$,
$\mathfrak{\Delta}^{0}\leq b_{4}$, $\mathfrak{\Delta}^{1}\leq b_{5}$, $d\leq
b_{6}$, $\lambda^{0}\leq b_{7}$, $\lambda^{1}\leq b_{8}$, $\lambda^{2}\leq
b_{9}$, $\mu^{0}\leq b_{10}$, $\mu^{1}\leq b_{11}$ means that $x^{in}_{jm}\leq
b_{1}$, $x^{jm}_{s}\leq b_{2}$, $x^{js}_{tk}\leq b_{3}$, $\delta^{in}_{g}\leq
b_{4}$, $\delta^{jm}_{g}\leq b_{5}$, $d_{jk}\leq b_{6}$, $\lambda^{0}_{i}\leq
b_{7}$, $\lambda^{1}_{jm}\leq b_{8}$, $\lambda^{2}\leq b_{9}$,
$\mu^{0}_{ing}\leq b_{10}$, $\mu^{1}_{jmg}\leq b_{11}$ for all $i,j,n,m,t,k$.
Then $\mathcal{K}_{b}$ is a bounded, closed, convex subset of
$R_{+}^{\mathcal{N}}$. Tangentially, the existence of $b$ is sensible from an
economic perspective. Thus, the VI expression

$\langle F(X^{b}),\ X-X^{b}\rangle\geq 0,\quad\forall X^{b}\in\mathcal{K}_{b}$
(38)

admits at least one solution. Following Kinderlehrer and Stampacchia (1980),
and Nagurney et al. (2002), it is straightforward to establish:

###### Lemma 1

The VI (37) admits a solution if and only if there exists a $b>0$ such that
the VI (38) admits a solution in $\mathcal{K}_{b}$, with

$\displaystyle Q^{0}<b_{1},\ Q^{1}<b_{2},\ Q^{2}<b_{3},\
\mathfrak{\Delta}^{0}<b_{4},\ \mathfrak{\Delta}^{1}<b_{5},\ d<b_{6},$ (39)
$\displaystyle\lambda^{0}<b_{7},\ \lambda^{1}<b_{8},\ \lambda^{2}<b_{9},\
\mu^{0}<b_{10},\ \mu^{1}<b_{11}.$

######  Theorem 2 (Existence)

The VI problem (1) admits at least one solution in $\mathcal{K}_{b}$.

Proof: By virtue of theorem 3.1 in Harker and Pang (1990), one can easily
verify that $\mathcal{K}_{b}$ is non-empty, compact, and convex, and that the
mapping $F$ representing (1) is continuous. Therefore, there exists a solution
to the problem (1) in $\mathcal{K}_{b}$. $\square$

Next, we provide a set of sufficient conditions for the uniqueness properties.
In general, uniqueness is often associated with the monotonicity of the $F$
that enters the VI problem. Here, we begin by redefining a set of well-known
monotonicities (definition 2.3.1 by Facchinei and Pang (2003)).

######  Definition 2 (Monotonicity)

A mapping $F:\mathcal{K}\subseteq R^{n}\rightarrow R^{n}$ is said to be
(a) monotone on $\mathcal{K}$ if

$\langle F(X^{1})-F(X^{2}),\ X^{1}-X^{2}\rangle\geq 0,\quad\forall
X^{1},X^{2}\in\mathcal{K}.$ (40)

(b) strictly monotone on $\mathcal{K}$, if

$\displaystyle\langle F(X^{1})-F(X^{2}),\ X^{1}-X^{2}\rangle>0,\quad\forall
X^{1},X^{2}\in\mathcal{K},\quad X^{1}\neq X^{2}.$ (41)

(c) strongly monotone on $\mathcal{K}$, if

$\langle F(X^{1})-F(X^{2}),\ X^{1}-X^{2}\rangle>\alpha\left\lVert
X^{1}-X^{2}\right\rVert^{2},\quad\forall X^{1},X^{2}\in\mathcal{K},$ (42)

where $\alpha>0$.

######  Definition 3 (Lipschitz Continuity)

$F(X)$ is Lipschitz Continuous on $\mathcal{K}$, if

$\langle F(X^{1})-F(X^{2}),\ X^{1}-X^{2}\rangle\geq L\left\lVert
X^{1}-X^{2}\right\rVert^{2},\quad\forall X^{1},X^{2}\in\mathcal{K},$ (43)

where L is the Lipschitz constant.

We will also use the following established lemma (theorem 1.6 in Nagurney
(1999)).

###### Lemma 2

Suppose that $F(X)$ is strictly monotone on $\mathcal{K}$, then the solution
to the $VI(F,K)$ problem is unique, if one exists.

To further characterize the nature of the network, we need to adopt an
additional set of notations. Following Melo (2018), we define the game
Jabocian777The game Jacobian is akin to the topology, equilibrium analyses,
strategic interaction, and comparative statics of the network. For more
details, see Bramoullé et al. (2014), Jackson and Zenou (2015), Parise and
Ozdaglar (2019), Melo (2018), and the reference therein. of $F(X)$ as an
$\mathcal{N}$-by-$\mathcal{N}$ matrix

$J(X)\equiv[\nabla_{q}F_{r}(X)]_{q,r}=\Bigg{[}\frac{\partial
F_{r}(X)}{\partial X_{q}}\Bigg{]}_{q,r},\quad\forall
q,r=1,...,\mathcal{N},X\in R_{+}^{\mathcal{N}}.$ (44)

Decomposing the Jacobian in terms of its diagonal and off-diagonal element
yields

$J(X)=D(X)+N(X),$ (45)

where $D(X)$ is a diagonal matrix, whose elements are

$D_{qr}(X)=\left\\{\begin{array}[]{ll}\frac{\partial F_{r}(X)}{\partial
X_{q}}&\quad if\ q=r\\\ 0&\quad otherwise,\end{array}\right.$ (46)

and $N(X)$ is an off-diagonal matrix, whose elements are

$N_{qr}(X)=\left\\{\begin{array}[]{ll}\frac{\partial F_{r}(X)}{\partial
X_{q}}&\quad if\ q\neq r\\\ 0&\quad otherwise.\end{array}\right.$ (47)

To accommodate the scenarios in which the Jacobian may be non-symmetric, we
expand the definition of Jacobian in (45) by rewriting it as

$\bar{J}(X)=D(X)+\bar{N}(X),$ (48)

where $\bar{N}(X)\equiv\frac{1}{2}[N(X)+N^{T}(X)]$.

Finally, we denote the lowest eigenvalue of a square matrix as
$\lambda_{min}(\cdot)$.888For more information on how the lowest eigenvalue of
a game Jacobian allows us to elicit insights on the interplay, neighboring
influences, aggregate effect, see Bramoullé et al. (2014). Clearly, both
$D(X)$ and $\bar{N}(X)$ have real-numbered eigenvalues because they are
symmetric. Now we are ready to present the main results for the uniqueness.

######  Theorem 3 (Sufficient conditions for Uniqueness)

Assuming the condition for Theorem 2 is satisfied, the solution $X^{*}$ to VI
(1) is unique, if

(i) $F$ is strictly monotone on $\mathcal{K}$, or

(ii) $F$ is strongly monotone on $\mathcal{K}$, or

(iii) $\mathfrak{U}$ is strictly concave, and the condition
$\lvert\lambda_{min}(\bar{N}(X))\rvert<\lambda_{min}(D(X))$ holds $\forall
X\in\mathcal{K}$, where $\nabla_{X}\mathfrak{U}(X)=F(X)$.

Proof:

(i) The proof is immediate with Lemma 2 and Theorem 2.

(ii) See the proof of theorem 1.8 in Nagurney (1999).

(iii) Because all cost, incentive, and demand functions in equation (1) are
continuously differentiable, all of their partial derivatives are well
defined. By virtue of proposition 4 in Melo (2018), $F(X)$ is strictly
monotone on $\mathcal{K}$. Combining Lemma 2 and Theorem 2, condition (iii) is
proved. $\square$

Remark

A few observations can be made. First, it is well-known that the monotonicity
conditions in Definition 2 are ranked in the ascending order of "strength"
(Facchinei and Pang, 2003), i.e., (c) implies (b), and (b) implies (a). In
Theorem 3, the uniqueness condition (i) and (iii), in essence, are established
under strict monotonicity, whereas (ii) is under strong monotonicity. Hence,
one can easily infer a relation of "(ii) $\Rightarrow$ (i)", as well as "(ii)
$\Rightarrow$ (iii)" in Theorem 3.

Second, to characterize the uniqueness of variational equilibrium (1), one
could also establish similar sufficient conditions by the semidefiniteness of
$J(X)$ as it pertains to the analogous monotonicity features (cf. Parise and
Ozdaglar (2019); Melo (2018)).

##  6 The algorithm

To solve a VI problem in standard form, we propose an algorithm with
theoretical measures of the result. The algorithm is extragradient method
first proposed by Korpelevich (1976), which is later promoted as the modified
projection method in Nagurney (1999) by setting the step length to 1. The
solution is guaranteed to converge as long as the function $F$ that enters the
standard form is monotone and Lipschitz continuous. The realization of the
algorithm for the cross-sector multi-product scarce resource supply chain
network model is given as follows.

The modified projection method:

Step 0. Initialization

Set
$X^{0}=(Q^{00},Q^{10},Q^{20},\mathfrak{\Delta}^{00},\mathfrak{\Delta}^{10},d^{0},\lambda^{00},\lambda^{10},\lambda^{20},\mu^{00},\mu^{10})\in\mathcal{K}$.
Set $\tau=:1$ and select $\varphi$ such that $0<\varphi\leq 1/L$, where $L$ is
the Lipschitz constant for function $F$.
Step 1. Construction and computation

Compute
$\bar{X}^{\tau-1}=(\bar{Q}^{0\tau},\bar{Q}^{1\tau},\bar{Q}^{2\tau},\bar{\mathfrak{\Delta}}^{0\tau},\bar{\mathfrak{\Delta}}^{1\tau},\bar{d}^{\tau},\bar{\lambda}^{0\tau},\bar{\lambda}^{1\tau},\bar{\lambda}^{2\tau},\bar{\mu}^{0\tau},\bar{\mu}^{1\tau})\in\mathcal{K}$
by solving the VI sub-problem

$\langle\bar{X}^{\tau-1}+\varphi F(X^{\tau-1})-X^{\tau-1},\
X-\bar{X}^{\tau-1}\rangle\geq 0,\quad\forall X\in\mathcal{K}.$ (49)

Step 2. Adaptation

Compute
$X^{\tau}=(Q^{0\tau},Q^{1\tau},Q^{2\tau},\mathfrak{\Delta}^{0\tau},\mathfrak{\Delta}^{1\tau},d^{\tau},\lambda^{0\tau},\lambda^{1\tau},\lambda^{2\tau},\mu^{0\tau},\mu^{1\tau})\in\mathcal{K}$
by solving the VI sub-problem

$\langle X^{\tau}+\varphi F(\bar{X}^{\tau-1})-X^{\tau-1},\
X-X^{\tau}\rangle>0,\quad\forall X\in\mathcal{K}.$ (50)

Step 3. Convergence verification

If $\lvert X^{\tau}-X^{\tau-1}\rvert\leq\epsilon$, for $\epsilon>0$, a pre-
specified tolerance, then, stop; otherwise, set $\tau=:\tau+1$ and go to step
1.

The algorithm converges to the solution of $VI(F,\mathcal{K})$ under the
following conditions.

######  Theorem 4 (Convergence)

Assume that F(X) is monotone, as is in expression (41), and that $F(X)$ is
also Lipschitz continuous, then the modified projection method converges to a
solution of VI (37).

Proof: See Theorem 2.5 in Nagurney (1999).

##  7 Small scale examples

In this section, we construct several numerical cases to illustrate our
model’s utility. Example 1 is an application to the medical PPE glove supply.
Example 2 broadens the application to an interconnected abstract resource-trio
supply chain. The aforementioned algorithm is implemented in MATLAB installed
on an ordinary ASUS VivoBook F510 personal laptop computer with an Intel Core
i5 CPU 2.5 GHz and RAM 8.00 GB. We exhibit and discuss the highlights of each
example.

Example 1.1: A medical gloves supply chain network (benchmark)

The COVID-19 pandemic of 2020 has reportedly caused a demand surge for medical
gloves (Finkenstadt et al., 2020). In a single 100-day wave during early 2020,
the estimated need for medical gloves was 3.939 billion (Toner, 2020),
followed by a subsequent official guideline on conservation and optimizing
usage of gloves during medical practice in the U.S. (CDC, 2020). It has become
clear that the scarcity of medical gloves calls for a boost in the supply
chain to the extent of better coordination and stimulus effort. Commonly used
medical glove materials include latex, made from natural rubber, and nitrile,
made from petroleum-based materials (Anedda et al., 2020; Henneberry, 2020).

This example illustrates a resource-duo supply chain network with natural
rubber and petroleum as the resources, and medical gloves as their end-
products. Specifically, the network contains 2 owners, 2 producers, and 2
retailers, as in Figure 2. The corresponding end-products, latex and nitrile
gloves, are shipped to in 2 demand markets, medical and residential
facilities, via 1 available transportation mode. We use this example as our
benchmark case, in which the supply chain network has unlimited resources and
is imposed no fiscal policies.

Figure 2: A resource-duo network with medical gloves supply chain

We will continue to use the same topology on example 1.2-1.5, with variations
of setting. We examine the output quantities of firms, market prices, and
welfare estimates.

The cost functions are constructed for all $i=1,...,I$ and $j=1,...,J$, with
$I=J=N_{i}=M_{j}=K_{j}=2$ and $T_{j}=1$, as the following.

$\displaystyle
f^{1n}(x^{1n})=2.5(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{1n}_{jm})^{2}+(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{1n}_{jm})(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{2n}_{jm})+2\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{1n}_{jm},$
$\displaystyle
f^{2n}(x^{2n})=0.5(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{2n}_{jm})^{2}+(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{1n}_{jm})(\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{2n}_{jm})+2\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{2n}_{jm},$
$\displaystyle c^{jm}_{s}(x^{jm}_{s})=0.5(x^{jm}_{s})^{2},\qquad
f^{js}(x^{js})=0.1(\sum^{M_{j}}_{m=1}x^{jm}_{s})^{2},\qquad\hat{c}^{js}_{tk}(x^{js}_{tk})=0.01x^{js}_{tk},$
$\displaystyle c^{11}_{11}=0.5(x^{11}_{11})^{2}+3.5x^{11}_{11},\quad
c^{11}_{12}=0.5(x^{11}_{12})^{2}+3.5x^{11}_{12},\quad
c^{12}_{11}=0.5(x^{12}_{11})^{2}+2x^{12}_{11},\quad
c^{12}_{12}=0.5(x^{12}_{12})^{2}+2x^{12}_{12},$ $\displaystyle
c^{21}_{11}=0.4(x^{21}_{11})^{2}+3.5x^{21}_{11},\quad
c^{21}_{12}=0.4(x^{21}_{12})^{2}+3.5x^{21}_{12},\quad
c^{22}_{11}=0.4(x^{22}_{11})^{2}+2x^{22}_{11},\quad
c^{22}_{12}=0.4(x^{22}_{12})^{2}+2x^{22}_{12}.$

All other costs are set to zero. The price-demand functions at the markets
are:

$\displaystyle p^{j}_{3k}(d_{jk})=-d_{jk}+300,\quad\forall j,k.$

In addition, the conversion rates of production by resource producers are
$\psi^{in}_{jm}=0.9$. The weights of resource commodities at the markets are
$w_{11}=0.5,w_{12}=0.5$. The parameters concerning the resource capacities and
policy instruments, $U_{i}$, $A^{i}_{g}$, $B^{i}_{g}$, are all set to a
sufficiently large number, given their absence.

We initialize the algorithm by setting all the flow quantity to be 1, the
step-size $\varphi$ to be 0.01 (unless noted otherwise), the convergence
tolerance $\epsilon$ to be $10^{-4}$. The computation process takes
approximately 2.0 seconds. We display the solution in Table
LABEL:table:eg1.999The complete result of all examples is available at:
https://github.com/Pergamono/SRSCN. Among the equilibrium solution, the zero
values of $\delta^{in}_{g}$ and $\delta^{jm}_{g}$ simply reaffirm the
incentives to be flat-rate, whereas the zero values of all Lagrange
multipliers suggest that the corresponding constraints are inactive.

Table 3: The equilibrium solution of example 1.1 Variable | Result | Variable | Result | Variable | Result | Variable | Result
---|---|---|---|---|---|---|---
$x^{11*}_{11}$ | 8.88 | $x^{11*}_{1}$ | 13.79 | $\delta^{11*}_{1}$ | 0.00 | $\lambda^{0*}_{1}$ | 0.00
$x^{11*}_{12}$ | 8.88 | $x^{11*}_{2}$ | 18.19 | $\delta^{12*}_{1}$ | 0.00 | $\lambda^{0*}_{2}$ | 0.00
$x^{11*}_{21}$ | 9.50 | $x^{12*}_{1}$ | 13.79 | $\delta^{21*}_{1}$ | 0.00 | $\lambda^{1*}_{11}$ | 247.28
$x^{11*}_{22}$ | 9.50 | $x^{12*}_{2}$ | 18.19 | $\delta^{22*}_{1}$ | 0.00 | $\lambda^{1*}_{12}$ | 247.28
$x^{12*}_{11}$ | 8.88 | $x^{21*}_{1}$ | 14.53 | $\delta^{11*}_{1}$ | 0.00 | $\lambda^{1*}_{21}$ | 247.29
$x^{12*}_{12}$ | 8.88 | $x^{21*}_{2}$ | 19.65 | $\delta^{12*}_{1}$ | 0.00 | $\lambda^{1*}_{22}$ | 247.29
$x^{12*}_{21}$ | 9.50 | $x^{22*}_{1}$ | 14.53 | $\delta^{21*}_{1}$ | 0.00 | $\lambda^{2*}_{11}$ | 266.59
$x^{12*}_{22}$ | 9.50 | $x^{22*}_{2}$ | 19.65 | $\delta^{22*}_{1}$ | 0.00 | $\lambda^{2*}_{12}$ | 263.64
$x^{21*}_{11}$ | 8.88 | $x^{11*}_{11}$ | 13.79 | $\mu^{0*}_{111}$ | 0.00 | $\lambda^{2*}_{21}$ | 267.64
$x^{21*}_{12}$ | 8.88 | $x^{11*}_{12}$ | 13.79 | $\mu^{0*}_{121}$ | 0.00 | $\lambda^{2*}_{22}$ | 264.97
$x^{21*}_{21}$ | 9.50 | $x^{12*}_{11}$ | 18.18 | $\mu^{0*}_{211}$ | 0.00 | $d^{*}_{11}$ | 15.98
$x^{21*}_{22}$ | 9.50 | $x^{12*}_{12}$ | 18.18 | $\mu^{0*}_{221}$ | 0.00 | $d^{*}_{12}$ | 15.98
$x^{22*}_{11}$ | 8.88 | $x^{21*}_{11}$ | 14.53 | $\mu^{1*}_{111}$ | 0.00 | $d^{*}_{21}$ | 17.10
$x^{22*}_{12}$ | 8.88 | $x^{21*}_{12}$ | 14.53 | $\mu^{1*}_{121}$ | 0.00 | $d^{*}_{22}$ | 17.10
$x^{22*}_{21}$ | 9.50 | $x^{22*}_{11}$ | 19.67 | $\mu^{1*}_{211}$ | 0.00 |  |
$x^{22*}_{22}$ | 9.50 | $x^{22*}_{12}$ | 19.67 | $\mu^{1*}_{221}$ | 0.00 |  |

Example 1.2: The commons101010A commons is where the natural resources are
owned and shared collectively (Ostrom, 1990). with a resource capacity limit

The natural rubber shipments from the commons of farming and harvesting, which
are concentrated in Southeast Asia, can be severely affected by external
shocks, such as natural disasters, geopolitical shifts, regulations, and
pandemics (Chou, 2020; Lee, 2020). Therefore, it bears merit to investigate
the resilience of these supply chains. We perform a sensitivity study on the
rubber capacity limit, by inheriting all settings from example 1.1, with the
additional imposition of resource capacity on natural rubber. Acknowledging
$U_{1}$’s level when such a limit can be reached, we vary $U_{1}$ between 10
and 100.

Figure 3: Owner profits under a resource capacity

Figure 4: Total profits and welfare under a resource capacity

As the natural rubber’s capacity increases, as expected, each glove producer,
supplier’s profit, as well as the consumer surplus, increases. The petroleum
miners, however, suffer from the abundance of natural rubber, as Figure 4
shows. The total profits or surplus of each supply chain tier are displayed in
Figure 4. It is worth noting that the overall height of each stacked bar
indicates social welfare of the entire network. Interestingly, we find that
social welfare peaks at a $U_{1}$’s level of around 60, likely owing to a
similar trend of the owners’ total profit that peaks at the similar level of
$U_{1}$.

Example 1.3: Who should get the incentive, owners or producers?

In this example, we compare the scenarios in which either the natural rubber
farmers or the latex glove producers receive a fairly small flat-rate
incentive on their production quantity. As such, we inherit all settings from
example 1.1, with the additional imposition of a fiscal policy in the form of
quantity incentives. Specifically, we separately incentivize the natural
rubber farmers and the latex glove producers with a flat-rate of
$\alpha^{1}_{0}$ and $\beta^{1}_{0}$, respectively.

(a) Latex glove market price

(b) Nitrile glove market price

Figure 5: Glove market prices by flat-rate incentive

We elect to examine only the glove prices at the markets, as the rest of the
equilibrium results can be examined invariably. In Figure 5, we observe that
with the flat-rate incentives administered to the latex glove supply chain,
the corresponding latex glove prices at the market are reduced, whereas the
prices of the substitute product, nitrile glove, change in opposite trends.

Figure 6: Total profits and welfare under the flat-rate incentives Figure 7:
Social-welfare gains and policy efficiency under the flat-rate incentives

From the standpoint of the supply chain participants, the farmers, producers,
and consumers each as a tier, benefit mildly, because of the relatively small
amount of incentives. See Figure 6. The suppliers, however, enjoy a
discernible gain in total profit when the incentives are given to the rubber
farmers.

From the standpoint of the incentive administer, it bears meaning to examine
such a fiscal policy’s social benefit and efficiency. We use the Benefit-Cost
ratio, i.e., the dollar amount of social-welfare gain for every $1 incentive
administered to the system, to represent the efficiency. In Figure 7, we show
the social-welfare gains, as well as the efficiency of the incentive. It can
be seen that the efficiency of the incentive given to the resource producer
suffers more severely.

Example 1.4: Regressive vs. flat-rate incentives

In this example, we examine how a regressive incentive policy differs from a
flat-rate one, as well as how the incentive bracket affects the system
performance. Once again, we inherit all settings from example 1.1, plus the
additional imposition of regressive incentive policy, with $\alpha^{1}_{0}=11$
and $\alpha^{1}_{1}=-2.2$ on both natural rubber farmers, to be selected to
provide a sensible comparison against a flat-rate, $\alpha^{1}_{0}=10$,
incentive policy. The only incentive bracket, $A^{1}_{1}$, will be left
varying as a sensitivity study parameter.

Figure 8: Total profits and welfare under the regressive incentives

Figure 9: Social-welfare gains and Benefit-Cost ratio under the regressive
incentives

The range of $A^{1}_{1}$ displayed in Figure 9 and 9 are selected to ensure
each farmer’s total output quantity exceeds such a bracket value for it to
take effect. In Figure 9, we observe that the change of the regressive
incentive bracket does not influence each supply chain tier’s profit and the
consumer welfare significantly. Such a result is consistent with the findings
with respect to the effect of the “tax brackets” on firm’s profits in a study
by Yu et al. (2018). In Figure 9, a mildly increasing gain of social welfare
can be gleaned as the bracket increases, though falling short of the
comparable flat-rate policy. The dollar amount of social benefit, embodied by
the Benefit-Cost ratio, on the other hand, trends up with the bracket level.

Example 1.5: Critical resource shortage relief

In this example, we use a flat-rate incentive to relieve a latex glove
shortage caused by a demand surge at the medical facilities. First, we
construct a distressed supply chain in which the natural rubber emerges to be
a critical shortage in its supplies at the medical facilities. In doing so, we
inherit all settings from example 1.1, except the price-demand function of
latex gloves at medical facilities, which is set as

$\displaystyle p^{1}_{31}(d_{11})=-d_{11}+420.$

Our algorithm returns the result including the shipments from two of the
suppliers to a market, $x^{11}_{12}$ and $x^{12}_{12}$ being 0. To illustrate,
Figure 10(a) displays the topology of this disrupted supply chain, in which
the residential facilities are completely cut from the supply of latex gloves.

(a) A latex glove shortage occurs at residential

(b) Shortage relieved by producer incentives

Figure 10: Critical resource shortage relief through production stimulus

To aid such circumstances, we impose a flat-rate incentive on both latex glove
producers, with $\beta^{1}_{0}=50$, and re-run the model. Immediately, the
previously disrupted supply can be restored. We use Figure 10(b) to display
the recovered supply chain status. Similar results of shortage relief can be
achieved by imposing a flat-rate incentive on both rubber farmers.

Example 2: A mixed fiscal policy in an abstract scarce resource supply chain

In many developed economies, governments tend to be concerned with levels of
income inequality, and are, therefore, interested in making redistribution of
societal wealth a substantial objective for economic development. Thus, it is
meaningful to evaluate the utility of a mixed fiscal policy in redistributing
social welfare. In this example, we construct an abstract resource-trio
network to illustrate a mixed fiscal policy with a combination of incentives
and taxes. In the network topology shown in Figure 11, the red nodes indicate
the firms that are being taxed whereas the blue ones incentivized.

Figure 11: A resource-trio supply chain network with a mixed fiscal policy

The cost functions are constructed, for all $i=1,...,I$ and $j=1,...,J$, with
$I=J=3,N_{i}=M_{j}=K_{j}=2$ and $T_{j}=1$, as the following.

$\displaystyle f^{11}=2.5q_{11}^{2}+q_{11}(q_{21}+q_{31})+2q_{11},\quad
f^{12}=2.5q_{12}^{2}+q_{12}(q_{22}+q_{32})+2q_{12},$ $\displaystyle
f^{13}=2.5q_{13}^{2}+q_{13}(q_{23}+q_{33})+2q_{12},\quad
f^{21}=0.5q_{21}^{2}+q_{21}(q_{11}+q_{31})+2q_{21},$ $\displaystyle
f^{22}=0.5q_{22}^{2}+q_{22}(q_{12}+q_{32})+2q_{22},\quad
f^{23}=0.5q_{23}^{2}+q_{23}(q_{13}+q_{33})+2q_{23},$ $\displaystyle
f^{31}=0.5q_{31}^{2}+q_{31}(q_{11}+q_{21})+2q_{31},\quad
f^{32}=0.5q_{32}^{2}+q_{32}(q_{12}+q_{22})+2q_{32},$ $\displaystyle
f^{33}=0.5q_{33}^{2}+q_{33}(q_{13}+q_{23})+2q_{33},\quad where,\quad
q_{in}=\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in}_{jm};$ $\displaystyle
c^{11}_{1}=0.5(x^{11}_{1})^{2},\quad c^{11}_{2}=0.25(x^{11}_{2})^{2},\quad
c^{12}_{1}=0.5(x^{12}_{1})^{2},\quad c^{12}_{2}=0.25(x^{12}_{2})^{2},$
$\displaystyle c^{21}_{1}=0.5(x^{21}_{1})^{2},\quad
c^{21}_{2}=0.25(x^{21}_{2})^{2},\quad c^{22}_{1}=0.5(x^{22}_{1})^{2},\quad
c^{22}_{2}=0.25(x^{22}_{2})^{2},$ $\displaystyle
c^{31}_{1}=0.5(x^{31}_{1})^{2},\quad c^{31}_{2}=0.25(x^{31}_{2})^{2},\quad
c^{32}_{1}=0.5(x^{32}_{1})^{2},\quad c^{32}_{2}=0.25(x^{32}_{2})^{2};$
$\displaystyle f^{11}=0.1(x^{11}_{1}+x^{12}_{1})^{2},\quad
f^{12}=0.1(x^{11}_{2}+x^{12}_{2})^{2},$ $\displaystyle
f^{21}=0.1(x^{21}_{1}+x^{22}_{1})^{2},\quad
f^{22}=0.1(x^{21}_{2}+x^{22}_{2})^{2},$ $\displaystyle
f^{31}=0.1(x^{31}_{1}+x^{32}_{1})^{2},\quad
f^{32}=0.1(x^{31}_{2}+x^{32}_{2})^{2};$ $\displaystyle
c^{11}_{11}=0.5(x^{11}_{11})^{2}+3.5x^{11}_{11},\quad
c^{11}_{12}=0.5(x^{11}_{12})^{2}+3.5x^{11}_{12},$ $\displaystyle
c^{12}_{11}=0.5(x^{12}_{11})^{2}+2x^{12}_{11},\qquad
c^{12}_{12}=0.5(x^{12}_{12})^{2}+2x^{12}_{12},$ $\displaystyle
c^{21}_{11}=0.4(x^{21}_{11})^{2}+3.5x^{21}_{11},\quad
c^{21}_{12}=0.4(x^{21}_{12})^{2}+3.5x^{21}_{12},$ $\displaystyle
c^{22}_{11}=0.4(x^{22}_{11})^{2}+2x^{22}_{11},\qquad
c^{22}_{12}=0.4(x^{22}_{12})^{2}+2x^{22}_{12},$ $\displaystyle
c^{31}_{11}=0.45(x^{31}_{11})^{2}+3.5x^{31}_{11},\quad
c^{31}_{12}=0.45(x^{31}_{12})^{2}+3.5x^{31}_{12},$ $\displaystyle
c^{32}_{11}=0.45(x^{32}_{11})^{2}+2x^{32}_{11},\qquad
c^{32}_{12}=0.45(x^{32}_{12})^{2}+2x^{32}_{12};$
$\displaystyle\hat{c}^{js}_{tk}(x^{js}_{tk})=0.1x^{js}_{tk}.$

All other costs are set to zero. The price-demand functions at the markets are

$\displaystyle p^{j}_{3k}(d_{jk})=-d_{jk}+300,\quad\forall j,k.$

Similar to example 1-5, we set the production conversion rates
$\psi^{in}_{jm}=0.9$, the market resource commodity weights
$w_{11}=0.5,w_{12}=0.5$, the parameters concerning the resource capacities,
$U_{i}$, sufficiently large. In contrast to previous examples, here we set the
step-size $\varphi$ to be $10^{-5}$, the convergence tolerance $\epsilon$ to
be $6\times 10^{-4}$ for this example.

To examine the efficacy of this mixed fiscal policy, we first establish the
equilibrium of the benchmark scenario, i.e., the setting without such policy.
As such, we present the results in Table LABEL:table:eg6. With the ex-ante
knowledge that the resource owners capture most of the supply chain profit, we
then impose such a policy in which the producers of resource 1 are given a
flat-rate incentive of $\beta^{1}_{0}=12$, whereas the owners of resource 1
and producers of resource 3 are charged a flat-rate tax of
$\alpha^{1}_{0}=-10$ and $\beta^{1}_{0}=-2$, respectively.

The projection method takes approximately 120 seconds for this problem of a
total of 99 variables to converge to the preset tolerance. We include all
equilibrium results in the supplemental file while displaying only the profit-
related outcome in Table LABEL:table:eg6 again. It is worth pointing out that
the "net incentive" is the total taxes collected net of the total incentive
disbursed by the government.

Table 4: Profit and welfare results of example 2 Profit |  | Benchmark | With policy | Welfare |  | Benchmark | With policy
---|---|---|---|---|---|---|---
Owner | $\pi_{11}$ | 2573.50 | 2359.58 | Consumer | $CS_{11}$ | 104.26 | 141.64
Owner | $\pi_{12}$ | 2573.50 | 2700.43 | Consumer | $CS_{12}$ | 120.11 | 97.84
Owner | $\pi_{21}$ | 2573.50 | 2700.43 | Consumer | $CS_{21}$ | 107.24 | 77.30
Owner | $\pi_{22}$ | 2573.50 | 2359.58 | Consumer | $CS_{22}$ | 104.26 | 141.64
Owner | $\pi_{31}$ | 2573.50 | 2700.43 | Consumer | $CS_{31}$ | 120.11 | 97.84
Owner | $\pi_{32}$ | 2573.50 | 2700.43 | Consumer | $CS_{32}$ | 107.24 | 77.30
Producer | $\pi_{11}$ | 248.25 | 340.20 | Tot. owner | $\pi_{total}$ | 15441.00 | 15520.90
Producer | $\pi_{12}$ | 433.20 | 345.63 | Tot. producer | $\pi_{total}$ | 1977.61 | 1963.28
Producer | $\pi_{21}$ | 307.36 | 295.82 | Tot. supplier | $\pi_{total}$ | 1825.59 | 667.13
Producer | $\pi_{22}$ | 248.25 | 340.20 | Tot. consumer | $CS_{total}$ | 663.21 | 633.57
Producer | $\pi_{31}$ | 433.20 | 345.63 | Soc. welfare | $SW$ | 19907.42 | 18784.87
Producer | $\pi_{32}$ | 307.36 | 295.82 | Net Incentive |  | 0.00 | 99.15
Supplier | $\pi_{11}$ | 323.04 | 175.60 | Soc. welfare gain | $\Delta SW$ | 0.00 | -1122.55
Supplier | $\pi_{12}$ | 302.42 | 60.78 | Benefit-to-cost | BC | - | -11.32
Supplier | $\pi_{21}$ | 255.43 | 95.10 |  |  |  |
Supplier | $\pi_{22}$ | 319.75 | 182.51 |  |  |  |
Supplier | $\pi_{31}$ | 291.83 | 58.54 |  |  |  |
Supplier | $\pi_{32}$ | 333.12 | 94.60 |  |  |  |

In Figure 12, we observe that under the mixed fiscal policy, the directly
affected firms, i.e., the top two tiers of the network, do not consistently
respond to the intended policy elements, as there are both increases and
decreases of profit on both the incentivized and taxed firms. On the bottom
two tiers of the network, the supplier profits drop unanimously, whereas the
distribution of consumer surplus across the markets becomes more uneven.

Figure 12: Profits and welfare under a mixed fiscal policy Figure 13: Total
profits and welfare under a mixed fiscal policy

Overall, with a positive net incentive, $99.15, administered to the network,
we observe, however, that social welfare gains a negative amount, owing to the
significant decrease in supplier profits. See Table LABEL:table:eg6 and Figure
13.

##  8 Managerial implications

Government plays a key role in maintaining a balance between the firm’s
profitability, consumer’s well-being, supply chain’s overall health, and
society’s sustainable growth. Yet in the face of widespread resource shortages
that stifle the economic growth in supply chains, balancing the tradeoffs
between the goals has been enormously challenging for governments. The
proposed cross-sector multi-product scarce resource supply chain network model
can serve as a support system for both the supply chain practitioners and
governments to make operational and strategic planning decisions. With a
particular focus on the practice that can be adopted by the government to help
address the insurgent scarcity and achieve overall sustainability, we herein
instill the following managerial implications.

  1. 1.

From the viewpoint of designing a stimulative fiscal policy, the legislator
will likely face a caveat between the choice of a flat-rate policy and a
regressive one. Our analysis shows that the flat-rate incentive generally
performs more effectively than its regressive counterpart when administered to
the resource owners in stimulating supply chain welfare. This result is in
general agreement with a dual study by Yu et al. (2018), in which they pointed
out the strength of flat-rate tax when imposed to discourage adverse
environmental activities. This result is also supported by the advantages of
approximately linear tax in maintaining welfare (Mirrlees, 1971). Furthermore,
as a case in point, we also find that if the legislator’s focus of policy
design shifts to the efficiency of it, i.e., the social-welfare gain on every
dollar of fiscal investment, then a carefully designed two-bracket regressive
incentive can be more advantageous in policy efficiency than its flat-rate
counterpart. That is, with a large enough “bracket”, the legislator may be
able to devise a regressive incentive scheme that can achieve a higher
Benefit-Cost ratio, and almost as high of a net welfare gain as the flat-rate
scheme. Though delicate, the value of such a bracket can be practically
narrowed down by determining the largest effective bracket, followed by
determining the largest achievable net welfare gain.

  2. 2.

For the choice of incentive recipients in a supply chain, i.e., the resource
owners or the resource producers, the decision-makers should anticipate a
tradeoff. When the incentives are administered to the owners, both the social
welfare gain and the policy efficiency will remain at a relatively high level.
When the incentives are administered to the producers, the supplier’s profit
will enjoy the most gains, whereas the market price of the affected products
will be reduced significantly. In practice, the market price of a scarce
resource not only influences the consumer’s behaviors, but also serves as a
key indicator for the broader economy. Under a supply chain disruption, we
suggest that the governments first administer an appropriate amount of
incentives to the producers so that the commodity prices can be reduced at the
market level in short term. We note that, at the same time, any commodities
experiencing shortage at the market level may also be relieved by the same
incentives. Once the cause for the disruption has subsided and the supply
chain’s performance measures stabilized, governments can then reelect the
appropriate supply chain tier(s) as the new recipients of the next round of
incentives, depending on the economic objectives and legislative priorities.

  3. 3.

“Income inequality” is believed to hinder economic growth. Using fiscal policy
to reduce income gaps has become the goal of many advanced economies (Coady
and Gupta, 2012). In our observation, the large profit difference across the
supply chain tiers entices a growth-minded government to pursue a
redistributive strategy for social surplus. It has been found, however, that
the redistribution of welfare among resource owners and producers via a mixed
fiscal policy may result in a net loss of welfare. While we acknowledge that
such a finding resembles the rhetoric of the opposition to the dominant social
policy notion that the resources generated by economic growth should be
redistributed to fund social programs (Midgley, 1999), it is pertinent to note
that our analytical takeaway is derived from a microeconomic framework with
standard assumptions. Nonetheless, we note further that the welfare loss
caused by our experimental mixed fiscal policy has been previously associated
with the elasticity of demand from the classic oligopoly theories (Worcester,
1975). In particular, it is also widely acknowledged in literature that
taxation can be ineffective in reducing wealth inequalities as opposed to what
conventional wisdom would have anticipated (Mirrlees, 1971). As is
demonstrated before, our proposed fiscal policy can effectively relieve
resource shortages and stimulate growth in supply chains. But for the
governments or practitioners who oversee the supply chain grand strategies in
a post-crisis stage, we caution that the well-intended fiscal interventionist
may derail the overall economic sustainability of society. We advise that the
use of a mixed fiscal policy in supply chains should follow the principle of
configuring a mild or less redistributive policy, as is recognized in Coady
and Gupta (2012).

  4. 4.

Finally, we note an interesting link between resource capacity and welfare
within a competitive supply chain. In related literature, Chen and Chou (2008)
imposed firm-wise resource capacity in their supply chain and found that
capacity limit restricts welfare. In both Nagurney et al. (2019b) and Hu and
Qiang (2013), though the shared capacity limit was modeled, neither study
associated such a limit with its impact on social welfare. The current study,
however, proffers further results. We find that the capacity limit of a given
type of scarce (more often, natural) resource does not strictly curtail social
welfare. Rather, an appropriate level of the limit can even benefit the social
outcome. In practice, the ownership and the right to use a natural resource as
a shared public good is often governed by its respective common laws or local
policies111111Public goods has been a widely discussed topic in studies of
economics and law. See Réaume (1988), Holcombe (1997) and the reference
therein.. Thus, if a natural resource of critical importance belongs to a
government’s jurisdiction, it is advisable for the government to legislate and
impose a mild restriction on the usage of such natural resources. The
restriction, if properly selected, will not only preserve the quantity of the
shared resource but also create a “sweet spot” that induces higher social
welfare.

Admittedly, the above managerial implications are based on our stylized
numerical experiments and thus, are limited to the extent of the
characteristics of the network, e.g., the competitive nature of the nodes, the
substitutability of the flows, the mix of policies, etc. Moreover, because of
the number of features incorporated in our model, it is probable that in the
presence of multiple features, their interactions could result in more
profound managerial implications than what we have uncovered. Nonetheless,
practitioners and decision-makers should carefully verify and validate the
premises of the model before expanding the aforementioned insights.

##  9 Conclusions

Prima facie, the munificence of scarce resources is akin to the sustainability
and growth of individual firms, societies, and the flourishing of humanity.
Yet, in this age of intensifying societal changes, shocks, crises, and inter-
connectivity, the conflict and competition for scarce resources and products
have become more fierce. Fiscal policy remains the most common governmental
policy instrument to relieve the shortage of supplies and stimulate economies.
In this paper, our contributions to the literature on scarce resources and
supply chain networks include the following.

We construct the first general decentralized cross-sector scarce resource SCNE
model with a unifying supply-side fiscal policy. We provide a rigorous VI
formulation for the governing equilibrium conditions of the network model.
Such a substitute network provides a versatile tool for the evaluation of
profit, welfare, policy instruments, cost structure, transportation,
conservation, competition, and interdependence of resources throughout the
supply chains. The generality of our model also allows for a variety of
extensions, i.e., dynamics, stochastic features, multi-criteria decision-
making, disequilibrium behaviors, etc, to be furnished.

Second, our model is also a general Nash equilibrium problem. We formulate the
GNE of the network model in VI. There are only a few GNEP studies that
incorporate fiscal policy in the SCNE literature. The utility of this model is
not limited to the scarce resource supply chains, but also eligible to any
resource commodity that pertains to the aforementioned characteristics of
scarce resources.

Third, from a technical aspect, we introduce a recently uncovered approach to
characterize the network equilibrium by adopting a novel set of theoretical
properties, including $\lambda_{min}$. To the best of our knowledge in the
supply chain network literature, such a means of characterization for the
uniqueness property of network equilibrium has yet to appear heretofore.

Lastly, we furnish the model with numerical studies and extract managerial
insights that provide governments, resource owners, and firms useful advice in
expansion, cost restructuring, resource conservation,
competition/collaboration strategy, shortage handling, and post-crisis
stimulation. In particular, we provide guidance on supply-side policy design
and administration in relieving and stimulating the PPE shortage caused by the
COVID-19 global pandemic. Our findings also enrich the political discussion on
public resource legislation, income inequality, and sustainable development.
We anticipate that the extension of this model can shed light on the
stimulation and relief effort on vaccine distribution and economic recovery.

## References

  * Alizamir et al. (2019) Alizamir, S., Iravani, F., and Mamani, H. (2019).  An analysis of price vs. revenue protection: Government subsidies in the agriculture industry.  Management Science, 65(1):32–49.
  * Anedda et al. (2020) Anedda, J., Ferreli, C., Rongioletti, F., and Atzori, L. (2020).  Changing gears: Medical gloves in the era of coronavirus disease 2019 pandemic.  Clinics in Dermatology, 38(6):734–736.
  * Arestis and Sawyer (2003) Arestis, P. and Sawyer, M. (2003).  Reinventing fiscal policy.  Journal of Post Keynesian Economics, 26(1):3–25.
  * Arifoğlu and Tang (2022) Arifoğlu, K. and Tang, C. S. (2022).  A two-sided incentive program for coordinating the influenza vaccine supply chain.  Manufacturing & Service Operations Management, 24(1):235–255.
  * Arrow and Kruz (2013) Arrow, K. J. and Kruz, M. (2013).  Public investment, the rate of return, and optimal fiscal policy.  Taylor & Francis.
  * Bai et al. (2016) Bai, Y., Ouyang, Y., and Pang, J.-S. (2016).  Enhanced models and improved solution for competitive biofuel supply chain design under land use constraints.  European Journal of Operational Research, 249(1):281–297.
  * Bakker et al. (2018) Bakker, C., Zaitchik, B. F., Siddiqui, S., Hobbs, B. F., Broaddus, E., Neff, R. A., Haskett, J., and Parker, C. L. (2018).  Shocks, seasonality, and disaggregation: Modelling food security through the integration of agricultural, transportation, and economic systems.  Agricultural Systems, 164:165–184.
  * Bazilian et al. (2011) Bazilian, M., Rogner, H., Howells, M., Hermann, S., Arent, D., Gielen, D., Steduto, P., Mueller, A., Komor, P., Tol, R. S., and Yumkella, K. K. (2011).  Considering the energy, water and food nexus: Towards an integrated modelling approach.  Energy Policy, 39(12):7896–7906.
  * Belhaj and Deroïan (2013) Belhaj, M. and Deroïan, F. (2013).  Strategic interaction and aggregate incentives.  Journal of Mathematical Economics, 49(3):183–188.
  * Bell et al. (2012) Bell, J. E., Autry, C. W., Mollenkopf, D. A., and Thornton, L. M. (2012).  A natural resource scarcity typology: Theoretical foundations and strategic implications for supply chain management.  Journal of Business Logistics, 33:158–166.
  * Bertsekas and Tsitsiklis (1989) Bertsekas, D. P. and Tsitsiklis, J. N. (1989).  Parallel and Distributed Computation: Numerical Methods.  Prentice Hall.
  * Besik and Nagurney (2017) Besik, D. and Nagurney, A. (2017).  Quality in competitive fresh produce supply chains with application to farmers’ markets.  Socio-Economic Planning Sciences, 60:62–76.
  * Biggs et al. (2015) Biggs, E. M., Bruce, E., Boruff, B., Duncan, J. M. A., Horsley, J., Pauli, N., McNeill, K., Neef, A., Van Ogtrop, F., Curnow, J., and Others (2015).  Sustainable development and the water-energy-food nexus: A perspective on livelihoods.  Environmental Science & Policy, 54:389–397.
  * Blinder (2021) Blinder, A. (2021).  Biden’s Plan Encourages True Supply-Side Economics.  https://www.wsj.com/articles/bidens-plan-encourages-true-supply-side-economics-11621982082.  Accessed: May 1, 2022.
  * Bramoullé et al. (2014) Bramoullé, Y., Kranton, R., and D’amours, M. (2014).  Strategic interaction and networks.  American Economic Review, 104(3):898–930.
  * Braunerhjelm (2022) Braunerhjelm, P. (2022).  Rethinking stabilization policies; Including supply-side measures and entrepreneurial processes.  Small Business Economics, 58:1–21.
  * Calvo-Armengol and Jackson (2004) Calvo-Armengol, A. and Jackson, M. O. (2004).  The effects of social networks on employment and inequality.  American Economic Review, 94(3):426–454.
  * Caniëls and Gelderman (2007) Caniëls, M. C. and Gelderman, C. J. (2007).  Power and interdependence in buyer supplier relationships: A purchasing portfolio approach.  Industrial Marketing Management, 36(2):219–229.
  * CDC (2020) CDC (2020).  Strategies for Optimizing the Supply of Disposable Medical Gloves.  https://www.cdc.gov/coronavirus/2019-ncov/hcp/ppe-strategy/gloves.html.  Accessed: May 1, 2022.
  * Chen et al. (2020) Chen, D., Ignatius, J., Sun, D., Goh, M., and Zhan, S. (2020).  Pricing and equity in cross-regional green supply chains.  European Journal of Operational Research, 280(3):970–987.
  * Chen and Chou (2008) Chen, H.-K. and Chou, H.-W. (2008).  Supply chain network equilibrium problem with capacity constraints.  Papers in Regional Science, 87(4):605–621.
  * Chen and Tang (2015) Chen, Y.-J. and Tang, C. S. (2015).  The economic value of market information for farmers in developing economies.  Production and Operations Management, 24(9):1441–1452.
  * Chintapalli and Tang (2021) Chintapalli, P. and Tang, C. S. (2021).  The value and cost of crop minimum support price: Farmer and consumer welfare and implementation cost.  Management Science, 67(11):6839–6861.
  * Chou (2020) Chou, B. (2020).  Glove Manufacturers: Conserve PPE During Widespread Shortage.  https://www.ehstoday.com/ppe/article/21127107/glove-manufacturers-conserve-ppe-during-widespread-shortage.  Accessed: May 1, 2022.
  * Coady and Gupta (2012) Coady, M. D. and Gupta, M. S. (2012).  Income inequality and fiscal policy (2nd edition).  IMF Staff Discussion Note, 2012(9):37.
  * Corbett (2013) Corbett, M. (2013).  Oil Shock of 1973–74.  https://www.federalreservehistory.org/essays/oil-shock-of-1973-74.  Accessed: May 1, 2022.
  * Cournot (1838) Cournot, A. (1838).  Researches into the mathematical principles of the theory of wealth, English translation.  Macmillan, London.
  * Dafermos and Nagurney (1984) Dafermos, S. and Nagurney, A. (1984).  Sensitivity Analysis for the General Spatial Economic Equilibrium Problem.  Operations Research, 32(5):1069–1086.
  * Davig and Leeper (2011) Davig, T. and Leeper, E. M. (2011).  Monetary–fiscal policy interactions and fiscal stimulus.  European Economic Review, 55(2):211–227.
  * Dutta and Nagurney (2019) Dutta, P. and Nagurney, A. (2019).  Multitiered blood supply chain network competition: Linking blood service organizations, hospitals, and payers.  Operations Research for Health Care, 23:100230.
  * Economist (2021) Economist (2021).  The world economy’s shortage problem.  https://www.economist.com/leaders/2021/10/09/the-world-economys-shortage-problem.  Accessed: May 1, 2022.
  * Facchinei and Kanzow (2007) Facchinei, F. and Kanzow, C. (2007).  Generalized Nash equilibrium problems.  Annals of Operations Research, 175(1):177–211.
  * Facchinei and Pang (2003) Facchinei, F. and Pang, J.-S. (2003).  Finite-dimensional variational inequalities and complementarity problems: Volume I.  Springer, Netherlands.
  * FDIC (1997) FDIC (1997).  Banking and the Agricultural Problems of the 1980s.  History of the Eighties - Lessons for the Future, pages 259–290.
  * Finkenstadt et al. (2020) Finkenstadt, D. J., Handfield, R., and Guinto, P. (2020).  Why the U.S. Still Has a Severe Shortage of Medical Supplies.  Harvard Business Review. Publication Date: September 17, 2020.
  * Frey and Jegen (2001) Frey, B. S. and Jegen, R. (2001).  Motivation crowding theory.  Journal of Economic Surveys, 15(5):589–611.
  * Friedman (2009a) Friedman, G. (2009a).  The Next 100 Years: A Forecast for the 21st Century.  Knopf Doubleday Publishing Group.
  * Friedman (2009b) Friedman, T. L. (2009b).  Hot, flat, and crowded 2.0: Why we need a green revolution-and how it can renew America.  Picador.
  * Gabay and Moulin (1980) Gabay, D. and Moulin, H. (1980).  On the uniqueness and stability of Nash-equilibria in noncooperative games.  In Applied stochastic control in econometrics and management science, pages 271–293. North-Holland Publ. Co., Amsterdam, The Netherlands.
  * Gasparro et al. (2020) Gasparro, A., Smith, J., and Kang, J. (2020).  Grocers Stopped Stockpiling Food. Then Came Coronavirus.  https://www.wsj.com/articles/grocers-stopped-stockpiling-food-then-came-coronavirus-11584982605.  Accessed: May 1, 2022.
  * Gilding (2012) Gilding, P. (2012).  The Great Disruption: How the Climate Crisis Will Transform the Global Economy.  Bloomsbury Publishing.
  * Gourinchas (2020) Gourinchas, P.-O. (2020).  Flattening the pandemic and recession curves.  In Mitigating the COVID Economic Crisis: Act Fast and Do Whatever, page 227. CEPR Press, London.
  * Guoyi et al. (2020) Guoyi, X., Caiquan, D., Yubin, Z., and Yunhui, Z. (2020).  Research on Multi-Period Closed-Loop Supply Chain Network Equilibrium Based on Consumers’ Preference for Products.  International Journal of Information Systems and Supply Chain Management (IJISSCM), 13(4):68–94.
  * Harker (1991) Harker, P. T. (1991).  Generalized Nash games and quasi-variational inequalities.  European Journal of Operational Research, 54(1):81–94.
  * Harker and Pang (1990) Harker, P. T. and Pang, J. S. (1990).  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications.  Mathematical Programming, 48(1-3):161–220.
  * Henneberry (2020) Henneberry, B. (2020).  How to Make Medical Gloves for Coronavirus/COVID-19.  https://www.thomasnet.com/articles/other/how-to-make-medical-gloves/.  Accessed: May 1, 2022.
  * Hoff (2011) Hoff, H. (2011).  Understanding the nexus: Background paper for the Bonn 2011 Nexus Conference.  Technical report, Stockholm University, Stockholm.
  * Holcombe (1997) Holcombe, R. G. (1997).  A theory of the theory of public goods.  The Review of Austrian Economics, 10(1):1–22.
  * Hu et al. (2019) Hu, M., Liu, Y., and Wang, W. (2019).  Socially beneficial rationality: The value of strategic farmers, social entrepreneurs, and for-profit firms in crop planting decisions.  Management Science, 65(8):3654–3672.
  * Hu et al. (2021) Hu, X., Jang, J., Hamoud, N., and Bajgiran, A. (2021).  Strategic Inventories in a Supply Chain with Downstream Cournot Duopoly.  International Journal of Operational Research, 42(4):524–542.
  * Hu and Qiang (2013) Hu, Y. and Qiang, Q. (2013).  An equilibrium model of online shopping supply chain networks with service capacity investment.  Service Science, 5(3):238–248.
  * Huck et al. (2012) Huck, S., Kübler, D., and Weibull, J. (2012).  Social norms and economic incentives in firms.  Journal of Economic Behavior & Organization, 83(2):173–185.
  * Hunt (2000) Hunt, S. D. (2000).  A General Theory of Competition.  Journal of Macromarketing, 20(1):77–81.
  * IMF (2021) IMF (2021).  Fiscal Monitor: Strengthening the Credibility of Public Finances.  Technical report, Washington, October.
  * Jackson and Zenou (2015) Jackson, M. O. and Zenou, Y. (2015).  Games on networks.  In Handbook of game theory with economic applications, volume 4, pages 95–163. Elsevier.
  * Jiang et al. (2021) Jiang, Z.-Z., He, N., and Huang, S. (2021).  Government penalty provision and contracting with asymmetric quality information in a bioenergy supply chain.  Transportation Research Part E: Logistics and Transportation Review, 154:102481.
  * Jones (2021) Jones, Z. (2021).  California records driest year since 1924.  https://www.cbsnews.com/news/california-drought-dry-water-year/.  Accessed: May 1, 2022.
  * Kazaz et al. (2016) Kazaz, B., Webster, S., and Yadav, P. (2016).  Interventions for an artemisinin-based malaria medicine supply chain.  Production and Operations Management, 25(9):1576–1600.
  * Keynes (1936) Keynes, J. M. (1936).  The General Theory of Employment, Interest and Money.  Harcourt, Brace.
  * Kinderlehrer and Stampacchia (1980) Kinderlehrer, D. and Stampacchia, G. (1980).  An Introduction to Variational Inequalities and Their Applications.  SIAM, New York.
  * Korpelevich (1976) Korpelevich, G. M. (1976).  The extragradient method for finding saddle points and other problems.  Ekonomika i Matematicheskie Metody, 12:747–756.
  * Krautkraemer (1998) Krautkraemer, J. A. (1998).  Nonrenewable resource scarcity.  Journal of Economic Literature, 36(4):2065–2107.
  * Krautkraemer (2005) Krautkraemer, J. A. (2005).  Economics of natural resource scarcity: The state of the debate.  Technical report, Resources for the Future, Washington, D.C.
  * Kulkarni and Shanbhag (2012) Kulkarni, A. A. and Shanbhag, U. V. (2012).  On the variational equilibrium as a refinement of the generalized Nash equilibrium.  Automatica, 48(1):45–55.
  * Kurian (2017) Kurian, M. (2017).  The water-energy-food nexus: Trade-offs, thresholds and transdisciplinary approaches to sustainable development.  Environmental Science and Policy, 68:97–106.
  * Lee and Billington (1993) Lee, H. L. and Billington, C. (1993).  Material management in decentralized supply chains.  Operations Research, 41(5):835–847.
  * Lee (2020) Lee, L. (2020).  Malaysia’s Top Glove sees supply shortages boosting latex glove prices.  https://www.reuters.com/article/us-health-coronavirus-top-glove-supplies/malaysias-top-glove-sees-supply-shortages-boosting-latex-glove-prices-idUSKBN2850XX.  Accessed: May 1, 2022.
  * Levi et al. (2022) Levi, R., Singhvi, S., and Zheng, Y. (2022).  Artificial shortage in agricultural supply chains.  Manufacturing & Service Operations Management, 24(2):746–765.
  * Liao et al. (2019) Liao, C.-N., Chen, Y.-J., and Tang, C. S. (2019).  Information provision policies for improving farmer welfare in developing countries: Heterogeneous farmers and market selection.  Manufacturing & Service Operations Management, 21(2):254–270.
  * Long (1973) Long, R. (1973).  Fiscal Policy and The Energy Crisis.  Technical report, the United States Senate, Washington, D.C.
  * Masoumi et al. (2012) Masoumi, A. H., Yu, M., and Nagurney, A. (2012).  A supply chain generalized network oligopoly model for pharmaceuticals under brand differentiation and perishability.  Transportation Research Part E: Logistics and Transportation Review, 48(4):762–780.
  * Matopoulos et al. (2015) Matopoulos, A., Barros, A. C., and van der Vorst, J. G. (2015).  Resource-efficient supply chains: A research framework, literature review and research agenda.  Supply Chain Management, 20(2):218–236.
  * Matsypura et al. (2007) Matsypura, D., Nagurney, A., and Liu, Z. (2007).  Modeling of electric power supply chain networks with fuel suppliers via variational inequalities.  International Journal of Emerging Electric Power Systems, 8(1).
  * Melo (2018) Melo, E. (2018).  A Variational Approach to Network Games.  Working paper, No. 005.2018, Fondazione Eni Enrico Mattei (FEEM), Milano.
  * Midgley (1999) Midgley, J. (1999).  Growth, redistribution, and welfare: Toward social investment.  Social Service Review, 73(1):3–21.
  * Mirrlees (1971) Mirrlees, J. A. (1971).  An exploration in the theory of optimum income taxation.  The Review of Economic Studies, 38(2):175–208.
  * Mun et al. (2021) Mun, K. G., Zhao, Y., and Rafique, R. A. (2021).  Designing hydro supply chains for energy, food, and flood.  Manufacturing & Service Operations Management, 23(2):274–293.
  * Nagurney (1999) Nagurney, A. (1999).  Network Economics: A Variational Inequality Approach.  Kluwer Academic Publishers, Dordrecht, The Netherlands, second edition.
  * Nagurney (2021a) Nagurney, A. (2021a).  Optimization of supply chain networks with inclusion of labor: Applications to COVID-19 pandemic disruptions.  International Journal of Production Economics, 235:108080.
  * Nagurney (2021b) Nagurney, A. (2021b).  Supply chain game theory network modeling under labor constraints: Applications to the Covid-19 pandemic.  European Journal of Operational Research, 293(3):880–891.
  * Nagurney et al. (2019a) Nagurney, A., Besik, D., and Dong, J. (2019a).  Tariffs and quotas in world trade: A unified variational inequality framework.  European Journal of Operational Research, 275(1):347–360.
  * Nagurney et al. (2019b) Nagurney, A., Besik, D., and Li, D. (2019b).  Strict quotas or tariffs? Implications for product quality and consumer welfare in differentiated product supply chains.  Transportation Research Part E: Logistics and Transportation Review, 129:136–161.
  * Nagurney et al. (2002) Nagurney, A., Dong, J., and Zhang, D. (2002).  A supply chain network equilibrium model.  Transportation Research Part E: Logistics and Transportation Review, 38(5):281–303.
  * Nagurney and Dutta (2021) Nagurney, A. and Dutta, P. (2021).  A multiclass, multiproduct Covid-19 convalescent plasma donor equilibrium model.  In Operations Research Forum, volume 2, pages 1–30. Springer.
  * Nagurney et al. (2017) Nagurney, A., Yu, M., and Besik, D. (2017).  Supply chain network capacity competition with outsourcing: a variational equilibrium framework.  Journal of Global Optimization, 69(1):231–254.
  * Nash (1951) Nash, J. (1951).  Non-cooperative games.  Annals of Mathematics, 54:286–295.
  * Nash (1950) Nash, J. F. (1950).  Equilibrium points in n-person games.  Proceedings of the National Academy of Sciences, USA, 36:48–49.
  * Olawuyi (2020) Olawuyi, D. (2020).  Sustainable development and the water-energy-food nexus: Legal challenges and emerging solutions.  Environmental Science & Policy, 103:1–9.
  * Ostrom (1990) Ostrom, E. (1990).  Governing the commons: The evolution of institutions for collective action.  Cambridge university press.
  * Parise and Ozdaglar (2019) Parise, F. and Ozdaglar, A. (2019).  A variational inequality framework for network games: Existence, uniqueness, convergence and sensitivity analysis.  Games and Economic Behavior, 114:47–82.
  * Pfeffer and Salancik (2003) Pfeffer, J. and Salancik, G. R. (2003).  The external control of organizations: A resource dependence perspective.  Stanford University Press.
  * Réaume (1988) Réaume, D. (1988).  Individuals, groups, and rights to public goods.  University of Toronto Law Journal, 38(1):1–27.
  * Romer and Romer (2021) Romer, C. D. and Romer, D. H. (2021).  The fiscal policy response to the pandemic.  Brookings Papers on Economic Activity, page 28.
  * Rosenberg (1973) Rosenberg, N. (1973).  Innovative responses to materials shortages.  The American Economic Review, 63(2):111–118.
  * Sadka (1976) Sadka, E. (1976).  On income distribution, incentive effects and optimal income taxation.  The Review of Economic Studies, 43(2):261–267.
  * Samuelson (1952) Samuelson, P. A. (1952).  Spatial price equilibrium and linear programming.  The American Economic Review, 42(3):283–303.
  * Shen and Sun (2021) Shen, Z. M. and Sun, Y. (2021).  Strengthening supply chain resilience during covid-19: A case study of jd.com.  Journal of Operations Management, pages 1–25. https://doi.org/10.1002/joom.1161.
  * Smith (2021) Smith, N. (2021).  Big Government Is the Answer to America_s Supply Problems.  https://www.bloomberg.com/opinion/articles/2021-06-16/big-government-is-the-answer-to-america-s-supply-problems.  Accessed: May 1, 2022.
  * Srai et al. (2022) Srai, J. S., Joglekar, N., Tsolakis, N., and Kapur, S. (2022).  Interplay between competing and coexisting policy regimens within supply chain configurations.  Production and Operations Management, 31(2):457–477.
  * Takayama and Judge (1964) Takayama, T. and Judge, G. G. (1964).  Spatial equilibrium and quadratic programming.  American Journal of Agricultural Economics, 46(1):67–93.
  * Tao et al. (2020) Tao, Y., Wu, J., Lai, X., and Wang, F. (2020).  Network planning and operation of sustainable closed-loop supply chains in emerging markets: Retail market configurations and carbon policies.  Transportation Research Part E: Logistics and Transportation Review, 144:102131.
  * Toner (2020) Toner, E. (2020).  Interim Estimate of the US PPE Needs for COVID-19.  Technical report, Johns Hopkins Center for Health Security, Baltimore, Maryland.
  * UN (2012) UN (2012).  Renewable Resources and Conflict.  Technical report, United Nations, New York.
  * UN (2019) UN (2019).  World Population Prospects 2019: Highlights.  Technical report, United Nations, Kenya.
  * Valinsky (2021) Valinsky, J. (2021).  Supply chain interrupted: Here’s everything you can’t get now.  https://www.cnn.com/2021/07/31/business/supply-chain-shortages-pandemic-july-2021/index.html.  Accessed: May 1, 2022.
  * Wagner and Light (2002) Wagner, L. A. and Light, H. M. (2002).  Materials in the economy - Material flows, scarcity, and the environment.  U.S. Department of the Interior, U.S. Geological Survey.
  * Weber and Friedrich (1929) Weber, A. and Friedrich, C. J. (1929).  Alfred Weber’s theory of the location of industries.  Italy: University of Chicago Press.
  * Willig (1976) Willig, R. D. (1976).  Consumer’s surplus without apology.  The American Economic Review, 66(4):589–597.
  * Winegarden (2021) Winegarden, W. (2021).  It’s Time For A Supply-Side Resurgence.  https://www.forbes.com/sites/waynewinegarden/2021/05/10/its-time-for-a-supply-side-resurgence.  Accessed: May 1, 2022.
  * Worcester (1975) Worcester, D. A. (1975).  On monopoly welfare losses: Comment.  The American Economic Review, 65(5):1015–1023.
  * Worland (2014) Worland, J. (2014).  California’s Drought Is Now the Worst in 1,200 Years.  https://time.com/3621246/california-drought-study/.  Accessed: May 1, 2022.
  * Wu et al. (2019) Wu, H., Xu, B., and Zhang, D. (2019).  Closed-Loop Supply Chain Network Equilibrium Model with Subsidy on Green Supply Chain Technology Investment.  Sustainability, 11(16):4403.
  * Wu et al. (2006) Wu, K., Nagurney, A., Liu, Z., and Stranlund, J. K. (2006).  Modeling generator power plant portfolios and pollution taxes in electric power supply chain networks: A transportation network equilibrium transformation.  Transportation Research Part D: Transport and Environment, 11(3):171–190.
  * Wu et al. (2018) Wu, X., Nie, L., Xu, M., and Yan, F. (2018).  A perishable food supply chain problem considering demand uncertainty and time deadline constraints: Modeling and application to a high-speed railway catering service.  Transportation Research Part E: Logistics and Transportation Review, 111:186–209.
  * Xu and Choi (2021) Xu, X. and Choi, T.-M. (2021).  Supply chain operations with online platforms under the cap-and-trade regulation: Impacts of using blockchain technology.  Transportation Research Part E: Logistics and Transportation Review, 155:102491.
  * Yang et al. (2021) Yang, Y., Goodarzi, S., Bozorgi, A., and Fahimnia, B. (2021).  Carbon cap-and-trade schemes in closed-loop supply chains: Why firms do not comply?  Transportation Research Part E: Logistics and Transportation Review, 156:102486.
  * Yang et al. (2022) Yang, Y., Goodarzi, S., Jabbarzadeh, A., and Fahimnia, B. (2022).  In-house production and outsourcing under different emissions reduction regulations: An equilibrium decision model for global supply chains.  Transportation Research Part E: Logistics and Transportation Review, 157:102446.
  * Young (2015) Young, M. (2015).  Fiscal instruments and water scarcity.  GGKP Research Committee on Fiscal Instruments, 18:2019.
  * Yu et al. (2020) Yu, J. J., Tang, C. S., Sodhi, M. S., and Knuckles, J. (2020).  Optimal subsidies for development supply chains.  Manufacturing & Service Operations Management, 22(6):1131–1147.
  * Yu et al. (2018) Yu, M., Cruz, J. M., and Li, D. M. (2018).  The sustainable supply chain network competition with environmental tax policies.  International Journal of Production Economics, 217:218–231.
  * Zhang (2006) Zhang, D. (2006).  A network economic model for supply chain versus supply chain competition.  Omega, 34(3):283–295.
  * Zhang et al. (2020) Zhang, G., Dai, G., Sun, H., Zhang, G., and Yang, Z. (2020).  Equilibrium in supply chain network with competition and service level between channels considering consumers’ channel preferences.  Journal of Retailing and Consumer Services, 57:102199.
  * Zhang et al. (2021) Zhang, G., Zhang, X., Sun, H., and Zhao, X. (2021).  Three-Echelon Closed-Loop Supply Chain Network Equilibrium under Cap-and-Trade Regulation.  Sustainability, 13(11):6472.
  * Zhang and Vesselinov (2016) Zhang, X. and Vesselinov, V. V. (2016).  Energy-water nexus: Balancing the tradeoffs between two-level decision makers.  Applied Energy, 183:77–87.

APPENDIX

##  Appendix A Proof of Theorem 1

Proof: First, we prove that an equilibrium according to Definition 1 coincides
with the solution of VI (1). The summation of (3.2), (3.3), (3.4), and (30),
after algebraic simplifications, yields (1).

Next, we prove the converse, that is, a solution to the VI (1) satisfies the
sum of conditions (3.2), (3.3), (3.4), and (30), and thereby, is a cross-
sector multi-product scarce resource SCNE pattern, in accordance with
Definition 1.

In (1), we begin by adding the term
$\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}(p^{in*}_{0jm}-p^{in*}_{0jm})$ to the first
summand expression over $i$ and $n$,
$\sum_{s=1}^{S_{j}}(p^{jm*}_{1s}-p^{jm*}_{1s})$ to the third summand
expression over $j$ and $m$, and lastly, $p^{js*}_{2tk}-p^{js*}_{2tk}$ to the
fifth summand expression over $i$, $s$, $t$, and $k$. Since these terms are
all equal to zero, (1) holds true. Hence, we obtain the following inequality:

$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\Bigg{[}\frac{\partial
f^{in}(x^{in*})}{\partial x^{in}_{jm}}+\frac{\partial
f^{jm}(x^{*}_{jm})}{\partial x^{in}_{jm}}+\frac{\partial
c^{in}_{jm}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}-\frac{\partial\alpha^{i}_{0}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}$
$\displaystyle+\lambda^{0*}_{i}-\psi^{in}_{jm}\lambda^{1*}_{jm}+\sum_{g=1}^{G}\mu^{0*}_{ing}+(p^{in*}_{0jm}-p^{in*}_{0jm})\Bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\alpha^{i}_{g}(\delta^{in*}_{g})}{\partial\delta^{in}_{g}}-\mu^{0*}_{ing}\bigg{]}$
$\displaystyle\times(\delta^{in}_{g}-\delta^{in*}_{g})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\Bigg{[}\dfrac{\partial
f^{js}(x^{js*})}{\partial x^{jm}_{s}}+\frac{\partial
c^{jm}_{s}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}-\frac{\partial\beta^{j}_{0}(x^{jm*}_{s})}{\partial x^{jm}_{s}}$
$\displaystyle+\lambda^{1*}_{jm}-\lambda^{2*}_{js}+\sum_{g=1}^{G}\mu^{1*}_{jmg}+(p^{jm*}_{1s}-p^{jm*}_{1s})\Bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\beta^{j}_{g}(\delta^{jm*}_{g})}{\partial\delta^{jm}_{g}}-\mu^{1*}_{jmg}\bigg{]}$
$\displaystyle\times(\delta^{jm}_{g}-\delta^{jm*}_{g})$
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}\bigg{[}\frac{\partial
c^{js}_{tk}(x^{js*}_{tk})}{\partial
x^{js}_{tk}}+\hat{c}^{js}_{tk}(x^{js*}_{tk})+\lambda^{2*}_{js}+(p^{js*}_{2tk}-p^{js*}_{2tk})\bigg{]}$
$\displaystyle\times(x^{js}_{tk}-x^{js*}_{tk})$
$\displaystyle-\sum^{I}_{j=1}\sum^{K}_{k=1}p^{j}_{3k}(d^{*})$
$\displaystyle\times(d_{jk}-d^{*}_{jk})$ (51)
$\displaystyle+\sum_{i=1}^{I}\bigg{[}U_{i}-\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}x^{in*}_{jm}\bigg{]}$
$\displaystyle\times(\lambda^{0}_{i}-\lambda^{0*}_{i})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}x^{in*}_{jm}\cdot\psi^{in}_{jm}-\sum_{s=1}^{S_{j}}x^{jm*}_{s}\bigg{]}$
$\displaystyle\times(\lambda^{1}_{jm}-\lambda^{1*}_{jm})$
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\bigg{[}\sum_{m=1}^{M_{j}}x^{jm*}_{s}-\sum_{k=1}^{K}\sum_{t=1}^{T_{j}}x^{js*}_{tk}\bigg{]}$
$\displaystyle\times(\lambda^{2}_{js}-\lambda^{2*}_{js})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}A^{i}_{g}-\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in*}_{jm}+\delta^{in*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{0}_{ing}-\mu^{0*}_{ing})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}B^{j}_{g}-\sum^{S_{j}}_{s=1}x^{jm*}_{s}+\delta^{jm*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{1}_{jmg}-\mu^{1*}_{jmg})\geq 0,$
$\displaystyle\forall(Q^{0},Q^{1},Q^{2},\mathfrak{\Delta}^{0},\mathfrak{\Delta}^{1},d,\lambda^{0},\lambda^{1},\lambda^{2},\mu^{0},\mu^{1})\in\mathcal{K}.$

Rearranging (A) yields:

$\displaystyle\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\frac{\partial
f^{in}(x^{in*})}{\partial x^{in}_{jm}}+\frac{\partial
c^{in}_{jm}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}-p^{in*}_{0jm}-\frac{\partial\alpha^{i}_{0}(x^{in*}_{jm})}{\partial
x^{in}_{jm}}$
$\displaystyle+\lambda^{0*}_{i}+\sum_{g=1}^{G}\mu^{0*}_{ing}\bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\alpha^{i}_{g}(\delta^{in*}_{g})}{\partial\delta^{in}_{g}}-\mu^{0*}_{ing}\bigg{]}$
$\displaystyle\times(\delta^{in}_{g}-\delta^{in*}_{g})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\frac{\partial
f^{jm}(x^{*}_{jm})}{\partial
x^{in}_{jm}}+p^{in*}_{0jm}-\psi^{in}_{jm}\lambda^{1*}_{jm}\bigg{]}$
$\displaystyle\times(x^{in}_{jm}-x^{in*}_{jm})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\bigg{[}\frac{\partial
c^{jm}_{s}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}-p^{jm*}_{1s}-\frac{\partial\beta^{j}_{0}(x^{jm*}_{s})}{\partial
x^{jm}_{s}}+\lambda^{1*}_{jm}+\sum_{g=1}^{G}\mu^{1*}_{jmg}\bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{g=1}^{G}\bigg{[}-\frac{\partial\beta^{j}_{g}(\delta^{jm*}_{g})}{\partial\delta^{jm}_{g}}-\mu^{1*}_{jmg}\bigg{]}$
$\displaystyle\times(\delta^{jm}_{g}-\delta^{jm*}_{g})$
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}\bigg{[}\frac{\partial
c^{js}_{tk}(x^{js*}_{tk})}{\partial
x^{js}_{tk}}-p^{js*}_{2tk}+\lambda^{2*}_{js}\bigg{]}$
$\displaystyle\times(x^{js}_{tk}-x^{js*}_{tk})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\sum_{s=1}^{S_{j}}\bigg{[}\dfrac{\partial
f^{js}(x^{js*})}{\partial x^{jm}_{s}}+p^{jm*}_{1s}-\lambda^{2*}_{js}\bigg{]}$
$\displaystyle\times(x^{jm}_{s}-x^{jm*}_{s})$
$\displaystyle+\sum_{j=1}^{I}\sum_{s=1}^{S_{j}}\sum_{t=1}^{T_{j}}\sum_{k=1}^{K}[p^{js*}_{2tk}+\hat{c}^{js}_{tk}(x^{js*}_{tk})]$
$\displaystyle\times(x^{js}_{tk}-x^{js*}_{tk})$
$\displaystyle-\sum^{I}_{j=1}\sum^{K}_{k=1}p^{j}_{3k}(d^{*})$
$\displaystyle\times(d_{jk}-d^{*}_{jk})$ (52)
$\displaystyle+\sum_{i=1}^{I}\bigg{[}U_{i}-\sum_{n=1}^{N_{i}}\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}x^{in*}_{jm}\bigg{]}$
$\displaystyle\times(\lambda^{0}_{i}-\lambda^{0*}_{i})$
$\displaystyle+\sum_{j=1}^{I}\sum_{m=1}^{M_{j}}\bigg{[}\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}x^{in*}_{jm}\cdot\psi^{in}_{jm}-\sum_{s=1}^{S_{j}}x^{jm*}_{s}\bigg{]}$
$\displaystyle\times(\lambda^{1}_{jm}-\lambda^{1*}_{jm})$
$\displaystyle+\sum_{j=I}^{I}\sum_{s=1}^{S_{j}}\bigg{[}\sum_{m=1}^{M_{j}}x^{jm*}_{s}-\sum_{k=1}^{K}\sum_{t=1}^{T_{j}}x^{js*}_{tk}\bigg{]}$
$\displaystyle\times(\lambda^{2}_{js}-\lambda^{2*}_{js})$
$\displaystyle+\sum_{i=1}^{I}\sum_{n=1}^{N_{i}}\sum_{g=1}^{G}\bigg{[}A^{i}_{g}-\sum^{I}_{j=1}\sum^{M_{j}}_{m=1}x^{in*}_{jm}+\delta^{in*}_{g}\bigg{]}$
$\displaystyle\times(\mu^{0}_{ing}-\mu^{0*}_{ing})$