Relief and Stimulus in A Cross-sector Multi-product Scarce Resource Supply Chain Network
RRelief and Stimulus in A Cross-sector Multi-product ScarceResource Supply Chain Network *Xiaowei Hu †a , Peng Li b , and Jaejin Jang ca,c Industrial and Manufacturing Engineering Department, College of Engineering and AppliedScience, University of Wisconsin-Milwaukee, 3200 N Cramer St, Milwaukee, WI 53211 b Department of Supply Chain Management, Rutgers University, 1 Washington Park, Newark, NJ07102
Abstract
In the era of a growing population, systemic change of the world, and rising risk of crises, humanity has beenfacing an unprecedented challenge of resource scarcity. Confronting and addressing the issues concerning the scarceresource’s conservation, competition, and stimulation by grappling their characters and adopting viable policy in-struments calls the decision-makers’ attention to a paramount priority. In this paper, we develop the first generaldecentralized cross-sector supply chain network model that captures the unique features of the scarce resources underfiscal-monetary policies. We formulate the model as a network equilibrium problem with finite-dimensional varia-tional inequality theories. We then characterize the network equilibrium with a set of classic theoretical properties,as well as some novel properties (with λ min ) that are new to the literature of network games application. Lastly, weprovide a series of illustrative examples, including a medical glove supply chain, to showcase how our model can beused to investigate the efficacy of the imposed policies in relieving the supply chain distress and stimulating welfare.Our managerial insights encompass the industry profit and social benefit vis-à-vis the resource availability and policyinstrument design. Keywords : networks; resource scarcity; game theory; supply chain management; variational inequalities * This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. † Corresponding author. Email: [email protected] a r X i v : . [ ec on . T H ] J a n Introduction
Humanity depends on the supply of an array of essential resources to survive. These resources, such as agricul-tural crops, fisheries, wildlife, forests, petroleum, metals, minerals, even air, soil, and water, are not only critical tothe flourishing of humanities but also considered assets for the operations of modern businesses (Rosenberg, 1973;Krautkraemer, 2005). As is pointed out by Population Reference Bureau, supply, demand, and structure can all induceresource scarcity (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 industrializationin the past two centuries has shifted the paradigm, by ending the “ever-cheaper” commodity resources and setting thefierce competition and rising prices for “ever-scarcer” resources as the new norm (Cf. Krautkraemer (1998)). Further-more, traumatic crises also induce the scarcity of specific supplies. For instance, in the COVID-19 global pandemic of2020, the demand for Personal Protective Equipment (PPE) and foods surges to the full capacity of supply chain (seeMurray (2020), Ranney et al. (2020), and WHO (2020), inter alia).As such, meeting the resource demands has becomes a substantive economic, commercial, and societal matterfor the decision-makers. For example, raw materials and mineral shortages have driven the product design initiatives(Duclos et al., 2010), and are becoming more of a strategic challenge for business and government (Wagner and Light,2002). In recent years, many popular books (see Friedman (2009a,b), Gilding (2012), inter alia) have highlighted theimpact of resource scarcity on potential societal collapse and humanity trajectory.A significant amount of scholarly and popular interest lies in the interconnectedness of scarce resources sincethe release of the World Economic Forum report introduced the water-energy-food (WEF) nexus as a novel concept(Hoff, 2011). Nearly all WEF related literature that we are aware of has sought integrated solutions or suggestedthe interconnection of resources. Hence, among the themes of WEF literature, the competition and interconnectionfor scarce resources, as well as the complex forms of the political negotiation across nations calls for the integratedsolutions (Daher and Mohtar, 2015).The broader studies on resource scarcity lie primarily in the field of economics and management, each of whichshapes the character of the scarce resources from a different angle. On one hand, the equilibrium and monopolytheories of Adam Smith and John Stuart Mill, and the rent and location theories of Alfred Marshall and Alfred Webertie the scarcity to its location and quality of resources needed for industrial and agricultural purposes (Weber andFriedrich, 1929). On the other hand, among the rising organizational theories, the resource dependence theory (Pfefferet al., 1978; Pfeffer and Salancik, 2003) claimed the importance of resource by which the owners, producers, andsuppliers are connected and interdependent (cf. Cannon and Perreault (1999), Caniëls and Gelderman (2007), interalia), whereas the resource advantage theory (Hunt, 2000) posited resources’ features, including demand heterogeneity,as well as their roles in a company. Overall, from an economic perspective, scarce resources can be characterized ashigh market power, whose markets are oligopolistic, and whose products are heterogeneous, low price-elasticity, andnon-substitution. From a management perspective, the possession of scarce resources indicates a firm’s competitiveadvantage and negotiation power. The coalescence of these two avenues of studies in resource scarcity has yet to occurin literature.Much of the current interdisciplinary research on scarce resources is seen in supply chain management (SCM),a proliferated scholarly area. The research works have focused on green strategies and sustainability (e.g. Pagell andWu (2009), Piercy and Rich (2015), Abdul-Rashid et al. (2017), inter alia), as well as meeting demand crises (Jüttnerand Maklan, 2011; Mehrotra et al., 2020; Rowan and Laffey, 2020). While most of the literature in SCM focuseson either a category of general products or one specific type, it remains largely sparse on the inter-dependence offirms on specific scarce resources (cf. Bell et al. (2012)). A systematic literature review by Matopoulos et al. (2015)has underscored a need for further research on understanding the implications of resource scarcity for supply chainrelationships and also its impact on supply chain configurations.In this paper, we construct a general supply chain network equilibrium model to better understand the scarceresources vis-à-vis conservation, competition, transportation, and policy intervention. The supply chain network ap-proach, initiated by Nagurney et al. (2002), permits one to represent the interactions between decision-makers in thesupply chain for general commodities in terms of network connections, flows, and prices. In our model, the supply For the scope, mission, significance, and critics of the concept, see Bazilian et al. (2011), Biggs et al. (2015), Endo et al. (2017), Cai et al.(2018), and the reference cited therein. In literature, different permutations of "W-E-F", with the possible addition of "L" (for Land) and "S" (forSystem) have been frequently used to represent the same concept. The term supply chain management is attributed to Oliver and Webber (1982), but the original focus tends to be on a firm acting independentlywith an emphasis on inventory management (e.g., Hu et al. (2020), inter alia.)
In the network, there are I types of scarce resources, denoted as i = , ..., I . Each type of resource is affiliated with N owners, M producers, and S suppliers. The unfinished or end-product throughout the entire supply chain network isassumed to be homogeneous by the types of resources. Later, when referring to a certain type of resource, we willomit the phrase “type” or “type of”. We assign i ’s alias j in order to use it in different tiers. Naturally, j = , ..., I .The first tier of nodes in the network represents the resource owner. For instance, a soybean-producing farmer isan owner of a resource type "soybean products". As the stake-holder of a scarce resource, an owner may supply suchresource to any producer. A typical resource owner is denoted as ( i , n ) , where n = , ..., N .The second tier of nodes represents the resource producer. An example of a resource producer can be a foodmanufacturer 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 selland ship its products to the suppliers who are specialized in the same type of resource. A typical resource producer isdenoted as ( j , m ) , where m = , ..., M .The third tier of nodes represents the resource supplier. A resource supplier can only obtain and supply one typeof resource to all markets. A soybean supplier, for instance, does not supply any other resources. A typical resourcesupplier is denoted as ( j , s ) , where s = , ..., S .The fourth tier of nodes represents the markets. A typical market, consuming some resources from their suppliers,is denoted as k , where k = , ..., K . 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 resource3esource i Resource 1 Resource I ResourceOwnerResourceProducerResourceSupplierMarket 1 , , n , N i , i , n i , N I , I , n I , N , , m , M j , j , m j , M I , I , m I , M , , s , S j , j , s j , S I , I , s I , S k K . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . T . . . t . . . T Figure 1: A cross-sector scarce resource supply chain network modelfrom different suppliers or transportation modes. We assume also all markets to be competitive, and are, 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, giventhe simultaneous best response of all competitors. All of the notations regarding the model are listed in Table 1.Table 1: Notations of the modelNotation DefinitionIndices: i , j The index of resource. j is an alias of i . i , j ∈ { , ..., I } .. n The index of resource owner. n ∈ { , ..., N } m The index of resource producer. m ∈ { , ..., M } s The index of resource supplier. s ∈ { , ..., S } t The index of transportation mode. s ∈ { , ..., S } k The index of market. k ∈ { , ..., K } g The index of incentive payment bracket. g ∈ { , ..., G } Parameters: U i The total amount of resource i . A ig The g ’th incentive payment bracket of resource owners ( i , n ) B jg The g ’th incentive payment bracket of resource producers ( j , m ) ψ injm 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 injm The flow between resource owner ( i , n ) and producer (j,m). x jms The flow between resource producer ( j , m ) and supplier ( j , s ) .4 jstk 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 injm , x jms , x jstk , respectively. d jk Demand of resource i at market k . p in jm The transaction price between resource owner ( i , n ) to producer ( j , m ) . p jm s The transaction price between resource producer ( j , m ) to supplier ( j , s ) . p js tk The sales price by supplier ( j , s ) for market k via transportation mode t . p js sk The transaction price at market k from supplier ( j , s ) via transportation mode t . δ ing Resource owner ( i , n ) ’s excess of output quantity to bracket g . δ jmg Resource producer ( j , m ) ’s excess of output quantity to bracket g . λ i The Lagrange multipliers associated with constraint (10). λ jm The Lagrange multipliers associated with constraint (21). λ js The Lagrange multiplier associated with constraints (29). µ ing The Lagrange multipliers associated with constraint (11). µ jmg The Lagrange multipliers associated with constraint (22). π in , π jm , π 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: α ig ( . ) The incentive payment to resource owners in bracket g . β jg ( . ) 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 injm = c injm ( x injm ) 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 jms = c jms ( x jms ) 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 jstk = c jstk ( x jstk ) The transaction cost incurred by supplier ( j , s ) for market k via transportation mode t .ˆ c jstk = ˆ c jstk ( x jstk ) The transaction cost incurred by market k with supplier ( j , s ) via transportation mode t . p j k = p j k ( d jk ) The price-demand function of resource j at the market k . Fiscal and monetary policies are popular strategic instruments for governments to relieve resource shortages andstimulate growths, especially during the critical times. Early in 2009, the U.S. Congress passed the $787 billionAmerican Recovery and Reinvestment Act, in addition to the $125 billion provided by the Economic Stimulus Actof 2008, to relieve the financial crisis (Davig and Leeper, 2011). In the COVID-19 pandemic of 2020, world-widegovernments adopted economic packages including fiscal, monetary, and financial policy measures to mitigate thenegative effects from the public health crisis on the economy and to sustain public welfare (Gourinchas, 2020). Thescholarly interest of monetary-fiscal policies such as economic incentives is often seen in behavioral economics. Ina decentralized supply chain network, where individuals and firms maximize their own objectives subject to theirown private constraints, the design of fiscal policies becomes a more delicate and complex matter, comparing to thatin a centralized one (cf. Arrow and Kruz (2013)). There have been attempts on using microeconomic analyses toassess the interplay of incentives and behaviors (see Kandel and Lazear (1992), Barron and Gjerde (1997), Frey andJegen (2001), Huck et al. (2012), inter alia). Incentives have been linked to the structure and design of networks (cf.Belhaj and Deroïan (2013), Jackson and Zenou (2015)). For instance, Calvo-Armengol and Jackson (2004) showthat education subsidies and other labor market regulation policies display local increasing returns due to the networkstructure. Yu et al. (2018) captured different environmental tax policies in a supply chain network competition context.Nonetheless, the theoretical literature of supply chain networks has been limited in assessing the efficacy of economicpolicies, a topic that would have provided invaluable managerial insights and should, therefore, merit attentions.In our model, a monetary incentive is applied to resource owners and producers. We design a piece-wise schemeto represent some of the most commonly used governmental incentive policies.We now define the incentive scheme. Let A ig denote the cutoff bracket of the incentive on resource i , with g enumerating between 1 , ... G . Let δ ing denote the resource owner ( i , n ) ’s output quantity excess to the bracket A ig . We5ssume the incentives given to each entity to be a series of linear functions, only affected by its output quantity. Assuch, we define the incentive payment function as α i ( . ) + G ∑ g = α ig ( . ) , (1)where α i ( . ) is usually non-negative, both α i ( . ) and α ig ( . ) are linear. For an arbitrary output quantity of x , the piece-wise payment scheme for the resource owners can be expressed as:If 0 ≤ x ≤ A i , then the incentive payment is α i ( x ) ;If A i ≤ x ≤ A i , then δ in = x − A i , and the incentive payment is α i ( x ) + α i ( δ in ) ;If A i ≤ x ≤ A i , then δ in = x − A i , and the incentive payment is α i ( x ) + α i ( δ in ) + α i ( δ in ) ;If A ig ≤ x ≤ A ig + , then δ ing = x − A ig , and the incentive payment is α i ( x ) + α i ( δ in ) + α i ( δ in ) + ... + α ig ( δ ing ) ;If x > A iG , then the incentive payment is α i ( x ) + α i ( δ in ) + α i ( δ in ) + ... + α ig ( δ inG ) = α i ( x ) + ∑ Gg = α ig ( δ ing ) .Finally, the general incentive payment function of the resource owners in terms of δ ing can be written as α i ( x ) + G ∑ g = α ig ( δ ing ) , i = , ..., I , n = , ..., N , (2)where, δ ing = max { x − A ig , } , g = , ..., G . (3)In particular, when α i ( . ) is set to be a constant and α ig ( . ) to be negative, expression (2) represents a regressiveincentive A typical resource i has a total capacity of U i , shared by all N owners of the same resource. The quantity of theresource the owner ( i , n ) can output should abide by the following flow capacity. N ∑ n = I ∑ j = M ∑ m = x injm ≤ U i , ∀ i . (4)We use Q , a column vector, to group all x injm from (4).An owner incurs an operating cost of f in = f in ( x in ) , ∀ i , n , (5)where x in ≡ ( x in , ..., x injm , ..., x inIM ) , and a transaction cost of c injm = c injm ( x injm ) , ∀ i , n , j , m . (6)The incentive payment received by the resource owner ( i , n ) is α i ( I ∑ j = M ∑ m = x injm ) + G ∑ g = α ig ( δ ing ) , ∀ i , n , (7)where, δ ing = max (cid:110) I ∑ j = M ∑ m = x injm − A ig , (cid:111) , ∀ g = , ..., G . (8)We group all δ ing into a column vector (cid:1) and assume all of the cost functions above, i.e., f in , c injm , to be continuouslydifferentiable. A regressive incentive is one whose marginal rate decreases (cf. Mirrlees (1971); Sadka (1976)). ( i , n ) to producer ( j , m ) is denoted by p in jm . We further denote theequilibrium of such price as p in ∗ jm . Hence, the total revenue of each owner is ∑ Ij = ∑ Mm = p in jm x injm .As a profit-maximizer, each typical owner ( i , n ) faces the following problem.Maximize: π in = I ∑ j = M ∑ m = p in jm x injm − f in ( x in ) − I ∑ j = M ∑ m = c injm ( x injm ) + α i ( I ∑ j = M ∑ m = x injm ) + G ∑ g = α ig ( δ ing ) (9)Subject to: N ∑ n = I ∑ j = M ∑ m = x injm ≤ U i , (10) I ∑ j = M ∑ m = x injm − δ ing ≤ A ig , ∀ g , (11) x injm ≥ , ∀ j , m , δ ing ≥ , ∀ g . In this decentralized supply chain, the resource owners compete in a non-cooperative fashion, a behavior of classicCournot-Nash (Cournot, 1838; Nash, 1950, 1951) stating that each owner determines the optimal output, given thesimultaneously optimal ones of all competitors. As such, the optimality condition expressed in variational inequality (VI) is: I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f in ( x in ∗ ) ∂ x injm + ∂ c injm ( x in ∗ jm ) ∂ x injm − p in ∗ jm − ∂ α i ( x in ∗ jm ) ∂ x injm (cid:35) × ( x injm − x in ∗ jm )+ I ∑ i = N ∑ n = G ∑ g = (cid:34) − ∂ α ig ( δ in ∗ g ) ∂ δ ing (cid:35) × ( δ ing − δ in ∗ g ) ≥ , ∀ ( Q , (cid:1) ) ∈ K , (12)where, K ≡ { ( Q , (cid:1) ) | Q ∈ R I MN + , (cid:1) ∈ R ING + , and ( ) , ( ) holds ∀ i , n } . Condition (12) proffers a readily interpretable mathematical form from an economic perspective. A resource ownerwill ship a positive amount of the product to a resource producer if the price that the producer is willing to pay for theproduct is exactly equal to, the owner’s marginal cost less the marginal incentive benefit. If the owner’s marginal costless the marginal incentive benefit, exceeds what the retailer is willing to pay for the product, then there will not beany shipments from the owner to the producer.Alternatively, (12) can be reformulated as general Nash equilibrium problem (GNEP). In doing so, we first, denote λ i and µ ing as the Lagrange multipliers associated with inequality constraints (10) and (11), respectively. Next, wegroup all λ i and µ ing into a column vector λ and µ , with λ ∈ R I + and µ ∈ R ING + , respectively. Now we reformulate(12) as: I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f in ( x in ∗ ) ∂ x injm + ∂ c injm ( x in ∗ jm ) ∂ x injm − p in ∗ jm − ∂ α i ( x in ∗ jm ) ∂ x injm + λ ∗ i + G ∑ g = µ ∗ ing (cid:35) × ( x injm − x in ∗ jm )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) − ∂ α ig ( δ in ∗ g ) ∂ δ ing − µ ∗ ing (cid:21) × ( δ ing − δ in ∗ g ) + I ∑ i = (cid:20) U i − N ∑ n = I ∑ j = M ∑ m = x in ∗ jm (cid:21) × ( λ i − λ ∗ i ) (13) + I ∑ i = N ∑ n = G ∑ g = (cid:20) A ig − I ∑ j = M ∑ m = x in ∗ jm + δ in ∗ g (cid:21) × ( µ ing − µ ∗ ing ) ≥ , ∀ ( Q , (cid:1) , λ , µ ) ∈ K , where, K ≡ { ( Q , (cid:1) , λ , µ ) | ( Q , (cid:1) , λ , µ ) ∈ R I MN + ING + I + } . Eaves (1971), Gabay and Moulin (1980) were among some of the earliest landmark works to establish the equivalence between the optimizationproblem and variational inequality problem. See also, Bazaraa et al. (1993); Nagurney (1999). Formally introduced by Debreu (1952), GNEP has been frequently used in common-pool resource studies as a mathematical-economic tool.For more information on GNEP, see Harker (1991), Facchinei and Kanzow (2007). For the details of such reformulation technique, see Bertsekas and Tsitsiklis (1989). , share a common pool of a resource.Moreover, with a VI formulation of the network equilibrium, we can resort to a set of well-established theoreticalproperties (more in Section 4) and algorithmic schemes than with a formulation by quasi-inequality, a formalism forGNEP (Facchinei and Kanzow, 2007). A typical resource producer is denoted as ( j , m ) . We introduce a non-negative conversion rate of production ψ injm . Thequantity of resource commodity produced by producer ( j , m ) must satisfy the following flow conservation: S ∑ s = x jms ≤ I ∑ i = N ∑ n = x injm · ψ injm , ∀ j , m . (14)We group all x jms from (14) into a column vector Q , with Q ∈ R IMS + . We assume that the producer’s operatingcost 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 ) , ∀ j , m , (15)where, x jm ≡ ( x jm , ..., x injm , ..., x INjm ) . The producer’s transaction cost is associated with the flow between each pair of producer and supplier c jms = c jms ( x jms ) , ∀ j , m , s . (16)Similar to the incentive scheme for the resource owners, we let B jg denote the cutoff bracket of the incentives, with g enumerating between 1 , ... G . Let δ jmg denote the excess of output quantity within bracket B jg . As such, we define theresource producer’s incentive payment function as β j ( . ) + G ∑ g = β jg ( . ) , (17)Specifically, the incentive payment received by the resource producer ( j , m ) is a function of its total output quantity,i.e., β j ( S ∑ s = x jms ) + G ∑ g = β jg ( δ jmg ) , ∀ j , m , (18)where, δ jmg = max (cid:110) S ∑ s = x jms − B jg , (cid:111) , ∀ g = , ..., G . (19)We group all δ jmg into a column vector (cid:1) , and assume f jm , c jms to be continuously differentiable, β j , and β jg to belinear.The price of products shipped from resource producer ( j , m ) to supplier ( j , s ) is denoted by p jm s . We further denotethe equilibrium of such price as p jm ∗ s . Hence, the total revenue of a producer is ∑ Ss = p jm s x jms .As a profit-maximizer, each typical producer ( j , m ) faces the following problem.Maximize: π jm = S ∑ s = p jm ∗ s x jms − I ∑ j = N ∑ n = p in ∗ jm x injm − f jm ( x jm ) − S ∑ s = c jms ( x jms ) + β j ( S ∑ s = x jms ) + G ∑ g = β jg ( δ jmg ) (20)Subject to: I ∑ i = N ∑ n = x injm · ψ injm ≥ S ∑ s = x jms , (21) It is worth noting that, the solution to problem (12) without constraint (10) and (11) is also a variational equilibrium. To the best of our knowledge, other than existence and local uniqueness theory, the theoretical study of GNEP remains partial (cf. Facchineiand Kanzow (2007), Dreves (2016)). ∑ s = x jms − δ jmg ≤ B jg , ∀ g , (22) x jms ≥ , ∀ s , δ jmg ≥ , ∀ g . The optimality condition is I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f jm ( x ∗ jm ) ∂ x injm + p in ∗ jm (cid:35) × ( x injm − x in ∗ jm ) + I ∑ j = M ∑ m = G ∑ g = (cid:34) − ∂ β jg ( δ jm ∗ g ) ∂ δ jmg (cid:35) × ( δ jmg − δ jm ∗ g )+ I ∑ j = M ∑ m = S ∑ s = (cid:34) ∂ c jms ( x jm ∗ s ) ∂ x jms − p jm ∗ s − ∂ β j ( x jm ∗ s ) ∂ x jms (cid:35) × ( x jms − x jm ∗ s ) ≥ , (23) ∀ ( Q , Q , (cid:1) ) ∈ K , where K ≡ { ( Q , Q , (cid:1) ) | Q ∈ R I MN + , Q ∈ R IMS + , (cid:1) ∈ R IMG + , and ( ) , ( ) holds ∀ j , m } . Similar to (12), condition (23) can also be reformulated as GNEP. We first, denote λ jm and µ jmg as the Lagrangemultipliers associated with inequality constraints (21) and (22), respectively. Next, we group all λ jm and µ jmg into acolumn vector λ and µ , with λ ∈ R IM + and µ ∈ R IMG + , respectively. Now we reformulate (23) as: I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f jm ( x ∗ jm ) ∂ x injm + p in ∗ jm − ψ injm λ ∗ jm (cid:35) × ( x injm − x in ∗ jm )+ I ∑ j = M ∑ m = S ∑ s = (cid:34) ∂ c jms ( x jm ∗ s ) ∂ x jms − p jm ∗ s − ∂ β j ( x jm ∗ s ) ∂ x jms + λ ∗ jm + G ∑ g = µ ∗ jmg (cid:35) × ( x jms − x jm ∗ s ) (24) + I ∑ j = M ∑ m = G ∑ g = (cid:20) − ∂ β jg ( δ jm ∗ g ) ∂ δ jmg − µ ∗ jmg (cid:21) × ( δ jmg − δ jm ∗ g ) + I ∑ j = M ∑ m = G ∑ g = (cid:20) B jg − S ∑ s = x jm ∗ s + δ jm ∗ g (cid:21) × ( µ jmg − µ ∗ jmg )+ I ∑ j = M ∑ m = (cid:20) I ∑ i = N ∑ n = x in ∗ jm · ψ injm − S ∑ s = x jm ∗ s (cid:21) × ( λ jm − λ ∗ jm ) ≥ , ∀ ( Q , Q , (cid:1) , λ , µ ) ∈ K , where, K ≡ { ( Q , Q , (cid:1) , λ , µ ) | ( Q , Q , (cid:1) , λ , µ ) ∈ R I MN + IMS + IMG + IM + } . A typical resource supplier is denoted as ( j , s ) . Each supplier has t modes to transport the resources to Market k . Thesupplier’s strategic variable is x jstk . The output quantity of resource commodity by the supplier ( j , s ) must abide by thefollowing flow conservation: K ∑ k = T ∑ t = x jstk ≤ M ∑ m = x jms , ∀ j , s . (25)We group all x jstk from (25) into a column vector Q , with Q ∈ R ISTK + . We assume the supplier incurs an operatingcost that is associated with the resource product flow from the producers, i.e., f js = f js ( x js ) , (26)where, x js ≡ ( x j s , ..., x jms , ..., x jMs ) , ∀ j , s . The supplier also incurs a transaction cost with each outgoing shipment tothe markets: c jstk = c jstk ( x jstk ) . (27)We assume f js and c jstk to be continuously differentiable.The price of goods from resource supplier ( j , s ) to market k via transportation mode t is denoted by p js tk . Wefurther denote the equilibrium of such price as p js ∗ tk . Hence, the total revenue of each supplier is ∑ Tt = ∑ Kk = p js tk x jstk .9s a profit-maximizer, each typical supplier ( j , s ) faces the following problem.Maximize: π js = T ∑ t = K ∑ k = p js ∗ tk x jstk − M ∑ m = p jm ∗ s x jms − f js ( x js ) − T ∑ t = K ∑ k = c jstk ( x jstk ) (28)Subject to: K ∑ k = T ∑ t = x jstk ≤ M ∑ m = x jms , (29) x jstk ≥ , ∀ t , k . The optimality condition is I ∑ j = I S ∑ s = T ∑ t = K ∑ k = (cid:20) ∂ c jstk ( x js ∗ tk ) ∂ x jstk − p js ∗ tk (cid:21) × ( x jstk − x js ∗ tk ) + I ∑ j = M ∑ m = S ∑ s = (cid:20) ∂ f js ( x js ∗ ) ∂ x jms + p jm ∗ s (cid:21) × ( x jms − x jm ∗ s ) ≥ , ∀ ( Q , Q ) ∈ K , (30)where K ≡ { ( Q , Q ) | Q ∈ R IMS + , Q ∈ R ISTK + , and ( ) holds ∀ j , s } . Again, (30) can also be reformulated as GNEP. By denoting λ js as the Lagrange multiplier associated with in-equality constraints (29), and grouping all λ js into a column vector λ with λ ∈ R IS + , we reformulate (30) as: I ∑ j = I S ∑ s = T ∑ t = K ∑ k = (cid:34) ∂ c jstk ( x js ∗ tk ) ∂ x jstk − p js ∗ tk + λ ∗ js (cid:35) × ( x jstk − x js ∗ tk )+ I ∑ j = M ∑ m = S ∑ s = (cid:20) ∂ f js ( x js ∗ ) ∂ x jms + p jm ∗ s − λ ∗ js (cid:21) × ( x jms − x jm ∗ s ) (31) + I ∑ j = I S ∑ s = (cid:34) M ∑ m = x jm ∗ s − K ∑ k = T ∑ t = x js ∗ tk (cid:35) × ( λ js − λ ∗ js ) ≥ , ∀ ( Q , Q , λ ) ∈ K , where, K ≡ { ( Q , Q , λ ) | ( Q , Q , λ ) ∈ R IMS + ISTK + IS + } . Scarce resources, such as clean water, energies, foods, industrial materials, are consumed at demand markets withunique needs. A market decides the purchase quantity from suppliers based on its willingness to pay. As such, weadopt the classic spatial price equilibrium model to represent the markets.For an arbitrary market k ( k = , ..., K ), without loss of generality, we assign a set of ex ante weight w ts as itsdemand for resource j ( j = , ..., I ). Hence, the demand for resource j at market k is d jk = T ∑ t = S ∑ s = w ts · x jstk , ∀ j , k . (32)Each of such demand corresponds with a market price p j k in the form of price-demand function p j k = p j k ( d jk ) . (33)Because the scarce resource is considered to be demand-heterogeneous, each resource in each market shall havea unique demand function, as is in (32). Further, we group all d jk from (32) into a column vector d , i.e., d ≡ ( d , ..., d jk , ..., d JK ) , where d ∈ R IK + . We further denote the equilibrium of such demands as d ∗ jk and d ∗ , respectively.Market k incurs a transaction cost from supplier ( j , s ) via transportation mode t ˆ c jstk = ˆ c jstk ( x jstk ) , ∀ j , s , k , t . (34) Spatial Price Equilibrium (SPE) model appeared in the early literature of Enke (1951); Samuelson (1952); Takayama and Judge (1964), interalia, and was thereafter extended with VI theories by Dafermos and Nagurney (1984).
10e assume ˆ c jstk to be continuously differentiable.The equilibrium condition is given by p js ∗ tk + ˆ c js ∗ tk ( x js ∗ tk ) (cid:40) = p j k ( d ∗ ) i f x js ∗ tk > > p j k ( d ∗ ) i f x js ∗ tk = ∀ j , s , t , k . (35)The intuition of equation (35) is, that the consumption of resource at the markets will remain positive if thesupplier’ purchase price of such 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, thenthere will be no shipments between the supplier and the market.The spatial price equilibrium in a VI form can be written as I ∑ j = S ∑ s = T ∑ t = K ∑ k = [ p js ∗ tk + ˆ c jstk ( x js ∗ tk )] × ( x jstk − x js ∗ tk ) − I ∑ j = K ∑ k = p j k ( d ∗ ) × ( d − d ∗ ) ≥ , ∀ ( Q , d ) ∈ K , (36)where, K ≡ { ( Q , d ) | Q ∈ R ISTK + , d ∈ R IK + } . To assess the social impact of the model elements in our supply chain network, we provide an estimate of consumersurplus and social welfare. The consumer surplus, termed to measure the aggregate of consumers’ benefits uponpurchasing a product (cf. Willig (1976)) of resource j at demand market k is given by: CS jk = (cid:90) d ∗ jk p j k ( z ) dz − p j ∗ k d ∗ jk , ∀ j , k . (37)We also denote social welfare of the network to be the aggregation of all firm profits and consumer surplus, i.e., SW = I ∑ i = N ∑ n = π in + I ∑ j = M ∑ m = π jm + I ∑ j = S ∑ s = π js + I ∑ j = K ∑ k = CS jk , ∀ i , j , n , m , s , k . (38) Definition 1 (Cross-sector multi-product scarce resource supply chain network equilibrium)
A product flow pattern ( Q ∗ , Q ∗ , Q ∗ , (cid:1) ∗ , (cid:1) ∗ , d ∗ , λ ∗ , λ ∗ , λ ∗ , µ ∗ , µ ∗ ) ∈ K is said to constitute a cross-sectormulti-product scarce resource supply chain network equilibrium if it satisfies condition (13), (24), (31), and (36). Theorem 1
The equilibrium conditions governing the cross-sector multi-product scarce resource supply chain network, accordingto Definition 1, are equivalent to the solution to the following VIs:Determine ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ K , satisfying I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f in ( x in ∗ ) ∂ x injm + ∂ f jm ( x ∗ jm ) ∂ x injm + ∂ c injm ( x in ∗ jm ) ∂ x injm − ∂ α i ( x in ∗ jm ) ∂ x injm + λ ∗ i − ψ injm λ ∗ jm + G ∑ g = µ ∗ ing (cid:35) × ( x injm − x in ∗ jm )+ I ∑ j = M ∑ m = S ∑ s = (cid:34) ∂ f js ( x js ∗ ) ∂ x jms + ∂ c jms ( x jm ∗ s ) ∂ x jms − ∂ β j ( x jm ∗ s ) ∂ x jms + λ ∗ jm − λ ∗ js + G ∑ g = µ ∗ jmg (cid:35) × ( x jms − x jm ∗ s )+ I ∑ j = I S ∑ s = T ∑ t = K ∑ k = (cid:34) ∂ c jstk ( x js ∗ tk ) ∂ x jstk + ˆ c jstk ( x js ∗ tk ) + λ ∗ js (cid:35) × ( x jstk − x js ∗ tk )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) − ∂ α ig ( δ in ∗ g ) ∂ δ ing − µ ∗ ing (cid:21) × ( δ ing − δ in ∗ g ) + I ∑ j = M ∑ m = G ∑ g = (cid:20) − ∂ β jg ( δ jm ∗ g ) ∂ δ jmg − µ ∗ jmg (cid:21) × ( δ jmg − δ jm ∗ g ) (39)11 I ∑ j = K ∑ k = p j k ( d ∗ ) × ( d − d ∗ ) + I ∑ i = (cid:20) U i − N ∑ n = I ∑ j = M ∑ m = x in ∗ jm (cid:21) × ( λ i − λ ∗ i )+ I ∑ j = M ∑ m = (cid:20) I ∑ i = N ∑ n = x in ∗ jm · ψ injm − S ∑ s = x jm ∗ s (cid:21) × ( λ jm − λ ∗ jm ) + I ∑ j = I S ∑ s = (cid:20) M ∑ m = x jm ∗ s − K ∑ k = T ∑ t = x js ∗ tk (cid:21) × ( λ js − λ ∗ js )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) A ig − I ∑ j = M ∑ m = x in ∗ jm + δ in ∗ g (cid:21) × ( µ ing − µ ∗ ing ) + I ∑ j = M ∑ m = G ∑ g = (cid:20) B jg − S ∑ s = x jm ∗ s + δ jm ∗ g (cid:21) × ( µ jmg − µ ∗ jmg ) ≥ , ∀ ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ K , where, K ≡ K × K × K × K = (cid:8) ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) | ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ R ( I MN + ING + IMG + IMS + ISTK )+ I ( + M + S + K )+ (cid:9) . Proof:
First, we prove that an equilibrium according to Definition 1 coincides with the solution of VI (39). Thesummation of (13), (24), (31), and (36), after algebraic simplifications, yields (39).Next, we prove the converse, that is, a solution to the VI (39) satisfies the sum of conditions (13), (24), (31),and (36), and thereby, is a cross-sector multi-product scarce resource supply chain network equilibrium pattern, inaccordance with Definition 1.In (39), we begin by adding the term ∑ Ij = ∑ Mm = ( p in ∗ jm − p in ∗ jm ) to the first summand expression over i and n , ∑ Ss = ( p jm ∗ s − p jm ∗ s ) to the third summand expression over j and m , and lastly, p js ∗ tk − p js ∗ tk to the fifth summand ex-pression over i , s , t , and k . Since these terms are all equal to zero, (39) holds true. Hence, we obtain the followinginequality: I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:34) ∂ f in ( x in ∗ ) ∂ x injm + ∂ f jm ( x ∗ jm ) ∂ x injm + ∂ c injm ( x in ∗ jm ) ∂ x injm − ∂ α i ( x in ∗ jm ) ∂ x injm + λ ∗ i − ψ injm λ ∗ jm + G ∑ g = µ ∗ ing + ( p in ∗ jm − p in ∗ jm ) (cid:35) × ( x injm − x in ∗ jm )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) − ∂ α ig ( δ in ∗ g ) ∂ δ ing − µ ∗ ing (cid:21) × ( δ ing − δ in ∗ g )+ I ∑ j = M ∑ m = S ∑ s = (cid:34) ∂ f js ( x js ∗ ) ∂ x jms + ∂ c jms ( x jm ∗ s ) ∂ x jms − ∂ β j ( x jm ∗ s ) ∂ x jms + λ ∗ jm − λ ∗ js + G ∑ g = µ ∗ jmg + ( p jm ∗ s − p jm ∗ s ) (cid:35) × ( x jms − x jm ∗ s )+ I ∑ j = M ∑ m = G ∑ g = (cid:20) − ∂ β jg ( δ jm ∗ g ) ∂ δ jmg − µ ∗ jmg (cid:21) × ( δ jmg − δ jm ∗ g )+ I ∑ j = I S ∑ s = T ∑ t = K ∑ k = (cid:20) ∂ c jstk ( x js ∗ tk ) ∂ x jstk + ˆ c jstk ( x js ∗ tk ) + λ ∗ js + ( p js ∗ tk − p js ∗ tk ) (cid:21) × ( x jstk − x js ∗ tk ) − I ∑ j = K ∑ k = p j k ( d ∗ ) × ( d − d ∗ ) (40) + I ∑ i = (cid:20) U i − N ∑ n = I ∑ j = M ∑ m = x in ∗ jm (cid:21) × ( λ i − λ ∗ i )+ I ∑ j = M ∑ m = (cid:20) I ∑ i = N ∑ n = x in ∗ jm · ψ injm − S ∑ s = x jm ∗ s (cid:21) × ( λ jm − λ ∗ jm )+ I ∑ j = I S ∑ s = (cid:20) M ∑ m = x jm ∗ s − K ∑ k = T ∑ t = x js ∗ tk (cid:21) × ( λ js − λ ∗ js ) I ∑ i = N ∑ n = G ∑ g = (cid:20) A ig − I ∑ j = M ∑ m = x in ∗ jm + δ in ∗ g (cid:21) × ( µ ing − µ ∗ ing )+ I ∑ j = M ∑ m = G ∑ g = (cid:20) B jg − S ∑ s = x jm ∗ s + δ jm ∗ g (cid:21) × ( µ jmg − µ ∗ jmg ) ≥ , ∀ ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ K . Rearranging (40) yields: I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:20) ∂ f in ( x in ∗ ) ∂ x injm + ∂ c injm ( x in ∗ jm ) ∂ x injm − p in ∗ jm − ∂ α i ( x in ∗ jm ) ∂ x injm + λ ∗ i + G ∑ g = µ ∗ ing (cid:21) × ( x injm − x in ∗ jm )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) − ∂ α ig ( δ in ∗ g ) ∂ δ ing − µ ∗ ing (cid:21) × ( δ ing − δ in ∗ g )+ I ∑ i = N ∑ n = I ∑ j = M ∑ m = (cid:20) ∂ f jm ( x ∗ jm ) ∂ x injm + p in ∗ jm − ψ injm λ ∗ jm (cid:21) × ( x injm − x in ∗ jm )+ I ∑ j = M ∑ m = S ∑ s = (cid:20) ∂ c jms ( x jm ∗ s ) ∂ x jms − p jm ∗ s − ∂ β j ( x jm ∗ s ) ∂ x jms + λ ∗ jm + G ∑ g = µ ∗ jmg (cid:21) × ( x jms − x jm ∗ s )+ I ∑ j = M ∑ m = G ∑ g = (cid:20) − ∂ β jg ( δ jm ∗ g ) ∂ δ jmg − µ ∗ jmg (cid:21) × ( δ jmg − δ jm ∗ g )+ I ∑ j = I S ∑ s = T ∑ t = K ∑ k = (cid:20) ∂ c jstk ( x js ∗ tk ) ∂ x jstk − p js ∗ tk + λ ∗ js (cid:21) × ( x jstk − x js ∗ tk )+ I ∑ j = M ∑ m = S ∑ s = (cid:20) ∂ f js ( x js ∗ ) ∂ x jms + p jm ∗ s − λ ∗ js (cid:21) × ( x jms − x jm ∗ s )+ I ∑ j = S ∑ s = T ∑ t = K ∑ k = [ p js ∗ tk + ˆ c jstk ( x js ∗ tk )] × ( x jstk − x js ∗ tk ) − I ∑ j = K ∑ k = p j k ( d ∗ ) × ( d − d ∗ ) (41) + I ∑ i = (cid:20) U i − N ∑ n = I ∑ j = M ∑ m = x in ∗ jm (cid:21) × ( λ i − λ ∗ i )+ I ∑ j = M ∑ m = (cid:20) I ∑ i = N ∑ n = x in ∗ jm · ψ injm − S ∑ s = x jm ∗ s (cid:21) × ( λ jm − λ ∗ jm )+ I ∑ j = I S ∑ s = (cid:20) M ∑ m = x jm ∗ s − K ∑ k = T ∑ t = x js ∗ tk (cid:21) × ( λ js − λ ∗ js )+ I ∑ i = N ∑ n = G ∑ g = (cid:20) A ig − I ∑ j = M ∑ m = x in ∗ jm + δ in ∗ g (cid:21) × ( µ ing − µ ∗ ing )+ I ∑ j = M ∑ m = G ∑ g = (cid:20) B jg − S ∑ s = x jm ∗ s + δ jm ∗ g (cid:21) × ( µ jmg − µ ∗ jmg ) ≥ , ∀ ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ K . Clearly, (41) is the sum of the optimality condition (12), (23), (30), and (36), and thereby, is, according to Definition13, a cross-sector multi-product scarce resource supply chain network equilibrium pattern. (cid:3)
It should be noted that variable p in ∗ jm , p jm ∗ s , and p js ∗ tk do not appear within the formulation of Theorem 1. They areendogenous to the model and can be retrieved once the solution is obtained. To retrieve p js ∗ tk , for all j , s , t , k , recallequilibrium condition (35). Since p j k ( d ∗ ) is readily available from (33), if x js ∗ tk > j , s , t , k , then p js ∗ tk can beobtained by the equality p js ∗ tk = p j ∗ k ( d ∗ ) − ˆ c jstk ( x js ∗ tk ) . (42)Invariably, if x jm ∗ s > j , m , s , then from the second summand in (31), one may immediately obtain p jm ∗ s = λ ∗ js . (43)And, if x in ∗ jm > i , n , j , m , then from the first summand of (24), p in ∗ jm = ψ injm λ ∗ jm − ∂ f jm ( x ∗ jm ) ∂ x injm . (44) We provide a few classic theoretical properties of the solution to VI (39), based on Rosen (1965), Gabay and Moulin(1980), Nagurney (1999), and Melo (2018), inter alia. In particular, we derive existence and uniqueness results byincorporating strategic measure of network games from the latest theoretical advancement.To facilitate the development in this section, we rewrite VI problem (39) in standard form as follows: determine X ∗ ∈ K satisfying (cid:104) F ( X ∗ ) , X − X ∗ (cid:105) ≥ , ∀ X ∗ ∈ K , (45)where, X ≡ ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) , with an indulgence of notation, F ( X ) ≡ ( F Q , F Q , F Q , F (cid:1) , F (cid:1) , F d , F λ , F λ , F λ , F µ , F µ ) , in which each component of F is given by each respective summand expression in (39),and, K ≡ K × K × K × K . The notation (cid:104)· , ·(cid:105) represents the inner product in a Euclidean space. Both F and X are N -dimensional column vectors, where N = ( I MN + ING + IMG + IMS + IST K ) + I ( + M + S + K ) .First, we provide the existence properties. While F in (45) is continuous, the feasible set K is not compact. Thiscauses the existence condition of (39) not readily available. But one can impose a weak condition to guarantee theexistence of a solution pattern, as in Nagurney et al. (2002). Let K b ≡ (cid:8) ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) | ≤ Q ≤ b , ≤ Q ≤ b , ≤ Q ≤ b , ≤ (cid:1) ≤ b , ≤ (cid:1) ≤ b , ≤ d ≤ b , ≤ λ ≤ b , (46)0 ≤ λ ≤ b , ≤ λ ≤ b , ≤ µ ≤ b , ≤ µ ≤ b (cid:9) , where b = ( b , b , b , b , b , b , b , b , b , b , b ) ≥ Q ≤ b , Q ≤ b , Q ≤ b , (cid:1) ≤ b , (cid:1) ≤ b , d ≤ b , λ ≤ b , λ ≤ b , λ ≤ b , µ ≤ b , µ ≤ b means that x injm ≤ b , x jms ≤ b , x jstk ≤ b , δ ing ≤ b , δ jmg ≤ b , d jk ≤ b , λ i ≤ b , λ jm ≤ b , λ ≤ b , µ ing ≤ b , µ jmg ≤ b for all i , j , n , m , t , k . Then K b is a bounded, closed, convex subsetof R N + . Tangentially, the existence of b is sensible from an economic perspective. Thus, the VI expression (cid:104) F ( X b ) , X − X b (cid:105) ≥ , ∀ X b ∈ K b (47)admits at least one solution. Following Kinderlehrer and Stampacchia (1980), and Nagurney et al. (2002), it is straight-forward to establish: Lemma 1
The VI (45) admits a solution if and only if there exists a b > such that the VI (47) admits a solution in K b , withQ < b , Q < b , Q < b , (cid:1) < b , (cid:1) < b , d < b , (48) λ < b , λ < b , λ < b , µ < b , µ < b . In VI literature, theoretical properties often refer to the existence, uniqueness, stability, and approximation, etc. heorem 2 (Existence) The VI problem (39) admits a least one solution in K b . Proof:
By virtue of theorem 3.1 in Harker and Pang (1990), one can easily verify that K b is non-empty, compact, andconvex, and that the mapping F representing (39) is continuous. Therefore, there exists a solution to the problem (39)in K b . (cid:3) Next, we provide a set of sufficient conditions for the uniqueness properties. In general, uniqueness is oftenassociated 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 : K ⊆ R n → R n is said to be(a) monotone on K if (cid:104) F ( X ) − F ( X ) , X − X (cid:105) ≥ , ∀ X , X ∈ K . (49) (b) strictly monotone on K , if (cid:104) F ( X ) − F ( X ) , X − X (cid:105) > , ∀ X , X ∈ K , X (cid:54) = X . (50) (c) strongly monotone on K , if (cid:104) F ( X ) − F ( X ) , X − X (cid:105) > α (cid:13)(cid:13) X − X (cid:13)(cid:13) , ∀ X , X ∈ K , (51) where α > . Definition 3 (Lipschitz Continuity) F ( X ) is Lipschitz Continuous on K , if (cid:104) F ( X ) − F ( X ) , X − X (cid:105) ≥ L (cid:13)(cid:13) X − X (cid:13)(cid:13) , ∀ X , X ∈ K , (52) 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 K , then the solution to the V I ( F , K ) problem is unique, if one exists. Furthermore, we need to adopt a set of notations to characterize the nature of the network. Following Melo (2018),we define the game Jabocian of F ( X ) as an N -by- N matrix J ( X ) ≡ [ ∇ q F r ( X )] q , r = (cid:34) ∂ F r ( X ) ∂ X q (cid:35) q , r , ∀ q , r = , ..., N , X ∈ R N + . (53)Decomposing the Jacobian in terms of its diagonal and off-diagonal element yields J ( X ) = D ( X ) + N ( X ) , (54)where D ( X ) is a diagonal matrix, whose elements are D qr ( X ) = (cid:40) ∂ F r ( X ) ∂ X q i f q = r otherwise , (55)and N ( X ) is an off-diagonal matrix, whose elements are N qr ( X ) = (cid:40) ∂ F r ( X ) ∂ X q i f q (cid:54) = r otherwise . (56) A common approach to prove existence of an variational equilibrium is via coercive assumptions of F (e.g., Nagurney (1999)). The 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.
15o accommodate the scenarios in which the Jacobian may be non-symmetric, we expand the definition of Jacobianin (54) by rewriting it as ¯ J ( X ) = D ( X ) + ¯ N ( X ) , (57)where ¯ N ( X ) ≡ [ N ( X ) + N T ( X )] .Finally, we denote the lowest eigenvalue of a square matrix as λ min ( · ) . Clearly, both D ( X ) and ¯ 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 (39) is unique, if(i) F is strictly monotone on K , or(ii) F is strongly monotone on K , or(iii) U is strictly concave, and the condition | λ min ( ¯ N ( X )) | < | λ min ( D ( X )) | holds ∀ X ∈ K , where ∇ X 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 (39) are continuously differentiable, all of theirpartial derivatives are well defined. By virtue of proposition 4 in Melo (2018), F ( X ) is strictly monotone on K .Combining Lemma 2 and Theorem 2, condition (iii) is proved. (cid:3) Remark
A few observations can be made. First, it is well-known that the monotonicity conditions in Definition 2 are rankedin the ascending order of "strength" (Facchinei and Pang, 2003), i.e., (c) implies (b), and (b) implies (a). In Theorem3, the uniqueness condition (i) and (iii), in essence, are established under strict monotonicity, whereas (ii) is understrong monotonicity. Hence, one can easily infer a relation of "(ii) ⇒ (i)", as well as "(ii) ⇒ (iii)" in Theorem 3.Second, to characterize the uniqueness of variational equilibrium (39), one could also establish similar sufficientconditions by the semidefiniteness of J ( X ) as it pertains analogous monotonicity features (cf. Parise and Ozdaglar(2019); Melo (2018)). To solve a VI problem in standard form, we propose an algorithm with theoretical measures of the result. The algorithmis extragradient method first proposed by Korpelevich (1976), which is later promoted as the modified projectionmethod in Nagurney (1999) by setting the step length to 1. The solution is guaranteed to converge as long as thefunction F that enters the standard form is monotone and Lipschitz continuous. The realization of the algorithm forthe cross-sector multi-product scarce resource supply chain network model is given as follows. The modified projection method:
Step 0. InitializationSet X = ( Q , Q , Q , (cid:1) , (cid:1) , d , λ , λ , λ , µ , µ ) ∈ K . Set τ = : 1 and select ϕ such that 0 < ϕ ≤ / L ,where L is the Lipschitz constant for function F .Step 1. Construction and computationCompute ¯ X τ − = ( ¯ Q τ , ¯ Q τ , ¯ Q τ , ¯ (cid:1) τ , ¯ (cid:1) τ , ¯ d τ , ¯ λ τ , ¯ λ τ , ¯ λ τ , ¯ µ τ , ¯ µ τ ) ∈ K by solving the VI sub-problem (cid:104) ¯ X τ − + ϕ F ( X τ − ) − X τ − , X − ¯ X τ − (cid:105) ≥ , ∀ X ∈ K . (58)Step 2. AdaptationCompute X τ = ( Q τ , Q τ , Q τ , (cid:1) τ , (cid:1) τ , d τ , λ τ , λ τ , λ τ , µ τ , µ τ ) ∈ K by solving the VI sub-problem (cid:104) X τ + ϕ F ( ¯ X τ − ) − X τ − , X − X τ (cid:105) > , ∀ X ∈ K . (59)Step 3. Convergence verificationIf | X τ − X τ − | ≤ ε , for ε >
0, a pre-specified tolerance, then, stop; otherwise, set τ = : τ + V I ( F , K ) under the following conditions. For 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). heorem 4 (Convergence) Assume that F(X) is monotone, as is in expression (50), and that F ( X ) is also Lipschitz continuous, then the modifiedprojection method converges to a solution of VI (45). Proof: see Theorem 2.5 in Nagurney (1999).
In this section, we construct several numerical cases to illustrate our model’s utility. Example 1 is an application to themedical PPE glove supplies and example 2 broadens the application to an interconnected abstract resource-trio supplychain. The aforementioned algorithm is implemented in MATLAB installed on an ordinary ASUS VivoBook F510personal laptop computer with an Intel Core i5 CPU 2.5 GHz and RAM 8.00 GB. We release all computational resultsin the supplemental data file, while exhibiting and discussing only the highlights in each of the examples.
Example 1.1: A medical gloves supply chain network (benchmark)
With an estimated 47.2 million cases through September 30th (Reese et al., 2020), the COVID-19 pandemic of2020 has caused a demand surge of medical gloves (Finkenstadt et al., 2020; Zubrow, 2020). In a single 100-daywave, as of April 18th, 2020, the estimated need for medical gloves was 3.939 billion (Toner, 2020), followed bya subsequent official guideline on conservation and optimizing usage of gloves during medical practice(CDC, 2020;Mahmood et al., 2020). It has become clear that the scarcity of medical gloves calls for a boost in the supply chainto the extent of better coordination and stimulus effort. Commonly used medical glove materials include latex, madefrom 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.See Figure 2. The corresponding end-products, latex and nitrile gloves, are shipped to in 2 demand markets, medicaland residential facilities, via 1 available transportation mode. We use this example as our benchmark case, in whichthe supply chain network has unlimited resources and is imposed no fiscal or monetary policies. We will continueFigure 2: A resource-duo network with medical gloves supply chainto 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 = , ..., I , j = , ..., J , n = , ..., N , m = , ..., M , k = , ..., K , and t = , ..., T , in which I = J = N = M = K = T =
1, as the following. f n ( x n ) = . ( J ∑ j = M ∑ m = x njm ) + ( J ∑ j = M ∑ m = x njm )( J ∑ j = M ∑ m = x njm ) + J ∑ j = M ∑ m = x njm , n ( x n ) = . ( J ∑ j = M ∑ m = x njm ) + ( J ∑ j = M ∑ m = x njm )( J ∑ j = M ∑ m = x njm ) + J ∑ j = M ∑ m = x njm , c jms ( x jms ) = . ( x jms ) , f js ( x js ) = . ( M ∑ m = x jms ) , c = . ( x ) + . x , c = . ( x ) + . x , c = . ( x ) + x , c = . ( x ) + x , c = . ( x ) + . x , c = . ( x ) + . x , c = . ( x ) + x , c = . ( x ) + x , ˆ c jstk ( x jstk ) = . x jstk . All other costs are set to zero. The price-demand functions at the markets are: ρ j k ( d jk ) = − d jk + , ∀ j , k . In addition, the conversion rates of production by resource producers are ψ injm = .
9. The weights of resourcecommodities at the markets are w = . , w = .
5. The parameters concerning the resource capacities and policyinstruments, U i , A ig , B ig , 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 ϕ to be 0.01 (unless noted other-wise), the convergence tolerance ε to be 10 − . The computation process takes approximately 2.0 seconds. We displaythe solution in Table (2). Among the equilibrium solution, the zero values of δ ing and δ jmg simply reaffirm the incen-tives to be flat-rate, whereas the zero values of all Lagrange multipliers suggest that the corresponding constraints areinactive. Table 2: The equilibrium solution of example 1.1Variable Result Variable Result Variable Result Variable Result x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ δ ∗ λ ∗ x ∗ x ∗ µ ∗ λ ∗ x ∗ x ∗ µ ∗ λ ∗ x ∗ x ∗ µ ∗ d ∗ x ∗ x ∗ µ ∗ d ∗ x ∗ x ∗ µ ∗ d ∗ x ∗ x ∗ µ ∗ d ∗ x ∗ x ∗ µ ∗ x ∗ x ∗ µ ∗ Example 1.2: The commons with a resource capacity limit The natural rubber shipments from the commons of farming and harvesting in Southeast Asia can be easily affectedby external shocks, such as natural disasters, geopolitical shifts, regulations, and pandemics (Chou, 2020; Lee, 2020).Therefore, it bears merit to investigate how the shipment limits can impact the entire supply chain. As such, weperform a sensitivity study on the rubber capacity limit, by inheriting all settings from example 1.1, with the additionalimposition of resource capacity on natural rubber. Acknowledging U ’s level when such a limit can be reached, wevary U between 10 and 100. As the natural rubber’s capacity increases, as expected, each glove producer, supplier’s The complete result of all examples is available at: https://github.com/Pergamono/SRSCN. A commons is where the natural resources are owned and shared collectively (Ostrom, 1990). U ’s level of around 60, owing to a similar trend of the owners’ total profit that peaksat the similar level of U . 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 producersreceive a fairly small flat-rate incentive on their production quantity. As such, we inherit all settings from example1.1, with the additional imposition of a monetary policy in the form of quantity incentives. Specifically, we separatelyincentivize the natural rubber farmers and the latex glove producers with a flat-rate of α and β , respectively.(a) Latex glove market price (b) Nitrile glove market priceFigure 5: Glove market prices by flat-rate incentiveWe select and examine only the glove prices at the markets, as the rest of the equilibrium results can be examinedinvariably. In Figure 5, we observe that with the flat-rate incentives administered to either the farmers or producersof the latex glove supply chain, the latex glove prices at the market are reduced, whereas the prices of the substituteproduct, nitrile glove, change inconsistently. It is worth noting, given the parameters of this example, that the gloveprices in either medical or residential facilities turn out to be coincidentally equal.19igure 6: Total profits and welfare under the flat-rate incentivesFigure 7: Social-welfare gains and policy efficiencies under flat-rate incentivesFrom the standpoint of the supply chain tiers, the farmers, producers, and consumers each as a group, benefitmildly, because of the relatively small amount of incentives. See Figure 6. The suppliers, however, enjoy a discerniblegain of total profit when the incentives are given to the rubber farmers.From the standpoint of the incentive administer, it bears meaning to examine such monetary policy’s social benefitand efficiency. In Figure 7, we show the social-welfare gains, as well as the efficiency of the incentive. The benefit-20ost ratio can be interpreted as the dollar amount of social-welfare gain for every $1 incentive administered to thesystem. In this example, we demonstrate that the efficiency of the incentive given to the resource producer suffersmore severely. Example 1.4: Regressive vs. flat-rate incentives
In this example, we study how a regressive incentive policy differs from a flat-rate one, as well as how the incentivebracket affects the system performance. Once again, we inherit all settings from example 1.1, with the additionalimposition of regressive incentive policy with α =
11 and α = − . A will be left varying as a sensitivity study parameter.Figure 8: Total profits and welfare under the regressiveincentives Figure 9: Social-welfare gains and benefit-cost ratio un-der the regressive incentivesThe aforementioned value of α and α are selected to provide a sensible comparison with a flat-rate α = A displayed in Figure 8 and 9 below are selected to ensure each farmer’s total outputquantity exceeds such bracket value for it to take effect.In Figure 8, we observe that the change of regressive incentive bracket does not influence each supply chain tier’sprofit and the consumer welfare significantly. In Figure 9, a mildly increasing gain of social welfare can be gleanedas 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 themedical facilities. First, we construct the distressed supply chain in which natural rubber emerges to be a criticalshortage in supplies at medical facilities as a market. In doing so, we inherit all settings from example 1.1, except theprice-demand function of latex gloves at medical facilities, which is set as ρ ( d ) = − d + . Our algorithm returns the result including the shipments from two of the suppliers to a market, x and x being0. To illustrate, Figure 10(a) displays the topology of this disrupted supply chain, in which the residential facilities arecompletely cut from the supply of latex gloves.To aid such circumstance, we impose a flat-rate incentive on both latex glove producers, with β =
50, and re-runthe model. Immediately, the previously disrupted supply can be restored. We use Figure 10(b) to display the recoveredsupply chain status.
Example 2: A mixed monetary-fiscal policy in an abstract scarce resource supply chain i = , ..., I , j = , ..., J , n = , ..., N , m = , ..., M , k = , ..., K , and t = , ..., T , in which I = J = , N = M = K = T =
1, as the following. f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , f = . q + q ( q + q ) + q , where , q in = J ∑ j = M ∑ m = x injm ;22 = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) , c = . ( x ) ; f = . ( x + x ) , f = . ( x + x ) , f = . ( x + x ) , f = . ( x + x ) , f = . ( x + x ) , f = . ( x + x ) ; c = . ( x ) + . x , c = . ( x ) + . x , c = . ( x ) + x , c = . ( x ) + x , c = . ( x ) + . x , c = . ( x ) + . x , c = . ( x ) + x , c = . ( x ) + x , c = . ( x ) + . x , c = . ( x ) + . x , c = . ( x ) + x , c = . ( x ) + x ;ˆ c jstk ( x jstk ) = . x jstk . All other costs are set to zero. The price-demand functions at the markets are ρ j k ( d jk ) = − d jk + , ∀ j , k . Similar to example 1-5, we set the production conversion rates ψ injm = .
9, the market resource commodity weights w = . , w = .
5, the parameters concerning the resource capacities, U i , sufficiently large. In contrast to previousexamples, here we set the step-size ϕ to be 10 − , the convergence tolerance ε to be 6 × − for this example.To examine the efficacy of this mixed fiscal-monetary policy, we first establish the equilibrium of the benchmarkscenario, i.e., the setting without such policy. As such, we present the results in table 3. With the ex-ante knowledgethat the resource owners captures most of the supply chain profit, we then impose such policy in which the producersof resource 1 are given a flat-rate incentive of β =
12, whereas the owners of resource 1 and producers of resource 3are charged a flat-rate tax of α = −
10 and β = −
2, respectively.The projection method takes approximately 120 seconds for this problem of total 99 variables to converge to thepreset tolerance. We include all equilibrium results in the supplemental file while displaying only the profit-relatedoutcome in Table 3 again. It is worth pointing out that the "net incentive" is the total taxes collected net the totalincentive disbursed by the government.Table 3: Profit and welfare results of example 2Profit Benchmark with policy Welfare Benchmark with policyOwner π CS π CS π CS π CS π CS π CS π π total π π total π π total π CS total π SW π π ∆ SW π π π π π .1 Discussion From these examples, we devise two supply chain networks to illustrate the utility of our model. We separate allfeatures across the examples and furnish with sensitivity analysis when appropriate, to examine the impact of thesefeatures. We herein distill the following insights.First, the limited resource amount sensibly restrains the directly affected resource owners’ profit and boosts theindirectly affected ones. Interestingly, the capacity limit does not strictly curtail social welfare. Instead, a certainlevel of resource limit can benefit the social outcome. Second, when given a fairly small flat-rate incentive, the totalsupplier profit responds better to the producer incentive, whereas the rest of the tiers respond to both the owner andthe producer’s incentive mildly. The society as a whole, however, benefits from the owner incentive better than theproducer one. Moreover, for a given amount of flat-rate government incentive, the social benefit is higher if or whenadministered to the owners than on the producers, and will remain more efficient as the incentive rate increases.Third, when applied to the resource owners, the regressive incentives are less efficient, in that for the same amountof stimulus expenditure, they do not stimulate social welfare as much as the comparable flat-rate ones. Fourth, theproducer incentive shown in our example can successfully relieve a shortage caused by a demand surge. Similarefficacy of the resource owner incentive can also be tested by our model. Finally, we show that the redistribution ofsocial wealth via a mixed fiscal-monetary policy may result in a net loss of welfare. Although our policy experimentdoes not return an optimistic insight, it indeed proffers a sensible case in line with the welfare loss associated with theelasticity of demand from the classic oligopoly theories (Worcester, 1975).Admittedly, these observations are based on stylized numerical experiments and thus, are limited to the extent ofthe characters of the network, e.g., the competitive nature of the nodes, the substitutability of the flows, the mix, andlevel of fiscal-monetary policy rates, 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 different outcomes. Hence,practitioners and decision-makers should carefully verify and validate the premises of the model before utilizing theaforementioned insights.
Prima facie, the munificence of scarce resources is akin to the sustainability and growth of individual firms, societies,and the flourish 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 and monetary policiesremain the most common governmental policy instruments to relieve the shortage of supplies and stimulate economicperformances. In this paper, our contributions to the literature of scarce resource and supply chain networks includethe following.We construct the first general decentralized cross-sector scarce resource supply chain network equilibrium modelwith multiple fiscal-monetary policy instruments. We provide a rigorous VI formulation for the governing equilibriumconditions of the network model. Such a substitute network provides a versatile framework for the evaluation ofprofit, welfare, policy instruments, cost structure, transportation, conservation, competition, and interdependence ofresources 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, and disequilibrium behaviors, etc, to be furnished.Second, our model is also a general Nash equilibrium problem. We establish the equivalence of a VI problem withthe GNEP. There are still very few GNEP studies that incorporate fiscal-monetary policy in the decentralized supplychain management literature. The utility of this model is not limited to the scarce resource supply chains, but alsoeligible for those of any commodity the pertains to the characters of the aforementioned scarce resource.Third, from a technical aspect, we characterize the network equilibrium solution by adopting a novel set of the-oretical properties, including λ min . To the best of our knowledge in supply chain network literature, such means ofcharacterization for the uniqueness property of network equilibrium has yet to be seen heretofore.Lastly, we furnish the model with numerical studies and extracted managerial insights that provide governments,resource owners, and firms associated with supply chains useful advice in expansion, cost restructuring, resourceconservation, coping with supply chain stressors, handling competition, and collaboration, and post-crisis stimulation.In particular, we show the relevance of our model in relieving and stimulating the PPE shortage caused by the COVID-19 global pandemic. We anticipate that the extension of this model can shed light on the stimulation and relief efforton the vaccine distribution and economic recovery. 25 eferences Abdul-Rashid, S. H., Sakundarini, N., Raja Ghazilla, R. A., and Thurasamy, R. (2017). The impact of sustainablemanufacturing practices on sustainability performance: Empirical evidence from Malaysia.
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