# Mechanistic Framework of Global Value Chains

Sourish Dutta  Centre for Development Studies
Trivandrum, Kerala

##  1 Motivation

Indeed, the global production (as a system of creating values) is eventually
forming like a gigantic and complex network/web of value chains that explains
the transitional structures of global trade and development of the global
economy. It’s truly a new wave of globalisation, and we term it as the global
value chains (GVCs), creating the nexus among firms, workers and consumers
around the globe. The emergence of this new scenario asks– how an economy’s
firms, producers and workers connect in the global economy. And how are they
capturing the gains out of it in terms of different dimensions of economic
development? This GVC approach is very crucial for understanding the
organisation of the global industries and firms. It requires the statics and
dynamics of diverse players involved in this complex global production
network. Its broad notion deals with different global issues (including
regional value chains also) from the top down to the bottom up, founding a
scope for policy analysis (Gereffi & Fernandez-Stark 2011). But it is true
that, as Feenstra (1998) points out, any single computational framework is not
sufficient to quantification this whole range of economic activities. We
should adopt an integrative framework for accurate projection of this dynamic
multidimensional phenomenon.

##  2 Introduction

The purpose of this paper is to propose a mechanistic structure to capture the
systemic mechanics of increasing and complex global production network or
global value chains11endnote: 1Antras (2015) uses a variety of terms
interchangeably throughout his book to refer the phenomenal hyper-
globalisation of economic activities, such as ”slicing of the value chain”,
”fragmentation of the production process”, ”disintegration of production”,
”delocalization”, ”vertical specialization”, ”global production sharing”,
”unbundling”, ”offshoring”, ”outsourcing” and many more.. Here I use the term
"mechanistic structure" to imply a simplified systemic mathematical
construction or description that is framed using comprehensive information
about the system of global value chains (Velten 2009). In this context, one
fundamental question will arise naturally that what did happen to the existing
theoretical works and their predictions? As Fort (2016) rightly indicates that
there’s been a lot of theoretical work, a lot of models that people have
written down, but it’s really hard to look at the plant level or firm level
and see exactly who is doing it and what factors are driving them. Similarly,
Antràs & Rossi-Hansberg (2009) draws attention towards a particular limitation
in the building blocks (consumer preferences, factor endowments, production
technologies) in traditional and new trade theory, where the specification of
technology treats the mapping between factors of production and final goods as
a black box (Input-output system, which transform given input parameters into
output parameters). Besides this, they also identify some gaps in the
literature. Firstly, most of the literature in this sphere is static22endnote:
2Some exceptions are the works of Rodríguez-Clare (2010) and Antràs (2005).
Rodríguez-Clare (2010)’s Ricardian model shows that although the rich country
always gains from a reduction in offshoring costs in the long run, the poor
country may reduce research effort and therefore suffer in the long run.
Whereas, Antràs (2005)’s Ricardian model of North-South trade shows that
incompleteness of international contracts leads to the emergence of product
cycles, and thus many fascinating dynamic questions have not been explored.
Secondly, many theories of organisations need a general equilibrium
implementation to study robust economic predictions. Thirdly, very little work
has been devoted to structurally estimate the existing models33endnote: 3An
exception is the work of Antràs et al. (2014). They structurally estimate the
multi-country global sourcing model.. And lastly, there is little concern for
welfare impact and policy implications of assessed (quantitatively) changes in
the international organisation of production.

Indeed, most of the theories are of stylised nature (stylised fact-driven or
gray box models), which in turn urges us to conceive a best systemic framework
(mechanistic or white box model) is required to realise all those gaps in the
global value chains literature. The best systemic framework will view this
hyper-globalisation phenomenon as a web of relationships among different
components that thereby constitute the larger system of value chains
throughout the world. Here we need the best one because it should be the
simplest framework that still serves its purpose, that is, which is still
complex enough to help us understand the system of global value chains and to
solve problems. Perceived in terms of a simple framework, the complexity of
this complex global value chains will no longer haze our perspective, and we
will virtually be able to observe through the complexity of this larger value
chains system at the heart of things (Velten 2009).

We can easily realise that we need a conceptual framework (having economical
reality) to build the purpose-oriented mechanistic framework for the complex
global value chains. We can get this conceptual framework from USAID (U.S.
Agency for International Development). Because USAID applies the broader value
chain perspective with the framework of inclusive market systems development
to drive economic prosperity through the integration of large numbers of
bottom level enterprises into increasingly competitive and complex value
chains. Besides this, we will rely on the systems and control theory to frame
the mechanistic structure.

##  3 Conceptual Framework

The value chain describes the full range of activities and services that firms
and workers perform to bring a product from its conception to sale in its end
use and beyond (final markets – whether local, national, regional or global).
This includes many economic agents such as inputs suppliers, producers,
processors or traders, and buyers with different activities such as design,
production, marketing, distribution and support to the final consumer. The
activities that comprise a value chain can be contained within a single firm
or divided among different firms. In the context of globalisation, the
activities that constitute a value chain have generally been carried out in
inter-firm and intra-firm networks on a global scale. As a result, these
international production networks are obviously highly complex in terms of
geography, technology, and the variety of types of firms involved – from large
retailers to highly mechanised large-scale manufacturers to small, even home-
based, production as well as a variety of intra-firm and inter-firm linkages.
Sometimes it may be impossible even to identify all the countries that are
involved or the extent of their involvement (Gereffi & Fernandez-Stark 2011).

Value chains are situated within broader economic systems. For a systematic
framework that expresses the context for value chains, we have to adopt an
unique market systems framework, which is proposed by Campbell (2014) for the
USAID.

###  3.1 Market Systems Framework

The systemic framework of market should, indeed, be fundamental enough to
facilitate inclusive economic prosperity (in terms of competitiveness,
inclusiveness, and resiliency)44endnote: 4Competitive – system actors are able
to effectively innovate, upgrade and add value to their products and services
to match market demand and maintain or grow market share; Inclusive –
delivering a sustainable flow of benefits to a range of actors, including the
poor and otherwise marginalised, as well as to society as a whole; and
Resilient – system actors are able to address, absorb and overcome shocks in
the market, policy environment, resource base or other aspect of the system.
at any level. Therefore, it should follow some meta-economic principles, by
studying deeper structural and functional aspects of the market systems,
understood as a holistic, dynamic, complex, regulated, evolving, and adaptive
system of value chains. In fact, these principles interdepend, overlap, and
cross-refer with one another (Mobus & Kalton 2015). In other words, market
systems should include value chains, households & communities, interconnect
with other systems, soft boundaries to become a self-organising complex
structure (Campbell 2014).

  * •

Market systems include value chains – The market system, shows how households
interact with multiple, possibly interconnected value chains. Value chains
catalyse and are impacted by broader economic change through multiplier
effects.

  * •

Market systems interconnect with other systems – Market systems interact with
other systems, such as health systems, education systems, socio-cultural
systems, and ecosystems. Changes in one system can affect the functioning of
other systems. Though no single intervention can be expected to simultaneously
transform multiple systems, such networking sometimes allows us to trigger
broad-scale change in the market system by targeting linkages with other
systems.

  * •

Market systems include households and communities – Households and communities
are also systems. Decisions about resource allocation are negotiated among
household and community members, influenced by individuals’ incentives and
expectations, status and decision-making power, a range of socio-cultural
norms and traditions, and physical factors that constrain available options.
Understanding household and community systems and how they interact with each
other and other systems can be important for achieving development objectives.

  * •

Market systems have "soft" boundaries – Since market systems contain sub-
systems (household systems, value chain systems, etc.), and are connected to
health systems, political systems, ecosystems, and the like, we face the
challenge of how to define the intervention space. How wide or focused should
the parameters be for the purposes of analysis, the design of interventions,
the monitoring of systems change, and the evaluation of impact?

  * •

Market systems are complex – Market systems contain many actors, institutions,
structures and influences that are both interconnected and independent. These
system elements interact in ways that are often unpredictable at the
transaction level—although at the aggregate level may be observed to follow
patterns over time. Consequently, the results of many developmental
interventions in market system cannot be predicted in advance.

  * •

Market systems are self-organising – Market systems evolve in response to many
external and internal factors and forces. It is extremely important for donors
and project implementers attempting to strengthen inclusive market systems to
understand the drivers that have resulted in a system’s current way of
operating. The vested interests of market actors generally need to be
addressed for widespread change to occur. Vested interests may not be apparent
at the outset of an intervention, but often emerge as market system changes
begin to take hold. Ongoing analysis and learning are therefore essential.

###  3.2 Value Chains Framework

The systemic framework of larger value chains requires to identify the
structure of the chain, including all individuals and firms that conduct
economic activities by adding value and helping move the product toward the
end markets, influences the dynamics of firm behaviour and the dynamics of the
value chain, which refers to the determinants of individual and firm behaviour
and their effect on the functioning of the chain, influence how well the value
chain performs.

####  3.2.1 Structural Factors

The structure of a value chain includes all the firms in the chain and can be
characterised in terms of five elements described below:

  * •

End markets – End markets are the focal point of the value chains framework.
The term end market is used to specify where the final instance of buying or
selling takes place in a value chain. Usually it is where the end-user is
located, such that the individual or organisation for whom the product or
service has been created, and who is not expected to resell that product or
service. For example, creating a consumer product may entail many transactions
between various value chain actors, but the end market is where the product
becomes available for purchase by the consumer. For a business-related product
or service, the end market is where the sale occurs to the organisation that
will use the product or service in its own operations. The terms destination
market, target market and final market are often used interchangeably with end
market.

The end market determines the characteristics—including price, quality,
quantity and timing—of a successful product or service. Depending on its size,
an end market can often be divided into smaller market segments according to
distinctive characteristics such as price, quality, buyer gender, and
geographic location (reflecting regional needs and/or preferences, for
example). The potential classifications are numerous and based on a group of
people or organisations sharing a combination of characteristics that leads
them to exhibit similar purchasing behaviour. If the resulting subgroup is
highly differentiated, it may be referred to as a niche market (a portion of a
market identified as having a special characteristic that is worth marketing
to). It is called market segmentation, and such categorisation of end markets
helps sellers focus resources on targeted buyers for their products or
services.

The end market for a product or service can be located anywhere in the world
depending on the value chain it flows through. As well, a product or service
can flow through multiple value chains reaching many different end markets.
For example, a product may simultaneously be sold in a village market by the
producer, in major cities of the producer’s country by retailers who purchased
the product from local traders, in neighbouring countries by regional
retailers who purchased the product from an exporter, and in a distant foreign
market by retailers who purchased the product from a wholesale importer.

However, local end markets are limited to areas surrounding the source of a
product or service. These are typically defined by towns and districts within
a country (although in some cases a nearby cross-border market may also be
considered a local end market). A national market suggests the product or
service is sold throughout a country, or at least in one or more venues beyond
local markets. Regional markets encompass an international territory defined
by geographic proximity, common language and culture, historic relations, or
other characteristic that facilitates trade between countries–examples include
Central America, West Africa and the Middle East. And, by definition, global
end markets include all potential markets, although in practice often refer to
large international markets that are outside of the producing country’s
region.

In the systematic value chains framework, it is important to explore each
existing and potential end market to determine which offers the greatest
benefits (profit margins, embedded services, competitive advantages, etc.) and
risks (competition, sizeable investment, fleeting business relationships,
etc.) for each actor in the value chain. We have to find a perfect balance
between the opportunities and risks of global markets, and the accessibility
and limitations of local and regional markets.

  * •

Business enabling environment – System of value chains operate in a business
enabling environment (BEE) that can be all at once global, national and local
and includes norms and customs, laws, regulations, policies, international
trade agreements and public infrastructure (roads, electricity, etc.). The
international enabling environment requires the investigation of conventions,
treaties, agreements and market standards. While trade agreements can open
opportunities for firms, international standards can build obstacles to the
same opportunities. Information is also needed at other levels, including
national and local policies, duties, business licensing procedures, enacted
regulations and the state of public infrastructure. The systemic analysis may
need to be further broken down in terms of firm size: there may be particular
constraints and opportunities facing micro- and small enterprises (MSEs), for
example. Overall, the analysis process must determine whether and how the
business enabling environment facilitates or hinders performance of the value
chain, and if it hinders, where and how can it be improved. BEE constraints
can be difficult to resolve—sometimes requiring considerable time, resources
and political capital.

In addition to these more formal factors, social norms, business culture and
local expectations can be powerful aspects of the business enabling
environment. Understanding these unwritten rules of society is essential if we
want to understand why value chain actors behave the way they do, and
reasonably predict how they will behave in response to value chain
interventions.

  * •

Vertical linkages – Linkages between firms at different levels of the value
chain are critical for moving a product or service to the end market. Vertical
cooperation reflects the quality of relationships among vertically linked
firms up and down the value chain. More efficient transactions among firms
that are vertically related in a value chain increase the competitiveness of
the entire industry. In addition, vertical linkages facilitate the delivery of
benefits and embedded services and the transference of skills and information
between firms up and down the chain. MSEs are vertically linked to a varied
range of market actors including wholesalers, retailers, exporters, traders,
middlemen, input dealers, suppliers, service providers and others. The nature
of vertical linkages—including the volume and quality of information and
services disseminated—often defines and determines the benefit distribution
along the chain and creates incentives for, or constrains, upgrading, defined
as “innovation to increase value added.” Moreover, the efficiency of the
transactions between vertically linked firms in a value chain affects the
competitiveness of the entire industry. An important part of systemic value
chain analysis is the identification of weak or missing vertical linkages.

Effective vertical linkages are generally characterised by:

    * –

Mutually beneficial relationships. Symbiotic relationships that benefit all of
the actors in a value chain are a major trait of effective vertical linkages.
In such a scenario various market actors focus on their own core competencies
and through collaborative action realise synergies that improve the
competitiveness of the entire chain. Trust long-term joint vision and mutual
respect usually form the foundations for developing such relationships.

    * –

Knowledge transfer. Upgrading of production processes technology equipment
management systems etc. is critical for the survival and growth of firms in a
competitive marketplace. It is often difficult for small firms to access
information about global best practices. Effective vertical linkages
facilitate the transfer of knowledge between firms and create the incentives
and knowledge platforms required for effective upgrading of MSEs. Prompt
information transfers and transparency between vertically linked firms help a
value chain respond effectively to changes in market demand.

    * –

Quality standards. Well-defined widely understood and constantly upgraded
quality standards are another defining element of effective vertical linkages.
Vertically linked firms are proactive not reactive: Large firms empower and
help small firms to understand and adopt the quality standards to meet market
demand.

    * –

Embedded services. The frequent provision of high-quality embedded services
(where a service is provided as part of the transaction at no extra cost)
typifies effective vertical linkages. Lead firms can provide a wide range of
embedded services to affiliated suppliers and buyers to ensure consistent
quality of end products and services. These embedded services are often seen
as an integral part of business transactions and considered a necessary cost
of doing business.

    * –

Financial flows. Effective vertical linkages are often accompanied by a high
volume and variety of financial flows. Larger firms may employ a variety of
financial instruments (supplier credit working capital loan leasing services
etc.) to support the operations of their linked suppliers.

Nonetheless, the nature of the vertical relationship between buyers and
sellers is typically varied and dynamic and affected by end market
requirements the business enabling environment product attributes technology
socio-economic conditions and competitive pressures.

  * •

Horizontal linkages – Horizontal linkages, both formal as well as informal,
between firms at all levels in a value chain can reduce transaction costs,
create economies of scale, and contribute to the increased efficiency and
competitiveness of an industry. In addition to lowering the cost of inputs and
services, horizontal linkages can contribute to shared skills and resources
and enhance product quality through common production standards. Such linkages
also facilitate collective learning and risk sharing, while increasing the
potential for upgrading and innovation. Value chain analysis also considers
competition between firms. While cooperation can help firms achieve economies
of scale and overcome common constraints to pursue opportunities, competition
can encourage innovation and drives firms to upgrade. The most successful
horizontal linkages maintain a balance between these two contrasting, but
critical and complementary concepts. One of the objectives of value chain
analysis is to identify areas where collaborative bargaining power could
reduce the cost or increase the benefits to small firms operating in the
chain.

  * •

Supporting markets – Supporting markets play an important role in firm
upgrading. They include financial services; cross-cutting services such as
business consulting, legal advice and telecommunications; and sector-specific
services, for example, irrigation equipment or handicraft design services. Not
all services can be provided as embedded services by value chain actors, and
so vibrant supporting markets are often essential to competitiveness. Service
providers may include for-profit firms and individuals as well as publicly
funded institutions and agencies. Support markets operate within their own
value chain—most service providers themselves need supplies, training and
financing in addition to strong vertical and horizontal linkages. Value chain
analysis should therefore seek to identify opportunities for improved access
to services for target value chain actors in such a way that the support
markets will be simultaneously strengthened, rather than undermined. Formal
supporting markets are likely to expand as the value chain develops.
Therefore, when analysing emerging value chains, or ones predominated by MSEs,
particular care should be taken to uncover informal sector service providers,
which often go unnoticed.

####  3.2.2 Dynamic Factors

The firms in an industry create the dynamic elements described below through
the choices they make in response to the value chain structure.

  * •

Value chain governance – A distinguishing characteristic of value chain
analysis is the emphasis not only on the dynamics of end markets but also on
the dynamics and shifts in relationships. Value chain governance refers to the
relationships among the buyers, sellers, service providers and regulatory
institutions that operate within or influence the range of activities required
to bring a product or service from inception to its end use. Governance is
about power and the ability to exert control along the chain – at any point in
the chain, some firm (or organisation or institution) sets and/or enforces
parameters under which others in the chain operate. Understanding how and when
lead firms set, monitor and enforce rules and standards can help MSEs and
other firms in the chain better integrate and coordinate their activities.
Governance is particularly important for the generation, transfer and
diffusion of knowledge leading to innovation, which enables firms to improve
their performance and sustain competitive advantage. Awareness of the
governance structure of a value chain can provide governments, donors and
development practitioners with information about how best to provide MSEs with
the training and technical assistance needed to upgrade their position in the
chain. When conducting systemic analysis of value chains, the type of
governance structure that exists must be identified since it will contribute
significantly to the selection of interventions to increase competitiveness.

  * •

Inter-firm relationships – This refers to the nature and quality of the
interactions between stakeholders in a value chain. Effective inter-firm
relationships are considered an essential component in creating and
maintaining value chain competitiveness. Even when other conditions are
favorable—end market demand is strong, quality inputs are affordable,
technologies are efficient, supporting markets function well, and so
on—ineffective relationships can jeopardize the competitiveness of a value
chain and its ability to generate economic growth, employment and incomes.
Hence, relationships can be supportive of industry competitiveness that
enhances MSE benefits or adversarial to it. Supportive relationships
facilitate collaboration; enable the transmission of information, skills and
services; and provide incentives for upgrading. They are based on a long-term
view of the industry, while adversarial relationships are structured to
maximise short-term profits. For larger firms at higher levels of the value
chain, both cooperation and competition contribute to upgrading and
competitiveness. For example, cooperation among exporters can help them to
identify common needs and address those needs, through lobbying, branding or
other joint activities. At the same time, healthy competition among exporters
can foster innovation and promote upgrading. With each firm vying to offer the
best quality, design, reliability or other competitive characteristic, the
result will be increased efficiency and competitiveness at the industry level.

  * •

Upgrading – In order to respond effectively to market opportunities, firms and
industries need to innovate to add value to products or services and to make
production and marketing processes more efficient. These activities, known as
firm-level upgrading, can provide MSEs with higher returns and a steady, more
secure income through the development of knowledge and the ability to respond
to changing market conditions. Upgrading at the industry-level focuses on
increasing the competitiveness of all activities involved in the production,
processing and/or marketing of a product or service and mitigating the
constraints that limit value chain performance. Upgrading needs to be a
continual process and can leads to national economic growth. In value chain
analysis, the objective is to identify opportunities and constraints to firm-
and industry-level upgrading; specifically the analysis looks for catalyst
firms with the incentives, resources and willingness to promote and facilitate
upgrading within the chain.

##  4 Mechanistic Framework

The system of value chains $\mathbf{S}_{l}$ can be represented as a 6-tuple:

$\mathbf{S}_{l}=\mathbf{\\{C,N,I,B,K,H\\}}_{l},\qquad\forall\ l=0,1,2,...,m$
(1)

where $l$ is an index related to the level of complexity of the value chains
system. $l=0$ indicates the top level of this system. For example, when
$\mathbf{C}$ is decomposed and each $c_{i}$ is treated as its own distinct
system of value chains, $S_{l,i}$. In fact, the definition recurses with
$l=1$, then $2$, and so on.

$\mathbf{C}$ is a multiset of component subsystems of the larger system of
value chains, $\\{\\{c_{1},n_{1}\\},\\{c_{2},n_{2}\\},...,$
$\\{c_{i},n_{i}\\},...,\\{c_{k},n_{k}\\}\\}$, where each $c_{i}$ is a type of
structural component of value chains and may also be a subsystem but at level
$l+1$. That is, each $c_{i}$ may itself be defined by a similar 6-tuple in a
recursive fashion. Nevertheless, the recursion can prevented from being
infinite (In essence, a decomposition stops at a level $l=L$ when the
components in $S_{L}$ are all defined as atomic for the particular kind of
system being analysed. Atomic, here, simply means the simplest component, not
needing further decomposition because its functions, etc. are already known).
Each $n_{i}$ in the component tuples is an integer enumeration of the number
of components of that particular type (multiplicity). These tuples,
$\\{c_{i},n_{i}\\}$, are multisets. As written here, their cardinality is one.
However, in very complex systems, we can aggregate any number of variations of
components of the same basic type (a category) into a multiset whose
cardinality is the number of distinct variations identified.

$\mathbf{N}$ is a network description in graph theoretical terms. A graph,
$\mathbf{G=\\{V,E\\}}$, is a tuple containing two sets, $\mathbf{V}$ , the
vertices or nodes, which in this case correspond to the components in
$\mathbf{C}$, and $\mathbf{E}$, the edges or connections between the nodes.
$\mathbf{N}$ provides a map of what components are connected to what other
components at this level of complexity

$\mathbf{I}$ is also a graph in which half of the nodes are not inside the
system per se but are either sources or sinks or other entity types in the
environment. That is, $\mathbf{I}$ contains elements of the environment that
are relevant to the system. The other nodes are component subsystems within
the system. $\mathbf{I}$ describes the connections a system has with objects
in its environment using the same format as $\mathbf{N}$ (which is strictly
internal connections).

$\mathbf{B}$ is a complex object that will vary based on details.
Fundamentally, it is a description of the set of boundary conditions that
maintain the system identity.

$\mathbf{K}$ is also a complex object. This object contains the descriptions
of how the various components interact with one another and with the
environment. That is, it contains the state transition rules for all
interactions. It is an augmentation to the $\mathbf{N}$ and $\mathbf{I}$
graphs. For instance, for each edge $e_{i}\in N$ , there is an object $k_{i}$
containing the parameters associated with $e_{i}$ . For example, $k_{i}$ could
contain a tuple representing the capacity of a flow, the substance flowing,
simply the strength of the connection, etc. A similar representation is
provided for the graph in $\mathbf{I}$ where augmentation of the connections
between external (environmental) elements and internal components is needed.
$\mathbf{K}$ is called the system knowledge. It is both self-knowledge
(internal structural and functional knowledge) and other knowledge to the
extent it details the external connections. Note that the external objects are
not, in this definition, modeled. That is, the system contains no explicit
knowledge of the internals of external objects, only how they behave. However,
this can be extended in complex adaptive systems to include knowledge of the
environmental objects.

Finally, $\mathbf{H}$ is a super complex object that records the history of
the system, or its record of state transitions, especially as it develops or
evolves. For example, brains learn from experience, and as such their internal
microstructures change over time. This is called memory and the current state
of $\mathbf{K}$ is based on all previous states. Some simple systems may have
a null $\mathbf{H}$; that is, there is no memory of past states. As just
mentioned, on the other hand, brains (and indeed all biological systems) have
very rich memories. $\mathbf{H}$ is an augment for $\mathbf{K}$.

Each one of these elements is a description of some aspect of the system and
all are needed to provide a complete description. Many of them, such as
$\mathbf{K}$, can be extremely complex. For example, in the case of the brain
mentioned above, $\mathbf{K}$ is actually distributed between the genetic
inheritance of the individual, which prescribes the major wiring diagram and
the major processing modules, and any culturally determined factors that push
the direction of development (learning). This is why $\mathbf{K}$ must also
contain augments for $\mathbf{I}$. It is also why $\mathbf{H}$ may become
extremely complex.

By having a formal definition such as this, it is now possible to apply a
great many mathematical tools to analyse many aspects of the system.

###### Notes

  1. 1 Antras (2015) uses a variety of terms interchangeably throughout his book to refer the phenomenal hyper-globalisation of economic activities, such as ”slicing of the value chain”, ”fragmentation of the production process”, ”disintegration of production”, ”delocalization”, ”vertical specialization”, ”global production sharing”, ”unbundling”, ”offshoring”, ”outsourcing” and many more.
  2. 2 Some exceptions are the works of Rodríguez-Clare (2010) and Antràs (2005). Rodríguez-Clare (2010)’s Ricardian model shows that although the rich country always gains from a reduction in offshoring costs in the long run, the poor country may reduce research effort and therefore suffer in the long run. Whereas, Antràs (2005)’s Ricardian model of North-South trade shows that incompleteness of international contracts leads to the emergence of product cycles
  3. 3 An exception is the work of Antràs et al. (2014). They structurally estimate the multi-country global sourcing model.
  4. 4 Competitive – system actors are able to effectively innovate, upgrade and add value to their products and services to match market demand and maintain or grow market share; Inclusive – delivering a sustainable flow of benefits to a range of actors, including the poor and otherwise marginalised, as well as to society as a whole; and Resilient – system actors are able to address, absorb and overcome shocks in the market, policy environment, resource base or other aspect of the system.

## References

  * (1)
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  * Gereffi & Fernandez-Stark (2011) Gereffi, G. & Fernandez-Stark, K. (2011), ‘Global value chain analysis: a primer’, Center on Globalization, Governance & Competitiveness (CGGC), Duke University, North Carolina, USA .
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# F-pure Threshold for the Symmetric Determinantal Ring

Justin Fong  Department of Mathematics, Purdue University, West Lafayette, IN
47907, USA<EMAIL_ADDRESS>
###### Abstract.

We give a value for the $F$-pure threshold at the maximal homogeneous ideal
$\mathfrak{m}$ of the symmetric determinantal ring over a field of prime
characteristic. The answer is characteristic independent, so we immediately
get the log canonical threshold in characteristic zero as well.

##  1\. Introduction

The $F$-pure threshold, introduced by Takagi and Watanabe in [TW04], is a
numerical invariant of a ring in positive characteristic associated to the
$F$-singularity class of $F$-purity. Suppose $R$ is a reduced $F$-finite
$F$-pure ring of characteristic $p>0$, and let $\mathfrak{a}\subseteq R$ be a
nonzero ideal. For a real number $t\in\operatorname{\mathbb{R}}_{\geq 0}$, the
pair $(R,\mathfrak{a}^{t})$ is called _$F$ -pure_ if for every $q=p^{e}\gg 0$,
there exists an element $f\in\mathfrak{a}^{\lfloor(q-1)t\rfloor}$ (where
$\lfloor(q-1)t\rfloor$ is the greatest integer less than or equal to the real
number $(q-1)t$) such that the inclusion $f^{1/q}R\hookrightarrow R^{1/q}$
splits as an $R$-module homomorphism. The _$F$ -pure threshold_ of
$\mathfrak{a}$ is defined to be

$\operatorname{fpt}(\mathfrak{a}):=\sup\\{t\in\operatorname{\mathbb{R}}_{\geq
0}\mid(R,\mathfrak{a}^{t})\ \text{is $F$-pure}\\}.$

When $\mathfrak{a}=\mathfrak{m}_{R}$ is the maximal homogeneous ideal of $R$,
we will simply refer to $\operatorname{fpt}(\mathfrak{m}_{R})$ as the $F$-pure
threshold of $R$.

The $F$-pure threshold has been calculated for the determinantal rings of
generic matrices [STV17, Proposition 4.3], Hankel matrices [CMSV18, Theorem
4.5], and is trivial for skew-symmetric matrices (see section 2.1 for an
explanation). However, to the author’s knowledge,
$\operatorname{fpt}(\mathfrak{m}_{R})$ has been unknown for the symmetric
matrix case. We provide an answer in Theorem 2.6, as well as a proof that the
symmetric determinantal ring is $F$-pure, a fact not clearly stated in the
literature.

### Note

At the time of writing this article, the author was notified of the recent
paper [CSV24], which includes a proof of the more general property of
$F$-regularity for symmetric determinantal rings, and mentions the same
conclusion of Proposition 2.5 in this article. Our work is independent of that
paper.

### Acknowledgements

I would like to thank Irena Swanson, Vaibhav Pandey, and my advisor Uli
Walther for providing helpful discussions. Partial support by NSF Grant
DMS-2100288 is gratefully acknowledged.

##  2\. Symmetric Determinantial Rings

Suppose that $X=(x_{ij})$ is a symmetric $n\times n$ generic matrix (i.e.,
$x_{ij}=x_{ji}$), let $\Bbbk[X]=\Bbbk[x_{ij}\mid 1\leq i\leq j\leq n]$ be the
polynomial ring over a field $\Bbbk$ with the entries of $X$ as its variables,
and $I_{t}(X)$ be the ideal generated by the $t$-minors of $X$. For an integer
$1\leq t\leq n$ let $R=\Bbbk[X]/I_{t}(X)$ be the _symmetric determinantal
ring_. A formula for the $a$-invariant of $R$ has been given by Barile in
[Bar94]. When $R$ is given the standard grading this is given as follows:

$a(R)=\begin{cases}-(t-1)\frac{n}{2},&\text{if $n\equiv t\ (\mathrm{mod}\
2)$}\\\ -(t-1)\frac{n+1}{2},&\text{if $n\not\equiv t\ (\mathrm{mod}\
2)$}.\end{cases}$ (2.1)

In [Con94] it is shown that the graded canonical module of $R$ is

$\omega_{R}=\begin{cases}R\left(-(t-1)\frac{n}{2}\right),&\text{if $n\equiv t\
(\mathrm{mod}\ 2)$}\\\ \mathfrak{p}\left(-(t-1)\frac{n-1}{2}\right),&\text{if
$n\not\equiv t\ (\mathrm{mod}\ 2)$},\end{cases}$ (2.2)

where $\mathfrak{p}$ is the prime ideal generated by the $t-1$ minors of the
first $t-1$ rows of $X$. In [Got79] and [Got77] Goto shows that $R$ is
Gorenstein if and only if $n\equiv t\ (\mathrm{mod}\ 2)$, and the divisor
class group of $R$ is $\operatorname{\mathbb{Z}}/2\operatorname{\mathbb{Z}}$
generated by $[\mathfrak{p}]$.

######  Lemma 2.1.

[DSMnNnB23, (4.2)] Suppose $S$ is a polynomial ring over a field of
characteristic $p>0$, and let $I\subset S$ be a prime ideal of
$h=\operatorname{height}(I)$. Then $I^{(h(p-1))}\subseteq(I^{[p]}:_{S}I)$,
i.e., $I^{(h(p-1))+1}\subseteq I^{[p]}$.

Proof.  To show the containment $I^{(h(p-1))+1}\subseteq I^{[p]}$, it is
enough to show it locally at all the associated primes of $S/I^{[p]}$.
However, since the Frobenius map is flat, the set of associated primes of
$S/I^{[p]}$ is the same as for $S/I$, which consists only of $I$ since it is
prime. In the regular local ring $(S_{I},IS_{I})$ the symbolic power of
$IS_{I}$ coincides with its ordinary power since $IS_{I}$ is a maximal ideal.
Hence it follows that

$(IS_{I})^{(h(p-1))}I=(IS_{I})^{h(p-1)}I=(IS_{I})^{h(p-1)+1}.$

Since $(S_{I},IS_{I})$ has dimension $h=\operatorname{height}(I)$, its unique
maximal ideal $IS_{I}$ is generated by $h$ elements $g_{1},\dots,g_{h}$. The
power $(IS_{I})^{h(p-1)+1}$ is generated by $g_{1}^{a_{i}}\cdots
g_{h}^{a_{h}}$ such that $a_{1}+\cdots+a_{h}=h(p-1)+1$. By the pigeonhole
principle, at least one $a_{i}\geq p$, hence it follows that
$(IS_{I})^{h(p-1)+1}\subseteq I^{[p]}S_{I}$. $\Box$

######  Proposition 2.2.

Over a field of characteristic $p>0$ the symmetric determinantal ring $R$ is
$F$-pure.

Proof.  Let $I_{t}=I_{t}(X)$, and $S=\Bbbk[X]$ be the polynomial ring in
$n(n+1)/2$ many variables. We will apply Fedder’s criterion: $R$ is $F$-pure
if and only if $\mathfrak{m}_{S}^{[p]}\nsubseteq(I_{t}^{[p]}:_{S}I_{t})$.
Thus, in order to show that $R$ is $F$-pure, we need to find an element
$f\in(I_{t}^{[p]}:_{S}I_{t})\setminus\mathfrak{m}_{S}^{[p]}$.

Let $\mathfrak{D}=d_{1}\cdots d_{n}$ be a product of $i$-minors $d_{i}$ of the
$i\times i$ submatrices in the bottom-left corner of $X$ (for $i=1,\dots,n$),
whose diagonals are the $n$ distinct diagonals of $X$ (see Example 2.3 below).
Our proposed element will be $f=\mathfrak{D}^{p-1}$. The leading term of
$\mathfrak{D}$ with respect to the diagonal term ordering

$x_{11}>x_{12}>\cdots>x_{1n}>x_{21}>\cdots>x_{n-1n}>x_{nn}$

is $\operatorname{in}(\mathfrak{D})=\prod_{1\leq i\leq j\leq n}x_{ij}$, which
is the product of all the distinct variables of $X$, hence it is a squarefree
monomial. This implies $\mathfrak{D}^{p-1}\notin\mathfrak{m}_{S}^{[p]}$.

Next, we need to show $\mathfrak{D}^{p-1}I_{t}\subseteq I_{t}^{[p]}$, i.e.,
$\mathfrak{D}^{p-1}\in(I_{t}^{[p]}:_{S}I_{t})$. It suffices to prove this
containment after localizing at the prime ideal $I_{t}$. If one can show that
$\mathfrak{D}$ belongs to the symbolic power $I_{t}^{(h)}$, where
$h=\operatorname{height}(I_{t})$, then it would follow

$\mathfrak{D}^{p-1}I_{t}S_{I_{t}}\subseteq\left((I_{t}S_{I_{t}})^{(h)}\right)^{p-1}I_{t}S_{I_{t}}=(I_{t}S_{I_{t}})^{h(p-1)+1}\subseteq
I_{t}^{[p]}S_{I_{t}},$

where $(I_{t}S_{I_{t}})^{(h)}=(I_{t}S_{I_{t}})^{h}$, because $I_{t}S_{I_{t}}$
is the (unique) maximal ideal of $S_{I_{t}}$, and the last containment follows
from Lemma 2.1. We will now show $\mathfrak{D}\in I_{t}^{(h)}$. In [Kut74,
Theorem 1], Kutz has shown that $\dim R=n(t-1)-\frac{1}{2}(t-1)(t-2)$, hence
it follows

$h=\dim S-\dim R=\frac{n(n+1)}{2}-n(t-1)+\frac{(t-1)(t-2)}{2}.$

In general, suppose $\Delta$ is a product $\delta_{1}\cdots\delta_{u}$ of
$a_{i}$-minors $\delta_{i}$ of $X$, where the vector
$(a_{1},\dots,a_{u})\in\operatorname{\mathbb{N}}^{u}$ is called the _shape_ of
$\Delta$. With the previous notation, the symbolic power $I_{t}^{(k)}$ is
described by

$\left[\Delta\in I_{t}^{(k)}\right]\ \Longleftrightarrow\
\left[\text{$a_{i}\leq n\enspace\forall i$
and}\enspace\gamma_{t}(\Delta):=\sum_{i=1}^{u}\max\\{0,a_{i}-t+1\\}\geq
k\right]$

(see [JMnV15, Proposition 4.3].) We must show $\gamma_{t}(\mathfrak{D})\geq
h$. Observe that the product $\mathfrak{D}=d_{1}d_{2}\cdots d_{n}$ has shape
$(1,2,\dots,n)$, and note that $i-t+1>0\Leftrightarrow i>t-1$, hence
$\max\\{0,i-t+1\\}=i-t+1$ for $t\leq i\leq n$. Thus, one has

$\displaystyle\gamma_{t}(\mathfrak{D})$ $\displaystyle=\sum_{i=t}^{n}(i-t+1)$
$\displaystyle=\sum_{i=t}^{n}i+\sum_{i=t}^{n}(1-t)$
$\displaystyle=\frac{n(n+1)}{2}-\frac{t(t-1)}{2}+(n-t+1)(1-t)$
$\displaystyle=\frac{n(n+1)}{2}-n(t-1)+\frac{(t-1)(t-2)}{2},$

i.e., $\gamma_{t}(\mathfrak{D})=\operatorname{height}(I_{t})=h$, so it follows
that $\mathfrak{D}\in I_{t}^{(h)}$. $\Box$

######  Example 2.3.

For the $3\times 3$ symmetric matrix

$X=\begin{pmatrix}x_{11}&x_{12}&x_{13}\\\ x_{12}&x_{22}&x_{23}\\\
x_{13}&x_{23}&x_{33}\end{pmatrix}$

the $i$-minors $d_{i}$ ($i=1,2,3$) of the $i\times i$ botom-left corner of $X$
are

$d_{1}=x_{13},\quad d_{2}=\begin{vmatrix}x_{12}&x_{22}\\\
x_{13}&x_{23}\end{vmatrix},\quad d_{3}=|X|,$

which have leading terms $\operatorname{in}(d_{1})=x_{13},\
\operatorname{in}(d_{2})=x_{12}x_{23}$, and
$\operatorname{in}(d_{3})=x_{11}x_{22}x_{33}$, hence
$\mathfrak{D}=d_{1}d_{2}d_{3}$ has squarefree leading term
$x_{11}x_{12}x_{13}x_{22}x_{23}x_{33}$.

######  Lemma 2.4.

[Gri, Lemma 1.71] Let $Q$ be a primary ideal of a Noetherian ring $R$. If $P$
is a prime ideal that contains $Q$, then $QR_{P}\cap R=Q$.

######  Proposition 2.5.

Over a field $\Bbbk$ of arbitrary characteristic, let $\mathfrak{p}$ be the
prime ideal of $R=\Bbbk[X]/I_{t}(X)$ generated by the $t-1$ minors of the
first $t-1$ rows of $X$. Then the second symbolic power $\mathfrak{p}^{(2)}$
is principally generated by the $(t-1)$-minor in the top left-hand corner of
$X$.

Proof.  Let $d$ represent the $(t-1)$-minor in the top left corner of $X$. By
Proposition (1) of [Got77] the radical of the principal ideal $(d)=dR$ is the
prime ideal $\mathfrak{p}$. To see that $(d)$ is primary note that $R$ is a
Cohen-Macaulay domain by [Kut74, Theorem 1], and $d\in R$ is a nonzero
divisor, hence $R/(d)$ is Cohen-Macaulay, so all associated primes of $(d)$
are minimal. Since $\sqrt{(d)}=\mathfrak{p}$ is prime, it follows that $(d)$
is $\mathfrak{p}$-primary.

By Proposition (2) of [Got77], the local ring $R_{\mathfrak{p}}$ is a discrete
valuation ring with valuation $v_{\mathfrak{p}}$, and $v_{\mathfrak{p}}(d)=2$.
Let $(d)R_{\mathfrak{p}}$ be the extension of $(d)$ in $R$ to
$R_{\mathfrak{p}}$. This is a nonzero proper ideal in $R_{\mathfrak{p}}$ by
the valuation of $d$. Since every proper nontrivial ideal of a DVR
$R_{\mathfrak{p}}$ is a power of the unique maximal ideal
$\mathfrak{p}R_{\mathfrak{p}}$, we have
$(d)R_{\mathfrak{p}}=\mathfrak{p}^{m}R_{\mathfrak{p}}=\\{a\in
R_{\mathfrak{p}}\mid v_{\mathfrak{p}}(a)\geq m\\}$ for some
$m\in\operatorname{\mathbb{N}}$. From $v_{\mathfrak{p}}(d)=2$, it follows that
$(d)R_{\mathfrak{p}}\subseteq\mathfrak{p}^{2}R_{\mathfrak{p}}$. Also,
$d\in\mathfrak{p}^{m}R_{\mathfrak{p}}$ implies $2=v_{\mathfrak{p}}(d)\geq m$,
so $m$ is either 1 or 2. However,
$\mathfrak{p}^{m}R_{\mathfrak{p}}\subseteq\mathfrak{p}^{2}R_{\mathfrak{p}}$
can only mean $m=2$, so it follows
$(d)R_{\mathfrak{p}}=\mathfrak{p}^{2}R_{\mathfrak{p}}$.

Since $d\in\mathfrak{p}$, one has $(d)\subseteq\mathfrak{p}$, and it follows
from Lemma 2.4 and our previous discussion that $(d)=(d)R_{\mathfrak{p}}\cap
R=\mathfrak{p}^{2}R_{\mathfrak{p}}\cap R=:\mathfrak{p}^{(2)}$. $\Box$

######  Theorem 2.6.

The $F$-pure threshold of the symmetric determinantial ring $R$ is

$\operatorname{fpt}(\mathfrak{m}_{R})=\frac{n(t-1)}{2}.$

In characteristic zero, this is equal to the log canonical threshold as well.

Proof.  The ring $R=\Bbbk[X]/I_{t}$ is Gorenstein if and only if $n\equiv t\
(\mathrm{mod}\ 2)$, hence it follows from [DSNnB18, Theorem 5.2] and equation
(2.1) that $\operatorname{fpt}(\mathfrak{m}_{R})=-a(R)=(t-1)\frac{n}{2}$, when
$n\equiv t\ (\mathrm{mod}\ 2)$. Now, suppose $R$ is non-Gorenstein, i.e.,
$n\not\equiv t\ (\mathrm{mod}\ 2)$. Its $a$-invariant and graded canonical
module are respectively $a(R)=-(t-1)\frac{n+1}{2}$ and
$\omega_{R}=\mathfrak{p}\left(-(t-1)\frac{n-1}{2}\right)$ (see (2.2)). Since
the class group $\operatorname{Cl}(R)$ is cyclic of order $2$, the ring $R$ is
$\operatorname{\mathbb{Q}}$-Gorenstein, so we will apply [STV17, Proposition
4.5]. We determine the generating degree $D$ of $\omega_{R}^{(2)}$. First,
note that $\mathfrak{p}^{(2)}$ is principally generated by a $(t-1)$-minor, by
Proposition 2.5, hence as graded modules $\mathfrak{p}^{(2)}\cong R(-(t-1))$.
The second symbolic power of $\omega_{R}$ is then

$\displaystyle\omega_{R}^{(2)}=\left(\mathfrak{p}\left(\frac{-(t-1)(n-1)}{2}\right)\right)^{(2)}$
$\displaystyle=\mathfrak{p}^{(2)}\left(2\cdot\frac{-(t-1)(n-1)}{2}\right)$
$\displaystyle=R\big{(}-(t-1)\big{)}\big{(}-(t-1)(n-1)\big{)}$
$\displaystyle=R\big{(}-(t-1)-(t-1)(n-1)\big{)}$
$\displaystyle=R\left(-(t-1)n\right),$

which is generated in degree $D=(t-1)n$. It follows that
$\operatorname{fpt}(\mathfrak{m}_{R})=D/c=n(t-1)/2$.

In characteristic zero, one has by [MTW05, Theorem 3.4] that the log canonical
threshold of $R$ is
$\operatorname{lct}(\mathfrak{m}_{R})=\lim_{p\to\infty}\operatorname{fpt}(\mathfrak{m}_{R_{p}})=n(t-1)/2$,
where $R_{p}$ is the prime reduction modulo $p$ of $R$. $\Box$

###  2.1. Ring of Pfaffians

Let $X=(x_{ij})$ be an $n\times n$ skew-symmetric matrix (i.e., $x_{ii}=0$,
$x_{ij}=-x_{ji}$ for $i<j$), $\Bbbk[X]$ be the polynomial ring in the entries
of $X$, and $\operatorname{Pf}_{2t}(X)$ be the ideal generated by the
$2t$-Pfaffians of $X$. The ring of Pfaffians
$R=\Bbbk[X]/\operatorname{Pf}_{2t}(X)$ is Gorenstein [Avr79] and strongly
$F$-regular [Bae01]. Its $a$-invariant has been computed in [BH92, Corollary
1.7], and assuming it is standard graded, is given by $a(R)=-n(t-1)$, if
$2t\leq n$. By [DSNnB18, Theorem 5.2] we immediately get the following.

######  Proposition 2.7.

For $2t\leq n$, the $F$-pure threshold of the ring of Pfaffians
$R=\Bbbk[X]/\operatorname{Pf}_{2t}(X)$ is
$\operatorname{fpt}(\mathfrak{m}_{R})=n(t-1)$.

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