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¡C An Equilibrium Model of Hierarchical Network*
Department of Economics, Monash University This Version: February 2002 * The author is grateful to John Freebairn,
Simon Grant, Jeff Borland, Wilig Spanjers, Yew-Kwang Ng, and the anonymous
referee for helpful comments. The financial support for this research
from Australian Research Council is gratefully acknowledged.
This paper develops a general equilibrium
model with endogenous specialization and endogenous pyramid structure
of wholesale and retail networks. It formalizes the tradeoff between
economies of specialization in producing goods and transaction costs
and the tradeoff between economies of specialization in producing
transaction services and transaction costs in delivering such services.
The comparative statics of the equilibrium, based on corner solutions,
indicate that the equilibrium will shift between several market structures
as parameters reach certain critical values. As the transaction cost
coefficient of transaction services falls and/or as the degree of
economies of specialization increases, the level of division of labor
in producing goods and transaction services and the number of layers
of a hierarchical structure of division of labor and transactions
increases. A small number of layers of the hierarchy emerges from
the division of labor with partially specialized production of goods
and transaction services. A larger number of layers of the hierarchy
emerges from a greater division of labor between wholesale and retail
middlemen and between specialist producers of goods. The purpose of the paper is twofold. First,
I extend the received partial equilibrium analysis of hierarchy to
a general equilibrium context. Second, the equilibrium model of hierarchy
will be used to explore the relationship between the hierarchical
structure of economic organization and the network size of division
of labor. There are three strands of the literature
on hierarchy. The first relates to multiple layers of principal-agent
relationships and related endogenous transaction costs caused by moral
hazard and other opportunism (see Tirole [1986], Laffont [1990], Bac
[1996], Bag [1997], MacLeod and Malcomson [1988], Calvo and Wellisz
[1978, 1979], Hart and Homstrom [1987], Sappington [1991], Homstrom
and Milgrom [1995], and others). Milgrom and Roberts [1992] provide
a comprehensive review of this literature up to 1991. Bolton and Scharfstein
[1998] and Gibbons [1998] provide recent surveys of the literature.
The second literature studies a series connection of elements that
involves the tradeoff between reliability of and gains from the series
connection (see Sah [1991] and Sah and Stiglitz [1988, 1991], Kremer
[1993]). The models in the two literatures do not necessarily involve
a choice of the optimum pyramidal structure with a series connection
of many layers and parallel connections between the elements in the
two layers. The third literature of hierarchy focuses on the tradeoffs
that determine the efficient number of layers in a pyramidal structure
and efficient spanning. This literature is reviewed by Borland and
Eichberger [1998], Spanjers [1997], and Zandt [1995]. In the current
paper, I will develop a general equilibrium model that bridges between
the third literature of hierarchy and the literature of endogenous
specialization. Pyramidal hierarchies can be classified into
two types. A library is an example of a one way hierarchy: a person
uses a library catalogue system to look for a book at the bottom layer
of this system, he searches from the top layer, such as, categories
to the bottom layer, such as, books. I refer to such a hierarchy as
an one way hierarchy because the search process is in a single direction
from top to bottom. Among other one way hierarchies are the military
command system, an English dictionary, a hierarchical screening process
in a tournament involving many contestants, and a packaging system
of some standard commodities. The top of the hierarchy of the dictionary
is its user. The next layer is divided into 26 letters used for the
first letter of words. Each of the letters is divided into 26 letters
used for the second letter of the words at the next layer, and so
on. In the tournament a large number of groups of contestants have
the first round tournament separately. The winners of the first round
tournament then compete separately in a smaller number of groups in
the second round of tournament, and so on. In the packaging system,
goods are packaged in a large number of small boxes which are packaged
in a smaller number of larger boxes which are put in a few containers,
which are then carried by a cargo ship. A postal system is a hierarchy based on a
network. The bottom of the hierarchy consists of many people whose
mail is delivered to one another through the higher layers of the
hierarchy (postal center in large cities and post offices in medium
cities and small towns). The function of such a hierarchy is to improve
the efficiency of the elements at the bottom in communication with
one another through the higher layers. I refer to such a hierarchy
as a network hierarchy since communications are two way, both from
bottom to top and from top to bottom. Among other network hierarchies
are a wholesale and retail distribution network, the telephone network,
and the management system of a large company. Another way to classify pyramidal hierarchies
is to look at the structure of decision making. If a decision-maker
at the top of an hierarchy can choose the number of layers and other
structural features of the hierarchy, then the hierarchy is referred
to as a centralized hierarchy. If no individual decision-maker can
consciously choose the number of layers of a hierarchy, then the hierarchy
is referred to as a decentralized hierarchy. A hierarchical structure
of cities and a wholesale and retail network in the market place are
typical decentralized hierarchies (Muller-Hagedorn [1995]). Business case studies, such as case studies
of McCormick harvesting machine company and Ford model company in
Dicke (1992), document the fact that understanding of the impact of
the number layers of a hierarchical wholesale and retail network on
the level of division of labor in production and thereby productivity
is crucial for a successful business. In many of such business cases,
the hierarchical structure of transactions is associated with the
symmetric market relationship between the franchisor and franchisee
rather than asymmetric ownership relationship between the employer
and employee. The intrinsic relationship between the hierarchical
structure of transactions and the hierarchical structure of the division
of labor is highlighted in the growing literature of endogenous specialization.
A recent survey of this literature is Yang and Ng [1998]. The papers of Carter [1995] and Bolton and
Dewatripont [1994] investigate the relationship between the level
of division of labor and a hierarchical structure of information communication.
In these very interesting papers, there is the tradeoff between economies
of specialization in handling information and communication costs
caused by the division of labor between professional managers, salesmen,
and producers or specialists processing different information. Economies
of specialization are not well specified as features of production
functions in Carter's model where benefits of specialization are represented
by values of some exogenously given parameters. Hence, the efficient
balance of the tradeoff between economies of division of labor and
communication costs is not achieved via choosing output and input
levels in production process (of goods or information). The logic
chain between the assumptions on values of parameters and the conclusion
based on comparative statics of choice of organization structure is
too short. In Bolton and Dewatriont's model, economies of specialization
in processing information are specified as a feature of processing
functions of information. But both models are decision models for
a firm where utility functions and consumption of goods are not specified.
A general equilibrium model is needed to investigate a mechanism that
simultaneously determines many interdependent endogenous variables:
the level of division of labor, the number of layers and number of
elements at each layer of a pyramidal hierarchy, transaction efficiency,
total transaction costs, quantities of goods and services that are
produced and consumed, and prices. As Allyn Young [1928] stated, not
only the division of labor is dependent on the extent of the market,
but also the extent of the market is determined by the level of division
of labor. Further, in a model with endogenous specialization and endogenous
structure of a pyramidal hierarchy, the equilibrium number of layers
of the hierarchical structure of wholesale and retail network is dependent
on the level of division of labor. As shown in the literature of endogenous
specialization (see Yang and Ng [1993] for instance), the equilibrium
level of division of labor in society is determined by the level of
division of labor in transacting activities that determine transaction
efficiencies. However, in a model of endogenous specialization and
endogenous hierarchy, the number of layers of hierarchical structure
of transactions affects the level of division of labor in transacting
activities and thereby transaction efficiencies. Therefore, not only
the number of layers of hierarchy is dependent on the level of division
of labor, but also the level of division of labor is determined by
the number of layers of hierarchy. The circular causation and infinite
feedback loops between many endogenous variables that characterize
organization structure in the market imply that the notion of general
equilibrium (a fixed point) is needed to identify a mechanism that
simultaneously determines all of the interdependent variables. This paper will develop a general equilibrium
model to simultaneously endogenize the network pattern of division
of labor and the number of layers in a pyramidal structure of transactions.
Because of the tradeoff between the degree of endogenization and tractability
of the model, I do not specify uncertainties in communication. The
models of endogenous specialization with transaction risk and moral
hazard and with an exogenous hierarchical structure of transactions
are discussed by Lio [1996, 1998]. In the model to be considered there are four
consumer goods, two types of transaction services, and many ex ante
identical consumer-producers. Each good is essential in consumption.
Each individual, as a consumer, prefers diverse consumption and, as
a producer, prefers specialized production of goods and transaction
services due to a fixed learning cost which generates economies of
specialization in each activity. Moreover, trade will incur transaction
costs. The tradeoff between economies of specialization and transaction
costs can be used to endogenize the level of division of labor. A
low level of division of labor involves only trade between neighbors
and a high level of division of labor involves trade between neighbors
as well as between non-neighbors. Assume that trade with an individual's
neighbor is a transacting activity that is distinctive from trade
with non-neighbors. Then, there are three possible levels of specialization
in transacting activities if the level of division of labor in producing
goods is sufficiently high: (i) Self-provision of two types of transaction
services by producers of goods; (ii) Supply of one type of transaction
service by partially specialized middlemen and self-provision of the
other type of transaction service by each person; and (iii) Supply
of each type of transaction service by completely specialized middlemen.
For the third case, a middleman may specialize in mediating transactions
between local middlemen who take care of trade between neighbors.
This implies that a hierarchical structure of wholesale and retail
distribution networks may emerge from a high level of division of
labor. In addition to the tradeoff between economies of specialization
in producing goods and transaction costs, a tradeoff between economies
of specialization in transacting activities and transaction costs
in delivering transaction services can then be used to simultaneously
endogenize the number of layers of a hierarchical structure of transactions
and the level of division of labor in production and transaction activities.
The first tradeoff entails endogenization of each individual's level
of specialization and the number of his trading partners. Hence, as
the number of trading partners increases with the level of specialization,
a new transacting activity for trade between non-neighbors endogenously
emerges and thereby creates scope for a hierarchical structure of
transactions to emerge from the second tradeoff. We draw the distinction between the quantities
of the same good or service that are self-provided, sold, and purchased.
This distinction leads to corner solutions of consumer-producer optimization
problems. Using the Kuhn-Tucker approach to corner solutions, it can
be shown that each individual sells at most one good and does not
simultaneously buy and sell nor simultaneously buy and self-provide
the same good. The combinations of individuals' corner solutions that
are consistent with this condition generate many candidates for the
general equilibrium which satisfy the market-clearing conditions but
not all conditions for full utility maximization of individuals. We can identify four types of market structures
as candidates for the general equilibrium. The first is autarky where
no market exists. The second is the partial division of labor where
each partially specialized producer of goods self-provides transaction
services. This structure of transactions is symmetric, and no hierarchy
exists. The third is associated with the division of labor between
partially specialized producers of goods and partially specialized
producers of one type of transaction service where each middleman
has two producers of goods as trading partners, but each producer
of goods has only one trading partner. The simplest hierarchy of transactions
with two layers emerges from a higher level of division of labor between
the production of goods and the production of transaction services.
The fourth is a complete division of labor where everybody trades
all four goods and all types of transaction services, and each professional
middleman specializes in a single transacting activity. A retail middleman
specializes in local business between neighbors and a wholesale middleman
specializes in trade between local communities. There are three layers
of the hierarchy of transactions: The top consists of the wholesale
agents; The middle layer consists of the retail agents; The bottom
consists of specialist producers of goods. In the decentralized hierarchy
of three layers the producers of goods are more peripheral than the
retail middlemen who are more peripheral than those wholesale middlemen. Comparative statics of equilibrium show that
the equilibrium is autarky if the degree of economies of specialization
is sufficiently small and if the transaction cost coefficient for
a unit of transaction service is sufficiently large, since in this
case transaction costs outweigh economies of specialization generated
by the division of labor. As the transaction cost coefficient is slightly
lowered and/or the degree of economies of specialization increases,
the general equilibrium shifts to the partial division of labor with
partially specialized producers of goods where there is no middleman.
If the transaction cost coefficient is further lowered and/or the
degree of economies of specialization further increases, the general
equilibrium entails a hierarchical structure of transactions with
two layers where there is a division of labor between producers and
middlemen. If the transaction cost coefficient is extremely small
and the degree of economies of specialization is great a hierarchy
of three layers emerges from a high level of division of labor between
wholesale and retail business and between producers and middlemen.
As the transaction cost coefficient is reduced, the division of labor
evolves, resulting in an increase in productivity, the extent of the
market, the degree of endogenous comparative advantage, the degree
of diversity of economic structure, and in trade dependence. Section 2 of this paper describes the model
which has consumer-producers, economies of specialisation, and transaction
costs. Section 3 investigates individuals' decision problems and different
market structures. In section 4 the comparative statics of the general
equilibrium are used to explore the implications of the model for
endogenizing the number of layers of a decentralized hierarchy and
the level of division of labor. Inframarginal Economics Society www.inframarginal.com
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