Modelling the Economy as an Evolving Space of Flows

Abstract

The spatial economy has increasingly come to be viewed, in the felicitous phrase of Manuel Castells (2000), as a space of flows. The mental picture we have of this economy is a motion picture, not a still shot. Moving along the links of various networks are ever greater quantities of people, goods, material, money, and information. Settlements, in turn, appear as increasingly interdependent nodes through which these vast quantities pass. The acceleration of flows through space can be accounted for largely by technological advances in communication and transportation and the emergence of far-flung value chains, which are driven by economizing behaviour, and abetted by increasingly liberal trade agreements and industrial deregulation (Wolf 2004).

Many authors have commented on how the spatial economy would seem to manifest characteristics of complex systems – and there are indeed similarities. Steven Durlauf, who has written extensively on economic complexity (both theoretical and empirical), defines complex systems as “those [systems composed] of a set of heterogeneous agents whose behaviour is interdependent and may be described as a stochastic process” (Durlauf 2005, p. 226). Durlauf sees the following four properties as distinguishing complex systems from other systems characterized by stochastic processes and interdependencies.

Appendix. A Dynamic Commodity Flow Model of Donaghy et al. (2006)

We adopt the following notation to characterize network flows. Nodes of the network through which goods are shipped are indexed by l and m. Links joining such nodes are indexed by a and routes comprising contiguous links are indexed by r. The length of some link a connecting two nodes is denoted by d _a. If link a is part of route r connecting nodes l and m, an indicator variable ${\delta _{lmr}^a}$ assumes the value 1.0. It is 0 otherwise. The length of a given route from some node l to another node m, D _tmr, is given by the sum of link distances along the route:

$${D_{lmr} \equiv \sum\limits_a {d_a \delta _{lmr}^a }.}$$

(12.1)

Turning to quantities shipped through the network, we index sectors engaged in production in the spatial economy by i and j. Types of final demand will be indexed by k. Let ${X_l^i }$ denote the total output (in dollars) of sector i produced at node l, ${x_{lm}^{ij} }$ denote interindustry sales from sector i at location l to sector j at location m, and ${{\rm{FD}}_{lm}^{ik} }$ denote final demand of type k at location m for sector i's product at location l. The physical flow of sector i's product from l to m along route r is ${h_{lmr}^i }$. This quantity is obtained by converting the value flow along route r from dollars to tons by means of the ratio of total annual interregional economic flow to total annual physical flow, ${q_x^i }$. The total physical flow of all commodities shipped on a link a via all routes using the link is given by

$${f_a \equiv \sum\limits_i {\sum\limits_{lmr} {h_{lmr}^i \delta _{lmr}^x } } ,}$$

(12.2)

and the periodic flow capacity of link a is denoted by k _a . Conditions that the network must satisfy at any point in time are as follows.

Material balance constraint

$${X_l^i = \sum\limits_m {\sum\limits_j {x_{lm}^{ij} + \sum\limits_m {\sum\limits_k {FD_{lm}^{ik} ,\forall i,\forall l} } } }.}$$

(12.3)

Conservation of flows constraint

$${\sum\limits_r {h_{lmr}^i = \sum\limits_j {x_{lm}^{ij} } } /q_x^i + \sum\limits_k {{\rm{FD}}_{lm}^{ik} } /q_x^i ,\forall i,\forall l,\forall m..}$$

(12.4)

Link capacity constraint

$${\sum\limits_i {\sum\limits_{lmr} {h_{lmr}^i } } \delta _{lmr}^a = f_a \le k_a ,\forall a.}$$

(12.5)

Non-negativity and feasibility conditions

$${fa \ge 0,\forall a;{h_{lmr}^i } } \ge 0,\forall i,\forall l,\forall m,\forall r; {x_{lm}^{ij} } > 0,\forall i,\forall j,\forall l,\forall m.$$

(12.6)

Equation (12.3) ensures that shipments from industry i in location l do not exceed production by the industry in that location, while (12.4) reconciles physical and value flows. Inequality (12.5) ensures that flows along links do not exceed capacities and the conditions given in (12.6) ensure that the distribution of goods throughout the network is feasible. ⁶

In the sequel we shall assume that at each location l the behavior of all establishments engaged in production in a given industrial sector can be characterized by a representative establishment. ⁷ Following Dixit and Stiglitz (1977), we further assume that firms operating the establishments act as monopolistic competitors of the Chamberlinian sort: they are output-level and input-price takers and they set output prices by a mark-up over marginal cost (which equals average cost in equilibrium). For a firm with an establishment producing in sector i at location l, the mark-up, ${\pi _l^i }$, is given in terms of the price-elasticity of demand for ${X_l^i }$, ${\sigma _l^i }$, as

$${\pi_l^i = [\sigma_l^i /(\sigma_l^i - 1)]}.$$

Under the assumption of Chamberlinian monopolistic competition, the spatial markets in which firms compete are sufficiently competitive – barriers to entry are sufficiently low – so as to drive to a very low margin, if not zero, profits earned by firms from production of commodities at all locations.

Each local representative establishment is assumed to produce its output according to a two-level C.E.S. – constant elasticity of substitution – technology (Sato 1967). This fungible output can be used in production of other commodities or absorbed in final demand (in the forms of household and government consumption, investment, and export). At the first level, inputs of each industrial type procured locally and non-locally are aggregated into input bundles:

$${c_m^{ij} = \gamma _m^{ij} \left[ \sum\limits_l {\theta _{lm}^{ij} (x_{lm}^{ij} )^{ - \varepsilon _m^{ij} } } \right] ^{ - 1/\varepsilon _m^{ij} } ,\forall i,\forall j,\forall m.}$$

(12.7)

In (12.7), ${c_m^{ij} }$ is a bundle of inputs produced by representative establishments operating in industry i at various locations l used by the representative establishment in industry j in its production activities at location m. The parameters ${\gamma _m^{ij} ,\theta _{lm}^{ij} ,\,{\rm{and }}\ \varepsilon _m^{ij} }$ have standard interpretations as scale, factor-intensity and substitution parameters (see Ferguson 1969).

At the second level of the production function, total output by a representative establishment in a given industry in a given location is produced from the commodity bundle aggregates at the first level and labor and capital services, ${L_m^j }$ and ${K_m^j }$. At the second level, we allow explicitly for the possibility of increasing returns to scale in production at the establishment, regardless of the number of varieties aggregated in the commodity bundles, by employing a generalized C.E.S. function in which ${\kappa _m^j \ge 1.0}$ is the scale parameter (see Henderson and Quandt 1980).

$${X_m^j = \beta _m^j \left[ \sum\limits_i \alpha _m^{ij} (c_m^{ij} )^{ - \rho _m^j } + \alpha _m^{Lj} (L_m^j )^{ - \rho _m^j } + \alpha _m^{Kj} (K_m^j )^{ - \rho _m^j } \right]^{ - \kappa _m^j /\rho _m^j}}.$$

(12.8)

Again, the parameters of this function have their standard interpretations. The marginal product (in terms of good j) at location m of a unit of good i produced at and shipped from location l is

$$\frac{{\partial X_m^j }}{{\partial x_{lm}^{ij} }} = \frac{{\partial X_m^j }}{{\partial c_m^{ij} }}\frac{{\partial c_m^{ij} }}{{\partial x_{lm}^{ij} }} = \frac{{\kappa _m^j \alpha _m^i }}{{(\beta _m^j )^{\rho _m^j /\kappa _m^j } }}\frac{{(X_m^j )^{(\kappa _m^j + \rho _m^j )/\kappa _m^j } }}{{(c_m^{ij} )^{(\rho _m^j + 1)} }}\frac{{\theta _{lm}^{ij} }}{{(\gamma _m^{ij} )^{\varepsilon _m^{ij} } }}\left( {\frac{{c_m^{ij} }}{{x_{lm}^{ij} }}} \right)^{\varepsilon _m^{ij} + 1}.$$

(12.9)

To make further progress with an explanation of economic behavior, we need to introduce prices as well as technology. Let ${p_m^j }$ denote the f.o.b. (or mill) price of a unit of industry js output at location m and ${p_{lm}^i }$ the delivered price of a unit of intermediate good i at m. Then, defining ${w_m^j }$ and ${{\rm{ucc}}_m^j }$ as the wage rate and user cost of capital in industry j at location m, the mill price of this good under Chamberlinian monopolistic competition is given by

$${p_m^j = \pi _m^j \left[ {\sum\limits_i {\sum\limits_l {p_{lm}^i \cdot x_{lm}^{ij} + w_m^j \cdot L_m^j + ucc_m^j \cdot K_m^j } } } \right]/X_m^j ,\forall j,\forall m.}$$

(12.10)

The delivered price at location m of a good i produced at location l, ${p_{lm}^i }$, includes the unit cost of transport by a carrier from location l to location m, ${\vartheta _{lm}^{ti} }$, which is set by the carrier. Collecting these various price components, the delivered price of a unit of good i at location m will be

$${p_{lm}^i = p_l^i + \vartheta _{lm}^{ti} ,\forall l,\forall m,\forall i.}$$

(12.11)

Defining several new variables for the time rates of change in installed capacity (net of depreciation), in interindustry and interregional commodity flows, in employment, and the f.o.b. goods price that is,

$${I_m^j = \dot K_m^j,\ a_{lm}^{xij} = \dot x_{lm}^{ij} ,\!\!\!\quad a_m^{Lj} = \dot L_m^j, \,\,\rm and \,\;{a_m^{pj} = \dot p_m^j },}$$

the intertemporal optimization decision of a representative establishment in sector j at location m is to choose ${I_m^j ,a_{lm}^{xij} ,{\rm{ }}a_m^{Lj} \ {\rm{ and }} \ a_m^{pj} }$ so as to minimize the present value of costs of operation at and adjustment to equilibrium levels of capital, intermediate goods, and labor: ⁸

$$\begin{array}{l} \int_{t_0 }^{t_1 } \rm{e}^{- \lambda _m^{sj} t}\left\{\sum\limits_i \sum\limits_l p_{lm}^i \cdot x_{lm}^{ij} + w_m^j L_m^j + {\rm{ucc}}_m^j K_m^j + q_m^j I_m^j + \frac {\omega _m^{Kj}} {2} (I - \upsilon _m^{Kj} (K_m^j * - K_m^j ))^2 \right.\\ \quad+ \sum\limits_i \sum\limits_l \frac {\omega _{lm}^{xij}}{2}(a_{lm}^{xij} - \upsilon _{lm}^{xij} (x_{lm}^{ij} * - x_{lm}^{ij}))^2 + \frac {\omega _m^{Lj}} {2}(a_m^{Lj} - \upsilon _m^{Lj} (L_m^j* - L_m^j ))^2 \\ \quad+ \frac {\omega _m^{pj}} {2}(a_m^{pj} -\upsilon _m^{pj} (p_m^j * - p_m^j ))^2 {\rm{d}}t,\end{array}$$

(12.12)

subject to the following identities

$${\dot K_m^j = I_m^j ,}$$

(12.13)

$${\dot x_{lm}^{ij} = a_{lm}^{xij} ,\forall i,\forall l,}$$

(12.14)

$${\dot L_m^j = a_m^{Lj} ,}$$

(12.15)

$${\dot{p}_m^j = a_m^{pj}},^9$$

(12.16)

and (12.3) and the non-negativity condition ⁹on ${x_{lm}^{ij} }$ in (12.6). In objective functional (12.12), ${\lambda _m^{sj} }$ denotes the temporal discount rate of representative establishment j in location m, the equilibrium price level is given by (12.10), and the (atemporal) equilibrium (cost-minimizing) levels of capital, intermediate goods, and labor are given by

$${x_{lm}^{ij} * = \left[ {\frac{{\theta _m^{ij} }}{{(\gamma _m^{ij} )^{\varepsilon _m^{ij} } }}\frac{{\kappa _m^j }}{{\pi _m^j }}\frac{{\alpha _m^{ij} }}{{(\beta _m^j )^{\rho _m^j /\kappa _m^j } }}\frac{{p_m^j }}{{p_{lm}^i }}\frac{{(X_m^j )^{(\kappa _m^j + \rho _m^j )/\kappa _m^j } }}{{(c_m^{ij} )^{(\rho _m^j + 1)} }}} \right]^{1/(1 + \varepsilon _m^{ij} )} c_m^{ij} ,}$$

(12.17)

$${L_m^j * = \left[ {\frac{{\kappa _m^j }}{{\pi _m^j }}\frac{{\alpha _m^{Lj} }}{{(\beta _m^j )^{\rho _m^j /\kappa _m^j } }}\frac{{p_m^j }}{{w_m^i }}} \right]^{1/(1 + \rho _m^j )} (X_m^j )^{(\kappa _m^j + \rho _m^j )/(\kappa _m^j + \kappa _m^j \rho _m^j )} ,}$$

(12.18)

$${K_m^j * = \left[ {\frac{{\kappa _m^j }}{{\pi _m^j }}\frac{{\alpha _m^{Kj} }}{{(\beta _m^j )^{\rho _m^j /\kappa _m^j } }}\frac{{p_m^j }}{{{\rm{ucc}}_m^i }}} \right]^{1/(1 + \rho _m^j )} (X_m^j )^{(\kappa _m^j + \rho _m^j )/(\kappa _m^j + \kappa _m^j \rho _m^j )}.^{10}}$$

(12.19)

We now make an assumption analogous to that made above concerning ¹⁰representative establishments: we assume that at each location l there is a representative carrier which (1) takes as given quantities of goods to be transported from l to other locations m and the prevailing cost structure of goods movement, and (2) sets prices of carriage by commodity, origin, and destination and determines the routing pattern. The intertemporal optimization decision of a representative carrier at location l is, then, to determine a time-varying schedule of prices, ${\vartheta _{lm}^{ti} }$, for shipping commodities from its respective location l to establishments and sources of final demand (households, government agencies, etc.) at all other locations m, ${x_{lm}^{ij} \;{\rm{and}}\;{\rm{FD}}_{lm}^{ik} }$, and time-varying flows of commodities along available routes r, ${h_{lmr}^i }$, so as to maximize the present value of its anticipated stream of net revenues over the time horizon ${t_0 \;{\rm{to}}\;t_1 }$,

$${\int_{t_0 }^{t_1 } {\rm{e}}^{ - \lambda _l^c t} \left\{ \sum\limits_i \sum\limits_m \vartheta _{lm}^{ti} (\sum\limits_j {x_{lm}^{ij} + \sum\limits_k {{\rm{FD}}_{lm}^{ik}}}) - \sum\limits_i \sum\limits_m \sum\limits_r h_{lmr}^i D_{lmr}p_{lmr}^{ti} \right\} {\rm{d}}t,}$$

(12.20)

subject to (12.4) and inequalities (12.5) and (12.6). In (12.20), ${\lambda _l^c }$ is the temporal discount rate of the representative carrier at location l. Also in (12.20), ${p_{lmr}^{ti} }$ denotes the cost to the carrier of delivering a ton of commodity i from location l to location m via route r, and is assumed to be determined by the following cost relationship,

$$p_{lmr}^{ti} = p^t \cdot D_{lmr} ^{\xi _{1i} - 1} h_{lmr}^{i ^{\xi _{2i}} - 1},\ \ {\rm{ where }}\ \ \xi _{1i} ,\xi _{2i} < 1.0,\forall i,\forall l,\forall m,\forall r,$$

(12.21)

where ${p^t }$ denotes the industry average ton-mile price of shipping a commodity. Cost relationship (12.21) implies that unit transport costs decline with distance and with total weight of shipment.

We shall further assume that volumes of final demand for goods at various locations are affected by the prices carriers set (through the delivered price) and that carriers are aware of this dynamic. The implied feedback relationship can be captured by defining final demand of type k at location m for good i produced at location l as

$${{\rm{FD}}_{lm}^{ik} = {\rm{F \tilde{D}}}_{lm}^{ik} (p_{lm}^i /\bar p_{lm}^i )^{ - b_{lm}^{ik} } ,}$$

(12.22)

in which ${{\rm{F \tilde{D}}}_{lm}^{ik} }$ is the volume of exogenously given final demand of type k at location m for good i produced at location l when the (normalized) delivered price is constant and ${\bar p_{lm}^i }$ is a period-average or reference delivered price. ¹¹

When taken over all producers and carriers, the first-order necessary conditions for the solution to the above joint intertemporal optimization problem – including the network constraints (12.3)–(12.6) – correspond to a non-cooperative (Nash) game in which each player takes all others' strategic behaviors as given (the first-order conditions are provided in the appendix to Donaghy et al. 2006). Given the curvature properties of the functional forms employed, a solution to the non-cooperative game should exist and should be unique (questions about the stability of the solution remain). Variations on the game set out above can also, and will be, investigated.

Note that the present set-up differs from the usual commodity-flow model formulation in that producers are minimizing transportation costs of inputs used in production along with other input costs, instead of minimizing shipping costs of supplying the market (cf. Boyce 2002). Carriers seek maximal profits through optimal route selection. The present set-up also differs from other formulations of dynamic games of shippers and carriers in that considerations of transportation costs influence production decisions (cf. Friesz and Holguin-Veras 2005).

Realism would dictate that in applied research on the evolution of goods movement and associated systems effects, transportation modes such as rail, air, or water should also be explicitly introduced, as Ham et al. (2005) have done for a static model of interregional commodity shipments and transportation network flows. This should not present great difficulties and would enable the basic model to support simulation and dynamic gaming exercises whose intent is to examine infrastructure policies.

A more satisfying and more complete modeling framework would account for the evolution of final demand components (including exports) and the evolution of labor markets. An expenditure system for a representative household could be introduced along the lines of a modified almost ideal demand system (MAIDS) (see Cooper and McLaren 1992). Capacity expansion of establishments should also be related to purchases of capital goods from other producers. Changes along these lines would bring the model within the ambit of spatial computable general equilibrium frameworks.