A theory of rational spatial agglomerations

We model the behavior of rational forward-looking agents in a spatial economy. The economic geography structure is built on Fujita et al. (1999)'s racetrack economy. Workers choose optimally what to consume at each period, as well as which spatial itinerary to follow in the geographical space. The spatial extent of the resulting agglomerations increases with the taste for variety and the expenditure share on manufactured goods, and decreases with transport costs. Because forward-looking agents anticipate the future formation of agglomerations, they are more responsive to spatial utility differentials than myopic agents. As a consequence, the emerging agglomerations are larger under perfect foresight spatial adjustments than under myopic ones.

The structure of the optimal income tax in the quasi-linear model

Existing numerical characterizations of the optimal income tax have been based on a limited number of model specifications. As a result, they do not reveal which properties are general. We determine the optimal tax in the quasi-linear model under weaker assumptions than have previously been used; in particular, we remove the assumption of a lower bound on the utility of zero consumption and the need to permit negative labor incomes. A Monte Carlo analysis is then conducted in which economies are selected at random and the optimal tax function constructed. The results show that in a significant proportion of economies the marginal tax rate rises at low skills and falls at high. The average tax rate is equally likely to rise or fall with skill at low skill levels, rises in the majority of cases in the centre of the skill range, and falls at high skills. These results are consistent across all the specifications we test. We then extend the analysis to show that these results also hold for Cobb-Douglas utility.

The unsteady flow of a weakly compressible fluid in a thin porous layer II: three-dimensional theory

We consider the problem of determining the pressure and velocity fields for a weakly compressible fluid flowing in a three-dimensional layer, composed of an inhomogeneous, anisotropic porous medium, with vertical side walls and variable upper and lower boundaries, in the presence of vertical wells injecting and/or extracting fluid. Numerical solution of this three-dimensional evolution problem may be expensive, particularly in the case that the depth scale of the layer h is small compared to the horizontal length scale l, a situation which occurs frequently in the application to oil and gas reservoir recovery and which leads to significant stiffness in the numerical problem. Under the assumption that $\epsilon\propto h/l\ll 1$, we show that, to leading order in $\epsilon$, the pressure field varies only in the horizontal directions away from the wells (the outer region). We construct asymptotic expansions in $\epsilon$ in both the inner (near the wells) and outer regions and use the asymptotic matching principle to derive expressions for all significant process quantities. The only computations required are for the solution of non-stiff linear, elliptic, two-dimensional boundary-value, and eigenvalue problems. This approach, via the method of matched asymptotic expansions, takes advantage of the small aspect ratio of the layer, $\epsilon$, at precisely the stage where full numerical computations become stiff, and also reveals the detailed structure of the dynamics of the flow, both in the neighbourhood of wells and away from wells.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

A unified theory of available potential energy

Traditional derivations of available potential energy, in a variety of contexts, involve combining some form of mass conservation together with energy conservation. This raises the questions of why such constructions are required in the first place, and whether there is some general method of deriving the available potential energy for an arbitrary fluid system. By appealing to the underlying Hamiltonian structure of geophysical fluid dynamics, it becomes clear why energy conservation is not enough, and why other conservation laws such as mass conservation need to be incorporated in order to construct an invariant, known as the pseudoenergy, that is a positive‐definite functional of disturbance quantities. The available potential energy is just the non‐kinetic part of the pseudoenergy, the construction of which follows a well defined algorithm. Two notable features of the available potential energy defined thereby are first, that it is a locally defined quantity, and second, that it is inherently definable at finite amplitude (though one may of course always take the small‐amplitude limit if this is appropriate). The general theory is made concrete by systematic derivations of available potential energy in a number of different contexts. All the well known expressions are recovered, and some new expressions are obtained. The possibility of generalizing the concept of available potential energy to dynamically stable basic flows (as opposed to statically stable basic states) is also discussed.

An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol'd's stability theorems

Disturbances of arbitrary amplitude are superposed on a basic flow which is assumed to be steady and either (a) two-dimensional, homogeneous, and incompressible (rotating or non-rotating) or (b) stably stratified and quasi-geostrophic. Flow over shallow topography is allowed in either case. The basic flow, as well as the disturbance, is assumed to be subject neither to external forcing nor to dissipative processes like viscosity. An exact, local ‘wave-activity conservation theorem’ is derived in which the density A and flux F are second-order ‘wave properties’ or ‘disturbance properties’, meaning that they are O(a2) in magnitude as disturbance amplitude a [rightward arrow] 0, and that they are evaluable correct to O(a2) from linear theory, to O(a3) from second-order theory, and so on to higher orders in a. For a disturbance in the form of a single, slowly varying, non-stationary Rossby wavetrain, $\overline{F}/\overline{A}$ reduces approximately to the Rossby-wave group velocity, where (${}^{-}$) is an appropriate averaging operator. F and A have the formal appearance of Eulerian quantities, but generally involve a multivalued function the correct branch of which requires a certain amount of Lagrangian information for its determination. It is shown that, in a certain sense, the construction of conservable, quasi-Eulerian wave properties like A is unique and that the multivaluedness is inescapable in general. The connection with the concepts of pseudoenergy (quasi-energy), pseudomomentum (quasi-momentum), and ‘Eliassen-Palm wave activity’ is noted. The relationship of this and similar conservation theorems to dynamical fundamentals and to Arnol'd's nonlinear stability theorems is discussed in the light of recent advances in Hamiltonian dynamics. These show where such conservation theorems come from and how to construct them in other cases. An elementary proof of the Hamiltonian structure of two-dimensional Eulerian vortex dynamics is put on record, with explicit attention to the boundary conditions. The connection between Arnol'd's second stability theorem and the suppression of shear and self-tuning resonant instabilities by boundary constraints is discussed, and a finite-amplitude counterpart to Rayleigh's inflection-point theorem noted

Convergent structure of multitrophic communities over three continents

Ecological theory predicts that communities using the same resources should have similar structure, but evolutionary constraints on colonization and niche shifts may hamper such convergence. Multitrophic communities of wasps exploiting fig fruits, which first evolved about 75MYA, do not show long-term “inheritance” of taxonomic (lineage) composition or species diversity. However, communities on three continents have converged ecologically in the presence and relative abundance of five insect guilds that we define. Some taxa fill the same niches in each community (phylogenetic niche conservatism). However, we show that overall convergence in ecological community structure depends also on a combination of niche shifts by resident lineages and local colonizations of figs by other insect lineages. Our study explores new ground, and develops new heuristic tools, in combining ecology and phylogeny to address patterns in the complex multitrophic communities of insect on plants, which comprise a large part of terrestrial biodiversity.

The vertical structure of oceanic Rossby waves: a comparison of high-resolution model data to theoretical vertical structures

Tests of the new Rossby wave theories that have been developed over the past decade to account for discrepancies between theoretical wave speeds and those observed by satellite altimeters have focused primarily on the surface signature of such waves. It appears, however, that the surface signature of the waves acts only as a rather weak constraint, and that information on the vertical structure of the waves is required to better discriminate between competing theories. Due to the lack of 3-D observations, this paper uses high-resolution model data to construct realistic vertical structures of Rossby waves and compares these to structures predicted by theory. The meridional velocity of a section at 24° S in the Atlantic Ocean is pre-processed using the Radon transform to select the dominant westward signal. Normalized profiles are then constructed using three complementary methods based respectively on: (1) averaging vertical profiles of velocity, (2) diagnosing the amplitude of the Radon transform of the westward propagating signal at different depths, and (3) EOF analysis. These profiles are compared to profiles calculated using four different Rossby wave theories: standard linear theory (SLT), SLT plus mean flow, SLT plus topographic effects, and theory including mean flow and topographic effects. Our results support the classical theoretical assumption that westward propagating signals have a well-defined vertical modal structure associated with a phase speed independent of depth, in contrast with the conclusions of a recent study using the same model but for different locations in the North Atlantic. The model structures are in general surface intensified, with a sign reversal at depth in some regions, notably occurring at shallower depths in the East Atlantic. SLT provides a good fit to the model structures in the top 300 m, but grossly overestimates the sign reversal at depth. The addition of mean flow slightly improves the latter issue, but is too surface intensified. SLT plus topography rectifies the overestimation of the sign reversal, but overestimates the amplitude of the structure for much of the layer above the sign reversal. Combining the effects of mean flow and topography provided the best fit for the mean model profiles, although small errors at the surface and mid-depths are carried over from the individual effects of mean flow and topography respectively. Across the section the best fitting theory varies between SLT plus topography and topography with mean flow, with, in general, SLT plus topography performing better in the east where the sign reversal is less pronounced. None of the theories could accurately reproduce the deeper sign reversals in the west. All theories performed badly at the boundaries. The generalization of this method to other latitudes, oceans, models and baroclinic modes would provide greater insight into the variability in the ocean, while better observational data would allow verification of the model findings.

Superhumps: confronting theory with observation

We review the theory and observations related to the "superhump" precession of eccentric accretion discs in close binary systems. We agree with earlier work, although for different reasons, that the discrepancy between observation and dynamical theory implies that the effect of pressure in the disc cannot be neglected. We extend earlier work that investigates this effect to include the correct expression for the radius at which resonant orbits occur. Using analytic expressions for the accretion disc structure, we derive a relationship between the period excess and mass ratio with the pressure effects included. This is compared to the observed data, recently derived results for detailed integration of the disc equations and the equivalent empirically derived relations and used to predict values for the mass ratio based on measured values of the period excess for 88 systems.

Local and average structure in zinc cyanide: towards an understanding of the atomistic origin of negative thermal expansion

Neutron diffraction at 11.4 and 295 K and solid-state 67Zn NMR are used to determine both the local and average structures in the disordered, negative thermal expansion (NTE) material, Zn(CN)2. Solid-state NMR not only confirms that there is head-to-tail disorder of the C≡N groups present in the solid, but yields information about the relative abundances of the different Zn(CN)4-n(NC)n tetrahedral species, which do not follow a simple binomial distribution. The Zn(CN)4 and Zn(NC)4 species occur with much lower probabilities than are predicted by binomial theory, supporting the conclusion that they are of higher energy than the other local arrangements. The lowest energy arrangement is Zn(CN)2(NC)2. The use of total neutron diffraction at 11.4 K, with analysis of both the Bragg diffraction and the derived total correlation function, yields the first experimental determination of the individual Zn−N and Zn−C bond lengths as 1.969(2) and 2.030(2) Å, respectively. The very small difference in bond lengths, of ~0.06 Å, means that it is impossible to obtain these bond lengths using Bragg diffraction in isolation. Total neutron diffraction also provides information on both the average and local atomic displacements responsible for NTE in Zn(CN)2. The principal motions giving rise to NTE are shown to be those in which the carbon and nitrogen atoms within individual Zn−C≡N−Zn linkages are displaced to the same side of the Zn···Zn axis. Displacements of the carbon and nitrogen atoms to opposite sides of the Zn···Zn axis, suggested previously in X-ray studies as being responsible for NTE behavior, in fact make negligible contribution at temperatures up to 295 K.

Information strategies and the theory of the Firm

Information costs play a key role in determining the relative efficiency of alternative organisational structures. The choice of locations at which information is stored in a firm is an important determinant of its information costs. A specific example of information use is modelled in order to explore what factors determine whether information should be stored centrally or locally and if it should be replicated at different sites. This provides insights into why firms are structured hierarchically, with some decisions and tasks being performed centrally and others at different levels of decentralisation. The effects of new information technologies are also discussed. These can radically alter the patterns and levels of information costs within a firm and so can cause substantial changes in organisational structure.

Reinterpreting ‘generic structure’: evolution, application and limitations of a concept

This paper traces the evolution of thegeneric structure concept in system dynamics and discusses the different practical uses to which they have been put. A review of previous work leads to the identification of three different views of what a ‘generic structure’ is and, hence, what transferability means. These different views are distinguishable in application as well as in theory. Examination of these interpretations shows that the assumptions behind them are quite distinct. From this analysis it is argued that it is no longer useful to treat ‘generic structure’ as a single concept since the unity it implies is only superficial. The conclusion is that the concept needs unbundling so that different assumptions about transferability of structure can be made explicit, and the role of generic structures as generalisable theories of dynamic behaviour in system dynamics theory and practice can be debated and clarified more effectively.

Social theory and system dynamics practice

This paper explores the social theories implicit in system dynamics (SD) practice. Groupings of SD practice are observed in different parts of a framework for studying social theories. Most are seen to be located within `functionalist sociology'. To account for the remainder, two new forms of practice are discussed, each related to a different paradigm. Three competing conclusions are then offered: 1. The implicit assumption that SD is grounded in functionalist sociology is correct and should be made explicit. 2. Forrester's ideas operate at the level of method not social theory so SD, though not wedded to a particular social theoretic paradigm, can be re-crafted for use within different paradigms. 3. SD is consistent with social theories which dissolve the individual/society divide by taking a dialectical, or feedback, stance. It can therefore bring a formal modelling approach to the `agency/structure' debate within social theory and so bring SD into the heart of social science. The last conclusion is strongly recommended.

Reinterpreting 'generic structure': evolution, application and limitations of a concept

This paper traces the evolution of the generic structure concept in system dynamics and discusses the different practical uses to which they have been put. A review of previous work leads to the identification of three different views of what a generic structure is and, hence, what transferability means. These different views are distinguishable in application as well as in theory. Examination of these interpretations shows that the assumptions behind them are quite distinct. From this analysis it is argued that it is no longer useful to treat generic structure as a single concept since the unity it implies is only superficial. The conclusion is that the concept needs unbundling so that different assumptions about transferability of structure can be made explicit, and the role of generic structures as generalisable theories of dynamic behaviour in system dynamics theory and practice can be debated and clarified more effectively.