Theory of Markets

Book titles, because they are compromises between concision and precision, provide but an approximate description of real content. For this book an al­ ternative and more comprehensive title would be: An investigation of spatial arbitrage as an introduction to the theory of commodity markets: trade and space-time patterns of price fluctuations. In this title, both the specificities and the limitations of our approach are emphasized. Firstly, our approach con­ centrates on the basic mechanisms of spatial arbitrage, leaving aside a number of accessory facets of international trade such as the impact of quotas or of ex­ change rates. Secondly, for the sake of simplicity we restrict ourselves to single commodity markets; the interrelationship of different goods on multi-commodity markets are only occasionally mentioned. The previous restrictions, however drastic they may at first appear delimit and define what can be considered as the core of the process of trade and of spatial transactions. Having thus simplified the object of our study, we are able to tackle the problem in a systematic way and to model spatial differentials along with their relationships to the volume of trade both in eqUilibrium and in non-equilibrium situations. As far as the subtitle of the book is concerned, we shall postpone the discussion of what is meant by the expression "analytical economics" until the concluding chapter.

I Prologue.- 1 Introduction.- 1 Smith’s “invisible hand” in commodity markets.- 2 Spatial interaction in economic theory.- 3 Spatial interaction in geographical analysis.- 4 Regional market integration and famines.- 5 Organization of commodity markets.- 5.1 The twentieth century wheat market.- 5.2 Which prices?.- 5.3 Long term evolution of ocean freight rates.- 6 Spatial price differentials.- 6.1 Three examples of spatial price differentials.- 6.2 Evolution of spatial price differentials.- 7 The concept of market integration.- 8 Defining and delimiting the problems to be investigated.- 9 The methodology of our approach: parsimony as a condition of testability.- 10 Empirical findings.- 10.1 Interdependence between markets.- 10.2 Price intercorrelations.- 10.3 Variations in trade with respect to transportation costs.- 10.4 The evolution of market integration.- 10.5 The evolution of price volatility.- 11 Outline of the book.- 2 Pricing models.- 1 Dynamic market models with exogenous price expectations.- 1.1 Cobweb models without inventories.- 1 Conservative price anticipation.- 2 Extrapolative price anticipation.- 3 Adaptative price anticipation.- 4 The problem of mixed time scales.- 1.2 Cobweb models with inventories.- 1 A linear model.- 2 An example: the FAO cocoa price model.- 3 Comparison with empirical evidence.- 4 Nonlinear models.- 2 Rational expectations models.- 2.1 Origins of the concept of rational expectations.- 2.2 Rational expectations in commodity markets without inventories.- 2.3 Rational expectation with inventories.- 2.4 More about expectional equations.- 3 Oligopoly theory and spatial competition.- 3.1 The monopoly optimum.- 1 The firm is able to sell all it wishes.- 2 The firm cannot sell all it wishes.- 3.2 The duopoly equilibrium.- 1 Cournot’s model.- 2 Nash equilibrium.- 3 Spatial competition: two marketplaces.- 4 Spatial competition: several marketplaces.- A Appendix A: Conditional expectation: a mathematical reminder.- A.1 Conditional expectation: two random variables.- 1 Definitions.- 2 Basic properties of conditional expectation.- A.2 Conditional expectation: generalization to n random variables.- B Appendix B: Consumption, closing stocks and prices of cocoa, sugar and wheat.- II Equilibrium models.- 3 The stochastic Enke-Samuelson arbitrage model.- 1 Defining the stochastic Enke-Samuelson model.- 1.1 The spatial price equilibrium model.- 1 General presentation.- 2 The spatial price equilibrium model for two markets.- 3 Algebraic solution.- 4 Variational solution.- 1.2 Possible generalizations to more than two markets.- 1 The algebraic solution.- 2 The variational solution.- 1.3 The stochastic Enke-Samuelson model.- 1 The rationale of a stochastic model.- 2 Smoothing and linearization of the model.- 3 Consistency tests of the model.- 4 Predictions of the model.- 2 The stochastic Enke-Samuelson model for two markets.- 2.1 Basic equations.- 2.2 Solutions of the linear model.- 1 Uncorrelated local shocks (identical means).- 2 Correlated local shocks (identical means).- 3 Correlated local shocks (different means).- 4 Linear versus nonlinear model.- 3 Chain of markets.- 3.1 Chain of markets: direct trade relations restricted to closest neighbours.- 1 Solving the linear model.- 2 Proof.- 3 Price differentials as a function of distance.- 4 Linear versus nonlinear model.- 3.2 Chain of markets with an arbitrary exchange pattern.- 1 Equations and results.- 2 Roots of reciprocal equation.- 3 Covariance function.- 4 Variance.- 5 Trade.- 6 Discussion.- 4 Market networks.- 4.1 Solving the linear Enke-Samuelson model.- 1 Equations of the model.- 2 Solution by Fourier transformation.- 3 Integral representation of the covariance function.- 4 Asymptotic expressions of the price covariance function.- 5 Approximation formula.- 4.2 Process of market integration.- 4.3 Price differentials as a function of inter-market distance.- A Appendix A: Covariance function of a network of markets.- A.1 Development for vanishing transportation costs.- A.2 Asymptotic expression for large transportation costs.- A.3 Approximation formula.- 4 Empirical evidence about transport costs.- 1 Transportation costs.- 1.1 European nineteenth century wheat markets.- 1 Inter-regional trade.- 2 International trade.- 1.2 Twentieth century commodity markets.- 1 Inter-regional trade in the United States.- 2 International trade.- 1.3 Long term evolution of transportation costs.- 1 Rail and waterways freight rates.- 2 Ocean freight rates.- 3 Tariffs.- 2 The spatial patterns of price differentials.- 2.1 European nineteenth-century wheat markets.- 1 Comparison between the evolution of price differentials and of transportation costs.- 2 Methodology for the observation of price differentials.- 3 Price differentials at the regional level.- 4 Price differentials at the national level.- 5 Price differentials at the international level.- 2.2 Twentieth-century commodity markets.- 1 Wheat market in the United States.- 2 Potato market in the United States.- 2.3 Is the spatial distribution of prices Gaussian?.- 1 ?2 test versus cumulant tests.- 2 The spatial distribution of prices.- 3 The reduction in spatial price differentials and its implications.- 3.1 Evidence of long term price convergence.- 1 How to measure spatial price dispersion?.- 2 Spatial price convergence.- 3.2 The relationship between price convergence and decrease in price volatility.- 3.3 The relationship between price convergence and trade development.- 1 Trade development at the level of single commodities.- 2 Trade development at the macroeconomic level.- 4 Estimation of the Enke-Samuelson trade model.- 4.1 Methodology.- 4.2 Results.- A Appendix A: Dispersion measures for spatial distributions.- A.1 The mean difference.- 1 Existence.- 2 Relation with Gini’s coefficient.- 3 Sampling properties.- A.2 The range of the sample.- 1 The limiting distributions.- 2 Sampling properties 156 B Appendix B: Trade and wheat differentials between England and Prussia 1828-1859.- C Appendix C: Conversion tables for volumes, weights and currencies.- 5 Grain markets and demographic phenomena.- 1 The green-belt model for city-size distributions.- 1.1 The finite Pareto distribution.- 1 Cumulated distribution of the finite Pareto distribution.- 2 Expectation of the finite Pareto distribution.- 3 Concentration of a finite Pareto distribution.- 1.2 Evolution of urban systems in the Pareto plane.- 1 The transportation constraint in the green-belt model.- 2 Graphical representation in the Pareto plane: possible trajectories.- 1.3 Confronting the implications of the model with empirical evi-dence.- 1 Sources.- 2 Empirical trajectories in the Pareto plane.- 3 Prices of commodities in small versus large cities.- 1.4 Conclusion.- 2 The impact of price fluctuations on vital rates.- 2.1 The methodology.- 1 Selecting the data.- 2 Alternative options for estimating the correlation.- 2.2 Results.- 1 Nineteenth century.- 2 Discussion of the period after World War I.- A Appendix A: Measure of concentration for a finite Pareto distribution.- A.1 Expression of Gini’s coefficient.- A.2 Application to finite Pareto distributions.- B Appendix B: First moments of a finite Pareto distribution.- III Dynamic Models.- 6 Interdependence between markets and autoregresslve modelling.- 1 Analysing market interdependence.- 1.1 From price differentials to correlation analysis.- 1.2 General methods for measuring market interdependence.- 1 Model-independent measures of market integration.- 2 Model-dependent measures of market integration.- 1.3 Simulations of autoregressive modelling.- 1 Adjusting ARMA processes to a simulated multivariate process.- 2 Estimation of a multivariate autoregressive process.- 3 Inadequate sampling time.- 2 Correlation analysis.- 2.1 Methodology.- 1 The influence of foreign trade.- 2 Structural versus temporary interdependence.- 3 The data.- 2.2 Local interdependence.- 1 Regional level.- 2 National level.- 3 International level.- 2.3 Global measure of interdependence: the correlation length.- 1 The correlation length.- 2 The correlation length of precipitation.- 3 Evolution of the correlation length during the nineteenth century.- 3 Autoregressive modelling: dominant markets and satellite markets.- 3.1 Multivariate autoregressive models: identification and estimation.- 3.2 Application of multivariate autoregressive models.- 1 Direction of interaction.- 2 Satellite markets.- 4 Conclusion.- A Appendix A: Technicalities of correlation analysis.- A. 1 Prewhitening or not.- A. 2 Differentiating or not.- 1 Confidence intervals and tests.- B Appendix B: Technicalities of autoregressive modelling.- B.1 Definition of the models.- B.2 Estimating the model.- C Appendix C: Wheat prices in England, Finland, France, Germany and the United States: 1801-1913.- 7 Spatial and space-time autoregressive processes.- 1 Spectral functions and covariance functions of spatial processes.- 1.1 Spatial versus time dependent autoregressive processes.- 1 Causality condition.- 2 Boundary conditions.- 1.2 Green’s functions of recurrence equations.- 1 Fundamental property.- 2 Green’s functions of first-order equations.- 1.3 Spectral theory of autoregressive processes.- 1 The Fourier formalism.- 2 Applications.- 2 Stationarity conditions for spatial processes.- 2.1 Time dependent processes.- 1 Recurrence reasoning.- 2 The Schur theorem.- 2.2 Spatial processes.- 1 An illustrative example.- 2 Stationarity conditions in terms of roots of the characteristic equation.- 3 Stationarity conditions in terms of parameters of the process: second order processes.- 4 Stationarity conditions in terms of parameters of the process: symmetric processes.- 3 Maximum likelihood estimation in spatial autoregressive processes.- 3.1 Time dependent processes.- 3.2 Spatial processes.- 1 The nonlinear equations for the estimates.- 2 Discussion.- 3 The variance of the disturbances is unknown.- 4 Simulation.- 4 Space-time autoregressive processes.- 4.1 Multivariate autoregressive processes.- 1 The Green’s matrix of a system of recurrence equations.- 2 Spectral theory: from Green’s matrices to covariance functions.- 3 Stationarity conditions.- 4.2 Bidimensional processes.- 1 Definitions.- 2 Stationarity of diffusion and propagation processes.- 3 Maximum likelihood parameter estimation for space-time processes.- A Appendix A: Validity of Fourier expansion for a system of finite size.- B Appendix B: Stability of partial difference equations.- B.1 Stability threshold in forward Euler’s method.- B.2 Stiff systems.- B.3 The diffusion equation.- B.4 The wave equation: Von Neumann’s method and Courant ratio.- 8 Time dependent Enke-Samuelson trade models.- 1 The equations of the dynamic Enke-Samuelson arbitrage models.- 1.1 Equations for two markets.- 1 The nonlinear model.- 2 The linear model.- 1.2 Equations for market networks.- 1 Connection to nearest neighbours.- 2 Long range interdependence.- 2. Stationary solutions of the Enke-Samuelson model.- 2.1 Stability in a set of spatially interdependent markets.- 2.2 Two markets.- 1 General expressions.- 2 Variance and coefficient of correlation.- 3 Behaviour of prices and of trade for large, respectively small transportation costs.- 4 Graphical representation.- 2.3 Three markets.- 1 General expressions.- 2 Developments to first and second order.- 3 Comment.- 2.4 Chain of markets.- 1 Slope of the covariance function in the vicinity of t = 0.- 2 Development to first order of the covariance function.- 3 Transient behaviour of trade and prices.- 3.1 Evolution of trade during the Great Depression.- 1 Qualitative discussion.- 2 Evolution of prices and trade expectations.- 3 The transient stochastic model.- 4 Statistical evidence.- 5 Clark’s analysis of spatial price dispersion before and after the crash of 1929.- 4 The ergodic assumption: ensemble averages versus time averages.- 4.1 Definitions and criteria of ergodicity.- 4.2 Ergodicity and non stationarity.- 4.3 Ergodicity and Mandelbrot’s scaling principle.- 9 Dynamic random field models.- 1 Introduction.- 1.1 From discrete to continuous space-time equations.- 1 The continuous Enke-Samuelson model.- 2 Trade and price differentials.- 3 Generalizations and comments.- 1.2 The spectral theory in the continuous case.- 1.3 A special case.- 1 The covariance function.- 2 Qualitative features of the covariance function.- 3 Covariance function of regional price averages.- 2 Field equations.- 2.1 Classification of spatial differential equations of the second order.- 2.2 Source terms and boundary conditions.- 2.3 Diffusion versus wave equations.- 1 Linear equations.- 2 Nonlinear diffusion: the porous media equation.- 3 Correlation function of hyperbolic and parabolic random fields.- 3.1 The passage theorems.- 1 White noise.- 2 Comments.- 3 Spatially autocorrelated noise.- 4 Numerical computation.- 3.2 One and two-dimensional hyperbolic fields.- 1 White noise.- 2 Spatially autocorrelated noise.- 3.3 One and two dimensional parabolic fields.- 1 One spatial dimension.- 2 Two spatial dimensions.- 3.4 Random fields on one and two-dimensional spheres.- 1 Economic rationale of the introduction of fields on compact manifolds.- 2 Dynamic random field on the circumference of a circle.- 3 Dynamic random field on the sphere.- 4 Estimating the random field model.- 4.1 Selecting appropriate price data.- 4.2 Identification: qualitative analysis of statistical evidence.- 1 Modifications of intercorrelation functions with distance.- 2 Wave models versus diffusion models.- 3 Specifying the disturbance term.- 4.3 Estimating a random wave equation.- 1 Procedure.- 2 Hypothesis tests of the model.- 3 Time evolution of estimated parameters.- 4 Epidemic velocities.- A Appendix A: Expression of c(x, 0) for a wave equation.- B Appendix B: Existence, continuity and differentiability of the Fourier integral.- B.1 Continuity.- B. 2 Continuity and differentiability in Rn+1 - (0,0).- C Appendix C: Green’s functions of wave equations on one and two dimensional spheres.- C.1 Green’s function on S1.- C.2 Green’s function on S2.- D Appendix D: Checking the correlation function as a solution of the field equation.- IV Epilogue.- 10 Conclusion and perspectives.- 1 “A study in analytical economics”.- 1.1 The two purposes of economics.- 1.2 “Collectors of facts”.- 2 Perspectives.- 2.1 Tests of quantitative models.- 2.2 Construction of qualitative models.- References.