Theory of Markets

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.