Choice and Allocation Models for the Housing Market

1 Introduction.- 1.1 The Problem.- 1.2 The Analytical Tools.- 1.3 A Summary of the Book.- A: General.- 2 Individual Choice Behaviour.- 2.1 Introduction.- 2.2 Additive Random Utility Models.- 2.2.1 Discrete Choices: A Stepwise Approach.- 2.2.2 Random Utility Theory.- 2.2.3 Additive Random Utility Models.- 2.2.4 Continuity and Differentiability of the Choice Probability Functions.- 2.2.5 Gross Substitutability.- 2.2.6 Price Sensitivity.- 2.3 Examples of Discrete Choice Models.- 2.3.1 Probit Models.- 2.3.2 Generalized Extreme Value Models.- 2.3.3 Discussion.- 2.4 Discrete Choice in a Dynamic Context.- 2.4.1 State Dependence.- 2.4.2 Deterministic Variation versus Stochastic Instability.- 2.4.3 Alternative Interpretations.- 2.4.4 Discussion.- 2.4.5 Inertia.- 2.5 Conclusion.- Notes.- 3 Uncertainty and the Consistency of Discrete Choice Models.- 3.1 Introduction.- 3.2 Discrete Choice under Uncertainty.- 3.2.1 Utility Maximization under Uncertainty.- 3.2.2 Three Conditions.- 3.2.3 Derivation of a Generalized Choice Model.- 3.2.4 Discussion.- 3.3 Evaluation.- 3.3.1 Comparison with Conventional Demand Theory.- 3.3.2 The Modified Logit Model.- 3.3.3 Discussion.- 3.3.4 Are there Alternative Models ?.- 3.4 The Welfare Economics of Discrete Choice Theory.- 3.4.1 Expected Maximum Utility.- 3.4.2 Invariance of Expected Utilities.- 3.4.3 The Case of Uncertainty.- 3.5 Conclusion.- Notes.- 4 From Micro to Meso Choice Analysis.- 4.1 Introduction.- 4.2 Complete Description of the Market.- 4.2.1 Introduction.- 4.2.2 One Class of Actors.- 4.2.3 Multiple Classes of Actors.- 4.2.4 Interaction between Classes.- 4.2.5 Origin-Dependent Choice Probabilities.- 4.2.6 Entry and Exit.- 4.2.7 Discussion.- 4.3 Aggregation.- 4.3.1 Stochastic Processes.- 4.3.2 The Markov Assumption.- 4.3.3 The Master Equation.- 4.3.4 Convergence to an Interactive Markov Chain.- 4.4 Relationship with Spatial Interaction Models.- 4.4.1 Introduction.- 4.4.2 Entropy Maximization.- 4.4.3 Discussion.- 4.5 Conclusion.- Notes.- 5 Allocation of Actors within the Model.- 5.1 Constraints on Aggregate Behaviour.- 5.1.1 Allocation Mechanisms.- 5.1.2 Preliminaries.- 5.2 Price Equilibrium.- 5.2.1 Introduction.- 5.2.2 Market Demand Function.- 5.2.3 Existence and Uniqueness of Price Equilibrium.- 5.2.4 A Necessary and Sufficient Condition for Uniqueness.- 5.2.5 Review of the Literature.- 5.2.6 Interpretation of the Price Equilibrium.- 5.2.7 Conclusion.- 5.3 Rationing.- 5.3.1 Introduction.- 5.3.2 Preliminaries.- 5.3.3 Individual Choice Behaviour and Market Demand.- 5.3.4 Uniform Rationed Equilibrium.- 5.3.5 Mixed Equilibrium.- 5.3.6 Class and Origin-Specific Realization Probabilities.- 5.3.7 An Additional Assumption.- 5.3.8 Uniqueness.- 5.3.9 Discussion.- 5.4 Some Final Remarks.- Notes.- 6 The Meso-Model and its Dynamics.- 6.1 Introduction.- 6.2 The General Model.- 6.2.1 Description.- 6.2.2 Movements from One State to Another.- 6.2.3 Further Specification.- 6.2.4 An Alternative Formulation.- 6.2.5 Discussion.- 6.3 Dynamics: Perron-Frohenius Theory and its Extensions.- 6.3.1 Introduction.- 6.3.2 The Linear Case.- 6.3.3 Extensions o the Original Theorem: Homogeneous Systems.- 6.3.4 Discussion.- 6.4 Dynamics: Related Approaches.- 6.4.1 Introduction.- 6.4.2 Brouwer's Fixed Point Theorem and Existence of Steady States.- 6.4.3 The Global Univalence Theorem and Uniqueness of Steady States.- 6.4.4 Stability of Steady States.- 6.4.5 Nonlinear Dynamics.- 6.5 Concluding Remarks.- Notes.- B The Housing Market.- 7 The Housing Market: Specification of a Model.- 7.1 Introduction.- 7.2 A Rationed Market with Homogeneous Demanders.- 7.2.1 Introduction.- 7.2.2 The Allocation Process.- 7.2.3 The Dynamic System.- 7.2.4 Steady-State Solution of the Model.- 7.2.5 Features of the Steady-State.- 7.2.6 Conclusion.- 7.3 Extensions of the Basic Model.- 7.3.1 Introduction.- 7.3.2 Heterogeneous Households.- 7.3.3 Changes in the Rationing Procedure.- 7.4 Simulation Exercises.- 7.4.1 Introduction.- 7.4.2 The One-Class Model