1. A Spatial Model of Social Interactions Multiplicity of Equilibria Pascal Mossay U of Reading, UK Pierre Picard U of Manchester, UK CORE, Louvain-la-Neuve Paris, July 2009

2. Agglomeration Economies Increasing Returns (New Economic Geography) Market Mechanism Social Interactions Non-Market Mechanism

3. Aim of the Paper Social Interactions Desire of face-to-face contacts Framework Communication Externality Issues Emergence of Multiple Cities Shape and Spacing of Cities Market for Land

4. Related Literature Beckmann ( 1976 ), Fujita & Thisse ( 2002 ) “Implication of Social Contacts on Shape of Cities” Wang, Berliant, ( 2006, JET ) “Only 1 Agglomeration in Equilibrium” Fujita & Ogawa ( 1980, 1982, RSUE ) “Multiple center configurations – Multiple Equilibria” Hesley & Strange ( 2007, J.Econ.Geogr. ) “Endogenous Number of Social Contacts” Tabuchi ( 1986, RSUE ) “First best city is more concentrated than the eqm”

5. Plan of the Talk Spatial Interaction Model Model along a Circle Spatial Equilibria Characterization and Pareto-ranking First-best Distribution Robustness of Equilibria Local vs. Global Spatial Interactions

6. Spatial Interaction Model Beckmann (1976), Fujita and Thisse (2002) Each agent located in x –Faces some residence cost –Benefits from face-to-face contacts with others –Faces some accessing cost λ(x) Space x

7. MaxU(z,s) z: consumption good s: land consumption r(x): rent in location x λ(x): population in location x A: social interaction benefit d(x,y): distance between x and y τ: travelling cost λ(x)

8. MaxU(z,s) = z + u(s) z: consumption good s: land consumption r(x): rent in location x λ(x): population in location x A: social interaction benefit d(x,y): distance between x and y τ: travelling cost z,s,xz,s,x λ(x)

9. Indirect Utility Spatial Equilibrium Trade-off Residence Cost Accessing Cost

10. Spatial Equilibrium Agents in x=-a: Low Residence Cost High Accessing Cost Agents in x=0 : High Residence Cost Low Accessing Cost Spatial Equilibrium: All agents achieve the same utility level Distribution X=0 -a a λ(x)

11. Proposition: No Multiple-City Configuration Consider some agent in x By moving to his right, lower residence cost lower accessing cost Incentive to relocate x City 2 City 3 City 1

12. Spatial Interactions Along a Circle Agent density Each agent - faces a residence cost - benefits from face-to-face contacts - faces an accessing cost λ(x)

13. A priori: Many Possible Configurations Large & Small CitiesUneven Spacing City 1 City 2 City 3 City 2 City 3 City 1

14. Characterization 1) Proposition: Cities can’t face each other “No Antipodal Cities” At location x, by moving to the right marginal residence cost > 0 => Pop(east)> Pop(west) At location x+1/2, by moving clockwise marginal residence cost > 0 => Pop(west)> Pop(east) x X+1/2 East West

15. Characterization 2) The number of Cities can’t be even x1 x2 x3 x4 P1 P2 P3 P4 At location x1: P4=P2+P3 At location x3: P4=P1+P2 But then P1=P3 Similarly, P4=P2 Thus P1=0

16. Characterization 3) Cities of equal size & evenly spaced Proposition: An odd number of Equal & Evenly Spaced is a Spatial Equilibrium.

17. x City 1 City 3 City 2

18. x

19. x

22. Robustness of Spatial Equilibria Proposition: Spatial Adjustments towards higher utility neighborhoods leads back to eqm

23. Pareto-Ranking Consider a Spatial Equilibrium with M cities V(M)= - (resid. Cost + intra-city cost + inter-city cost) decreases with M increases with M V(M) decreases with M Proposition: M=1 is the “best” equilibrium

24. Equilibrium First-Best Social Optimum

25. In the decentralized equilibrium, agents do not internalize the other agents’ interaction cost [see Tabuchi (1986), Fujita-Thisse(2002)] The Spatial Planner will build a city that is more concentrated than the equilibrium allocation

26. Localized Interactions (x-n,x+n) Consider an agent in location x moving to the right - faces a higher residence cost - gets closer to people at his right - further away from people at his left - gets access to “new” agents - looses access to some agents x X+nX-n

27. Local vs Global Spatial Interactions Global Inter. Local Inter. Multiple Spatial Scales Single Spatial Scale

28. Conclusion Along a circle, multiple cities emerge Characterization: equal-size, evenly spaced Pareto-ranking: 1-city > 2-city > 3-city >… Robustness wrt small initial perturbations Local Interactions=>multiple spatial scales